Reconfigurable microwave network

Abstract

A method of configuring a reconfigurable microwave network includes obtaining a first matrix; determining, based on the first matrix, an admittance matrix, where elements of the admittance matrix indicate admittance values for use in the reconfigurable microwave network, the microwave network having controllable admittances between ports of the microwave network; controlling the controllable admittances of the reconfigurable microwave network based on the admittance matrix; applying, by a plurality of voltage sources, respective microwave frequency voltages to respective ports of the microwave network, the respective microwave frequency voltages indicative of elements of an input vector; and measuring respective output voltages at the ports of the microwave network to obtain an output vector based on the input vector and the first matrix.

Description

RECONFIGURABLE MICROWAVE NETWORKBACKGROUND

[0001] Computing devices are widespread in modern society. Computing devices are used continuously to process data, automate tasks, solve problems, and exchange information.Modern computers involve digital electronic circuits operating on discrete values (binary) to encode and manipulate data. When compared with analog circuits, digital circuits offer improved versatility, precision, noise robustness, flexibility in storing and processing data, and security. However, despite their popularity, digital computers face two key limitations that may hinder their ability to meet the ever increasing demand for computational power. First, they are characterized by high power consumption, especially due to the presence of power-hungry analog-to-digital converters (ADCs) and digital-to-analog converters (DACs). Second, their speed is limited by the clock of the digital processors and by the rate of ADCs and DACs. To overcome these two limitations, analog computing has been considered as it enables energyefficient and massively parallelized computations. Analog computing refers to manipulating analog signals in real time to perform specific operations. Modern analog computing devices have been proposed in recent years based on optical systems, metamaterials and metasurfaces, and resistive memory arrays.

[0002] Although optical analog computers have demonstrated significant advantages in low-power and ultra-fast processing, they typically rely on large-scale architectures, as their size cannot be reduced below the operational wavelength. To realize miniaturized analog computers at scales smaller than the operational wavelength, metamaterials have emerged as artificial materials designed to have unusual electromagnetic (EM) properties. Because of their artificially induced properties, metamaterials allow the manipulation of EM signals within subwavelength scales.

[0003] Analog computers based on metamaterials have demonstrated the possibility of solving mathematical equations, including multiple equations in parallel exploiting input signals at different frequencies. Particular interest has been dedicated to the 2-dimensional counterpart of metamaterials, namely metasurfaces, further reducing the dimensions of the computing devices. Metasurfaces may have application in wireless communications due to their ability to perform computations directly in the wave domain. Possible applications of computing metasurfaces inwireless communications include performing sensing tasks and enhancing physical layer security.

[0004] Resistive memory arrays have also emerged as a technology for building analog computing devices. Particularly noteworthy are crosspoint memory arrays, in which resistive memory cells are located at the intersections of horizontal and vertical lines in a matrix shape. The values stored in these memory cells can naturally represent the entries of a matrix, enabling the memory array to perform matrix operations, such as matrix-vector multiplications, directly in the analog domain. This approach, referred to as in-memory computing or analog matrix computing, has demonstrated the capability to execute matrix-vector multiplications with arbitrarily high precision while consuming less energy compared to digital computing.SUMMARY OF INVENTION

[0005] In one aspect, a method of configuring a reconfigurable microwave network, the method includes obtaining a first matrix, determining, based on the first matrix, an admittance matrix, where elements of the admittance matrix indicate admittance values for use in the reconfigurable microwave network, the microwave network having controllable admittances between ports of the microwave network, controlling the controllable admittances of the reconfigurable microwave network based on the admittance matrix, applying, by a plurality of voltage sources, respective microwave frequency voltages to respective ports of the microwave network, the respective microwave frequency voltages indicative of elements of an input vector, and measuring respective output voltages at the ports of the microwave network to obtain an output vector based on the input vector and the first matrix.

[0006] The method may also include where each port of the microwave network is electrically connected with a voltage source of the plurality of voltage sources via a respective series admittance, or is electrically connected to ground via a respective series admittance.

[0007] The method may also include where the microwave network includes a respective controllable admittance element (i) between each respective pair of the ports of the microwave network, and (ii) between each port and ground, wherein the controllable admittance elements form the controllable admittances..

[0008] The method may also include where the controllable admittances are purely imaginary.

[0009] The method may also include where when the first matrix is P and the admittance Ymatrix is Y, P = (Y / Y0) + IP, where Y0is the series admittance at each port of the microwave network, and I is a P×P identity matrix.

[0010] The method may also include where the ports of the microwave network comprise first ports, where the microwave frequency voltages are applied to the first ports, and where the ports of the microwave network comprise second ports when the number of ports of the microwave network exceeds the number of applied microwave frequency voltages, the second ports includes ports of the microwave network that lack an applied microwave frequency voltage, where the elements of the output vector are based on respective voltages measured at the first ports, respective voltages measured at the second ports, or respective voltages measured at the first ports and the second ports.

[0011] The method may also include where at least one of (i)= (P^1— Pn1Pi2 X (p2iPri1Pi2- p22)-1p2iPri1)u, (n) v2= ((p2iPri1Pi2- p22)-1p2iPri1)u, (m) = -(P12P221P21 - P11)-1U, or (iv) v2= (P221P2i(Pi2P221P2i- Pn)-1)u, where vi is a vector with elements corresponding with respective voltages measured at ports to which a voltage is applied, where v2is a vector with elements corresponding with respective voltages measured at ports to which no voltage is applied, P is the first matrix, P = [pn P121 and u is a vector lP2iP22.having elements corresponding with the voltages applied to the ports.

[0012] The method may also include where the first matrix is indicative of parameters of a minimum mean square error, MMSE, estimation problem, the applied microwave frequency voltages correspond with elements of an observation vector, and the output voltages measured at ports are indicative of elements of a LMMSE estimator for x, or indicative of elements of a column of a covariance matrix Ce of the LMMSE estimator for x.

[0013] The method may also include where the first matrix includes: a submatrix corresponding with a constant matrix H, a submatrix based on a covariance matrix C of a vector x, a submatrix based on a covariance matrix Cn of a vector n, and a submatrix corresponding with the a conjugate transpose of H, HH, an observation vector y = Hx + n, and (i) the applied microwave frequency voltages correspond with elements of y, and output voltages measured at ports that do not have an applied voltage correspond with elements of a linear minimum mean square error, LMMSE, estimator for x, or (ii) the applied microwavefrequency voltages correspond with elements of an / / th column of an N N identity matrix, Iv, and output voltages measured at ports that have an applied voltage correspond with elements of an wth column of a covariance matrix Ce of the LMMSE estimator for x.

[0014] The method may also include where ||CX||F» || Cn||F, where ||A||Fis the Frobenius norm of A, such that the LMMSE estimator corresponds with a generalized least squares, GLS, estimator, or ||CX||F« ||Cn||F, such that the LMMSE estimator corresponds with a generalized matched filtering, GMF, estimator, or Cx=XIXand Cn=nIy, such that the LMMSE estimator corresponds with a regularized least squares, RLS, estimator, or Cx— ^x, Cn—nIy, and Ax» Ansuch that the LMMSE estimator corresponds with an ordinary least squares, OLS, estimator, or Cx=XIX, Cn= AnIy, and Ax« Ansuch that the LMMSE estimator corresponds with an ordinary matched filtering, OMF, estimator.

[0015] A method for MIMO wireless communication, comprises at least one of: implementing regularized zero-forcing beamforming, R-ZFBF, or a minimum mean square error, MMSE, receiver by obtaining the RLS estimator by the method above, or implementing zero-forcing beamforming, ZFBF, or a ZF receiver by obtaining the OLS estimator by the method above, or implementing matched beamforming, MBF, or a matched beamforming receiver by obtaining the OMF estimator by the method above.

[0016] The method may also include where the first matrix is an N N invertible matrix P, the applied microwave frequency voltages correspond with elements of an / 7th column of an N N identity matrix, Lv, and the output voltages measured at the ports corresponds with elements of an / / th column of the inverse matrix of P, P1.

[0017] The method may also include where the first matrix is indicative of parameters of a state transition model of a Kalman filter, the applied microwave frequency voltages correspond with elements of a column of an identity matrix, and the output voltages measured at the ports are indicative of a column of an a priori estimate error covariance matrix.

[0018] The method may also include, where the voltage sources are electrically connected with respective ports of a first set of the ports, and a plurality of antennas are respectively electrically connected with ports of a second set of ports, where the voltage sources are based on a vector of symbols to be transmitted by the plurality of antennas and the first matrix is based on a precoding matrix relating the symbols to target transmitted signals for transmissionby the plurality of antennas, such that output signals at respective ports of the second set of ports cause the plurality of antennas to transmit the target transmitted signals.

[0019] The method may also include where the first matrix includes a submatrix corresponding with a precoding matrix for beamforming a multiple-input multiple-output, MIMO, transmission, the voltage sources are indicative of a vector of symbols to be transmitted, and the respective output voltages correspond with signals to be provided to respective antennas to transmit the beamformed MIMO transmission.0N×N

[0020] The method may also include where the first matrix P =, where Iwis— W I / Vyan AXA identity matrix, 0\ \ / is an N^M matrix of zeros, and W is the precoding matrix.

[0021] The method may also include where the voltage sources are antennas electrically connected with respective ports of a first set of the ports, and a second set of ports are electrically connected with ground, where the first matrix is based on a combining matrix and signals measured at the second set of ports correspond with a result of the combining matrix operating on symbols received at the first set of ports via the antennas.

[0022] The method may also include where the voltage sources correspond with a beamformed multiple-input multiple-output, MIMO, signal, the first matrix includes a submatrix corresponding with a combining matrix for processing the beamformed MIMO signal, and the output voltages correspond with the result of the combining matrix operating on the beamformed MIMO signal.

[0023] The method may also include where the first matrix P =, where INis— Gan N^N identity matrix, 0 \- \- is an N^N matrix of zeros, and G is the combining matrix.

[0024] An aspect provides one or more programming instructions, where the one or more programming instructions, when executed, cause a processor to perform the obtaining the first matrix and determining the admittance matrix in the method described above.

[0025] A processor-readable storage medium, the processor-readable storage medium including the one or more programming instructions described above.

[0026] An aspect provides a system for configuring a reconfigurable microwave network, the system comprising: the computer readable medium described above; the processor; and the reconfigurable microwave network.

[0027] A system for MIMO communication, the system comprising: the system for configuring a reconfigurable microwave network described above; and a plurality of antennas in electrical communication with respective ports of the reconfigurable microwave network.

[0028] Other technical features may be readily apparent to one skilled in the art from the following figures, descriptions, and claims.BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0029] FIG. 1 illustrates a model of an analog computer.

[0030] FIG. 2 illustrates a Microwave Linear Analog Computer (MiLAC).

[0031] FIG. 3A illustrates a circuit representing admittance components associated with diagonal entries of an admittance matrix in an example of a MiLAC according of FIG. 2.

[0032] FIG. 3B illustrates a circuit representing admittance components associated with off-diagonal entries of an admittance matrix in an example of a MiLAC according of FIG. 2.

[0033] FIG. 4 illustrates an example of a MiLAC having inputs on all ports.

[0034] FIG. 5 illustrates a method of configuring a microwave network.

[0035] FIG. 6 illustrates a comparison of the complexity of computing the linear minimum mean square error (LMMSE) estimator with a MiLAC and by digital computing.

[0036] FIG. 7 illustrates a comparison of the complexity of inverting a matrix with a MiLAC and by digital computing.

[0037] FIG. 8 illustrates a comparison of the complexity of computing the Kalman filter with a MiLAC and by digital computing.

[0038] FIG. 9 illustrates a MiLAC implemented using a lossless microwave network.

[0039] FIG. 10A illustrates device including a MiLAC for use in a transmitter.

[0040] FIG. 10B illustrates device including a MiLAC for use in a receiver.

[0041] FIG. 11A illustrates an example arrangement for digital beamforming.

[0042] FIG. 11B illustrates example of an arrangement for analog beamforming.

[0043] FIG. 11C illustrates an example of a system for hybrid beamforming.

[0044] FIG. 1 ID an arrangement for RIS-aided beamforming.

[0045] FIG. HE illustrates a system for SIM-aided beamforming.

[0046] FIG. 1 IF illustrates an aspect of the subject matter in accordance with one embodiment.

[0047] FIG. 12 illustrates an example of the sum rate versus the signal-to-noise ratio (SNR) achieved by digital and MiLAC-aided beamforming.

[0048] FIG. 13 illustrates an example of the bit error rate (BER) versus the SNR obtained with digital and MiLAC-aided beamforming.

[0049] FIG. 14 shows the computational complexity of digital and MiLAC-aided beamforming, as a function of the number of antennas.

[0050] FIG. 15 A shows the achieved sum rate and computational complexity for MiLAC-aided and digital RZFBF beamforming with SNR=0.

[0051] FIG. 15B shows the achieved sum rate and computational complexity for MiLAC-aided and digital RZFBF beamforming with SNR=10.

[0052] FIG. 15C shows the achieved sum rate and computational complexity for MiLAC-aided and digital RZFBF beamforming with SNR=20.

[0053] FIG. 16 illustrates an example of a device according to some examples.

[0054] FIG. 17 illustrates a system for configuring a reconfigurable microwave network.

[0055] FIG. 18 illustrates a system for MIMO communication.

[0056] FIG. 19 illustrates a device for MIMO communication.

[0057] FIG. 20 illustrates an operation flow of a MiLAC.

[0058] FIG. 21 A illustrates an example of a MiLAC structure.

[0059] FIG. 21B illustrates an example of a characterising board for a traveling- wave line of the MiLAC of FIG. 21 A.

[0060] FIG. 21 C illustrates an example of a characterising board for a standing- wave line of the MiLAC of FIG. 21 A.

[0061] FIG. 22A is a schematic view of a MiLAC matrix invertor.

[0062] FIG. 22B is a top view of a circuit for implementing the MiLAC matrix invertor of FIG. 22A.

[0063] FIG. 22C is a back view of the circuit of FIG. 22B.DETAILED DESCRIPTIONNotation

[0064] Vectors and matrices are denoted with bold lower and bold upper letters, respectively. Scalars are represented with letters not in bold font. ℜ{a}, ℑ{a}, and |a| refer to the real part, imaginary part, and absolute value of a complex scalar a, respectively. aT, aH, [a]i, and ||a|| refer to the transpose, conjugate transpose, zth element, and / 2-norm of a vector a, respectively. AT, AH, [A]i,k, [A]:,k, and ||A|| refer to the transpose, conjugate transpose, (z, )th element, th column, and Frobenius norm of a matrix A, respectively. ℝ and ℂ denote the real and complex number sets, respectively, j = √−1 denotes the imaginary unit. INand 0Ndenote the identity matrix and the all-zero matrix with dimensions N x N, respectively, anddenotes the allzero matrix with dimensions M x N.I. MODEL OF A MICROWAVE LINEAR ANALOG COMPUTER

[0065] FIG. 1 illustrates a model 100 of a class of analog computers, that is referred to herein as a MiLAC (Microwave Linear Analog Computer). Herein, a microwave analog computer as an analog computer that processes microwave EM signals, e.g., with frequencies from 300 MHz to 300 GHz. Herein, a linear analog computer is an analog computer that satisfies the superposition principle, i.e., the output caused by multiple inputs is the sum of the outputs caused by each input individually. Thus, a MiLAC is an analog computer that is both microwave and linear.

[0066] A MiLAC can be modelled in general as a linear microwave network having multiple terminals, or ports port 106, where input signals are applied and output signals are read. To enable the MiLAC to compute different functions, its microwave network is reconfigurable, which may be achieved through the use of controllable, or tunable, impedance components, or, equivalently, controllable, or tunable, admittance components. A MiLAC may be modelled in terms of a microwave network comprising tunable admittance components, along with input signals, and output signals.A. Microwave Network

[0067] A linear microwave network having a certain number of ports P can be characterized by its impedance matrix Z ∈ CPxP, which linearly relates the voltages and currents at the P ports throughv = Zi, (1)where v = [v1,...,vP]T∈ CPx1and i = [i1,... iP]T∈ CPx1are, respectively, the voltage and current vectors at the ports 106, with vpE C and ipE C being the voltage and current at the / ?th port, for p = 1,..., P. Alternatively, a linear microwave network with P ports 106 can also be represented by its admittance matrix Y ∈ CPxP, given by Y = Z-1, relating the currents and voltages at the P ports asi = Yv, (2)which is equivalent to (1). Note that impedance and admittance matrices are equivalent representations of a linear microwave network, allowing its analysis regardless of its internal circuit topology.B. Input Signals

[0068] Herein, it is assumed that the MiLAC receives input on the first N ports 106a of the reconfigurable microwave network 102, with N < P. Note that this comes with no loss of generality, as the ports can always be reordered to ensure the input is applied on the first N ports. In addition, if an input needs to be applied on an internal node of the microwave network 102, an additional port can be opened, making this model general. The input is applied by N voltage sources 108, each with series impedance Zo, e.g., Zo = 50 Q, or, equivalently, series admittance 104a Y0= Z0-1. Denoting the voltage of the wth voltage source as unG C, for n = 1,..., N, the input vector is given by u =[{•£,... iN]TE CNxl. Following Ohm’s law, the input voltage unis related to the voltage vnand current inat the wth port by in= Yo(un- vn), for n = 1,..., N, givingi1= Y0(u - v1), (3)where v1= [v1,...,vN]T∈ CNx1and i1= [i1,..., iN]T∈ CNx1as the vectors of the voltages and currents at the first N ports of the network, respectively.

[0069] The input signals may be applied to one or more input lines 110 that are electrically connected with the first N ports 106a via the series admittance 104a.

[0070] In the examples herein, each series admittance 104a is assumed to be the same, but in some examples, the series admittances 104a may have different values.C. Output Signals

[0071] The output of the MiLAC is read on all the P ports of its reconfigurable microwave network 102, i.e., is given by the vector v = [vi,...,vp]T. If an output needs to be read on an internal node of the microwave network 102, an additional port can be added, confirming the generality of the model. Since the number of input ports 106a N is never higher than the number of total ports 106 P, P = N + M, where M is the number of ports 106b with no input signal. To read the output voltage on these M ports 106b without causing undesired reflections within the micro wave network 102, these M ports 106b may be perfectly matched to Zo, e.g., by being terminated to ground via an impedance Zo, or, equivalently, an admittance Y0= Z0-1, as shown in FIG. 1. Thus, according to Ohm’s law, the voltage and current at the (N + m)th port are related by iN+m= −Y0vN+m, for m = 1,..., M, where the negative sign arises from the direction of the current iN+m. This yieldsi2= -y0v2, (4)wwhere v2= [vN+1,..., vN+M]T∈ CMx1and i2= [iN+1,..., iN+M]T∈ CMx1as the vectors of voltages and currents at the last M ports 106b of the network 102, respectively.

[0072] Accordingly, in some examples, each port of the microwave network may be connected with a voltage source (e.g., applied to a respective input line 110) of the plurality of voltage sources via a respective series admittance, or may be electrically connected to ground via a respective series admittance. As will be described in more detail below, ports may be electrically connected with respective antennas, instead of being connected to ground.D. Controllable Admittance Components

[0073] The microwave network 102 of the MiLAC may be implemented with controllable admittance components 202, also referred to herein as tunable admittance components 202, such that its impedance matrix Z and admittance matrix Y are reconfigurable, allowing the computation of different operations in the analog domain. As impedance and admittance matrices are equivalent representations of a micro wave network 102, the following uses the admittance matrix Y representation, since it can be directly related to the tunable components 202 of the micro wave network 102, as described in detail below.

[0074] A MiLAC having an admittance matrix Y that can be reconfigured to any complex matrix with dimension P P can be implemented as a microwave network 102 with P2tunable admittance components 202 connecting each port 106 to ground and to all other ports 106. Specifically, port k is connected to ground through an admittance Yk kE C, for k = 1,..., P, and port k is connected to port i through an admittance Yi k6 C, Vi V. An example of such a MiLAC 200 is illustrated in FIG. 2. A MiLAC 200 having an admittance matrix Y that can be reconfigured to any complex matrix with dimension P^P may be described as having maximum flexibility.

[0075] The example MiLAC 200 of FIG. 2 has P = 4 ports and N= 2 input signals. This topology of tunable admittance components 202 allows the admittance matrix Y to be arbitrarily reconfigured, as it can be shown from the definition of Y in (2). According to (2), [Y]i kis given by applying a voltage Vk to port k, short-circuiting all the other ports, i.e., vp= 0, Vp P k, measuring the current 6 entering at port z, and computing the ratio[Y]u= (5)Vfcbp = 0, Vp*fcfor i, k = 1,..., P. From (5), the entries [Y];t can be computed as a function of thepadmittances of the tunable admittance components 202 {k}. by separately considering the two cases of off-diagonal entries [Y];t with i k and diagonal entries [Y] / ,* with i=k. When i k, the ratio in (5) can be determined by considering the circuit in FIG. 3A, since all the other controllable admittance components 202 in the microwave network 102 do not impact the current ii. Thus, the off-diagonal entry \ }i ^-Ykk, where the negative sign arises from the direction of the current it. When i=k, the ratio in (5) can be determined by considering the circuit in FIG. 3B, since all the other admittance components do not influence ik. In this case, the diagonal entry [Y]*,* is given by the parallel of the / ’ admittance componentsi e.,[Y]kk= Yp k. Summarizing these two cases givesf — Yi ki V k|Y|t,fc=kyPp iY(6)2j =Ip,k1for i,k=l,..., P, giving the expression of each entry of the admittance matrix Y as a pfunction of the tunable admittance components 202 {Ki k].fc-i. By inverting (6), the expressionof the tunable admittance components 202 may be derived as a function of any given admittance matrix Y as( -[Y]iki * k|Y|“ =k1[Y =*(7)for i, k =showing that the P2tunable admittance components 202kj.k-1can be adjusted to allow Y to assume any arbitrary value. Equation (7) clarifies the direct relationship between the entries of the admittance matrix Y and the tunable components 202 of the micro wave network 102. Note that closed-form expressions are not available to relate the entries of the impedance matrix Z and the tunable impedance components 202.

[0076] The example of FIG. 2 has P=4, with N=2 and M=2. However, it is to be understood that other values of P, N and AT (with P, N > 0, and M > 0) could be selected without changing the analysis.

[0077] It is also possible to implement a MiLAC 200 with a reduced number of tunable admittance components 202 compared with the example having P2tunable admittance components 202. However, such a MiLAC may have reduced flexibility, e.g., by the microwave network 102 being unable to be reconfigured to take any arbitrary admittance matrix Y, as a consequence of the reduced circuit complexity. For the purposes of illustration, the following examples assume a MiLAC 200 having P2tunable admittance components 202. However, it is to be understood that fewer tunable admittance components could be used, depending on a desired trade-off between circuit complexity and flexibility.

[0078] As shown in FIG. 2, the microwave network may comprise a respective controllable admittance element 202 (i) between each respective pair of the ports of the microwave network, and (ii) between each port and ground, wherein the controllable admittance elements form the controllable admittances. Each controllable admittance element may include one or more components for controllably changing the admittance of the admittance element. For example, the admittance element may be a varactor, or a PIN diode, for example. The controllable admittances may be complex-valued admittances, or as described further in Section V they may be purely imaginary admittances (i.e., having ideally zero real part).II. ANALYSIS OF A MICROWAVE LINEAR ANALOG COMPUTER

[0079] As described above, a MiLAC 200 comprises a reconfigurable micro wave network 102 with multiple ports 106 where input signals may be applied and output signals may be read. In this section, the output v is characterized as a function of the input u and the admittance matrix of the microwave network Y. In the following cases where the input is applied only to some ports (N < P) and to all ports (N = P) are considered separately.A. MiLAC with Input on Some Ports (N< P)

[0080] In the case the input is applied only to some ports 106a of the reconfigurable microwave network 102, i.e., at least some of the ports 106 do not have an input applied to them, such that N0, the MiLAC is described by the system of linear equations (2), (3), and (4), given by( i = Yvii = L0(u - Vi), (8)( i2= -r0v2which characterizes the MiLAC in FIG. 1. Equation (8) may be rewritten it in a more compact form asby using v = [viT, V2T]Tand i = [iiT, izT]T, and introducing the vector u 6 CPxlas u = [uT, 0pxl]7'. Equation (9) yieldsYv = E0(u -v), (10)and so the output v may be expressed asv = P-1u, (11)where the matrix P G CPxPis defined asP = ^ + Ip, (12)yogiving the output v as a function of the admittance matrix Y and the input u. By denoting the inverse of P asQ = P1, (13)partitioned asQ = [§n! H, <i4)lV21 V221with Qu G < CNxN, Q12G < CNxM, Q21G < CMxN, Q22G < CMxM, the output vector v=[viT, v2T]Tgiven by (11) can be further expressed assolely depending on the blocks Qu and Q21 and the input u. Qu and Q21 in (15) may be explicitly expressed as a function of the admittance matrix Y using the relationships set out in (16) to (24), below.

[0081] Where P G < CPxPbe an invertible matrix, and P-16 < CPxPbe its inverse, partitioned asHe £] ■where A, A' G CNxN, B, B' G < CNxM, C, C' G < CMxN, D, D' G < CMxM, and P=N + M, it can be shown that:1) If A is invertible, P invertible implies that CAXB-D is invertible andA' = A-1— A-1B (CA-1B — D)-1CA-1, (17)B' = A B (CA B D)1. (18)C' = (CA-1B - D)-1CA-1, (19)D' = - (CA-1B - D)-1, (20)2) If D is invertible, P invertible implies that BD-1C-A is invertible andA' = (BD C - A)-1BD-1, (21)B' = (BD C - A)-1BD-1, (22)C' = D1C (BD1C -A)1, (23)D' = D-1— D-1C(BD-1C — A)-1BD-1, (24)3) If A and D are both invertible, (17)-(20) and (21)-(24) are equivalent.

[0082] Equation (15) may be rewritten as a function of Y by using (16)-(24). P and Y are partitioned asp= [ 1p*1211 P *1222]1'Y= [ 1Y*211 Y *1222]1 ’(25)where P YltG CNxN, P12, Y12G CNxM, P21, Y21G CMxN, P22, Y22G < CMxM, givingP11 P121=[Yn / Ko + k Y12 / YOP21 P22. Y2I / Y0Y22 / Y0 + 1«.as a consequence of (12). Thus, assuming P to be invertible, with Pn or P22 invertible, the following results hold as a consequence of (15):1) IfPn is invertible,vi = (pr? - pr^ x (P^P?^ - P22)-1P21P1-11X (27)v2= ((PziPr^Piz - p22)_1p2ipri1)u, (28) 2) If P22 is invertible, we havevi= —(Pi2P221P2i—Pn) X (29)V2=(P221P21(P12P221P21—Pll)1)u, (30)3) If Pn and P22 are both invertible, (27)-(28) and (29)-(30) are equivalent.

[0083] Notably, (27)-(28) and (29)-(30) contain the expressions of the output vectors vi and V2 as a function of the input u and the blocks of P, which are directly related to the blocks of Y by (26), when M > 0.

[0084] Assuming that the admittances of the tunable components 202 {areallowed totake any arbitrary complex value, a MiLAC can compute (27)-(28) and (29)-(30) given any blocs Pn, P12, P21, and P22. This can be achieved by first computing Y from any arbitrary P as Y = Y0P — Y0Ip, (31)which is obtained by inverting (12), and then by setting the tunable components 202 {Fj,k}k_ ^as a function of Y as in (7). By substituting (31) into (7), the tunable admittance components 202 can be reconfigured as a function of any arbitrary P asf -Y0[P]iki #= ki = «(32)for z, k = 1,..., P. Accordingly, the MiLAC may compute (27)-(28) and (29)-(30) for any given Pn, P12, P21, and P22.

[0085] Note that (27)-(28) and (29)-(30) would require computationally expensive matrixmatrix products and matrix inversion operations if computed through a digital computer.However, they can be quickly computed by a MiLAC as the EM signal propagates within the microwave network at light speed. In other words, after properly setting the input u and theadmittances of the tunable components 202 {Ki kj.fc-i, (27)-(28) and (29)-(30) can be computed with complexity 0(1), independently from the input size N and output size P = N + M.Remarkably, the expressions in (27)-(28) and (29)-(30) are particularly relevant for computing the linear minimum mean square error (LMMSE) estimator and the covariance matrix of its error, as described below in Section III.B. Analysis of a MiLAC with Input on All Ports

[0086] In the case the input is applied on all the ports 106 of the reconfigurable microwave network 102, i.e., N = P and M = 0, as shown in FIG. 4, v = vi and i = ii. Thus, this MiLAC is characterized by the system[!* v(ii = r0=r(uYV-1VJ v’ <«)which yieldsYV1= y0(u -V1), (34)which gives the output vi as= P-1u, (35)whereP = ^ + IN, (36)yogiving the output vector vi as a function of the input u and the matrix Y, when M = 0.

[0087] Assuming that the tunable components 202 ( rKi kA].fc-ican take any arbitrary complex value, a MiLAC can compute (35) given any matrix P. To this end, Y is first computed from any arbitrary P asY = rop - (37)which follows from (36), and then the tunable components 202 u r UJ A may be set as afunction of Y as in (7). By substituting (37) into (7), we obtain the expression of the admittance values of the tunable admittance components 202 as a function of any arbitrary P asf -r0[P]ui #= k(38)[PU - ^O i = k’for i, k= 1,..., A, enabling the MiLAC to compute (35) for any given P

[0088] Note that computing (35) with a digital computer would involve the expensive computation of the matrix inverse P-1. However, (35) can be computed instantly by a MiLAC, regardless of the number of inputs N and without requiring any matrix inversion operation. Remarkably, this enables an efficient computation of the matrix inverse, as described below in Section III.C. Primitive Operations of a MiLAC

[0089] In light of the analysis carried out in Sections II-A and II-B, a MiLAC can be used as a computing device able to perform computing operations in the analog domain. These operations are computed as the EM signal propagates within the micro wave network 102 at light speed and can be read as output on the network ports 106. As a digital computer can compute a set of primitive operations contained in its instruction set architecture (ISA), also a MiLAC can compute a set of primitive operations which can be sequentially used to evaluate more complicated functions. The following identifies three primitive operations that a MiLAC with P ports 106 can compute as a function of its input u applied on N ports 106a and the admittances of the tunable admittance components 202 {Kj The first primitive, denotedasv^ MiLAC^uJ^J^J (39)returns in vi the output on the first N ports 106a of the MiLAC. Mathematically, this primitive is expressed as (27) or (29) depending on the invertibility of Pn and P22, where P is defined as in (12). The second primitive operation, denoted asv2= MILAC2(u, {Ku}[fc=1), (40)gives in V2 the output on the last AT ports 106b of the MiLAC, with M = P - N. This primitive is mathematically given by (28) or (30) depending on the invertibility of Pn and P22. Last, the third primitive is denoted asv = MILAC3(U, {yu}[k=1), (41)and returns the output on all the P ports 106 of the MiLAC, i.e., the concatenation of (39) and (40). The primitive (41) coincides with (39) when the input is applied on all the ports 106, i.e., N = P and M = 0. The sequential use of different primitive operations computed in the analog domain can enable the efficient analog computation of complex algorithms. Thefollowing section shows how these three primitives may be exploited together with low-complexity digital operations to efficiently compute functions with practical interest, namely the LMMSE estimator, the matrix inversion, and the Kalman filter.

[0090] Accordingly, as shown in FIG. 5, a reconfigurable microwave network may be configured by obtaining 502 a first matrix, e.g., P and determining 504, based on the first matrix, an admittance matrix, e.g., Y, where elements of the admittance matrix indicate admittance values for use in the reconfigurable microwave network, where the microwave network has controllable admittances between ports of the microwave network. For example, complex admittance values of the controllable admittances may be set equal to complex values of the admittance matrix. The first matrix may be obtained by receiving the first matrix as an input. In other examples, the first matrix may be obtained by based on input information. The input information may include one or more input matrices or vectors. Obtaining the first matrix [pn P12’may include determining submatrices Pn, P12, P21, and P22, where P =LP21 P22.

[0091] The controllable admittances of the reconfigurable microwave network may be controlled 506 based on the admittance matrix. For example, the admittances of the micro wave network may be set to have admittance values equal to respective elements of the admittance matrix.

[0092] Respective microwave frequency voltages may be applied 508 by a plurality of voltage sources, to respective ports of the microwave network, the respective microwave frequency voltages may be indicative of elements of an input vector, e.g., u. For example, complex voltages of the voltage sources may be equal to values of respective elements of the input vector.

[0093] The respective output voltages at the ports of the microwave network may be measured 510 to obtain an output vector based on the input vector and the first matrix. The elements of the output vector may correspond with respective ports of the microwave network. For example, complex values of the elements of the output vector may be equal to the complex voltage measured at the respective port.

[0094] In some examples, when the first matrix is P and the admittance matrix is Y, P and Y Ymay be related by P = — Up, where Yo is the series admittance at each port of the microwave h)network, and I is a PM identity matrix, where P is a PM matrix. This relation may be used to set elements of Y for a given P.

[0095] In some examples, the ports of the microwave network may be considered to comprise first ports, being ports to which the microwave frequency voltages are applied. When the number of ports of the microwave network exceeds the number of applied microwave frequency voltages, the ports of the microwave network may also comprise second ports, the second ports being ports that lack an applied microwave frequency voltage. The output vector are based on (i) respective voltages measured at the first ports, (ii) respective voltages measured at the second ports, or (iii) respective voltages measured at the first ports and the second ports.

[0096] In some examples, when vi is a vector with elements corresponding with respective voltages measured at ports to which a voltage is applied, v2is a vector with elements corresponding with respective voltages measured at ports to which no voltage is applied, P is the first matrix, P = [pn P121, and u is a vector having elements corresponding with the lP2iP22.voltages applied to the ports, at least one of the following relationships may hold:(i) = (pr? - pr^Piz x (p^pr / p^ - p22)-1p2iPri1)u,(11) v2= ((P21P1’11P12- P^^PnPn1)!!,(iii) vx= — (Pi2P221P2i— Pn)1u, or(iv) v2= (P221P2i(Pi2P221P2i—Pn)1)u.Ill - ANALOG COMPUTING FOR THE LMMSE ESTIMATOR AND MATRIX INVERSION

[0097] The preceding sections introduced a MiLAC as an analog computer that linearly processes microwave signals and its output was characterized as a function of its input and microwave network 102 admittance matrix. This section shows that a MiLAC can be used to compute the LMMSE estimator for linear observation processes and invert a matrix with significantly reduced computational complexity compared with a digital computer.A. LMMSE Estimator for Linear Observation Processes

[0098] Consider a linear observation process in which the known random vector y 6 Crxl, also referred to as the observation vector, is given byy = Hx + n (42)where H 6 < CrxXis a known constant matrix, x 6 < CXxlis the unknown random vector with mean x = 0Xxland covariance matrix Cx, and n 6 Crxlis the random noise vector with mean n = 0yxland covariance matrix Cn. We assume that Cx and Cn are known and invertible, and that the cross-covariance matrix of x and n is Cxn = 0X*Y. According to estimation theory, an estimator x 6 CXxlof x is any function of y, its estimation error vector may be written as e = x — x, and the mean square error (MSE) is given by E || x — x||, where E[.] denotesthe expectation operator. The LMMSE estimator xLMMSEof x is defined as the linear estimator minimizing the MSE, i.e.,XLMMSE s t. x = Wy + b (43)where W 6 < CXxrand b 6 < CXxl. The expression of the LMMSE estimator for a linear observation process is available in closed form, as given in the following.

[0099] Considering the observation process in (42), the LMMSE estimator xLMMSEof x is equivalently given byXLMMSE,1 ((H^C-^ + Cx1)-1HHC„1)y, (44)XLMMSE, 2 = (CXHH(HCXHH+ Cn)-1)y, (45)where it holds that xLMMSE 1= xLMMSE 2. Furthermore, the LMMSE estimation error vector e = xLMMSE— x has mean e = 0Xxland covariance matrix Ce equivalently given by Ce,i = cx- CXHH(HCXHH+ Cn)-1HCX, (46)Ce 2= (H^C^H + C-1)-1, (47)where Ce,i = Ce,2.

[0100] There is a remarkable similarity between the LMMSE expressions in (44)-(47) and the expressions characterizing the output of a MiLAC, as highlighted in Table 1. Specifically, there is a similarity between the LMMSE estimator expressions in (44) and (45) and the expressions of V2 in (28) and (30), respectively. Likewise, (46) and (47) have the same structure as the matrix multiplying u in (27) and (29), respectively. Thus, the LMMSE estimator and thecovariance matrix of its error can be computed with a MiLAC, as described in more detail in the following.Table 1SIMILARITY BETWEEN MILAC OUTPUT AND LMMSE ESTIMATOR LMMSE estimator and itsMiLAC output P11 P12 covariance matrix P21 P22.r±cnHV =XLMMSE,1 =2 C(P21P111P12—P22)1P21 Pll1)11+ HH+CXXLMMSE, 2 = (CXHHr±cnHV =(HCP 2 P 1 PXH2 ( 212 ( 12P221P21—Pll)1)U H+Cn)-1)y HH+CXvi = (pr / - P^P^Ce l = Cx- CXHH(HCXHHCx1HHX(P^ll^l?+ Cn)-1HCX. H -cj - PzzL'PziPr,1)<•V1= —(P12P221P21—Pll)l uCe 2= (H^C^H + C-1)-1B. Computing the LMMSE Estimator with a MiLAC

[0101] The expressions of the LMMSE estimator in (44) and (45) can be obtained on the output V2 of a MiLAC, i.e., v2= xLMMSE;1and v2= xLMMSE 2, by exploiting (28) and (30), respectively. The input vector of the MiLAC may be set as u = y, and the tunable admittance i r — components 202 {Kjare setaccording to (32), with P = PLMMSE, where S X 1I*- [±cnH 1 s PLMMSE. HH+C’1.are two values of PLMMSE giving the desired output. Note that this MiLAC has N = Y input ports 106a and M = X output ports 106b with no input, giving a total of P = X+ Y ports 106. The steps necessary to compute the LMMSE estimator in the analog domain with a MiLAC are summarized in Algorithm 1. It is Noteworthy that analog computers have been proposed to compute generalized regression, ridge regression, and linear regression through resistive memory arrays. Interestingly, all these operations can be seen as special cases of the LMMSE estimator, confirming the generality of the present model.1 LMMSE estimator with MiLAC: y, H, Cx1and C n.Output: XLMMSE- l: Set by (32) with P=PLLMSE.LMMSE = MiLAC, (y, fo)' )

[0102] A MiLAC can compute the LMMSE estimator with significantly reduced computational complexity compared to a digital computer, as no matrix-matrix product or matrix inversion operation is required. A MiLAC computes the LMMSE estimator entirely in the analog domain requiring no digital operation. However, a digital computer may be used to compute the values of the tunable admittance components 202 of the MiLAC. Thus, the computational complexity of computing the LMMSE estimator with a MiLAC is given by the complexity of digitally computing the admittance values of the tunable admittance components 202 in (32) for use in the microwave network 102.

[0103] By exploiting the fact that P in (48) is Hermitian and precomputing offline the computations involving Cx and Cn, (32) may be involve approximately 2XY and 4XY real operations to be digitally computed for the two cases z k and i = k, respectively, giving 6XY real operations in total. Conversely, the number of real operations required to compute (44) and (45) through digital computing is 8 AK2+8 A2K+8A' / 3 and 8X2T+8XT2+8T3 / 3, respectively, due to the two matrix-matrix products and the matrix inversion (the complexity of which is assessed in Section XII). As (44) and (45) are equivalent, the complexity of computing the LMMSE estimator digitally is min{8XT2+8A2T+8X3 / 3, 8X2T+8XT2+8T3 / 3} i.e.,8(XT2+X2T+min{A3, T3} / 3). The complexity of computing the LMMSE estimator with a MiLAC and by digital computing are compared in EIG. 6, showing a gain of 2.5 x 104times in the case X = Y= 8192. In particular, EIG. 6 shows the computational complexity (number of real operations) for computing the LMMSE given an observation vector with size Y, for various sizes, A, of unknown vector, using a digital computer and using a MiLAC. Plots 602, 604, and 606 respectively show the computational complexity using a digital computer for X=Y, X=8192, and X=1024. Plots 608, 610, and 612 respectively show the computational complexity for calculating admittance values for a MiLAC for X=Y, X=8192, and X=1024.C. Computing the Covariance Matrix of the LMMSE Estimator Error with a MiLAC

[0104] In addition to the LMMSE estimator, a MiLAC can also efficiently compute the covariance matrix of the LMMSE estimation error in (46) and (47) by exploiting (27) and (29), respectively. To this end, the admittances of the tunable admittance componentsp202 {Ejfcj. are set as in (32) with P=PcOv wheref-l uH 'pCov = J j, (49)using a MiLAC with N = X input ports 106a and M = Y output ports 106b with no input, i.e., a total of P =X+ Y ports 106. Thus, by setting the input vector as u = [IN]: n, the / 7th column of the desired covariance matrix Ce is returned on the output vi following (27) and (29), i.e.,V1= [Cel] or, equivalently,= [Ce>2]. By iteratively considering all the input vectors u = [IN]:n, for n = 1,..., N, all the N columns of the matrix Cecan be obtained, as summarized in Algorithm 2.Algorithm 2 Covariance matrix of the LMMSE estimator error with MiLAC _Input: H, C“1and Cn.Output: Ce.l: Set by (32) with P=PcOv.2: for 77=1 to N do3: [Ce]: n= MiLAC! ([IN]:,n, {Yu}[fc=1).4: end for

[0105] By using a MiLAC to compute the covariance matrix of the LMMSE estimation error, the same benefit in terms of computational complexity can be obtained as described above for the computation of the LMMSE estimator. As a MiLAC computes the desired covariance matrix entirely in the analog domain requiring no digital operation, the required computational complexity is driven by the complexity of digitally computing the values of its tunable admittance components 202 as in (32). Specifically, the covariance matrix can be computed by a MiLAC with only 6XY real operations (digitally computed), while it would require 8(XT2+X2T+min{A3, T3} / 3) real operations to compute (46) or (47) by a digital computer.

[0106] Accordingly, the first matrix may be indicative of parameters of a minimum mean square error, MMSE, estimation problem, the applied microwave frequency voltages maycorrespond with elements of an observation vector, and the output voltages measured at the ports may be indicative of elements of a LMMSE estimator for x, or indicative of elements of a column of a covariance matrix Ce of the LMMSE estimator for x.

[0107] In some examples, the first matrix may include a submatrix corresponding with a constant matrix H, a submatrix based on a covariance matrix C of a vector x, a submatrix based on a covariance matrix Cn of a vector n, and a submatrix corresponding with the a conjugate transpose of H, HH, an observation vector y = Hx + n. In some examples, the applied microwave frequency voltages correspond with elements of y, and output voltages measured at ports that do not have an applied voltage correspond with elements of a linear minimum mean square error, LMMSE, estimator for x. In other examples the applied microwave frequency voltages correspond with elements of an wth column of an N N identity matrix, Lv, and output voltages measured at ports that have an applied voltage correspond with elements of an wth column of a covariance matrix Ce of the LMMSE estimator for x.

[0108] For a known constant matrix H 6 < C.YxX, an unknown vector x with mean x = 0Xxiand covariance matrix Cx, noise vector n 6 Crxl, with mean n = 0Yxiand covariance matrix Cn, an ±C Hobservation vector y = Hx + n, the first matrix may be P =n. The appliedHHFCx1microwave frequency voltages correspond with elements of y, and output voltages measured at ports that do not have an applied voltage may correspond with elements of a linear minimum mean square error, LMMSE, estimator for x.

[0109] For a known constant matrix H 6 < CrxX, an unknown vector x with mean x = 0Xxiand covariance matrix Cx, noise vector n 6 Crxl, with mean n = 0Yxiand covariance matrix Cn, an FC-1Hobservation vector y = Hx + n, the f '-xH]irst matrix may be P =. The appliedH -C nJmicrowave frequency voltages may correspond with elements of an wth column of an A / A identity matrix, Lv, and output voltages measured at ports that have an applied voltage may correspond with elements of an wth column of a covariance matrix Ce of the LMMSE estimator for x.D. Inverting a Matrix with a MiLAC

[0110] A MiLAC with input on all ports 106 (N= P and M = 0), whose input-output relationship is analyzed in Section II-B, can be used to efficiently compute the inverse of anarbitrary matrix P 6 CNxN. To this end, the tunable admittance components 202 of the MiLAC may be set according to (38), such that the output vector is vi=P-1u, asdescribed in Section II-B. Thus, by setting the input vector as u = [IN]:n, the / / th column of P-1is given on the output vi, i.e., vt= [P-1]:,n. To obtain all the N columns of P-1, all the input vectors u = [IN]:n, for n = 1,..., N, may be iteratively considered, as described in Algorithm 3.3 Matrix inversion with MiLACOutput: P1.l: Set {Yu}.^iby (38).: for 77=1 to N do3: [P-1]:,n = MiLAC3([IN]:,n,: end for[OHl] The computational complexity of this matrix inversion approach is driven by the number of operations required to compute the admittance values of the tunable admittance components 202 according to (38), given by approximately 4V2real operations in total. Once the tunable admittance components 202 are reconfigured, a MiLAC can compute the matrix inverse with only N measurements, where each measurement returns a column of the matrix inverse. Thus, a MiLAC can compute the inverse of an arbitrary N N matrix with complexity O(7V2), which is significantly less than the complexity of matrix inversion with digital computing given by 8 / V? / 3 (as described in Section XII).

[0112] The complexity of inverting a matrix with a MiLAC and by digital computing are compared in FIG. 7, showing a gain of 5.5 103times in the case N=8192. In particular, FIG. 7 shows the computational complexity of inverting a matrix with size N using a digital computer (plot 702) and for calculating admittance values for a using a MiLAC (plot 704).

[0113] In some examples, the first matrix is an N N invertible matrix P, the applied microwave frequency voltages correspond with elements of an / 7th column of an N N identity matrix, Lv, and the output voltages measured at the ports 106 correspond with elements of an / 7th column of the inverse matrix of P, P1.IV. ANALOG COMPUTING FOR THE KALMAN FILTER

[0114] This section shows that a MiLAC can efficiently perform Kalman filtering, which has not previously been unexplored in the context of analog computing, and is a more general operation than the LMMSE estimator.A. Kalman Filter

[0115] Consider a discrete-time dynamical process has a state at time t, denoted as xt6 CXxl, is a linear function of the state at time t~l, denoted as xt-t6 CXxl, given byxt= Atxt-x+ mt,(50)where At6 CXxXis a known state transition matrix and mt6 CXxlis the random state noise with mean m = 0Xxiand covariance matrix M. At time t, an observation yt6 Crxlof the state xr is acquired according toyt= Htxt+ nt, (51)where Ht6 < CrxXis a known observation matrix and nt6 < Crxlis the random observation noise with mean n = 0yxland covariance matrix N. Note that, herein, X and T have been used to denote the dimensions of both the LMMSE estimator and the Kalman filter, with the meaning may being clear from the context. The goal of the Kalman filter is to provide at time t an estimate of the state x? as a function of the estimate of the state x?-i at time t~l and the noisy observation yt. To this end, the Kalman filter operates in two phases, namely “prediction” and “correction”.

[0116] In the “prediction” phase, an a priori estimate of the current state is computed depending on the previous state estimate. The a priori estimate of xt, denoted as xt|t-15is given byxt|t- 1=AtXt.-qt-i, (52)

[0117] where is the a posteriori estimate of x?-i. In addition, the Kalman filter computes the a priori estimate error covariance matrix Rt|t-i G CXxXasRt|t- 1=AfRt-iit-i A / + M, (53)

[0118] as a function of the a posteriori estimate error covariance matrix at time t~l denoted as Rr-i|r-i. In the “correction” phase, the current state estimate is refined based on the current apriori estimate and the current observation. This refined estimate is named the a posteriori estimate of the state. After computing the so-called Kalman gain KtG CX*YasKt= Rt|t-1H^(HtRt|t-1H^ + N)’1, (54)

[0119] the a posteriori estimate of xt, denoted as xt\t, is given byxt\t = + Kt(yt- (55)

[0120] as a function of the a priori estimate xt|t-xand the observation yt. Besides, the a posteriori estimate error covariance matrix is also computed asRt|t= (I - KtHt)Rt|t-15(56)

[0121] refining the a priori estimate error covariance matrix Rr|r-i. The two phases “prediction” and “correction” are sequentially repeated at each time step, as described in in Algorithm 4. This process ensures that the Kalman filter output xt|tis the LMMSE estimator of xt. The complex-valued Kalman filter described herein implicitly assumes circularly symmetric initial state, state noise, and observation noise. To treat noncircularly symmetric random variables, augmented complex-valued Kalman filters have been proposed.Algorithm 4 Kalman filterInput: Ar, Hr, M, N, X^^^, and Rr-i|r-i.Output: xt|tand Rr|r- Prediction:1:xt\t-i=2'- Rt|t- 1=+ M.Correction:S^^ R^^H^HtR^H^ + N)-1.4: xt|t= Xtn-i + Kt(yt- HiX^-i).

[0122] Note that there is a similarity between the operations involved in the Kamlan filter (52)-(56) and the expressions computable with a MiLAC. Specifically, three similarities are highlighted in the following, and are summarized in Table 2. First, the inverse of (53) has the same structure as the matrix multiplying u in (29). Second, xt|t— xt^t-1given by (55) has thesame structure as (30). Third, (56) has the same structure as the matrix multiplying u in (27). Thus, the Kalman filter can be efficiently computed with a MiLAC, as described below.Table 2SIMILARITY BETWEEN MILAC OUTPUT AND KALMAN FILTER.MiLAC output Kalman filter equationsV1= —(P12P221P21—Pll)l uRqt-i = (AfRt-nt-iA? + M)-1xt|t- = (R^H^R^H" + N)-1) (ytV2=(P221P21(P12P221P21—Pll)1)U—Htxt|t-i )vi = (pr? - pr / p^CPziPr / Piz Rt|t= Rt|t-i -Rt|t-iH? (HtR^H?- p22)“1P2iPri1)uB. Computing the Kalman Filter with a MiLAC

[0123] The Kalman filter in Algorithm 4 can be computed by performing the most computationally expensive steps in the analog domain through a MiLAC, as shown in Algorithm 5. Note that Algorithm 5 takes in input the matrix inverseand returns the matrix inverse Rj|, differently from Algorithm 4, but without altering the logic of the Kalman filter. In the “prediction” phase, Step 1 in Algorithm 4 has low computational complexity, i.e., 8A2real operations (see Section XII), and can be digitally performed also in Algorithm 5.Besides, Step 2 in Algorithm 4 involves two matrix-matrix products, each with complexity 8A3(see Section XII), leading to a complexity of 16A3. Thus, it is replaced by step 2 in Algorithm 5, requiring the same complexity as Algorithm 2, i.e., 6A2real operations. In the “correction” phase, the computation of xt|tin Algorithm 4 through Steps 3 and 4 requires two matrix-matrix products and a matrix inversion, leading to a complexity of 8(AK2+A2K+F73), as for the LMMSE estimator described in Section III-B. To reduce complexity, these steps are replaced by Steps 3, 4, and 5 in Algorithm 5, having a respective complexity of 8 A2(matrix-vector product, see Section XII), 6AT (same as Algorithm 1, see Section III-B), and 2X (sum of complex vectors, see Section XII). Finally, Step 5 in Algorithm 4 includes two matrix-matrix products, and has complexity 8(A2K +X3). Thus, it may be replaced by Steps 6 and 7 in Algorithm 5 having lower complexity, i.e., 6XY (same as Algorithm 2) and 4A2(same as Algorithm 3), respectively.5 Kalman filter with MiLACOutput: xt|tand Rt||.Prediction:1:xt\t-i — A-txt-i\t-i-: Set R -1t|t-i by Algorithm 2 with Input: A / , M, and Rf-Sqt-vCorrection:3: yt= yt -Hf Xtit-!.Set xt|tby Algorithm 1 with Input: yt, H N, and Rt||_x5: xt|t= xt|t+xt|t-1.6: Set Rt|tby Algorithm 2 with Input: A2 / , R^q^, and N.7: Set R -1tit by Algorithm 3 with input Rt|t.

[0124] A MiLAC can compute the Kalman filter with much lower computational complexity compared to conventional digital computing. Specifically, the computational complexity of Algorithm 5 is given by adding the complexity of its Steps 1-7, i.e., 8A2+6A2+8A2+6A1 / +2A+6A} / +4A2, giving approximately 26A2+ 12 AT real operations. In contrast, the complexity of digitally computing the Kalman filter via Algorithm 4 is given by adding the complexity of its Steps 1-5, namely 24X3+16X2Y+8XY2+8Y3 / 3 real operations, approximately. The complexity of computing the Kalman filter with a MiLAC and by digital computing are compared in FIG. 8, which shows the computational complexity of computing the Kalman filter given an observation vector with size Y, for various sizes X of the state vector using a digital computer and using a MiLAC. Plots 802, 804, and 806 respectively show the computational complexity using a digital computer for X=Y, X=8192, and X=1024. Plots 808, 810, and 812 respectively show the computational complexity for calculating admittance values for a MiLAC for X=Y, X=8192, and X=1024. A MiLAC may provide a gain of 1.1 * 104 times when X=Y=8192. This decrease in complexity could have implications in applications where Kalman filters are extensively used for prediction with stringent latency requirements.

[0125] Accordingly, the first matrix may be indicative of parameters of a state transition model of a Kalman filter, the applied microwave frequency voltages may correspond with elements of a column of an identity matrix, and the output voltages measured at the ports may be indicative of a column of an a priori estimate error covariance matrix.

[0126] In some examples, the first matrix includes: a submatrix corresponding with a state transition matrix At at time t, a submatrix based on a covariance matrix M of a vector m(at time t, a submatrix based on an a posteriori estimate error covariance matrix at time t-1, and a submatrix corresponding with the a conjugate transpose of At, A. A state vector at time t is related to a state vector at time Z-l by x = Af^t-i + nt, and the applied microwave frequency voltages correspond with elements of an wth column of an N N identity matrix, Iv, and output voltages measured at ports that have an applied voltage corresponding with elements of an wth column of an a priori estimate error covariance matrix.

[0127] In some examples, where xt6 CXxldescribes a state at time t as xt= Atxt-t+ mt, AtG CXxXis a state transition matrix, nt6 CXxlis state noise with mean m = 0Xxland a covariance matrix M, — Rt-1|t-1is an a posteriori estimate error covariance matrix at time Z-l,Atthe first matrix may be P = _ p i The applied microwave frequency voltages maycorrespond with elements of an wth column of an A / A identity matrix, Iv, and output voltages measured at ports that have an applied voltage may correspond with elements of an wth column of an a priori estimate error covariance matrix Rt|t-i G CXxXof a prediction operation of a Kalman filter.

[0128] In some examples for calculating xt|t, an a posteriori estimate of a state x at time t for use in a correction operation of a Kalman filter, the first matrix includes a submatrix based on an observation matrix at time t, Hr, a submatrix based on a conjugate transpose of the observation matrix at time t, H^, a submatrix based on a covariance matrix, N, of random observation noise, and a submatrix based on an a priori estimate error covariancematrix, Rt|t-i. The applied microwave frequency voltages may correspond with elements of an observation vector at time t, yt, an observation matrix, Hr, and an a priori estimate of xr, xt|t-15

[0129] In some examples for calculating xt|t, an a posteriori estimate of a state xr at time t for Huse in a correction operation of a Kalman filter, the first matrix may be P = ±Rt|t-itHHtwhere Rt|t-i is an a priori estimate error covariance matrix, Hr is an observation matrix at time t, and N is a covariance matrix of random observation noise. The applied microwave frequencyvoltages may correspond with elements of a vector y = yt— Htxt|t_!, where yt is an observation vector at time t, and xt|t-xis an a priori estimate of x?.

[0130] In some examples for calculating an / / th column of an a posteriori estimate error covariance matrix Rt|tfor use in a correction operation of a Kalman filter, the first matrix may include: a submatrix corresponding with a conjugate transpose of a state transition matrixat time t, a submatrix based on an a posteriori estimate error covariance matrix at time t-1, and a covariance matrix of random observation noise, N. The applied microwave frequency voltages may correspond with elements of an / / th column of an N N identity matrix, Lv, and output voltages measured at ports that have an applied voltage corresponding with elements of the / / th column of Rt|t.

[0131] In some examples for calculating an / / th column of an a posteriori estimate error covariance matrix Rt|tfor use in a correction operation of a Kalman filter, the first matrix may AtKt-l|t- 1be P =, where Atis a state transition matrix, — is an a posteriori. A-Ht — Nestimate error covariance matrix at time Z-l, and N is a covariance matrix of random observation noise. The applied microwave frequency voltages may correspond with elements of an wth column of an N N identity matrix, I \-, and output voltages measured at ports that have an applied voltage may correspond with elements of an / 7th column of Rt|t.V. MILAC WITH A LOSSLESS MICROWAVE NETWORK

[0132] In the previous section, a MiLAC made of tunable admittance components 202 is allowed to take any complex value, with arbitrary real part (conductance) and imaginary part (susceptance). However, in some applications it may be advantageous to avoid the use of tunable admittance components 202 with non-zero real parts in a MiLAC for the following two reasons. First, positive conductance values dissipate power, causing losses and increasing power consumption. Second, negative conductance values require active components, which are costly and require additional power supply.

[0133] In this section, it is shown that it is possible to realize a fully capable MiLAC with purely imaginary tunable admittance components 202, i.e., with a lossless microwave network 902.

[0134] FIG. 9 illustrates a MiLAC implemented through a lossless microwave network 902 with an admittance matrix Y 6 < 2px2preceiving the input on the first 2N ports 106a, where P = N + M, M> 0. Since it is made of purely imaginary admittance components, Y is purely imaginary, and is given by Y = jB, where B 6 IR.2Px2Pis the susceptance matrix of the lossless microwave network 902. The voltage of the «th voltage source 108 is denoted as un, for n=l,...,2N, and the input vector as u = [u1(...,u2N]TE C2Nxl. The voltage at the th port 106 of the microwave network 902 is denoted as vp, for p=l,...,2P, and the output vector is denoted as v = [vr,..., v2P]TG C2Pxl, which can be partitioned as v = v1(v2], where vx=[v1(..., V2N]TE C2NX1and v2= [v2W+i> ■ ■ ->v2N+2M]TE C2Mxl. According to the analysis in _ r _ _ -1 T _Section II-A, the output v = v1(v2] is given as a function of the input u and the susceptance matrix B asVi(57)v2where Q ^•1,1, and Q ^2711 are blocks of the matrixQll Q12(58)Q21 Q22with Q ^•11 G < C2Nx2N', Q ^1127G < C2Nx2M', Q ^2711 G < C2Mx2N', Q ^2727G < C2Mx2M’, where(59)It is possible to prove that the output of this lossless MiLAC is given by vx=[— jvf, vf]Tand v2= [— jv2, vf]T, where vi and V2 are given by (15), when the input is set as u = [uT, juT]Tand the susceptance matrix is set asBn B12(60)B21 B22whereWil) SfYnll ININ1Bn (61)-3{YX1} 5H{Yn}J ° [-IJv V’ W22} 3{Y22}1 IMIMB22 (62)L-3{Y22} 5H{Y22}J ° L-IM W3W) 3{Yik}'Bit —l-3{Yik}for ikE {12, 21}, as represented in FIG. 9. Thus, it is possible to implement the MiLAC analyzed in Section II-A and compute the functions in (27)-(28) and (29)-(30) by only employing lossless tunable admittance components 202. To prove this, we show that it holds(64)when B is set as (60)-(63). Equivalently, the permutation matrix II6 C2Px2Pcanintroduced asIJV ONXM ONXM OMXN OMXJV IM 0M(65)ON IN ONXM ONXM-OMXN OMXN 0MIM.and rewrite (64) as-jvi -u- ju2P -J ■2] = n (66)VJ0V20— r— T —TIinc IITI =T~s e I I2P. Since v = [v1(v2] and u = [uT, 0^xl]7(66) becomes(67)where the term n(jB / Y0+ I2p)lITcan be rewritten asn(^ + i2P)nr^nBir + i2P3{Y}(68)1-3{Y} 5H{Y}. ’2Psince niT = i2Pand B is set as in (60)-(63). Substituting (68) into (67) and executing the products, (67) simplifies aswhich holds true given the relationship in (11 )-( 12), proving that a fully capable MiLAC as analyzed in Section II-A can be implemented with lossless microwave network 902.

[0135] When using lossless admittance components, which have purely imaginary values, the number of components increases to (2 / *)2, compared to the P2components used in the MiLAC with complex admittance components.

[0136] The reasoning above is valid for a MiLAC based on a lossless microwave network 902 with input applied on all the ports 106, i.e., with M=Q. Considering a MiLAC implemented with a lossless network as in FIG. 9 but with M=Q, the analysis carried out in Section II-B gives that Vi = P-1u, (70)whereP = ^yo + I2W, (71)Thus, it is possible to show that= [— jvf, vf ]T, where vi is given by (35)-(36), when the input is set as u = [uT, juT]Tand the susceptance matrix is set as- _ 9i{Y} 3{Y}1 ININ1(72)1-3{Y} 5H{Y}J °L-ITV VThe proof of this similar to the proof for the case M>0, and so is omitted for conciseness.

[0137] Accordingly, the MiLAC analyzed in Section II-B to compute (35)-(36) may be implemented using lossless tunable admittance components 202, and in particular, may be implemented using only lossless tunable admittance components 202.VI. USE OF MICROWAVE LINEAR ANALOG COMPUTER IN BEAMFORMING

[0138] Analog-domain operations offer a promising solution to accelerating signal processing and enabling future multiple-input multiple-output (MIMO) communications, for example with an increased number of antennas, e.g., thousands of antennas. The following sections describe the application of a MiLAC to wireless communications. According to some examples, use of a MiLAC may enable gigantic MIMO beamforming entirely in the analog domain. MiLAC-aided beamforming offers improved flexibility and performance of digital beamforming, while allowing a reduction in hardware costs by, for example, reducing the number of radiofrequency (RF) chains, and using low-resolution analog-to-digital converters (ADCs) anddigital-to-analog converters (DACs). In addition, per-symbol operations may be eliminated by avoiding digital-domain processing and offers reduced computational complexity of zeroforcing (ZF), which scales quadratically with the number of antennas instead of cubically. It may also process signals with fixed matrices, e.g., the discrete Fourier transform (DFT), directly in the analog domain. Numerical results show that it can perform ZF and DFT with a computational complexity reduction of up to 1.5 104and 4.0x107, respectively, compared to digital beamforming.

[0139] The use of analog operations within the field of signal processing may facilitate the use of an increased number of antennas, which supports higher data rates and simultaneously serving a larger number of users. Massive multiple-input multiple-output (MIMO) systems typically deploy 64 or more antennas, this number could reach several thousands in ultra-massive or gigantic MIMO systems, which have been recently proposed to meet the stringent requirements of 6G networks. With such a large number of antennas, digital beamforming faces two critical challenges in terms of hardware and computational complexity. First, digital beamforming typically requires a dedicated radio-frequency (RF) chain per antenna element, each including ADCs / DACs and mixers, which are costly and power-hungry components. Second, as the number of antennas increases, the data volumes to be processed in real-time grow accordingly. For example, to precode the transmitted symbol vector at the transmitter (or combine the received symbol vector at the receiver) a matrix-vector product is required on a per-symbol basis. These two challenges result in high hardware costs and computational complexity when digital beamforming is applied in massive or gigantic MIMO systems. For these reasons, alternatives to digital beamforming have been explored to enable beamforming operations fully or partially in the analog domain.

[0140] To address the high hardware cost associated with digital beamforming, alternative strategies such as analog beamforming and hybrid analog-digital beamforming have been proposed.

[0141] Analog beamforming steers the beam pattern by solely requiring a single RF chain. This is achieved by connecting the RF chain to the antennas via phase shifters or time delay elements, which impose a constant modulus constraint on the beamforming weights. Thus, while analog beamforming offers low hardware complexity, it is characterized by limited flexibility.

[0142] To balance hardware cost and flexibility by bridging analog and digital beamforming, hybrid analog-digital beamforming techniques have been investigated. In hybrid beamforming, the beamforming process is divided into two stages. First, the transmitted symbols are digitally processed in the baseband domain with a low-dimensional digital beamforming matrix, and fed to a reduced number of RF chains. Second, the RF chains are connected to the transmitting antennas through a network of phase shifters, operating analog beamforming in the RF domain. The reversed process is applied at the receiver.

[0143] Another method to efficiently perform beamforming in the analog domain is given by reconfigurable intelligent surface (RIS). This technology controls the wireless channel through surfaces made of multiple passive elements with reconfigurable scattering properties.Beamforming in the analog domain can be achieved through RIS by deploying a RIS in close proximity to an active transceiving device equipped with a reduced number of antennas. In this way, the effective radiation pattern of the active device can be adjusted by reconfiguring the RIS elements. Two extensions of this technology have been proposed to achieve additional flexibility over conventional RIS. First, beyond diagonal RIS (BD-RIS) include more general RIS architectures, characterized by a scattering matrix not restricted to be diagonal. By deploying a BD-RIS close to an active device, its additional flexibility over conventional RIS in manipulating the radiation pattern can enable higher performance. Second, stacked intelligent metasurface (SIM) technology considers multiple stacked transmissive RISs deployed close to the active device. The additional flexibility provided by the multiple layers has shown increased performance over just deploying a single RIS.

[0144] In the reviewed beamforming strategies operating in the analog domain, the number of RF chains is commonly lower bounded by the number of symbols, or streams, transmitted in parallel. Nevertheless, other beamforming strategies have been developed to also perform symbol modulation in the analog domain, allowing the transmission of multiple symbols with a single RF chain. First, in electronically steerable parasitic array radiators (ESPARs), the symbols are modulated and the radiation pattern reconfigured by adjusting the mutual coupling between one active antenna element and multiple “parasitic” elements having tunable loads. Second, in load modulated arrays (LMAs), a single RF source is connected to multiple antennas through tunable impedance components, which are reconfigured to achieve symbol modulation. Third, symbol modulation can be performed in the analog domain by employing a RISdeployed close to an active antenna transmitting a carrier signal. In addition to alleviating the hardware cost, analog-domain strategies may also reduce the computational complexity of future wireless networks, by performing specific mathematical operations in the analog domain. To this end, in-memory computing has been proposed to accelerate ridge regression with applications to massive MIMO. SIM has been used to compute the discrete Fourier transform (DFT) of the incident signal in the analog domain, which is particularly beneficial for direction-of-arrival (DoA) estimation. Moreover, over-the-air computing emerged as a form of analog computing that leverages the superposition principle characterizing the electromagnetic (EM) propagation over the wireless channels. Over-the-air computing allows efficient computation of the so-called nomographic functions. Various technologies have been proposed to reduce the hardware cost of future wireless transceivers through analog beamforming.However, operating in the analog domain typically decreases the flexibility compared to digital beamforming, with a consequent reduction in performance.

[0145] The following sections introduce a beamforming strategy enabled by MiLACs, denoted as MiLAC-aided beamforming. This beamforming strategy may maintain the same flexibility as digital beamforming while allowing operation entirely in the analog domain. Because of its fully analog nature, it allows a reduction in the number of RF chains and may be implemented using only low-resolution ADCs / DACs, reducing the hardware cost. By leveraging the computational capabilities of MiLACs, it may also reduce the computational complexity required for beamforming.

[0146] The following sections demonstrate that a MiLAC can efficiently compute five mathematical operations of practical interest in MIMO communications, such as least squares (LS) and matched filtering (MF). To this end, it is shown that these five operations can be seen as special cases of the linear minimum mean square error (LMMSE) estimator for linear observation processes, and then describe how the microwave network 102 of the MiLAC may be reconfigured to compute them. These operations can be computed with a MiLAC with significantly reduced computational complexity compared to conventional digital computing.

[0147] A MiLAC-aided beamforming strategy is described, that may be scalable to gigantic MIMO systems. MiLAC-aided beamforming can perform precoding at the transmitter and combining at the receiver with arbitrary matrices, i.e., with the same flexibility as digital beamforming. While being flexible, MiLACaided beamforming allows specific operations to beperformed in the analog domain, such as zero-forcing (ZF) precoding at the transmitter and ZF combining at the receiver, with gains in terms of computational complexity. In addition, it allows precoding or combining a signal with a fixed matrix, such as the DFT matrix, entirely in the analog domain, requiring no digital operations.

[0148] MiLAC-aided beamforming is compared with existing beamforming strategies, namely digital, analog, hybrid, RIS-aided, and SIM-aided beamforming, in terms of flexibility of the beamforming matrix, hardware complexity, and computational complexity. As a result of this comparison, five benefits of MiLAC-aided beamforming are identified: 1) it provides flexibility (e.g., enabling the maximum flexibility), 2) it allows a reduction in the number of RF chains, and may minimize a required number of RF chains, 3) it may be implemented using (only) low-resolution ADCs / DACs, 4) computation on a per-symbol basis may be avoided, and 5) it allows a reduction in a number of computations per coherence block for ZF (e.g., it may require the minimum number of computations).

[0149] Numerical results are provided to assess the performance of MiLAC-aided beamforming in terms of sum rate and bit error rate (BER), confirming that it can achieve the same performance as digital beamforming. The benefits of MiLAC-aided beamforming over digital beamforming are assessed in terms of computational complexity, showing that it can perform ZF, MF, and the DFT with a gain in computational complexity of 1.5xl04, 2.0xl02, and 4.0xl07times, respectively. Thus, in a gigantic MIMO system with thousands of antennas, MiLAC-aided beamforming may achieve the same performance as digital ZF beamforming at just 1 / 15000th of the computational cost. From a different perspective, a 4096-antenna MiLAC serving 4096 users via ZF beamforming requires the same computational complexity as a 256-antenna digital transmitter serving 256 users. Accordingly, MiLAC-aided beamforming may be applied in gigantic MIMO, allowing for new transceiver architectures that can massively scale the number of antennas and users.VII. ANALOG COMPUTING FOR THE SPECIAL CASES OF THE LMMSE ESTIMATOR

[0150] Previous sections shown how a MiLAC can be used to compute the LMMSE estimator of a linear observation process with reduced computational complexity. The following derives and five special cases of the LMMSE estimator for linear observation processes, of particularinterest in wireless communications. These five special cases can be efficiently computed by a MiLAC.A. LMMSE Estimator for Linear Observation Processes

[0151] As described in Section III. A, xLMMSE;1and xLMMSE 2are given by (44) and (45) for the linear observation process y=Hx+n, and xLMMSE>1= xLMMSE 2whenXH +Cx1and HCXHH+ Cnare both invertible, which holds true in general.B. Special Cases of the LMMSE Estimator for Linear Observation Process

[0152] The LMMSE estimator in Section II- A was obtained with no assumptions on the covariance matrices Cx and Cn beyond their invertibility. With specific assumptions on C and Cn, it can be shown that it includes five special cases of practical interest.

[0153] 1) Generalized Least Squares: In generalized least squares (GLS), also known as generalized linear regression, the variance on the a priori information on x is infinitely higher than on n, i.e., |CX|F» |Cn|F. Thus, C^1is negligible in (44) and, equivalently, Cn is negligible in (45), yieldingXGLS,1 = (H^C^^^H^C-^, (73)XGLS,2 = CxHH(HCxHH)-1y, (74)respectively. Note that, depending on the dimensions of H, only one matrixamong H^C^H and HCXHHmay be invertible. Thus, only one among theestimates xGLS;1and xGLS 2may exist.

[0154] 2) Generalized Matched Filtering'. Opposite to GLS, it may be assumed that the variance on the a priori information on x is infinitely smaller than on n, i.e., II CXIIF« II Cn IIF. In this case, denoted as generalized matched filtering (GMF), both (44) and (45) simplify to XGMF = CxHHCnXy, (75)

[0155] 3) Regularized Least Squares'. In regularized least squares (RLS), also known as ridge regression, the entries of x and n are uncorrelated and have equal variance, i.e., Cx=Axlvand Cn=XnIy, with W G IR. and Xx, Xn>0. Thus, by introducing A G K. such that A=An / Ax, (44) and (45) simplify asxRLS,i = (HHH + AIx)-1HHy, (76)XRLS,2=HH(HHH+ AIy)-1y, (77)respectively.

[0156] The expressions in (76) and (77) are widely used in MIMO wireless communications to realize the so-called regularized zero-forcing beamforming (R-ZFBF) and the minimum mean square error (MMSE) receiver, as effective linear transmission and reception techniques, respectively.

[0157] 4) Ordinary Least Squares'. In ordinary least squares (OLS), also simply known as LS, or linear regression, the entries of x and n are uncorrelated and have equal variance, i.e., Cx=AxIv and Cn=XnIy, and the variance on the a priori information on x is much higher than on n, i.e., X »X.n. Thus, since A,=An / A,x« 1, (76) and (77) further simplify asx0LS,i = (HHH)-1HHy, (78)XOLS,2= H^HH^^y, (79)respectively.

[0158] OLS is a special case of both GLS and RLS as it includes both their assumptions on Cx and Cn. As discussed for the GLS, since only one matrix among fV H and HHflmay be invertible depending on the dimensions of H, so only one among x0LS;1and x0LS 2may exist, unless H is a square matrix.

[0159] The expressions of the OLS are widely employed in MIMO wireless communications to implement zero-forcing beamforming (ZFBF) and the ZF receiver.

[0160] 5) Ordinary Matched Filtering'. In ordinary matched filtering (OMF), also simply known as MF, the entries of x and n are uncorrelated and have equal variance, i.e., Cx=AxIv and Cn=XnIy, and the variance on the a priori information on x is much smaller than on n, i.e., Xx«X.n. Thus, since A=An / Ax»l, both (76) and (77) simplify toXQMF =-1HHy. (80)

[0161] OMF is a special case of both GMF and RLS as it jointly accounts for their assumptions on Cx and Cn. In addition, it is utilized in MIMO wireless communications to realize matched beamforming (MBF) and the MF receiver.

[0162] The five special cases of the LMMSE estimator for linear observation processes and their assumptions on Cx and Cn are summarized in Table 3.Table 3SPECIAL CASES OF THE LMMSE ESTIMATOR AND THEIR ASSUMPTAssumptions on Cx and CnLMMSE NoneGLS l|Cx||F» ||Cn||FGMF IICxllF«IICnllFRES C\=L\Iv, Cn=XnIyOLS C\=L\Iv, Cn=XnIy, Xxz’z’XnOMF C\=L\Iv, Cn=XnIy, Lx « AnC. Computing the Special Cases of the LMMSE Estimator with a MiLAC

[0163] The expressions of the five special cases of the LMMSE estimator can be efficiently returned on the output vector V2 of a MiLAC having N=Y input ports 106a and M=X output ports 106b with no input, i.e., P=X+Y total ports 106. This is achieved by exploiting (28) and (30), as discussed for the LMMSE estimator. The input vector of the MiLAC is set as u=y and the admittance values of the tunable admittance components [T(areset according to (32)such that the blocks of the matrix P give the desired expression in V2. In detail, the expressions of the special cases in (73)-(80) can be computed by setting the matrix P as illustrated in Table 4 and Table 5, depending on the invertibility of the blocks Pn and P22. When the block Pn is invertible, V2 is given by (28), as reported in Table 4, while when the block P22 is invertible, V2 is given by (30), as reported in Table 5. Note that GLS and OLS are computed via a matrix P whose blocks Pn and P22 are not both invertible. Thus, for GLS and OLS the expressions in Table 4 and Table 5 are not equivalent in general, while they are equivalent for the LMMSE estimator and the other special cases.Table 4MILAC OUTPUT v2FOR DIFFERENT P WITH Pn INVERTIBLEv2Pll P12P21 P22 ±CnH LMMSE (H / / CnH + C-^’WCn1!!HH+C;1r±cnHl GLS (H^C^HrWCn1!![ H" oj +CnOyxX GMF C^L^UHH+C;1±Iy H RLS (HHH + AIX)-1HHUHH+AIyOLS (HHH)-1HHU+Iy OyxX OMF A H' uHH+AIyTable 5MILAC OUTPUT v2FOR DIFFERENT P WITH P22INVERTIBLEv2Pll P12P 1+2S1 P22±CnH - LMMSE CXHH(HCXHH+ Cn)-1uHH+C O EC71* _ 0YH GLS CXHH(HCXHH)-1UHH+C71+CnOyxX - GMF CxH'^^uHH+C71•±AIrH RLS H"(HH" + / Jj uH" +IX.Oy HOLS H"(HH") uHH+IX.- + AIy Oyxx - OMF A H' uHtf+1*

[0164] The five special cases of the LMMSE estimator can be efficiently computed using a MiLAC, which offers significantly reduced computational complexity compared to a digital computer. This is because a MiLAC directly performs matrix-matrix multiplications and matrix inversion operations in the analog domain. On the one hand, the computational complexity of computing the special cases with a MiLAC is driven by the complexity of computing (32) for agiven matrix P. Thus, the complexity required to compute GLS, RLS, and OLS with a MiLAC is the same as the complexity of computing the LMMSE estimator, i.e., 6AK real operations, as described in Section III. B. Computing (32) for the GMF and OMF require approximately 2XY real operations for both cases i #= k and i=k, resulting in a total of 4 T real operations, where it is assumed that the operations involving C and Cn are precomputed offline. On the other hand, a digital computer calculates the GLS with the same complexity as the LMMSE estimator due to the two matrix-matrix products and matrix inversion, i.e., 8(XE2+X2E+min{A3, Y3} / 3), as described in Section III. B. The RLS and OLS can be computed with reduced complexity as they require only one matrix-matrix product and a matrix inversion, namely 8min [A^+A'G, XY2+Y3 / 3}. The GMF and OMF only require three and one matrix-vector products, respectively, resulting in a complexity of 8(X2+AT+F2) and 8 7 real operations, respectively.

[0165] In some examples, ||CX||F» ||Cn||F, where ||A||Fis the Frobenius norm of A, such that the LMMSE estimator corresponds with a generalized least squares, GLS, estimator. In other examples, ||CX||F« || Cn||F, such that the LMMSE estimator corresponds with a generalized matched filtering, GMF, estimator. In other examples, Cx=XIXand Cn= AnIY, such that the LMMSE estimator corresponds with a regularized least squares, RLS, estimator. In other examples, Cx= AXIX, Cn= AnIy, and Ax» Ansuch that the LMMSE estimator corresponds with an ordinary least squares, OLS, estimator. In other examples, Cx— ^x, Cn— AnIy, and Ax« Ansuch that the LMMSE estimator corresponds with an ordinary matched filtering, OMF, estimator. The first matrix may correspond with P described in Table 4 or Table 5.VIII. MILAC-AIDED BEAMFORMING

[0166] This section describes an arrangement that uses a MiLAC to implement beamforming, referred to herein as MiLAC-aided beamforming, at the transmitter as well as receiver side. Embodiments of MiLAC-aided beamforming may have the same flexibility as digital beamforming, while offering circuit and computational complexity benefits by allowing operation that is fully in the analog domain.

[0167] In a MIMO communication system between NT transmitting antennas and NR receiving antennas through the wireless channel H6 transmitted signal is denoted as x 6 CNrX1and the received signal as y G CNRX1such that y=Hx+n, where n G CNRX1is the additivewhite Gaussian noise (AWGN). The system here is a general MIMO system, such as single-user MIMO or multi-user with one symbol per user. Single- and multi-user systems are specifically described below.A. Arbitrary Beamforming

[0168] MiLAC aided beamforming can be used to implement an arbitrary beamforming matrix.

[0169] It is assumed that a transmitter linearly precodes a vector of Ns symbols s 6 CNsX1to obtain the transmitted signal as x=Ws, where W 6 NTN js an arbitrary precoding matrix. Such a transmitter can be implemented purely in the analog domain by a MiLAC having N = Ns ports 106a receiving the symbol vector in input and M=NT ports 106b delivering the output to antennas matched (e.g., perfectly matched) to the reference impedance Zo= yo-1, as represented in FIG. 10A. Where the antennas are perfectly matched, their Thevenin equivalent circuit is an impedance Zo= Fo-1connected to ground. Thus, the MiLAC in FIG. 10A behaves as if its last NT ports 106b are connected to ground through an impedance Zo= Fo-1, i.e., as analyzed in relation to FIG. 1. Accordingly, MiLAC returns x=Ws on its last Ar ports 106b when the signal s is fed as input on the first Ns ports and the tunable admittance components 202 are set such that the matrix P is®NSXNP =T-W (81)1NT. ’according to (28) and (30). To this end, the tunable admittance components 202 are set based on (32) where P is given by (81), yieldingNs< i,k < Ns(82)0 Otherwise

[0170] for iyk. andYk,k= -KoZXi tWUk^Ns (83)0 k > Ns’for A=1,..., S+ T’. From (82) and (83), the tunable admittance components 202 of a MiLAC may be reconfigured in closed form to implement any arbitrary beamforming W matrix in the analog domain. The computational complexity of this beamforming technique, denoted as MiLAC-aided beamforming, is given by the complexity of designing W (depending on the adopted precoding strategy) and computing (82) and (83) (requiring in total 4 s r realoperations). Note that these operations may be executed at each coherence block, while no operation is required on a per-symbol basis, since the matrix-vector product Ws is performed in the analog domain.

[0171] As described on the transmitter side, MiLAC-aided beamforming can also be introduced on the receiver side. Specifically, a MiLAC can be used to linearly combine the received signal y G CNRX1to obtain the signal used for detection z G CNsX1as z=Gy, where G G (^NSXNR-s an arbitraiy combining matrix. This can be obtained with MiLAC having N=NR ports 106a acquiring the input from the receiving antennas and M=Ns ports 106b providing the output vector z, as illustrated in FIG. 10B. In FIG. 10B it is assumed that the antennas are perfectly matched to the reference impedance Zo= Ko-1, such that their Thevenin equivalent circuit is a voltage generator (inducing a voltage ynat port n, for n=l,..., N) in series to an impedance Zo= yr-io •

[0172] Among the possible linear precoding and combining strategies, MiLAC-aided beamforming can efficiently perform are RZFBF, ZFBF, and MBF at the transmitter and MMSE, ZF, and MF at the receiver by exploiting the capability of a MiLAC to compute the corresponding special case of the LMMSE estimator. This leads to significant computational complexity benefits, as illustrated below.

[0173] Accordingly, in some examples, the voltage sources may be connected with respective ports of a first set of the ports, and a plurality of antennas may be respectively connected with ports of a second set of ports. The voltage sources may be based on a vector of symbols to be transmitted by the plurality of antennas and the first matrix may be based on a precoding matrix relating the symbols to target transmitted signals for transmission by the plurality of antennas, such that output signals at respective ports of the second set of ports cause the plurality of antennas to transmit the target transmitted signals.

[0174] The first matrix may include a submatrix corresponding with a precoding matrix for beamforming a multiple-input multiple-output, MIMO, transmission, the voltage sources may be indicative of a vector of symbols to be transmitted, and the respective output voltages may correspond with signals to be provided to respective antennas to transmit the beamformed MIMO transmission.0N×N

[0175] In some examples, the first matrix may be P =, where INis an N*N— Widentity matrix, 0\ \ / is an N*M matrix of zeros, and W is the precoding matrix.

[0176] In some examples, the voltage sources may be antennas electrically connected with respective ports of a first set of the ports, and a second set of ports may be electrically connected with ground. The first matrix may be based on a combining matrix and signals measured at the second set of ports may correspond with a result of the combining matrix operating on symbols received at the first set of ports via the antennas.

[0177] In some examples, the voltage sources may correspond with a beamformed multipleinput multiple-output, MIMO, signal. For example, the voltage sources may represent signals generated at the output of RF chains of a MIMO transmitter. The first matrix may include a submatrix corresponding with a combining matrix for processing the beamformed MIMO signal. The output voltages may correspond with the result of the combining matrix operating on the beamformed MIMO signal.

[0178]

[0179] In some examples the first matrix may be P =, where INis an N*N— Gidentity matrix, 0 \- \- is an N^N matrix of zeros, and G is the combining matrix.B. LMMSE-Inspired Beamforming at the Transmitter: R-ZFBF, ZFBF, and MBF Transmitters

[0180] Assume that the previously considered MIMO system y=Hx+n is a multi-user system between an A / -antenna transmitter and NR single-antenna users, with NR< NT. The signal received at the NR users y is used to detect the symbols s 6 CNsX1, with NS=NR, and the transmitted signal is given by x=Ws, where W 6NMR -Sprecociing matrix implemented by a MiLAC, as in FIG. 10A. By exploiting the operations included in Table 5, it is possible to reconfigure the microwave network 102 of the MiLAC to efficiently execute three popular linear transmitters. First, a MiLAC computing RLS can realize R-ZFBF, with precoding matrixWR -ZFBF=H / / (HH / / +ZI\)1, where is an arbitrary regularization factor. Second, ZFBF can be implemented by a MiLAC computing OLS, whose precoding matrix is WZFBF=H / / (HH / / )~I. Third, a MiLAC computing OMF can realize MBF, with precoding matrix given by WMBF=-1H / Z. These three transmitters are obtained by reconfiguring the tunable admittancecomponents 202 of the MiLAC according to the channel H on a per coherence block basis, and do not require any operation on a per-symbol basis.

[0181] As described in Section VII. C, the R-ZFBF and ZFBF transmitters can be achieved with only 6NTNR real operations, while the MBF transmitter with only 47VT VR real operations, which are necessary to set the tunable admittance components 202 of the MiLAC. Conversely, the computational complexity of digitally calculating the R-ZFBF and ZFBF transmitters is 8(NTNR + lVp / 3) real operations, while for the MBF transmitter is SNTNR. This confirms the significant benefits of MiLAC-aided beamforming in terms of computational complexity, especially in implementing the R-ZFBF and ZFBF transmitters.

[0182] Accordingly, the first matrix may be based on a precoding matrix that implements a beamforming method selected from regularized zero-forcing beamforming, R-ZFBF, zeroforcing beamforming, ZFBF, and matched beamforming, MBF.C. LMMSE-Inspired Beamforming at the Receiver: MMSE, ZF, and MF Receivers

[0183] Assume that the MEMO system y=Hx+n is a single user system between an A / -antenna transmitter and a A / ?-antenna receiver, with NR> NT. The transmitted signal x contains the symbols to be detected at the receiver, i.e., x=s, and the receiver uses the signal z = Gy G CNrX1to accurately detect the symbols, where G G C / VT’X / VRis the combining matrix imposed by the MiLAC, as in FIG. 10B. By using the operations in Table 4, it is possible to reconfigure the microwave network 102 of the MiLAC to efficiently implement three widely used receivers. First, a MiLAC computing RLS can realize the MMSE receiver with combining matrix GMMSE=(H / 7H+ EV7’ )-1H77. If the covariance matrix of the transmitted signal x is Cx= PT / NTINT, where PT is the transmit power, and the covariance matrix of the noise n is Cn= &nINR, where cf is the noise power, the optimal regularization factor X is given by A = NTcf / PT. Second, the ZF receiver can be realized by a MiLAC computing OLS, whose combining matrix is GZF=(H / 7H)-1H / 7. Third, a MiLAC computing OMF can implement the MF receiver, with combining matrix GMF= CIH / / , where the optimal A is A = NTa^ / PTif Cx= PT / NTINTand Cn=cf. These three receivers may be realized solely reconfiguring the MiLAC on a per coherence block basis, while no operation is required on a per-symbol basis.

[0184] As described in Section VIII. B, the MMSE and ZF receivers can be obtained through a MiLAC with only 6NTNR real operations and the MF receiver with only 47V 7V / ? real operations,which are required to set the tunable admittance components 202. In contrast, the computational complexity of digitally calculating the MMSE and ZF receiversis 8(NRNR+ lV^ / 3) real operations, and for the MF receiver is SNTNR, showing significant benefits in using MiLAC-aided beamforming to implement the MMSE and ZF receivers.

[0185] Accordingly, in an example, a method for MIMO wireless communication may include implementing regularized zero-forcing beamforming, R-ZFBF, or a minimum mean square error, MMSE, receiver by obtaining the RLS estimator as described herein. In other examples, a method for MIMO wireless communication may include implementing zero-forcing beamforming, ZFBF, or a ZF receiver by obtaining the OLS estimator as described herein. In other examples, a method for MIMO wireless communication may include implementing matched beamforming, MBF, or a matched beamforming receiver by obtaining the OMF estimator as described herein. The first matrix may be as described in Table 4 or Table 5.

[0186] In some examples, the combining matrix may implement a minimum mean square error, MMSE, receiver, a ZF receiver, or a matched beamforming receiver.D. Computing the DFT

[0187] MiLAC-aided beamforming may also be beneficial when used to precode or combine a signal with a fixed beamforming matrix. In this case, since the admittance components of the MiLAC are optimized offline, no operation is required online with MiLAC-aided beamforming, while a matrix-vector product is needed on a per-symbol basis in the case of digital beamforming. Combining the received signal y with a fixed matrix G is particularly useful for DoA estimation, where the received signal is typically multiplied by the DFT matrix. This operation performed at the receiver can be formalized as z=Gy, where G 6 CWRX / VRis the DFT matrix, given byNr, (84)for i,k=\,..., NR. This operation can be implemented purely in the analog domain by a MiLAC having N=NR ports 106a acquiring the input y from the receiving antennas and M=NR ports 106b returning the output, as represented in FIG. 10B. To perform such a DFT operation, the admittance components of the MiLAC may be set offline following (32) where the matrix P is given byp = I1":OVX'’“], (85)L-l* 1JVRwith G being the DFT matrix in (84). Thus, by substituting (84) and (85) into (32), we obtain that the admittance componentsxare set asvY- \^=e NrNR< i,k < Ni,k ~ j / NRR’ (86)v 0 Otherwisefor i k, andYk,k =k = 1, (87)t 0 k #= 1for k=l,...,2NR.

[0188] By employing a MiLAC with fixed admittance components set as in (86) and (87), the DFT may be performed instantly in the analog domain requiring no digital operations.Conversely, the DFT requires approximately 34 / 97V / ?log2(A / ?) real operations to be digitally computed, which could become prohibitive as the number of antennas NR grows large. A onedimensional DFT has been assumed for simplicity, while a similar discussion also holds for performing a two-dimensional DFT in the analog domain, as described in J. An, C. Yuen, Y. L. Guan, M. Di Renzo, M. Debbah, H. Vincent Poor, and L. Hanzo, “Two-dimensional direction-of-arrival estimation using stacked intelligent metasurfaces,” IEEE J. Sei. Areas Commun., vol.42, no. 10, pp. 2786-2802, 2024.IX. COMPARISON WITH EXISTING BEAMFORMING STRATEGIES

[0189] In this section the performance of MiLAC-aided beamforming is compared with state-of-the-art beamforming strategies, showing that in some embodiments MiLAC-aided beamforming may be implemented with fewer RF chains, lower-resolution ADCs / DACs, and / or reduced computational complexity. These examples assume a transmitter with NT antennas and A RF chains transmitting Ns symbols, where Ns< NRF< NR. The symbol vector is denoted as s 6 CNsX1and the transmitted signal is denoted as x 6 (Cwrxi y|iefonowing discussion for a transmitter readily applies also for a receiver.

[0190] Each RF chain may be a series of components, such as amplifiers, filters, mixers, attenuators and detectors, that process RF signals for transmission or reception.A. Digital Beamforming

[0191] FIG. 11 A shows an example of an arrangement 1104 for digital beamforming. Ns symbols are received at inputs 1106. The transmitted signal is given by x=W / >s, where WD6 CNTXNSdescribes the operation of the digital precoder 1110, subject only to the power constraint. To allow flexibility, there are NRF=NT RF chains 1102 equipped with high resolution ADCs / DACs, each carrying an entry of the transmitted signal x to the corresponding antenna 1108. In addition, the computational complexity is driven by the matrix vector product Wps executed on a per-symbol basis and by the optimization of WD at each coherence block based on the channel realization, complexity of which depends on the specific digital precoding strategy.

[0192] Here, a high resolution ADC / DAC may have, for example, from 7 to 13 bits (or more) of resolution. For example, an analog value may be converted into a digital value with 27=128 to 213=8192 discrete possible values. In contrast, a low resolution ADC / DAC may have a lower resolution than a high resolution ADC / DAC. For example, a resolution as low as 1 bit. For example, an analog value may be converted into a digital value that can take only 21=2 possible values.B. Analog Beamforming

[0193] As example of an arrangement 1116 for analog beamforming is illustrated in FIG. 1 IB. In analog beamforming, only one symbol s G C is transmitted (Ns = 1), and the transmitted signal is given by x = WAS, where wAG CNrX1describes the operation of the analog precoder 1112. The symbol 5 is received at input 1114. Since the symbol 5 reaches the NT antennas by passing through NT phase shifters, the analog precoder is subject to wAE TN7’X1,where T = {c G C: | c | = 1], due to the unit-modulus constraint imposed by the phase shifters. In this case, a single RF chain 1102 is sufficient to carry the symbol s, which can be equipped with low-resolution ADCs / DACs since the symbol 5 lies in a constellation with finite cardinality. The computational complexity of analog beamforming is given by the complexity of optimizing wi at every coherence block as a function of the channel realization. Note that no operation takes place on a per-symbol basis since the product WAS is performed in the analog domain as the signal flows through the phase shifters connected to all antennas.C. Hybrid Beamforming

[0194] FIG. 11C illustrates an example of a system 1118 for hybrid beamforming with Ns symbols provided at inputs 1120. In hybrid beamforming, the transmitted signal is x=WiW / >s, where wAG CNTXNRFdescribes the analog precoder 1124, subject to specific constraints depending on the architecture of phase shifters and switches, and wDG CNRFXNSis the digital precoder 1126, subject only to the power constraint. Thus, the effective precoding matrix is given by W = W^WD. Numerous hybrid beamforming architectures have been proposed, each with different constraints on the matrix W^. For example, in the fully-connected architecture, each RF chain 1102 is connected to all the antennas through a phase shifter, and each entry of W i is subject to the unit-modulus constraint, i.e.,GNT*NRFybecompUtational complexity of hybrid beamforming is given by the complexity of computing the matrix-vector product WDS on a per-symbol basis, and optimizing W i and WD at each coherence block.D. RIS-Aided Beamforming

[0195] FIG. 1 ID illustrates an arrangement 1130 for RIS-aided beamforming. RIS-aided beamforming exploits a RIS 1128 deployed in close proximity with an active transmitting device. The RIS 1128 can work in either reflective or transmissive mode, without changing the system model. Ns symbols provided at inputs 1122. Denoting the digitally precoded signal transmitted by the active device as X -WDS, the received signal is given by y=HEFFx', where HEFFG CNKXNKF is the RIS-aided channel, given by HEFF=H0HRIS, with H GQE(£NTXNT,anj HRISG CNrxNRF being the channel from the RIS 1128 to the receiver, the scattering matrix of the RIS 1128, and the channel from the active transmitting device to the RIS 1128, respectively. Thus, effective precoding matrix is W=0HRISWD, which is optimized by reconfiguring 0, subject to 0H0 =in the case of a lossless RIS 1128, and WD, subject only to the power constraint. As in hybrid beamforming, the matrix- vector product W / >s needs to be computed on a per-symbol basis, while W / > and 0 are optimized per coherence block.E. SIM-Aided Beamforming

[0196] An example of a system 1132 for SIM-aided beamforming is illustrated in FIG. 1 IE. RIS-aided beamforming is extended by stacking multiple transmissive RISs 1128 at the transmitter, which is denoted in the literature as a SIM 1136. Given the additional flexibilityprovided by the multiple metasurfaces, SIM-aided beamforming does not include any digital precoding, and for this reason it only requires NRF=NS RF chains equipped with low-resolution ADCs / DACs. Similar to RIS-aided beamforming, SIM-aided beamforming exploits the fact that the effective channel between the transmitter and receiver can be engineered by reconfiguring the scattering matrices of the stacked RISs. Nssymbols are received and processed via Ns RF chains 1102 for transmission by NRF=NS antennas 1108. Assuming a SIM made of L stacked RISs each with NT elements, the effective channel between the active device transmitting the symbols s and the receiver is given byHEFF = H0LHSIM L...02HSIM 201HSIM 1, (88)where H 6 I£NRXNT, g ^NTXNT,ancj HSIMJ G i£fTxNT are channeifromthe Ath RIS to the receiver, the scattering matrix of the / th RIS, and the channel from the / -1th to the / th RIS (where the Oth RIS is the transmitting device and HSIM>16 CNTXNRF), respectively. Thus, the effective precoding matrix isW = 0LHSIMt...02HSIM;201HSIM;1, (89)which is optimized per coherence block by reconfiguring the scattering matrices of the L stacked RISs 0 / , for

[0197] SIM-aided beamforming is entirely performed in the analog domain. The flexibility of the beamforming matrix is constrained as in (89) and cannot be an arbitrary matrix.F. MiLAC-Aided Beamforming

[0198] FIG. 1 IF illustrates an arrangement for MiLAC-aided beamforming. MiLAC-aided beamforming brings five unique benefits over current beamforming strategies. Specifically, it offers high flexibility with low hardware complexity (in terms of the number of RF chains and resolution of ADCs / DACs) and low computational complexity (in terms of operations per-symbol basis and per coherence block).

[0199] 1) Flexibility. Embodiments of MiLAC-aided beamforming has the same flexibility as digital beamforming since it can apply any arbitrary beamforming matrix W, as described in Section VIII. Given its full flexibility, MiLAC-aided beamforming can obtain the same performance as digital beamforming, outperforming other beamforming strategies that impose limiting constraints on W, i.e., analog, hybrid, RIS-aided, and SIM-aided.

[0200] 2) Number of RF Chains'. In MiLAC-aided beamforming, the symbols are input to the MiLAC, which computes the transmitted signal in the analog domain. Thus, the full flexibility of digital beamforming may be achieved with only NRF=NS RF, which is the minimum number of RF chains required by the reviewed beamforming strategies. Note that a fully-connected hybrid beamforming architecture requires NRF 2NS RF chains to realize any digital beamforming matrix, while the other beamforming strategies are not proven to achieve full flexibility with any NRF< NT.

[0201] 3) Low -resolution ADCs / DACs: Since MiLAC-aided beamforming processes the symbols purely in the analog domain, each RF chain carries an individual symbol, which lies in a constellation with finite cardinality. Thus, low-resolution ADCs / DACs can be adopted without sacrificing performance. Low-resolution ADCs / DACs are less expensive and less power-hungry than high-resolution ADCs / DACs. High-resolution ADCs / DACs are necessary when beamforming is performed in the digital domain (entirely or in part), i.e., in digital beamforming, hybrid beamforming, and RIS-aided beamforming.

[0202] 4) No Computations on a Per-Symbol Basis: If beamforming is performed in the digital domain (entirely or in part), the matrix-vector product W / >s needs to be computed on a per-symbol basis. The complexity of this operation is SNRFNS real operations, as described in Section XII, and can become prohibitive in massive / gigantic MIMO systems. MiLAC-aided beamforming solves this issue since it does not require any computation on a per-symbol basis, as the beamforming is purely analog. This reduces the beamforming computational complexity, especially when the coherence time is much longer than the symbol time.

[0203] 5) Computations Per Coherence Block for RLS and OLS: MiLAC-aided beamforming offers further gains in terms of computational complexity by exploiting the capability of a MiLAC to efficiently compute RLS and OLS in the analog domain. Specifically, these gains can be obtained when MiLAC-aided beamforming is used to compute the R-ZFBF and ZFBF transmitters or the MMSE and ZF receivers, as discussed in Section VIII.

[0204] Table 6 summarizes and compares the beamforming strategies described in this section.Table 6BEAMFORMING STRATEGIES COMPARISONNumber Resolution Operations Operations BeamforminPrecoding of of per per g ConstraintsMatrix RF ADCs / symbol coherence strategychains DACs basis block Compute Optimize Digital WD wDe cNrXNsNRF=NT HighWDS WD NRF=NS=Analog VfA W4 e lNrX1Low None Optimize wx wDe CNRFXNS Compute Optimize Hybrid WAWD WAE JNTXNRF NRF> NS HighWDS WD, W I wDe CNRFXNSHRISe cN7’XN«F,Compute Optimize RIS-aided 0HRISWDfixed NRF> NS High0e GNTXNT)QH 0 WDS WD, 0HSIM, I 6 CNTXNRR,fixedHSIMJe C^XJVuSIM-aided 0LHSIM, L. ■ ■ NRF=NS Low None Optimize 0(■ ■ ■ ®1HSI l>2, fixedM,10; e(£NTXNT'0H0IMiLAC- \NAE CNTXNSLow None Optimize NRF=NSaidedX. NUMERICAL RESULTS

[0205] In this section, the performance and computational complexity of MiLAC-aided beamforming is evaluated.A. Performance Evaluation

[0206] Since MiLAC-aided beamforming can implement any arbitrary beamforming matrix, it is able to perform as digital beamforming and outperform other beamforming strategies that are limited by constrained beamforming matrices. The specific performance gains over the reviewed beamforming strategies match those of digital beamforming, which depend on the performance metric and scenario, such as whether the transmission is single- or multi-stream, and whether the system is single- or multi-user. The following analyzes cases where MiLAC-aided beamforming leverages the computing capabilities of a MiLAC to implement theLMMSE-inspired beamforming matrices, i.e., R-ZFBF, ZFBF, and MBF at the transmitter, and the MMSE, ZF, and MF at the receiver.

[0207] To evaluate the performance of R-ZFBF, ZFBF, and MBF realized with digital and MiLAC-aided beamforming, we consider a multi-user multiple-input single-output (MISO) system with an A / -antenna transmitter and NR single-antenna receivers, as presented in Section VIII-B. Here, the transmitter precodes the NR symbols s 6 CNRX1intended to the NR receivers through R-ZFBF or ZFBF to suppress inter-stream interference, or through MBF to maximize the channel gains. With digital beamforming, the R-ZFBF transmitter is realized by computing FR-ZFBF= HH(HHH+ and setting the precoding matrix WR-ZFBF such that [WR-ZFBF]: nR_ [FR-ZFBF]:, HR, for nR=l,..., NR, assuming uniform power / II >'‘R H2allocation to the users. For large NR and if the signal-to-noise ratio (SNR) is the same at all receiving antennas, the optimal X is A = ^R<7nis the transmit power, i.e., s hascovariance matrix Cs=T / ]JRI,an is the noise power, and it is assumed the channel has unit path gain. Similarly, digital beamforming can realize ZFBF by computing FZFBF= H'^HH")’1and setting WZFBF as [WZFBF].nfl= ^FzrB^:'nR / |r„,,, for nR=l,..., NR, and / || L zFBFJ:,nR||2MBF by using FMBF=H / Zand WMBF such that [WMBF].nR= ^MBFI IR / , for / || L MBFJ:, H^ H2nR= 1,..., NR. When a MiLAC is used to perform R-ZFBF, ZFBF, and MBF, one option is to reconfigure the MiLAC to implement an arbitrary beamforming matrix, as described in Section VIII-A. Thus, its tunable admittance components can be set according to (82) and (83), where W is the desired beamforming matrix WR-ZFBF, WZFBF, or WMBF with uniform power allocation, which may be digitally computed. A second option is to use the MiLAC to directly compute the precoding matrices of these LMMSE-inspired beamforming techniques in the analog domain, as explained in Section VIII-B. In this case, the columns of WR-ZFBF, WZFBF, and WMBF are not individually normalized, and the resulting precoding matrices are given by WR-ZFBF=WZFBFII FR- ZFBF II IIFZFBFIIFAND WMBFIIFMBF IIF’where the normalization factor has been included to ensure that the transmit power is the same as with digital beamforming.

[0208] FIG. 12 shows the sum rate versus the SNRachieved by digital and MiLAC- °naided beamforming, where NT=NR=4 and the channels are independent and identically distributed (i.i.d.) Rayleigh distributed with unit path gain.

[0209] Plot 1202 shows the result for R-ZFBF using digital beamforming, plot 1204, which coincides with plot 1202 shows the result for R-ZFBF using a MiLAC that is reconfigured based on the arbitrary beamforming matrix computed digitally, and plot 1206 shows the result for R-ZFBF using a MiLAC computes LMMSE-inspired beamforming matrix in the analog domain.

[0210] Plot 1208 shows the result for ZFBF using digital beamforming, plot 1210, which coincides with plot 1208 shows the result for ZFBF using a MiLAC that is reconfigured based on the arbitrary beamforming matrix computed digitally, and plot 1212 shows the result for ZFBF using a MiLAC that computes LMMSE-inspired beamforming matrix in the analog domain.

[0211] Plot 1214 shows the result for MBF using digital beamforming, plot 1216, which coincides with plot 1214 shows the result for MBF using a MiLAC that is reconfigured based on the arbitrary beamforming matrix computed digitally, and plot 1218, which coincides with plots 1214 and 1216, shows the result for MBF using a MiLAC that computes LMMSE-inspired beamforming matrix in the analog domain.

[0212] R-ZFBF is the best precoder, which corresponds with ZFBF at high SNR and with MBF at low SNR, for both digital and MiLAC-aided beamforming, as expected from MEMO theory. MiLAC-aided beamforming performs exactly as digital beamforming when the MiLAC is reconfigured based on the arbitrary beamforming matrix computed digitally, denoted as MiLAC (arbit.) in FIG. 12 (plot 1204). MiLAC-aided beamforming suffers a slight performance degradation at high SNR when the LMMSE-inspired beamforming matrix is computed in the analog domain, referred to as MiLAC (LMMSE) in FIG. 12. This occurs because, in this case, MiLAC-aided beamforming does not perform uniform power allocation, which is preferable at high SNR. At the cost of this slight performance degradation, computing the LMMSE-inspired beamforming matrices in the analog domain substantially reduces therequired computational complexity, as described below. To compare MiLAC-aided beamforming and digital beamforming at the receiver, a single-user MIMO system is considered, with an A / -antenna transmitter and an A / ?-antenna receiver, where the receiver combines the received signal through MMSE, ZF, or MF, as introduced in Section VIII-C. In this case, the LMMSE-inspired combining matrices GMMSE, GZF, and GMF defined in Section VIII-C for MiLAC-aided beamforming are also the optimal expressions of the MMSE, ZF, and MF receivers realized with digital beamforming.

[0213] FIG. 13 shows the BER versus the SNR I2obtained with digital and MiLAC-aided°nbeamforming with 7VT=7VR=4 and quadrature phase shift keying (QPSK) modulation.

[0214] Results for MMSE implemented digitally and using a MiLAC are shown in plots 1302 and 1304, respectively. Results for ZF implemented digitally and using a MiLAC are shown in plots 1306 and 1308, respectively. Results for MF implemented digitally and using a MiLAC are shown in plots 1310 and 1312, respectively.

[0215] Since MiLAC-aided beamforming can implement the same combining matrices as digital beamforming, the two beamforming strategies achieve exactly the same BER, and the corresponding digital and MiLAC-aided plots coincide with each other.B. Computational Complexity Evaluation

[0216] To quantify the benefits of MiLAC-aided beamforming in terms of computational complexity, a MIMO system with NT transmitting antennas and NR receiving antennas, with NT=NR, is considered, where MiLAC-aided beamforming is used at the transmitter to perform R-ZFBF, ZFBF, and MBF, at the receiver for MMSE, ZF, and MF, and for DFT, as described in Section VIII-B, VIII-C, and VIII-D. In the considered system, the beamforming matrix is reconfigured per coherence block, where each coherence block includes T symbol transmissions. The computational complexity is defined as the number of real operations per coherence block, as analyzed in the following.• Zero-forcing. For R-ZFBF and ZFBF at the transmitter, or MMSE and ZF at the receiver, MiLAC-aided beamforming includes 6N^ operations per coherence block, as specified in Sections VIII-B, and VIII-C, and no operation is required on a per-symbol basis. Conversely, with digital beamforming, these strategies require 8 I NR +N3 / \'3 ) °Perati°nsto design the beamforming matrix at each coherence block (see SectionsVIII-B, and VIII-C). In addition, a matrix-vector product (Ws or Gy) is performed on a per- symbol basis, requiringreal operations per symbol time and making the complexity per coherence block of digital beamforming 8 I NR + / 3 ) + BN^.• Matched filtering. For MBF at the transmitter, or MF at the receiver, MiLAC-aided beamforming includes 4NR operations per coherence block, as given in Sections VIII-B, and VIII-C, with no operation on a per-symbol basis. Performing these strategies with digital beamforming requires BNR real operations per symbol time to perform the matrixvector product Ws or Gy, while no computation is needed to design the beamforming matrix, making the complexity per coherence block BN^t.• DFT. Performing the DFT with MiLAC-aided beamforming requires no operation, since the MiLAC is not reconfigured at run-time (see Sections VIII-D). Conversely, computing the DFT digitally requires 34 / 91VRlog2(lVR) operations per symbol time, giving 34 / 91VRlog2(lVR)Toperations per coherence block.

[0217] FIG. 14 shows the computational complexity of digital and MiLAC-aided beamforming, as a function of the number of antennas NT=NR, by fixing T=100 symbols per coherence block.

[0218] Plots 1402 and 1404 show the results for ZF using digital and MiLAC-aided beamforming, respectively. Plots 1406 and 1408 show the results for MF using digital and MiLAC-aided beamforming, respectively. Plots 1410 and 1412 show the results for DFT using digital and MiLAC-aided beamforming, respectively.

[0219] For the cases of zero-forcing and matched filtering, MiLAC-aided beamforming demonstrates up to 1.5 104and 2.0 102times lower computational complexity compared to digital beamforming, respectively, when the number of antennas is 7V / ?=8192. There are two reasons for this significant reduction. First, MiLAC-aided beamforming does not require the computation of the product Ws, or Gy, on a per-symbol basis, which has significant complexity given the high value of T. Second, for zero-forcing, the complexity of MiLAC-aided beamforming scales with O( / V / ) instead of C(V / ) since the expensive computation of the beamforming matrix is not required. In the case of DFT, MiLAC-aided beamforming does notrequire any computation, saving up to 4.0 107operations per coherence block when V / ?=8192. Given this significant reduction in computational complexity, we expect a proportional reduction in central processing unit (CPU) energy consumption and processing time.C. Relationship Between Performance and Computational Complexity

[0220] This section addresses the relationship between achieved performance and the required computational complexity in digital and MiLAC-aided beamforming. A multi-user MISO system is considered with an Ar-antenna transmitter serving NR single-antenna receivers through R-ZFBF. For digital beamforming, the RZFBF beamforming matrix is considered to have uniform power allocation, as in Section X-A. For MiLAC-aided beamforming, the LMMSE-inspired design described in Section VIII-B is considered, where the computation of the R-ZFBF beamforming matrix is done in the analog domain. FIG. 15A to FIG. 15C shows the achieved sum rate and computational complexity for MiLAC-aided and digital RZFBF beamforming in the case of different values of NR and NT, where NR, NT G {256, 1024, 4096} and NR< NT, and SNR, where SNR G {0, 10, 20} dB. FIG. 15A shows results for SNR=0dB, with MiLAC-aided beamforming results to the left of line 1502, and the results for digital beamforming to the right of line 1502. FIG. 15B shows results for SNR=10dB, with MiLAC-aided beamforming results to the left of line 1504, and the results for digital beamforming to the right of line 1504. FIG. 15C shows results for SNR=20dB, with MiLAC-aided beamforming to the left of line 1506, and the results for digital beamforming to the right of line 1506.

[0221] The following three observations may be made. First, the sum rate increases with the number of transmit antennas NT, the number of receivers NR, and the SNR, for both MiLAC-aided and digital beamforming. The computational complexity increases with the number of transmit antennas NT and receivers NR, for both MiLAC-aided and digital beamforming, while is independent of the SNR. Second, with a high number of transmit antennas, i.e., NT>256, the performance achieved by a MiLAC computing the R-ZFBF matrix in the analog domain as per Section VIII-B is the same as with digital beamforming with uniform power allocation. The computational benefits of MiLAC-aided beamforming may be exploited without sacrificing performance. Third, MiLAC-aided beamforming offers the same performance as digital beamforming for all the considered configurations of NR, NT, and SNR, with significant gains in computational complexity. Under a different perspective, MiLAC-aided beamforming can significantly improve the sum rate over digital beamforming given the same computationalcomplexity. For example, MiLAC-aided beamforming can operate R-ZFBF in a system with NR=NT=4Q96 requiring approximately the same complexity as digital R-ZFBF in a system with NR=NT= 256.XI. IMPLEMENTATION

[0222] Methods and systems as described herein may be implemented by, or make use of, one or more processors that process program code that is retrieved from a storage medium, such as a non-transitory storage medium. FIG. 16 shows an example of a system or device 1602 comprising a computer-readable storage medium 1606 (also referred to as a processor readable storage medium, a memory device, etc.) coupled to at least one processor 1604. The computer-readable medium 1606 can be any media that can contain, store, or maintain programs and data for use by or in connection with an instruction execution system. Computer-readable medium 1606 can comprise any one of many physical media such as, for example, electronic, magnetic, optical, electromagnetic, or semiconductor media. More specific examples of suitable machine-readable media include, but are not limited to, a hard drive, a random-access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory, or a portable disc. The processor 1604 may be a digital processor.

[0223] In FIG. 16, the computer-readable storage medium 1606 comprises (e.g., stores) one or more programming instructions 1608 (e.g., program code) to perform a method 1610. The method 1610 may be any of the methods described herein.

[0224] For example, the programming instructions 1608 may, when executed, cause the processor 1604 to perform the obtaining a first matrix and determining an admittance matrix, as described herein.

[0225] In some examples, the programming instructions 1608 may cause the processor 1604 to cause the admittance values of the controllable admittances based on the admittance matrix, cause the respective microwave frequency voltages to be applied to respective ports of the microwave network; and cause the respective output voltages to be measured.

[0226] In some examples, the one or more programming instructions, when executed, may cause the processor to: obtain a first matrix; determine, based on the first matrix, an admittance matrix, the admittance matrix such that when admittance values of controllable admittances between ports of a reconfigurable microwave network are set based on the admittance matrix, andrespective microwave frequency voltages are applied to respective ports of the microwave network, output voltages at the ports of the microwave network correspond with an output vector, the output vector based on an input vector and the first matrix, wherein the respective microwave frequency voltages are indicative of elements of the input vector.

[0227] FIG. 17 illustrates a system 1700 for configuring a reconfigurable micro wave network, the system 1700 comprising a computer readable medium 1606 and processor 1604 as described in relation to FIG. 16, and a reconfigurable micro wave network 102 as described herein. The processor may be arranged to control admittance values of controllable admittances of the microwave network 102.

[0228] In some examples, the processor 1604 may be arranged to control voltage sources 108 to apply voltages to respective input ports 106a of the micro wave network 102. In some examples, the applied voltages may be applied to input lines 110. The applied voltages may be applied to the input ports 106a via a series impedance 104a.

[0229] In some examples, the processor 1604 may be arranged to receive measured voltage values of ports 106 of the micro wave network 102.

[0230] FIG. 18 illustrates a system for MIMO communication according to an example. The system comprises a system as described in relation to FIG. 17, and a plurality of antennas 902 / 904 in electrical communication with respective ports of the reconfigurable microwave network.

[0231] Examples of the reconfigurable microwave network described herein may comprise a plurality of ports 106, wherein each pair of ports 106 is electrically interconnected by a respective admittance unit 202 of a plurality of admittance units, and wherein each port 106 is electrically connected with ground via a respective admittance unit 202 of the plurality of admittance units. An admittance of each admittance unit 202 of the plurality of admittance units may be individually controllable. According to some examples, a microwave device, or a microwave system, may include such a reconfigurable microwave network.

[0232] In some examples, a device may comprise the micro wave network 102 and a plurality of microwave voltage sources 108. Each micro wave voltage source 108 may be connected with a respective port 106 in a first set of the ports 106a via a series admittance 104a. Each port in a second set of the ports 106 may be electrically connected to ground via a respective series admittance 104b.

[0233] In some examples, for a predetermined reference admittance value, each series admittance 104 has an admittance value that is equal to the predetermined reference admittance value.

[0234] In some examples, a device may comprise the micro wave network 102 and a plurality of microwave voltage sources 108. Each micro wave voltage source 108 may be connected with a respective port 106 in a first set of the ports 106a via a series admittance 104a. Each port in a second set of the ports 106 may connected to an antenna.

[0235] FIG. 19 illustrates a device for MIMO communication, the device comprises a controller 1902, a plurality of radio frequency, RF, chains 1102, a plurality of antennas 1108, a microwave network 102 having a plurality of ports 106, the micro wave network 102 arranged to receive input from the plurality of RF chains 1102 at a first set of ports 106a of the plurality of ports 106, and output signals for transmission by the plurality of antennas 1108 at a second set of ports 106b of the plurality of ports 106, each pair of the plurality of ports 106 electrically interconnected by a respective admittance unit 202, and each port 106 of the plurality of ports electrically connected to ground by a respective admittance unit 202, wherein each of the admittance units 202 has a controllable admittance value. The controller 1902 may be arranged to control admittance values of the admittance units 202, such that the input is beamformed by the micro wave network 102 to generate the output. The controller 1902 may be implemented by a processor 1604 or other computing device, e.g., as described in relation to FIG. 16XII. COMPUTATIONAL COMPLEXITY OF MATRIX OPERATIONS

[0236] This section reviews the computational complexity of relevant matrix operations performed with classical digital computing. The computational complexity of an operation is defined as the number of arithmetic real operations required to complete it, where these operations include addition, subtraction, multiplication, and division. Following this definition, it is assumed that computing the transpose or the conjugate transpose of a matrix requires no operations. For operations on complex numbers, addition and subtraction require two real operations, while multiplication requires six real operations.1) Scalar Product'. Given two vectors a 6 < ClxNand b 6 (CNxl, the scalar product ab requires TV— 1 complex additions and N complex multiplications, since it is givenby [a]i[b]i+... +[a]N[b]N. Considering that complex additions and multiplications require twoand six real operations, respectively, the complexity of the scalar product is approximately 8N real operations.2) Matrix-Vector Product'. Given a matrix A 6 < CMxNand a vector b 6 (CNxl, the matrixvector product Ab can be expressed as Ab = a1b, a^b^1, where am6 ClxNis the / / th row of A, for m= \,..., M. Thus, it requires AT times the complexity of the scalar product ab, which is approximately 8 V real operations.3) Matrix-Matrix Product'. Given two matrices A 6 < CMxNand B 6 < CNxL, the matrixmatrix product AB can be expressed as AB = [Ab1(..., AbL],where b| 6 CNxlis the / th column of B, for l=l,..., L. Thus, it requires L times the complexity of the matrix-vector product Ab, which is approximately 13LMN real operations.4) Matrix Inversion '. Given an invertible matrix A 6 < CNxN, the algorithm commonly used to compute the matrix inverse A-1is the Gauss’s method. Thus, we regard the complexity of matrix inversion as equal to the complexity of the Gauss’s method algorithm. It can be shown that the Gauss’s method requires 7V(7V+l) / 2 divisions, (2N3+3N2~5N) / 6 multiplications, and (2N3+3N2~5N) / 6 subtractions. Consequently, the algorithm’s complexity is dominated by approximately N3 / 3 multiplications and N3 / 3 subtractions, which require a total of 8A' / 3 real operations.XIII. EXAMPLE EMBODIMENTS

[0237] According to some examples, a MiLAC circuit with reconfigurable response at microwave frequencies may comprise components that can be categorized into three design layers based on their functionality. These layers are: 1) the guides, 2) tunable elements (TEs), and 3) setup circuitry (SC). The guides may be microstrip lines, waveguides, parallel plates, etc. The TEs may be varactor diodes, PIN diodes, graphene, piezoelectric sensors, etc. The SC may be DC routing, wires, etc.

[0238] For low-frequency circuits, a reconfigurable response can also be granted by employing a set of tunable lumped elements where the guide would naturally be eliminated, allowing conventional voltag e / curr ent analysis. However, the following examples describe a hardware implementation at RF frequencies, making use of distributed elements.

[0239] In such a hierarchical design layer, the guide will have a specific response in the absence of components from other layers, this unloaded RF response may be indicated as(RFUL) herein. When the second and third layers of components are attached to the guides, the RF response can be reconfigured for different states of TEs, defined by the SC. Assume the case where all layers are connected together, but TEs are not set to alter the RF response. The RF reading response in this case may be referred to as loaded response RFL). In an ideal design, RFUL) = RFL, such that the response of the guides should ideally not be influenced by the physical proximity of other layers while cooperating with each other. In addition, this response should ideally be insensitive to environmental situations. In practice however, typically (RFUL) (RFL), as the physics of layers 2 and 3 components will generally perturb the field distribution of the guides. Even where closed structures, such as waveguides, form the building blocks of layer 1 components, attaching the components from other layers will generally involve some modifications of the geometry of the layer 1 components that will alter the response. Additionally, applying a wiring system in layer 3 may alter (RFL), and may also affect the repeatability and robustness of the response, particularly for arrangements having a large number of TEs in layer 2.

[0240] According to some examples, a design method may characterize (RFL), and then systematically use this information to design the MiLAC structure. The TEs, in this context, are to shift the phase in the guide. Hence, they TEs may also be referred to as “phase shifter (PS)” herein to refer to the function of the TEs. However, the TEs could also alter the amplitude of the guided waves as well, and this may also be taken into account.

[0241] The operation flow of a MiLAC according to some examples is shown in FIG. 20. The elements of the MiLAC include the guide and TEs, which together provide a tunable RF response. The control unit (CU) sets the state (e.g., controls the admittance) of the TEs, via the SC.

[0242] According to the example of FIG. 20, “characterising boards (CBs)” may be used to characterize, i.e. determine (RFL) for, each guide in the network. Each of the CBs is a circuit that is based on a respective portion of the MiLAC layout, as described below. The characterization of the guides may be performed via offline measurements. In examples, the MiLAC layout is designed before the characterization of the guides using the CBs.

[0243] For each guide there is a complex propagation constant, y=a+j0, with a and 0 representing attenuation rate and phase constant, respectively. The CBs may be used to determine y for each separate guide in the MiLAC board while that guide is connected tocomponents of other design layers of the network. With this, the RF response of the loaded guide of each section may be studied and characterised. This information may be used to translate online readings of the MiLAC circuit to a mathematical operation of interest.

[0244] According to an example, a microstrip line may be employed as the guide (representative of layer 1). The guided waves on this transmission line may be influenced by the surrounding environment. Associated with each line, there is a PS (layer 2) to alter the response of the MiLAC. This may be obtained by changing 0 and consequently, the electrical length of the line for different states of the PS. To specify different states of the PS, a set of DC bias lines (layer 3) connect to the PS to a control unit (CU). In this example, CU is not categorized as a design layer, as its properties do not influence the design mechanism of the MiLAC.

[0245] FIG. 21 A shows an example of a MiLAC structure. For the purpose of analysis, the RF layout may be split into several RF paths. In the layout, there are generally two types of paths: those that are to support a traveling wave and those that are to support standing waves. Two of these paths are marked by dashed lines in the structure illustrated in FIG. 21 A. The marked path that includes PS3 is a traveling-wave path, and the marked path that includes PS2 is a standing-wave path. The corresponding CBs for these two paths are shown in FIG. 21B and FIG. 21 C, respectively. These CBs may have their own excitation (denoted by Pci and Pc2 for the CB of FIG. 21B, and pci for the CB of FIG. 21C). The CBs may be arranged to support the form of guided waves in their desired format on the MiLAC (either standing or travelling waves). For a traveling-wave CB (as in FIG. 21B, for example), the S parameters of the corresponding 2-port network may be measured at the desired state of its PS. This will be a complex matrix as in equation (90):li(90)21

[0246] The ABCD matrix T = j may be extracted from the S-parameter (see DavidPozar, Microwave Engineering, 4th ed, John Wiley& Sons, 2011).

[0247] In a uniform line,A = D = cosh (yL) (91)1,Z.4 + L\y = -acosh (— ) (92)a = Re{y,f> = Im{y (93)

[0248] To characterise the standing-wave paths, Sn of the CB (e.g., as in FIG. 21C) may be measured, then:Zin= Zc^- (94)1-511

[0249] With Zin and Zcstates for input impedance and characteristic impedance of the line. Then,yL = atanh (Zc / In) (95)a =^Ijg =^ (96)

[0250] In these examples, the guide under study is connected to PSs (layer 2) and SCs (layer 3) similar to the PSs and SCs of the MiLAC original layout. This will determine (RFL) of the guide which is to characterise a and 0 of the guide when it is loaded by PS and SC.

[0251] The information (e.g., a and 0) that is extracted from the CBs may be stored in an offline process. Then, the S-parameters of the MiLAC (e.g., as in FIG. 21A) are measured for different states of the PSs in an online fashion. The offline stored information may then be used to interpret the S-parameter readings while the MiLAC is operating.

[0252] According to a design example, a MiLAC may be to compute the inverted value of a matrix. In this scenario, a 2-port network may be utilized to generate a 2x2 matrix. To have a moderate level of flexibility in response reconfigurability, a 7r-network may be applied, composed of two open stubs and a transmission line in between, each equipped with a PS. In this example a symmetrical network is assumed, for simplicity. The schematic view of this network is shown in FIG. 22 A with corresponding Y-matrix as:YS + ~ coth (ytLt) - csch (ytLt)Y =c(97)- — csch (ytLt) Ys+ — coth (ytLt)Z> C,t ^c,tYs=— tanh (ysLs) (98)

[0253] Where Zc,t, Zc,sare the characteristic impedance values of the interconnecting line and stubs respectively, with Lt, and Lsbeing their corresponding physical lengths. The length ofthese lines can be electrically changed for different states of the PSs. Note that ytin (97) may be determined using (90) to (93) as its corresponding guide supports traveling wave whereas ysof (98) may be specified by (94) to (96) due to the standing waves of the stubs.

[0254] With any obtained Y matrix, P may be defined:P = I + Z0Y (99)

[0255] Which is the MiLAC input matrix to be inverted.

[0256] The PSs may then be set to specific values, the S-parameters of the corresponding MiLAC may be read, and use the obtained data (P) may be used to obtain P-1. FIG. 22B and 22C illustrate a circuit for fulfilling this operation. In the example of FIG. 22B and FIG. 22C, two substrates are bonded together. RF guides are printed on the outer face of the bonded substrates, and a DC bias line is sandwiched between the substrates. This reduces the response dependence on the surrounding environment, improving repeatability of measurements.

[0257] To specify the inverted matrix, the reading may be rearranged as follows:S = (7 + Z0^)-1G - Z0Y) = 2 (1 + ZQY)-1- I (100)' - FT - ■P-1= 0.5(S + / ) (101)

[0258] According to an example, varactor diodes in their reverse bias may be used as TE to change the phase of the corresponding line. With a capacitance value of C=3pf, the input P is p > [2.6 + 1.08i 0.09 — 0.7i] rinoi-1.0.09 - 0.7i 2.6 + 1.08i-l ’

[0259] Applying this to an EM simulation of the arrangement shown in FIG. 22B and FIG.22C yieldsp-1 > r 04 — o.l 5i 0.06 + 0.09il“ 1-0.06 + 0.09i 0.33 - 0.14( }

[0260] The mathematical inverse value of P may be used to obtain the error of calculation:Inv(P) = [0,3" 0.05 + 0.07il O4)}[0 05 + 0 07j 0.3 — 0.12i J1’||p-1- / nv(P)||[L0261]JWith this data, the normalized Frobenius error is — - — — —L« 24% with the real- l| / nv(P)||Fpart-only and imaginary-part-only normalized error of about 23% and 18% respectively. This error is mainly rooted in numerical computation of EM simulators due to differences in themesh pattern of the simulated CBs and the MiLAC. Hence, it is expected that this error will be reduced in actual measurement.XIV. CONCLUSION

[0262] The concept of MiLAC as a generic analog computer that linearly operates with microwave signals has been introduced, and its application to signal processing has been analyzed. A MiLAC can generally be modelled as a multiport microwave network 102 made of tunable impedance (or admittance) components. The input signals are applied to the ports 106 of such a microwave network 102 via respective reference impedances 104a, and the output signals are read. The expression of the output of a MiLAC as a function of its input and the values of its admittance components has been derived using rigorous microwave theory. A MiLAC can efficiently compute the LMMSE estimator and invert an arbitrary matrix in the analog domain, with significantly reduced computational complexity over conventional digital computing. In particular, matrix inversion can be performed with complexity growing with the square of the matrix size, rather than the cube as with digital computing. The primitive operations computable with a MiLAC can serve as building blocks for implementing more complex algorithms, such as the Kalman filter, directly in the analog domain, achieving a substantial decrease in computational complexity. A MiLAC may be based on lossless admittance components, avoiding losses and costly active RF components.

[0263] A MiLAC may be used in wireless communications, potentially supporting communications with thousands of antennas by enabling analog-domain computations and beamforming. Five special cases of the LMMSE estimator have been derived, each of practical interest in wireless communications, such as the LS and MF, and it has been shown that they can be computed by a MiLAC in the analog domain. MiLAC-aided beamforming has been introduced as a novel beamforming strategy. Compared to previous beamforming strategies, MiLAC-aided beamforming offers the following benefits in terms of performance, hardware complexity, and computational complexity. First, it may provide the same flexibility as digital beamforming. Second, it does not require a large number of costly RF chains, e.g., using a number of RF chains equal to the number of transmitted symbols. Third, embodiments may be realized with low-resolution ADCs / DACs since the symbols may be precoded fully in the analog domain. Fourth, operations on a per-symbol basis may be avoided, since precoding inthe digital domain may be avoided. Fifth, it may offer a significant computational complexity reduction in the case of ZF operations at the transmitter or at the receiver, which can be performed with complexity growing with the square of the antenna number, rather than the cube. MiLAC-aided beamforming can offer a remarkable reduction in computational complexity. For example, ZF can be achieved with a complexity reduction of 1.5 104times

[0264] Throughout the description and claims of this specification, the words “comprise”, “contain”, and “include” and variations of them mean “including but not limited to”, and they are not intended to (and do not) exclude other components, integers or steps. Throughout the description and claims of this specification, the singular encompasses the plural unless the context otherwise requires. In particular, where the indefinite article is used, the specification is to be understood as contemplating plurality as well as singularity, unless the context requires otherwise.

[0265] Features, integers, characteristics, etc. described in conjunction with a particular aspect or example are to be understood to be applicable to any other aspect or example described herein unless incompatible therewith. All of the features disclosed in this specification (including any accompanying claims, abstract and drawings), and / or all of the steps of any method or process so disclosed, may be combined in any combination, except combinations where at least some of such features and / or steps are mutually exclusive. Examples are not restricted to the details of any foregoing examples. The Examples may extend to any novel one, or any novel combination, of the features disclosed in this specification (including any accompanying claims, abstract and drawings), or to any novel one, or any novel combination, of the steps of any method or process so disclosed.

[0266] The reader's attention is directed to all papers and documents which are filed concurrently with or previous to this specification in connection with this application and which are open to public inspection with this specification, and the contents of all such papers and documents are incorporated herein by reference

Claims

1. CLAIMSWhat is claimed is:

1. A method of configuring a reconfigurable microwave network, the method comprising:obtaining a first matrix;determining, based on the first matrix, an admittance matrix, where elements of the admittance matrix indicate admittance values for use in the reconfigurable microwave network, the microwave network having controllable admittances between ports of the microwave network;controlling the controllable admittances of the reconfigurable microwave network based on the admittance matrix;applying, by a plurality of voltage sources, respective microwave frequency voltages to respective ports of the microwave network, the respective microwave frequency voltages indicative of elements of an input vector; andmeasuring respective output voltages at the ports of the microwave network to obtain an output vector based on the input vector and the first matrix.

2. The method of claim 1, wherein each port of the micro wave network is electrically connected with a voltage source of the plurality of voltage sources via a respective series admittance, or is electrically connected to ground via a respective series admittance.

3. The method of claim 1 or 2, wherein the microwave network comprises a respective controllable admittance element (i) between each respective pair of the ports of the microwave network, and (ii) between each port and ground, wherein the controllable admittance elements form the controllable admittances.

4. The method of any one of claims 1 to 3, wherein the controllable admittances are purely imaginary.

5. The method of any one of claims 1 to 4, wherein when the first matrix is P and theadmittance matrix is Y, P = — I- IP, where Ko is the series admittance at each port of themicrowave network, and I is a P P identity matrix.

6. The method of any one of claims 1 to 5, wherein the ports of the microwave network comprise first ports, wherein the microwave frequency voltages are applied to the first ports, and wherein the ports of the microwave network comprise second ports when the number of ports of the microwave network exceeds the number of applied microwave frequency voltages, the second ports comprising ports of the microwave network that lack an applied microwave frequency voltage, wherein the elements of the output vector are based on:respective voltages measured at the first ports,respective voltages measured at the second ports, orrespective voltages measured at the first ports and the second ports.

7. The method of any one of claims 1 to 6, wherein at least one of:(I)V1= (pr / - pr / p^ x (P21P1-1IP12- p^r^Pn1)!!,(II) v2= ((p^pr / p^ - p22)“1P2iPri1)u,(iii) vx= — (Pi2P221P2i—Pn)1u, or(iv) v2= (P221P2i(Pi2P221P2i—Pn)1)u,where vi is a vector with elements corresponding with respective voltages measured at ports to which a voltage is applied, where V2 is a vector with elements corresponding with respective voltages measured at ports to which no voltage is applied, P is the first matrix, P = [pn P12’, and u is a vector having elements corresponding with the LP21 P22.voltages applied to the ports.

8. The method of any one of claims 1 to 7, wherein:The first matrix is indicative of parameters of a minimum mean square error, MMSE, estimation problem, the applied microwave frequency voltages correspond with elements of an observation vector, and the output voltages measured at ports are indicative of elements of a LMMSE estimator for x, or indicative of elements of a column of a covariance matrix Ce of the LMMSE estimator for x.

9. The method of any one of claims 1 to 7, wherein:the first matrix includes:a submatrix corresponding with a constant matrix H,a submatrix based on a covariance matrix C of a vector x,a submatrix based on a covariance matrix Cn of a vector n,and a submatrix corresponding with the a conjugate transpose of H, HH, an observation vector y = Hx + n, and:(i) the applied microwave frequency voltages correspond with elements of y, and output voltages measured at ports that do not have an applied voltage correspond with elements of a linear minimum mean square error, LMMSE, estimator for x, or(ii) the applied microwave frequency voltages correspond with elements of an / 7th column of an N N identity matrix, Iv, and output voltages measured at ports that have an applied voltage correspond with elements of an / 7th column of a covariance matrix Ceof the LMMSE estimator for x.

10. The method of claim 8 or 9, wherein:||CX||F» llcn||F, where || A ||Fis the Frobenius norm of A, such that the LMMSE estimator corresponds with a generalized least squares, GLS, estimator, or||CX||F« II Cn||F, such that the LMMSE estimator corresponds with a generalized matched filtering, GMF, estimator, orCx=XIXand Cn=nIy, such that the LMMSE estimator corresponds with a regularized least squares, RLS, estimator, orCx=XIX, Cn=nIy, and Ax» Ansuch that the LMMSE estimator corresponds with an ordinary least squares, OLS, estimator, orCx=XIX, Cn=nIy, and Ax«nsuch that the LMMSE estimator corresponds with an ordinary matched filtering, OMF, estimator.

11. A method for MIMO wireless communication, the method comprising at least one of:implementing regularized zero-forcing beamforming, R-ZFBF, or a minimum mean square error, MMSE, receiver by obtaining the RLS estimator by the method of claim 10, orimplementing zero-forcing beamforming, ZFBF, or a ZF receiver by obtaining the OLS estimator by the method of claim 10, orimplementing matched beamforming, MBF, or a matched beamforming receiver by obtaining the OMF estimator by the method of claim 10.

12. The method of any one of claims 1 to 7, wherein the first matrix is an N N invertible matrix P, the applied microwave frequency voltages correspond with elements of an wth column of an N N identity matrix, Iv, and the output voltages measured at the ports corresponds with elements of an wth column of the inverse matrix of P, P1.

13. The method of any one of claims 1 to 7, wherein the first matrix is indicative of parameters of a state transition model of a Kalman filter, the applied microwave frequency voltages correspond with elements of a column of an identity matrix, and the output voltages measured at the ports are indicative of a column of an a priori estimate error covariance matrix.

14. The method of any one of claims 1 to 7, The method of any one of claims 1 to 7, wherein the voltage sources are electrically connected with respective ports of a first set of the ports, and a plurality of antennas are respectively electrically connected with ports of a second set of ports, wherein the voltage sources are based on a vector of symbols to be transmitted by the plurality of antennas and the first matrix is based on a precoding matrix relating the symbols to target transmitted signals for transmission by the plurality of antennas, such that output signals at respective ports of the second set of ports cause the plurality of antennas to transmit the target transmitted signals.

15. The method of any one of claims 1 to 7, wherein the first matrix includes a submatrix corresponding with a precoding matrix for beamforming a multiple-input multiple-output, MIMO, transmission, the voltage sources are indicative of a vector of symbols to be transmitted, and the respective output voltages correspond with signals to be provided to respective antennas to transmit the beamformed MIMO transmission.

16. The method of claim 15, The method of claim 15, wherein the first matrix P =®NSXNT, where INis an N*N identity matrix, 0\ \ / is an N^M matrix of zeros, and W isthe precoding matrix.

17. The method of any one of claims 1 to 7, wherein the voltage sources are antennas electrically connected with respective ports of a first set of the ports, and a second set of ports are electrically connected with ground, wherein the first matrix is based on a combining matrixand signals measured at the second set of ports correspond with a result of the combining matrix operating on symbols received at the first set of ports via the antennas.

18. The method of any one of claims 1 to 7, wherein the voltage sources correspond with a beamformed multiple-input multiple-output, MIMO, signal, the first matrix includes a submatrix corresponding with a combining matrix for processing the beamformed MIMO signal, and the output voltages correspond with the result of the combining matrix operating on the beamformed MIMO signal.

19. The method of claim 18, wherein the first matrix P =, where INis an N*N— Gidentity matrix, 0 \- \- is an AXA matrix of zeros, and G is the combining matrix.

20. One or more programming instructions, wherein the one or more programming instructions, when executed, cause a processor to:perform the obtaining the first matrix and determining the admittance matrix of any one of claims 1 to 19.

21. A processor-readable storage medium, the processor-readable storage medium including the one or more programming instructions of claim 20.

22. A system for configuring a reconfigurable microwave network, the system comprising:the computer readable medium of claim 21;the processor; andthe reconfigurable microwave network.

23. A system for MIMO communication, the system comprising:the system of claim 22; anda plurality of antennas in electrical communication with respective ports of the reconfigurable microwave network.

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