Single input single output (SISO) physical layer key exchange

Abstract

A processor coupled to a first communication device produces and transmits a first encoded vector and a second encoded vector to a second communication device via a communication channel that applies a channel transformation to the encoded vectors during transmission. A processor coupled to the second communication device receives the transformed signals, constructs a matrix based on the transformed signals, detects an effective channel thereof, and identifies left and right singular vectors of the effective channel. A precoding matrix is selected from a codebook of unitary matrices based on a message, and a second encoded vector is produced based on a second known vector, the precoding matrix, a complex conjugate of the left singular vectors, and the right singular vectors. A first symbol of the second encoded vector and a second symbol of the second encoded vector are sent to the first communication device for identification of the message.

Description

1. SINGLE INPUT SINGLE OUTPUT (SISO) PHYSICAL LAYER KEY EXCHANGE

[0001]

[0002] STATEMENT REGARDING FEDERAL GOVERNMENT INTEREST

[0003] This United States Government holds a nonexclusive, irrevocable, royalty-free license in the invention with power to grant licenses for all United States Government purposes. TECHNICAL FIELD

[0004] The present disclosure relates to systems and methods for transmitting wireless signals for electronic communications and, in particular, to increasing the data rate of, and reducing the computational complexity of, wireless communications. CA 3164606 Date reçue / Received date 2025-04-03 7_ 7_ CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 BAl'lffiROUND !0005] In multiple access communications, multiple user devices transmit signals over a given communications channel to a receiver. These signals are superimposed, fonning a combined signal that propagates over that channel. TI1e receiver lhen performs a separation operation on the combined signal to recover one or more individual signals from the combined signal. For example, each user device may be a cell phone belonging to a different user and the receiver may be a cell tmver. By separating signals transmitted by different user devices, the different user devices may share the same communications channel without interference.

[0006] A transmitter may transmit different symbols by varying a state of a carrier or subcarrier, such as by varying an amplitude, phase, and / or frequency of the carrier. Each symbol may represent one or more bits. TI1ese symbols can each be mapped to a discrete value in the complex plane, thus producing Quadrature Amplitude Modulation, or by assigning each symbol to a discrete frequency, producing Frequency Shift Keying. The symbols are then sampled at lhe Nyquist rate, which is at least twice the symbol transmission rate. TI1e resulting signal is converted to analog through a digitalto- analog converter, and then up-converted to the carrier frequency for transmission. When different user devices send symbols at the same time over the communications channel, the sine waves represented by those symbols are superimposed to fom1 a combined signal that is received at the receiver.

[0007] A known approach to wireless signal communication is orthogonal frequencydivision mulliplexing (OFDl\ii), which is a method of encoding digital data on multiple carrier frequencies. OFDl\!1 methods have been adapted to permit signal communications that cope with severe conditions of communication channels such as attenuation, interference, and frequency-selective fading. Such an approach, hovvever, does not address a desire for a physical layer of security of signal transmission. Fmthennore, the OFDM signal includes signal amplitudes over a very large dynamic range, often involving transmitters that can handle a high peak-to-average-power ratio. Tirns, a need exists for improved systems, apparatuses and methods for a secure, power efficient approach to \vireless communication of signals. SUMMARY [00081 Tn some embodiments, a processor coupled to a first communication device produces and transmits a first encoded vector and a second encoded vector to a second 2. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 communication device via a communication channel that applies a channel transformation to the encoded vectors during transmission. A processor coupled to the second communication device receives the transfom1ed signals, constrncts a matrix based on the transfom1ed signals, detects an effective channel thereot~ and identifies left and right singular vectors of the effective channel. A precoding matrix is selected from a codebook of unitary matrices based on a message, and a second encoded vector is produced based on a second known vector, the precoding matrix, a complex conjugate of the left singular vectors, and the right singular vectors. A first symbol of the second encoded vector and a second symbol of the second encoded vector are sent to the first communication device for identification of the message. [00091 Tn some embodimenls, a communication method using UBDM or OFDM with physical layer security includes receiving, via a first communication device and at a first processor, a first signal that represents a first symbol of a first encoded vector and a channel transfomiation. The method also mcludes receiving, via the first communication device and at lhe firsl processor, a second signal that represents a second symbol of the first encoded vector and a channel transformation. The first processor detects a representation of an effective channel based on the first signal and the second signal. The first processor pe1fom1s a singular value decomposition of the representation of the effective channel to identify a left singular vector of the representation of the effoctive channel and a right singular vector of the representation of the effective channel. "foe first processor selects a precoding matnx from a codebook of unitary matrices, the precoding matrix associated with an index for a message for transmission. TI1e first processor produces a second encoded vector based on a second known vector, the precoding matrix, a complex conjugate of the left singular vector, and the right singular vector of the representation of the effective channel. The method also includes lransmitting (]) a signal that represents a first symbol of the second encoded vector and (2) a signal that represents a second symbol of the second encoded vector, through a communication channel, to a second commumcation device, for identification of lhe message at a second processor associated with the second communication device. BRlEF DES(-:RlP110N OF THE DRAWINGS [00101 FIG. l is a schematic illustration of a secure and efficient Unitary Braid Divisional Multiplexing (UBDM) system, according to some embodiments. 3. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 rmn l] FIG. 2 is a schematic representation of a signal transmilter within a UBDl\ii system, according to some embodiments.

[0012] FTG. 3 is a schematic representation of a signal receiver within a UBDM system, according to some embodiments. [00131 FIG. 4 is a schematic representation of a communication system using UBDM or OFDM with Single lnput Single Output (SISO) implemented physical layer security (PLS), according to some embodiments. rmH4] FIG. 5 is a flowchart illustrating a firsl method of performing UBDM or OFDl\ii with SISO-implemented PLS, according to some embodiments.

[0015] FTG. 6 is a flowchart illustrating a second method of perfonning UBDM or OFDM with SISO-implemented PLS, according to some embodimenls. [00161 FIG. 7A is a flowchart illustrating a method of operating a Lll3DM system, according to an embodiment.

[0017] FIG. 7B JS a flowchait illustrating a method of operating a UBDM system, according to an embodiment. !0018] FIG. 8A is a schematic representation of a processing of a signal at a signal transmitter of an OFDM system.

[0019] FTG. 8B JS a schematic representation of a processing of a signal at a signal transmitter of a UBDM system, according to an embodiment. [00201 FlG. 8C is a schematic representation of a processing of a signal at a signal transmitter of a UBDM system, according to an embodiment. DETAILED JJE5;'CRIPTJON

[0021] "l11e present disclosure sets fcnth a Unitary Braid Divisional Multiplexing (UBDM) system, also referred to herein as a generalized UBDM (gUBDM) system, for modulation-based communications securily, followed by a UBDM or OFDM system implementation that includes Physical Layer Security (PLS) that is implemented via a Single Input Single Output (SISO) configuration. The PLS can be referred to as "enhanced-MOPRO," and includes a modified version of a key exchange algorithm referred to as I\.,rnv10-0FDM Precoding with Rotation (MO PRO).

[0022] In some embodiments, a communication method that uses unitary braid divisional multiplexing {UBDM) with SISO-implemented PLS includes receivmg, via a first communication device and at a first processor, a first signal that represents a first svmbol of a first encoded vector and a channel transformation. TI1e method also 4. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 includes receiving, via the first communication device and at the first processor, a second signal that represents a second symbol of the first encoded vector and a channel transfonnation. A representation of an effective channel is detected, via the first processor, based on the first signal and the second signal. The first processor perfom1s a singular value decomposition of the representation of the effective channel to identify a kft singular vector of the representation of the effective channel and a right singular vector of the representation of the effective channel. The first processor selects a precoding matrix from a codebook of unitary mauices, the precoding matrix associated with a.n index for a message for transmission. The first processor produces a second encoded vector based on a second known vector, the precoding matrix, a complex conjugate of the left singular vector, and the right singular vector of the representation of the effective channel. TI1e method also includes transmitting O) a signal that represents a first symbol of the second encoded vector and (2) a signal that represents a second symbol of the second encoded vector, through a commumcation channeL to a second communication device, for identification of the message at a second processor associated with the second communication device. l0023] In some embodiments, a communication method using UBDM or OFDM with SISO-implemented PLS includes generating, at a processor of a first communication device, a first encoded vector using a known vector and a unitary matrix. A first signal representing a first symbol of the first encoded vector and a second signal representing a second svmbol of the first encoded vector are transmitted to a second communication device through a communication channel that applies a channel transfonnation to the first signal and to the second signal during transmission. A third signal representing a first symbol of a second encoded vector and the channel transformation, and a fomth signal representing a second symbol of the second encoded vector and the channel transfonnation are received at the processor and from the second communication device. 111e processor detects a representation of an effoctive channel based on the third signal and the fourth signal. The processor perfonns a singular value decomposition of the representation of the effective channel to identify a right singular vector of the representation of the effective channel. The method also includes querying a codebook of unitary matrices to identify a message associated with the third signal and the fomth signal based on the singular vector of the representation of the effective channel and the unitary matrix. 5. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 rml24] In some embodiments set forth herein, a UBDM system with STSOimplemented PLS includes a modified Orthogonal frequency Divisional Multiplexing (OFDM) system. The modified OFDM system can include some components common to an unmodified OFDl\ii system, but also includes a generalized version of an OFDM component (e.g., a subset of the functionality of the OFDIV1). The UBDM system can be designed to implement (e.g., in hardware and / or software executed by or stored in hardware) a modified OFDM step during operation, to execute a paired operation including perfom1ing an inverse Fast Fourier Transfonn (iFFT) (or a Fast Fourier Transform FFT) of signals at a signal transmitter to generate transformed signals that are transmitted, and then perfom1ing a Fast Fourier Transfonn (FFT) (or an inverse Fourier Transfonn iFFT) on lhe transformed signals at a receiver lO recover the signals. The modification includes generalizing the iFFT / FFT performed by the transmitter to an arbitrary transformation (represented by an arbitrary matrix, for example an arbitrary unitary matnx) rmu5] Embodiments of a UBDM system with SISO-implemented PLS, as described herein, and including embodiments \vith the above modification of an OFDM system, can impart exceptional security and efficiency in transmission of signals over wireless communication channels. Other benefits of embodiments of the UBDM as described herein include an ability to use non-linear transformations, as well as a generalized implementation involving equiangular tight fran1e (ETF) transfomiations or nearly equiangular tight frame (NETF) transformations as an example. Standard OFDM doesn't allmv for a generalization to ETF / NETF "overloading." !0026] Generalizing to an arbitrary unitary as implemented in a UBDM system as described herein can also have the effect of spreading the energy of each symbol or vector in a signal to be transmitted across the different subcarriers. Spreading the energy of each symbol or vector in a signal lO be transmitted can reduce the Peak-to-AveragePower- Ratio (PAPR) of the signal, and provide a degree of spreading (and, therefore, interference rejection) that is comparable to systems such as Direct Sequence Spread Spectrum (DSSS) systems. Spreading the energy of each symbol or vector in a signal to be transmitted can also provide an extra degree of freedom in multiplexing. in other words, in addition to standard frequency division multiplexing and time division multiplexing, a UBDM system mtroduces code diviswn multiplexing, which adds a powerful degree of freedom for multiplexing in a signal lransmission system. 6. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 rmu7] "Physical Layer Security" (PLS) refers to the leveraging of physical properties of a communications channel bet\veen users of a communications system for the purposes of exchanging secret mfonnation. Although some of the foregoing UBDM implementations describe the application of security at the physical layer, they do not, in a strict sense, incorporate PLS, which involves the exploitation ofa physical property of the shared channel between two users. For example, in PLS, users generate a secret key for a symmetric cryptologic / security scheme (e.g., Advanced Encryption Standard (AES)), based on the physical properties of a communications channel, for the secret infom1ation. Unless an eavesdropper has a receiver that is sufficiently close to one of the users to directly measure (or to gather sufficient information to approximate) the physical properties of the communications channel, the eavesdropper will be unable to access the shared secret. According to embodiments set forth below, PLS can be implemented in combination with UBDM (or generalized UBDM), OFDM, or any other communication system, to enhance security of the commumcations. rmu8] As used herein, a "transmitter" (or "signal transmitter") refors to any collection of components that are used in the transmission of signals, including any combination o±: but limited to, one or more: antennas, amplifiers, cables, digital-to-analog conve1ters, filters, up-conve1ters, processors (e.g., for reading bits and / or mapping of bits to a baseband), etc. Similarly, as used herein, a "'receiver" (or "signal receiver") refers to any collection of components that are used in receiving signals, including any combination o±: but limited to, one or more: antennas, amplifiers, cables, analog-todigital conveners, filters, dovvn-conve1ters, processors, etc. Sending and Receiving Signals in SISO-Implemented Enhanced-MOPRO

[0029] FTG. 1 is a schematic illustration of a secure and efficient, Unitary Braid Divisional Multiplexing system 100, also referred to herein as a "UBDM system" or "a system," according to an embodiment. "11le UBDM 100 is configured to send and / or receive wireless electronic communications m a secure and efficient manner. ·111e UBDM system 100 includes a signal transmitter JO 1 and a signal receivers 103, and a communication network 106, as illustrated in FlG. l. TI1e UBDM system 100 optionally includes a signal transmitter 102 and a signal receivers 104. The UBDM system 100 is configured to process and transmit a signal from the signal transmitter 101 and / or optionally from the signal transmitter l 02 via one or more communication channels defined via the communication network to the signal receiver 103 a.nd / or optionally to 7 SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 the signal receiver 104. Given a signal lO be lransmitted from a signal transmitter ] 0 l and / or 102 and to a signal receiver 103 and / or 104, the UBDM system 100 is configured such that the signal transmitter 101 and / or l 02 can process the signal by applying an arbitrary transfonnation to generate a transfom1ed signal that is transmitted to lhe signal receivers l 03 and / or 104.111e arbitrary tra.nsfonnation can be applied using one or more of hardware (e.g., a field-programmable gate array) and / or software. The signal transmitters 10 l and / or 102 also send to the signal receivers 103 and / or l 04 {e.g., before lransmitting the signal) an indication of the arbitrary transfi.mnation that was applied. TI1e signal receivers l 03 and / or l 04 are configured to receive the transfom1ed signal and the indication of the arbitrary transfomrntion applied by the signal transmilter(s) and apply an inverse of the arbitrary transfom1ation to recover the signal from the transformed signal. \.:Vhile the system 100 is illustrated to include two signal transmitters 101 and 102, and two signal receivers 103 and 104, a similar UBDM system can include any number of signal transmitters and / or signal receivers. rmno] In some embodiments, the communication network l 06 (also referred lO as "the network") can be any suitable communication network that includes one or more wired and / or wireless communication channels configured for transferring data, operating over public and / or pnvate networks. Although not shown, in some implementations, the signal transmitters 101,102 a.nd signal receivers 103, 104 ( or portions thereof) can be configured to operate within, for example, a data center (e.g., a cloud computing environment), a computer system, one or more server / bost devices, and / or so forth. In some implementations, the signal transmitters 10] ,102 and signal receivers l 03, 104 can function within various types of netvvork environments that can include one or more devices and / or one or more server devices. For example, the net\vork 106 can be or can include a private neti.vork, a Virtual Private Network (VPN), a Multiprotocol Label Svvitching (MPLS) circuit, the Internet, an intranet, a local area nelwork (LAN), a wide area net\vork (WAN), a metropolitan area network (JvV\N), a worldwide interoperability for microwave access netvvork (WiMAX®), a Bluetooth® network, a virtual network, and / or any combination thereof. In some inslances, the communication network l 06 can be a wireless network such as, for example, a Wi-Fi or wireless local area network ("'WLA . N"), a wireless wide area network (''WWAN"), and / or a cellular network. [0031 l 111e communication netvvork 106 can be, or can include a wired netvvork and / or a wireless network implemented using, for example, gateway devices, bridges, 8. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 switches, and / or so fo1th. 1l1e vvired network or the wireless network can use one or more communication channels, for example, a radio frequency (RF) communication channel(s), an extremely low frequency (ELF) communication channel(s), an ultra-low frequency (UIS) communication channel(s), a low frequency (LF) communication channel(s), a medium frequency (l\ff) communication channel(s), a.n ultra-high frequency (UHF) communication channel(s), an extremely high frequency (EHF) communication channel(s), a fiber optic commination channel{s), an electronic communication channel(s), a satellite communication channel(s), and / or lhe like. The network 106 can include one or more segments and / or can have portions based on various protocols such as Internet Protocol (IP) and / or a proprietary protocol. The communication network 106 can include at least a portion of lhe lnlemet. In some instances, the comrrnmication network 106 c,m include multiple networks or subnetworks operatively coupled to one another by, for example, net\vork bridges, routers, svvitches, gateways and / or the like (not shovvn). rmn2] FIG. 2 is a schematic block diagran1 of an example signal lransmitter 201 that can be a part of a UBDM system such as the UBDM system 100 described above with reference to FlG. l, according to an embodiment. The signal transmitter 201 can be stmcturally and functionally similar to the signal transmitter 10 l and signal transmitter 102 of the system 100, shmvn and describe above with respect to FIG. 1. In some embodiments, the signal transmitter 201 can be, or can include, a processor configured to process instructions stored in a memory. The signal transmitter 20 l can be a hardware-based computing device and / or a multimedia device, such as, for example, a server, a desktop compute device, a smartvhone, a tablet, a wearable device, a laptop and / or the like. The signal transmitter 201 includes a processor 211, a memory 212 (e.g., including data storage), and a communication interface 213. [00331 The processor 21 l can be, for example, a hardware based integrated circuit (TC) or any other suitable processing device configured to nm and / or execute a set of instmctions or code. For example, the processor 211 can be a general purpose processor, a central processing unil (CPU), an accelerated processing unil (APU), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a programmable logic array {PLA), a complex programmable logic device (CPLD), a programmable logic controller (PLC), a graphics processmg unit (GPU), a neural network processor (N'NP), and / or the like. The processor 21 l is operatively coupled to l / SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 the memory 212 through a system bus (for example, address bus, data bus and / or control bus, not shown).

[0034] T11e processor 21 J can be configured to receive a signal to be transmitted and to perfi.mn processing to transfom1 the signal into a transfom1ed signal by applying an arbitrary transformation. In some implementations, the processor 211 can apply an arbitrary transfomiation that is defined to be a unitary transformation such that the transfom1ed signal can be transmitted ma secure and efficient manner using the UBDM system. !0035] The processor 211 can include a set of components including a converter 214, an arbitrary transfom1 selector 215, and an arbitrary transfonn applier 216. The processor 21 l can receive a set of signals 221 and 222, perfonn a sel of arbitrary transformations 231 and 232, and send a set of transformed signals 241 and 242. [00361 ln some embodiments, each of the conve1ter 214, an arbitrary transfom1 selector 215, and an arbitrary transfonn applier 216 can be software stored m the memory 212 and executed by processor 211. For example, each of the above mentioned ponions of the processor 211 can be code to cause the processor 211 to execute the converter 214, the arbitrary transform selector 215, and the arbitrary transform applier 216. The code can be stored in the memory 212 and / or a hardware-based device such as, for example, an ASIC, an FPGA, a CPLD, a PLA, a PLC and / or the like. In other embodiments, each of the converter 214, the arbitrary transform selector 215, and the arbitrary transfom1 applier 216 can be hardware configured to perfom1 the respective functions. In some embodiments, each of lhe components can a combination of softvvare and hardware. In some embodiments one or more of the components (e.g., converter 214, the arbitrary transform selector 215, the arbitrary transfom1 applier 216) of the processor 211 can be configured to operate based on one or more platforms (e.g., one or more similar or difforenl platfi.mns) that can include one or more 1ypes of hardware, sotlware, fi.nmvare, operating systems, mntime libraries, and / or so forth. In some implementations, the components of the signal transmitter can be configured to operate within a cluster of devices (e.g., a server fam1). In such an implementation, the functionality and processing of the components of the signal transmitter 201 can be distributed to several devices of the cluster of devices. The components of the signal transmitter 201 and a signal receiver {such as the signal receiver 301 shown and described in FIG. 3) can be, or can include, any type of hardware and / or soflware configured to process attributes. 10. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 rmn7] The converter 214 can be configured to receive a signal to be transmitted and prepare the signal in a fom1 that can be transformed by the processor 211 using an arbitrary transformation. For example, m some embodiments, the processor 21 J can recerve a signal in the fom1 of a parallel set of symbols hn. The converter 214 can be configured to perfonn a parallel-to-serial computation (e.g., using a shift register) on the set of symbols bn to convert the parallel set of symbols bn to a serial set of symbols. In some other embodiment, the converter 214 can include a configuration to perfom1 a serial-to-parallel computation (e.g., using a shift register) on a serial set of symbols to a parallel set of symbols. ln some embodiments, the converter 214 can generate a plurality of vectors (e.g., representing the set of signals 221 and 221) based on the set of symbols. In some implementations, the converter 214 can receive a signal in the form of a plurality of input bits. The converter 214 can be configured to generate a plurality of symbols based on the plurality of input bits. The converter 214 can be forther configured to generate a plurality of blocks based on the plurality of symbols where each block from the plurality of blocks represents a vector from a plurality of vectors (e.g., representing the set of signals 221 and 222). Alternatively, the converter 214 can be further configured to generate multiple pluralities of blocks based on the plurality of symbols where each plurality of blocks from the multiple pluralities of blocks represents a vector from a plurality of vectors ( e.g., representing the set of signals 221 and 222). [00381 111e arbitrary transfonn selector 215 can be configured to select, based at least partly on the signal to be transmitted or the plurality of vectors generated by the converter 214, an arbitrary transfomiation (e.g., arbitrary transformation 231 and 232) to be applied on the plurality of vectors (e.g., representing the set of signals 221 and 221) to securely and efficiently transmit the vectors from the signal transmitter 201 to one or more receivers associated with the UBDM system. The arbitrary transfom1ation (e.g., arbitrary transformation 231 and 232) can include one of, or a combination of any of, a non-linear transfomiation, a unitary transfomiation, an ETF transfonnation, or a NETF transfonnation. In some embodiments, the arbitrary transfom1 selector 215 can have access to a library of arbitrary transfomiations that are unitary by design (e.g., arbitrary transformation 231 and 23 2) from ,vhich one can be selected for transmitting a signal. The arbitrary transfom1 selector 215 can select the arbitrary transfom1ation based, for example, on a transfom1ation type and / or a criteria negotiated bet\veen two communicating entities via a telecommunications handshake or otherwise input by a 11. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 participant in the communications system The criteria can include, for example, one or more of: a desired security level, a latency threshold, an error rate threshold, a minim um data rate, a maximum data rate, etc. Notably, unitary transfom1ation is the largest class of transfom1ations that can be perfom1ed on a vector of symbols that leaves the tolal power of the signal unchanged. If a non-unitary transformation is used, then the inverse transformation at the receiver will necessarily ampli(y noise in some of the received symbols, vvhereas this is not the case of unitary transfom1ations. rmn9] In some instances, the arbitrary transfonnation selector 215 can be configured to select a transformation that is not an identity matrix, a discrete Fourier matrix, or is any other direct sum of Fourier matnces. For example in some implementations the arbitrary transfonnations selector 215 can have a library of unitary transfom1ations and based on a set of guidelines select one unitary transformation U and perform computations to check if U is an identity matrix, or a discrete Fourier matrix, or is any other direct sum of a set of Fourier matrices. lf U is one of the three above categories, in some embodiments the arbitrary lransfi.mn selector 215 can discard U and select another transformation that can meet the guideline of not being any of the above three categories. If the arbitrary transformation selector 215 picks a transfomiation U that is not an identity matrix, a discrete Fourier matrix, or is any other direct sum of Fourier matrices it can then assign U as the arbitrary transfom1ation A to be used for an instance of transfom1ing a signal to be transmitted using a UBDM system according to that embodiment. roo40] In some implementations, the arbitrary transfom1 selector 215 can perfom1 the selection based on a set of inputs received by the processor 211. In some implementations, the arbitrary transfom1 selector 215 can perform the selection based on a set of parameters associated with the signal, the plurality of vectors, the nature of signal transmission (e.g., a security requirement, sensitivity of infom1ation content in the signal, path of signal transmission, etc.). In some implementations, the arbitrary transfom1 selector 215 can be configured to define and generate an arbitrary transfom1ation according a set of inputs received by the processor 21 l (e.g., a sel of user inputs received by the processor 211). l0041] "llle arbitrary transform applier 216 can apply the selected arbitrary transfonnation on the plurality of vectors (e.g., vectors 221 and 222) to generate a plurality of transfom1ed vectors (e.g., transfom1ed vectors 241 and 242). In some implementations, the plurality of transformed vectors can have a total magnitude that 12. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 substantially equals a total magnil11de of the plurality of vectors. Tn some implementations, for example, the arbitrary transform applier 216 can be configured to perfonn matrix operations to apply a transformation matrix A on a set of vectors to generate transfom1ed vectors. In some implemenlations, lhe arbitrary transfom1 applier 216 can be configured to perform any suitable number of procedures (e.g. signal processing procedures, suitable matrix operations) on a set of vectors before applying an arbitrary transfonnation. The plurality of transfom1ed vectors can then be sent to the signal transmitter antenna 217 and optionally sent to the signal transmitter anlenna 218 included in the communicator 213 to be sent to one or more signal receivers associated with a signal receiver. While illustrated to include two signal transmitter antennas 2 l 7 and 218, as described above, a similar signal transmitter could include and use a single transmitter antenna according (e.g., signal transmitter antenna 217) to some embodiments, configured to perform Single lnput Single Output (SISO) operation. A similar signal transmitter could include any suitable higher number of signal transmitter antennas (i.e., more than two transmitter antennas) according to still other embodiments. In some embodiments, the signal transmitter 201 can include a plurality of antenna arravs configured to perfom1 Multiple Input Multiple Output (MlMO) operations. [00421 TI1e memory 212 of the signal transmitter 201 can be, for example, a random access memory (RAM), a memory buffer, a hard drive, a read-only memory (ROM), an erasable programmable read-only memory (EPROM), a flash drive, a secure digital (SD) memory card, an embedded multi-lime programmable (MTP) memory, and / or the like. l11e memory 212 can store, for example, one or more software modules and / or code that can include instructions to cause the processor 211 to perfom1 one or more processes, functions, and / or the like (e.g., functions associated with the converter 214, the arbitrary transfonn selector 215, and / or the arbitrary transfom1 applier 216). In some embodiments, the memory 212 can include extendable storage units that can be added and used incrementally In some implementations, the memory 212 can be a portable memory (e.g., a flash drive, a portable hard disk, and / or the like) lhat can be operatively coupled to the processor 211. In other instances, a memory can be remotely operatively coupled with the signal transmitter 201. For example, a remote database server can serve as a memory and be operatively coupled to the signal transmitter 201. [00431 111e commtmication interface 2 l 3 can be a hardware device operatively coupled to the processor 211 and memory 212 and / or software stored in the memory 212 13. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 executed by the processor 2 l l. 1l1e comm uni cation interface 213 can include a signal transmitter antenna 217 and optionally a signal transmitter antenna 218. \.:Vhile a second transmitter antenna 218 in addition to the transmitter antenna 217 is shown in FIG. 2, a signal transmilter similar to the signal transmitler 201 can have just a single lransmitter antenna, according to some embodiments, or any number of transmitter antennas, according to some other embodiments. The communication interface 213 can be, for example, a netvvork interface card (NTC), a Wi-FiTM module, a Bluetooth@ module and / or any other suitable wired and / or wireless communication device. Furthem10re, the communication interface 213 can include a switch, a router, a hub and / or any other network device. The communicator 213 can be configured to connect the compute device 20 J to a communication network (such as the communication netvvork 106 shown above with respect to FIG. 1 ). In some instances, the communication net\vork 213 can be configured to connect, via one or more communication channels, to a commumcation netvvork such as, for example, the Internet, an intra.net a local area network (LAN), a wide area network (WAN), a melrnpolitan area network (MAN), a worldwide interoperability for microwave access network (WiMAX1~)), an optical fiber (or fiber optic)-based network, a Bluetooth® network, a virtual network, and / or any combination thereof. [00441 In some instances, the communication interface 213 can facilitate rece1vmg and / or transmitting a file or a set of files via one or more communication channels through a commumcation network (e.g., the communication network 106 shown and desc1ibed with respect to FIG. l). Tn some inst.ances, a received file can be processed by the processor 211 a.nd / or stored in the memory 212 as described in further detail herein. ln some instances, as described previously, the communication interface 213 can be configured to send a plurality of transfonned vectors, via the signal transmitter antenna 217, lO at least one signal receiver antenna associated with at least one signal receiver connected to the communication network. 111e communication interface 213 can also be configured to send and / or receive data associated with a library of arbitrary transfom1ation systems. !0045] FIG. 3 is a schematic block diagram of an example signal receiver 301 that can be a part of a UBDM system such as the UBDM system 100 described above with reference to FIG. l, according to an embodiment. The signal receiver 301 can be stmclurally and functionally similar to the signal receiver 10 l and signal lransmitter 102 of the svstem 100, shmvn and describe above with respect to FIG. 1. In some 14. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 embodiments, the signal receiver 30] can be, or can include, a processor 31 l configured to process instructions stored in a memory 312. TI1e signal receiver 30 l can be a ha:rdvvare-based computing device and / or a multimedia device, such as, for example, a server, a desktop compute device, a smartphone, a tablet, a wearable device, a laptop and / or the like. The signal receiver 301 includes the processor 311, the memory 312, and a communication interface 313.

[0046] The processor 31 l can be, for example, a hardware based integrated cJrcuit {IC) or any other suitable processing device configured lO run and / or execute a set of instrnctions or code. For example, the processor 311 can be, a hardware based integrated circuit (TC) or any other suit1.ble processing device configured to nm and / or execute a set of instmctions or code. For example, lhe processor 31] can be a general purpose processor, a central processing unit (CPO), an accelerated processing unit (APU ), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a: programmable logic array (PLA), a complex programmable logic device (CPLD), a programmable logic controller (PLC), a graphics processing unit (GPU), a neural network processor (NNP), and / or the like. TI1e processor 311 is operatively coupled to the memory 312 through a system bus (for example, address bus, data: bus and / or control bus, not shovvn). [00471 TI1e processor 311 can be, for example, a hardware based integrated circuit (IC) or any other suitable processing device configured to nm and / or execute a set of instrnctions or code. For example, the processor 31 l can be a: general purpose processor, a central processing unil (CPU), an accelerated processing unil (APU), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a programmable logic array (PLA), a complex programmable logic device (CPLD), a programmable logic controller {PLC), a graphics processmg unit (GPU), a neural network processor (N'NP), and / or the like. The processor 31 l is operatively coupled to the memory 312 through a svstem bus (for example, address bus, data bus a:nd / or control bus, not shown). roo48] The processor 31 l can be configured to receive a signal to be transmitted and to perform processing to transform the signal into a transformed signal by applying an arbitrary tra:nsfonnation. The processor 311 can also, or alternatively, be configured to receive a transfom1ed signal that is securely transmitted via: one or more communication channels defined in a communication network (e.g., netvvork 106 of FIG. l), obtain information associated with an arbitrary transfom1ation that was used to generate the 15. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 transfom1ed signal, and based on the information process lhe transfi.mned signal to recover an original signal (e.g., by applying an inverse of the arbitrary transfomiation) such that the original signal can be received by a destination in a secure and efficient manner using the gUBDM system. In some implementations, the processor 311 can apply an arbitrary transformation that is defined to be a unitary transfom1ation such that the transfom1ed signal can be transmitted in a secure and eflicient manner using the UBD1\1 system. roo49] The processor 311 can include a set of components including a converter 314, an arbitrary transform identifier 315, and a.n arbitrary transform reverser 316. l11e processor 31 l can include, or access from memory 312, a plurality of transfonned vectors 34 l and 342, representing transfom1ed signals, received from one or more transmitter antennas of a signal transmitter ( e.g., transmitter ,mtenna 2 l 7 or transmitter antenna 218 of signal transmitter 201 as shown and describe with respect to FIG. 2) that is part of the UBDM system that the signal receiver 301 is pait of. The processor 311 can include or access in memory 3 l 2 a set of arbitrary transfonnation 33 l and 332, identified based on infom1ation associated with a signal received from a signal transmitter, and a set of reverse transformations 3 51 and 352, computed based on the identified arbitrary transfonnations, and a plurality of vectors 3 21 and 322 representing a set of original signals. [00501 The arbitrary transfom1 identifier 315 can be configured to receive information associated with a transfom1ed signal (e.g., transfonned signal represented by transfom1ed vectors 341 and 342) received via the signal receiver antenna 317 and optionally via the signal receiver a.ntenna 318, the information including a.n indication of the identity of an arbitrary transformation that was used in generating the transfom1ed signals. The arbitrary transform identifier 315 is configured to, based on the information, identify the arbitrary transfom1ation that can be used to recover an original signal (e.g., original signal represented by plurality of vectors 321 and 322) from the transfom1ed signal (e.g., transfom1ed signals 34 land 342). roo5l] The arbitrary transform reverser 316 generates, based on the identity of the arbitrary transfomiation, a.n inverse of the identified arbitrary transformation, also referred to as a reverse transformation (e.g., reverse transfomiations 351 and 352) configured to reverse the effects of the identified arbitrary transfonnation to recover the original signal from a transfom1ed signal For example, in some embodiments, the arbitrary transfom1 reverser 316 generates a reverse transformation (A') 351 configured 16. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 to be applied on a plurality of transfom1ed vectors 341 and 342, represenling a transformed signal, and received by the signal receiver 301, so that the reverse transfonnation (A') 351 can reverse the effects of an arbitrary transfom1ation (A) 33 l and recover a plurality of vectors 321 and 322 representing an original signal. In anolher example, in some embodiments, the arbitrary transform reverser 316 constructs a matrix and generates the reverse transfommtion (A') based on the matrix configured to be applied to the plurality of transformed vectors 341 and 342 and recover the plurality of vectors 321 and 322. !0052] The converter 3 l 4 can be configured to receive a recovered plurality of vectors (e.g., 321 and 322) representing an original signal and regenerate the original signal from the recovered plurality of vectors. For example, in some embodiments, the processor can receive a parallel set of symbols bn. The converter 314 can be configured to perfom1 (e.g., using a phase register) a serial-to-parallel computation on the set of symbols b71 to convert the serial set of symbols b71 to a parallel set of symbols that can be similar to the original signal. ln one instance, the converter 314 can include a configuration (e.g., using a shift register) to perfom1 a parallel-to-serial computation. In some embodiments, the converter 314 can receive a plurality of recovered vectors (e.g., vectors 321 and 322) and generate, based on the vectors, an original signal including a set of symbols. In some embodiments, the converter 314 can receive a plurality of recovered vectors (e.g., vectors 321 and 322) and generate, based on the recovered vectors pluralities of blocks each plurality of blocks represenling a vector of the plurality of vectors. The converter 314 can then regenerate, based on the pluralities of blocks, a plurality of input bits from which it can recover an original signal.

[0053] The memory 312 of the signal receiver 301 can be, for example, a random access memory (RAM), a memory buffer, a hard drive, a read-only memory (ROM), an erasable programmable read-only memory (EPROM), a flash drive, a secure digital (SD) memory card, an embedded multi-time programmable (MTP) memory, and / or the like. The memory 312 can store, for example, one or more sofuvare modules and / or code that can include instrnctions to cause the processor 311 lO perfom1 one or more processes, fimctions, and / or the like (e.g., fimctions associated with the converter 314, the arbitrary transform identifier 315, and / or the arbitrary transform reverser 316). In some embodiments, lhe memory 212 can include extendable storage units lhal can be added and used incrementally. In some implementations, the memory 312 can be a p01tabk memory (e.g., a flash drive, a portable hard disk, and / or the like) that can be 17. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 operatively coupled to the processor 31 l. Tn olher instances, a memory can be remotely operatively coupled with the signal receiver 301. For example, a remote database server can serve as a memory and be operatively coupled to the signal receiver 301. [00541 111e commtmication interface 3] 3 can be a hardware device operatively coupled to the processor 311 and memory 312 and / or software stored in the memory 312 executed by the processor 311. T11e communication interface 313 can include a signal receiver antenna 3 J7 and optionally a signal receiver antenna 318. Wfole a second receiver antenna 3] 8 in addition to lhe receiver antenna 317 is shown in FIG. 3, a signal receiver similar to the signal receiver 30 l can have just a single transmitter antenna, according to some embodiments, or any number of transmitter antennas, according to some olher embodimenls. The communication interface 313 can be, for example, a network interface card (NIC), a Wi-F?M module, a BluetoothCR. 1 module and / or any other suitable wired and / or wireless communication device. Furthem10re, the communication interface 313 can include a switch, a router, a hub and / or any other network device. 111e communicator 213 can be configured to connect the compute device 301 to a communication network (such as the communication network 106 shown above with respect to FlG. l). In some instances, the communication network 313 can be configured to connect, via one or more communication channels, to a communication network such as, for example, the internet, an intranet, a local area network (LAN), a wide area network (WAN), a metropolitan area network (MAN), a vvorldwide interoperability for microwave access network (WiMAX@), an optical fiber (or fiber optic)-based netvvork, a BluetootM-1.) nelwork, a virtual netvvork, and / or any combination thereof. l0055] In some instances, the communication interface 313 can facilitate rece1vmg and / or transmitting a file and / or a set of files via one or more communication channels defined in a communication netvvork (e.g., the communication network ] 06 in the UBDM system 100 of FIG. 1). In some instances, a received file can be processed by the processor 311 and / or stored in the memory 312 as described in fwther detail herein. in some instances, as desc1ibed previously, the communication interface 313 can be configured such that the signal receiver antenna 317 and optionally signal receiver antenna 318 include one or more antennas tuned to receive transfom1ed signals of a particular predetennined center frequency within a predetem1med bandwidth, to receive lransfrmned signals securely and efficiently transmitted by one or more signal transmitter antennas associated with one or more signal transmitters connected to a 18. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 communication network as part of a UBDM system. 111e communication interface 313 can also be configured to send and / or receive data associated with a library of arbitrary transfonnation systems. In some embodiments the signal receiver 30 l can mclude a plurality of antenna arrays configured to perform Multiple Input Multiple Output (MIMO) operations. Introduction to SISO-Implemented, Enhanced-MOPRO roo56] In an example embodiment of the present disclosure, lhe first communication device receives an indication to transmit a symbol b using 4 subcarriers, [00571 After transmitting the symbol b through a communication channel having a channel vector h, a channel vector h tra.nsfrmns the symbol b to a transformed symbol. The transformed symbol received at the second communication device is a Hadamard product of the symbol b and the channel vector h. "foe second communication device receives the transfonned symbol as a 4- x 1 matrix: ( h1 b1 \ h;b_ 2 ) h3b1 h4 b2.1

[0058] 111e second communication device arranges the transfonned symbol (e.g., usmg the converter 314 shown and desc1ibed with respecl lO FIG. 3) into a 2 x 2 matrix,

[0059] "11le second communication device further performs a matrix decomposition to represent the 2 x 2 matrix as product of a 2 x 2 channel transformation matrix and a 2 x 2 symbol matrix.

[0060] By doing so, the second communication device transfom1s the speciallydesigned 4-component vector for Single Inpul Single Output (SISO) operation into a 19. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 2 x 2 matrix, vvhile benefiting efficiency and physical layer security of MOPRO operation or enhanced-1\iiOPRO operation. Enhanced-MOPRO - MIMO

[0061] In one embodiment, MO PRO operation or enhanced-MOP RO operation can be perfi.mned on a MIMO system. In one example, the MTMO system can be a 2 x 2 MIMO system that is used by users "Bob" and ''A lice." This example will be perfonned for a single subcarrier. The following procedure can be performed on each subcarrier in a system having multiple subcarriers. Alice initially chooses an arbitrary unitary matrix G E U (2), where U represents a unitary matrix. Alice then multiplies a publicly known / agreed-upon training sequence B of 2 symbols b1 and b2. (in two separate symbols) by G, to produce an encoded value for transmission to Bob: [00621 Alice then sends the encoded value to Bob. After going through the channel H, which has SVD H = BDAt (vvhere Dis the diagonal and positive definite matrix of the channel singular values, B are singular vectors on Bob's "side" of the channel (letl singular values), and A are the singular vectors on Alice's "side" of the channel (right singular values)), Bob will receive: r =Ht= HGb = BDAtc;b. Bob has knowledge of the training values in b (e.g., ln = 1, b2 = -L etc.) and right multiplies the received r by b-·1 to isolate the matrix HG= BDAl"G = BD(GtAyt·. In some implementations, the training values in b correspond to, or are based on, elements from one or more constellation diagrams of a signal modulated by one or more digit1.l modulation schemes. Examples of digital modulation schemes include, but are not limited to: binary phase shift keying (BPSK), quadrature phase shitl keying (QPSK), eight phase shift keying ( 8-PSK), and quadrature amplitude modulation (QAM) formats such as 16-QAM, 3 2-QA M, 64-QAlVL etc.

[0063] When Bob performs a singular value decomposition of HG, he obtains {B, D, Gl"A}. Bob then responds to Alice with t' = B* r; (Gi"A)'Tb, SUBSTITUTE SHEET (RULE 26) 21.

[0064] where ⋆ denotes complex conjugate but no transpose, and 𝐹􀯡 is one of the elements of the public codebook of unitary matrices (Bob may select this matrix, for example, as a way of encoding shared secret bits). It is desired that Eve should not be able to determine which matrix 𝐹􀯡 is being transmitted.

[0065] After transmitting 𝑡􁇱 back to Alice, Alice receives 𝑟􁇱, which is a version of 𝑡􁇱 that has been distorted by the transpose of the channel (assuming channel reciprocity): 𝑟􁇱 􀵌 𝐻􀯍𝑡􁇱 􀵌 𝐻􀯍𝐵⋆𝐹􀯡𝐴􀮱𝐺𝑏 􀵌 􁈺𝐵𝐷𝐴􀮱􁈻􀯍𝐵⋆𝐹􀯡𝐴􀮱𝐺𝑏 􀵌 𝐴⋆𝐷𝐵􀯍𝐵⋆𝐹􀯡𝐴􀮱􀯚􀯘􀯥𝐺𝑏 􀵌 𝐴⋆𝐷𝐹􀯡𝐴􀮱𝐺𝑏.

[0066] Because Alice knows the (public) training sequences b, she removes it by right multiplying this matrix by b–1, isolating the matrix 𝐴⋆𝐷𝐹􀯡𝐴􀮱𝐺. She can then perform a singular value decomposition (SVD) of this matrix, to obtain: 𝐴⋆𝐷𝐹􀯡𝐴􀮱𝐺 􀵌 𝐴⋆𝐷􁈺𝐺􀮱𝐴𝐹􀯡 􀮱􁈻􀮱 ⟶ 􁈼𝐴⋆, 𝐷, 𝐺􀮱𝐴𝐹􀯡 􀮱􁈽 .

[0067] Because Alice knows G, she multiplies the right singular vectors by G, leaving 𝐴𝐹􀯡 􀮱. She then multiplies 𝐴𝐹􀯡 􀮱 by the transpose of the left singular vectors, leaving 𝐹􀯡 􀮱. This leaves 𝐹􀯡 􀮱, from which Alice can determine which of the matrices in the codebook this is, and recover the shared secret bits.

[0068] The foregoing is an example of how enhanced-MOPRO works. In regular MOPRO, Bob does not include the factor of 𝐴􀮱𝐺 in his response to Alice, and as such, Alice does not need to remove it. MOPRO and enhanced-MOPRO systems described in U.S. Patent Application No. 16 / 527,240 operate using a MIMO system, as opposed to a SISO system, for example because if the 2 x 2 matrices are reduced to scalars, the SVD will no longer be applicable, and the algorithms shown above can become infeasible. As set forth herein, MOPRO and enhancedMOPRO can be performed using a SISO system, using the systems and methods set forth herein. SISO-Implemented MOPRO and Enhanced-MOPRO

[0069] In one embodiment, MOPRO operation or enhanced-MOPRO operation can be performed on a Single Input Single Output (SISO) system. In one example the SISO system can process a 2 􀵈 2 matrix and perform MOPRO and / or enhanced-MOPRO operations, however the process can also be implemented using any matrix size. Alice (e.g., via signal transmitter 201 as shown and describe with respect to FIG. 2) starts with a publicly-known training sequence of symbols b1 and b2, builds an arbitrary CA 3164606 Date reçue / Received date 2025-04-03 CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 unitary matrix G E U(2), and computes the product of lhe unitary matrix G and the publicly knovvn training sequences: l0070] All four of components of the product of the unitary matrix G and the publiclyknown training sequences are transmitted to Bob (e.g., the signal receiver 301 as shown and described with respect to FIG. 3) via a communication channel (also referred to herein as 'the channel'). So, Alice will split the four components of Cb lO two symbols f1 and l2 (tvvo separate "bauds" sent one after the other in a sequential manner), !0071] In each symbol from the two symbols, each component is in one frequency bin (also referred to herein as 'subcarrier' ). 111is means that, after going through the channel and applymg a set of coefficients to the tvvo symbols, Bob receives: From these two vectors, Bob builds the following 2 x 2 matrix, r12 + 121' r14 + r23) = (·h1lh + h2g3 J13g1 + h4g3 hzg4 + h1B2) h4g4 + h3gzJ = HGb.

[0072] where the product Gb is used to get the second to last equality. There are several other ways that Alice and Bob could break up the components. Tn some 22. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 implementations, Alice can pennute the components in any way, and Bob can in return still constrnct a 2 x 2 matrix, as described above.

[0073] At this point, the problem is almost identical to the MTMO version oflv1OPRO or enhanced-MO PRO. The only difference is lhat rank-2 nature of this example has been contrived by splilting multiple frequency subcarrier values into a 2 x 2 matrix, rather lhan relying on mulliple antennas to get a 2 x 2 matrix. Mathematically, however, they are identical problems. Therefore, Bob right multiplies the matrix HGb by b 1 to calculate HG, and then does a smgular value decomposition (SVD) on matrix HG. If lhe SVD of His H = BDAt, then the SVD Bob will get is: HG= BDAtc = BD(GtA)t-; {B, D. (GtA)} Bob then builds the matrix t' as: vvhere each matrix has the same meaning it had m the above example of enhancedMOPRO on the MIMO system. roo74] From lhe built t' matrix, Bob builds the two symbols (corresponding to tvvo separate bauds, sent one after the other), to be transmitted over the same frequency subcarriers that Alice used,

[0075] Because f 1 and f 2 are being transmitted over the same frequency subcarriers, and under an assumption that Bob responds quickly enough that the channel has not changed (or has not substantially changed), then the channel will apply the expected set of channel coefficients, and Alice will receive:

[0076] Depending on the implementation, the channel can be considered static (i.e., not subst1.ntially changed) if Bob responds vvithm 10-20 milliseconds for local area network (LAN) protocol iEEE 802. l Lor within 500 microseconds (e.g., within 250- 500 microseconds) for the Long-Term Evolution (LTE) 4G mobile telecommunications 23. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 standard. A channel may be slatic for a period of time during which lhe factors impacting or interfering with the channel are unchanged. Such factors can include, but are not limited to, vveather conditions (e.g., humidity, fog, rain, etc.), the presence and properties of fixed objects, the presence and properties of moving objects, the presence and properties of terrain, and the stationarity of the transmitting device and / or receiving device. When the channel is static, the correction applied to a received signal to remove the effects of that channel from the received signal {i.e., to "equalize" the channel) can remain constant. !0077] From the above vectors, Alice then builds the 2 x 2 matrix as: R' = ('r' 11 + r'. 23 r' 12 + r' 21- = { h1 \h.2 r I 13 -;; - r I2 1 ) , ' f r H -r- r 22· = HT B'f~i(c;t A)Tb = (BDA't)I'B'F;i(GtA)1·b = A*DBTB*f;(c;1A)'tb = A*DF;,(GtA)tb [0078l 111e matrix A*DF17 (GtA)Tb calculated in this SiSO system operation exan1ple is identical to the matrix A* Dr;, ( c; 1·A }i b calculated above with respect to the example of the MlMO system operation. Thus, Alice follows the san1e procedure as enhancedMOPRO operated in MIMO system (i.e., removes b, takes a SVD, isolates F~, and recovers the shared secret bits). "llierefore, Alice and Bob can fully imitate the functionality of enhanced-MO PRO by operating a SiSO system, as described above. [00791 Although the example presented above shows an implementation to perform a 2 x 2 MIMO system's MOPRO or enhanced-MOPRO operation by using a SISO system, there is no limitation as to size, and any n x n MIMO system's MO PRO or enhanced-MO PRO operation can be implemented on a SISO system. in some instances, a 2 x 3, a 3 x 2, a 3 x 3, a 17 x 48, or any n x n MIMO system can be implemented. The larger the value of n in an n x n system implemented in SISO, the more subcarriers may be used. Tn one example, a 3 x 3 MIMO syslem's l\iiOPRO or enhanced-MO PRO operation implemented in a SISO system may use a total of 9 subcarriers. In another 24. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 example, a 17 x 4B MIMO system's l\iiOPRO or enhanced-MOPRO operation implemented in a SlSO system may use 17 x LtB = 816 total subcarners. roo80] In some implementations, the SiSO system set fo1th above can include additional physical security in the vicinity of Alice, making it difficult for Eve to get dose to Alice. Alternatively or in addition, the two intended communicating entities may each play the roles of Alice and Bob (as described above) at different times. For example, entity 1 may perform the steps outlined above for Alice, while entity 2 may perform the steps outlined above for Bob (resulting in the secret bits originating from entity 2 and being shared with entity 1). Then, entity 2 can perfom1 the steps outlined above for Alice, and entity l can perfl.mn the steps outlined above for Bob (resulting in the secret bits originating from entity l and being shared with entity 2). TI1e entities may can continue in this alternating fashion, such that each entity has produced roughly half of the total number of shared bits. In this case, if Eve is only in the vicmity of one of the entities, she only recovers half of the secret bits. If a sufficient number of bils were produced by both of the entities, and the secret is (for example) the hash of both sets, then Eve has no hope of recovering the secret.

[0081] While in above examples, Alice and Bob broke their encoded vectors into two separate 4-component symbols of what appeared to be neighboring subcarriers: doing so may not always be necessary. First, Alice and Bob can choose to use any 4 subcarriers. Using 4 subcarriers that are not near each other can be desirable, because doing so increases the chance of higher vaiiance between the channel coefficients, and thus increases the probability of getting a foll rank channel matrix with large singular values, which is desirable. For example, Alice and / or Bob may make subcarriers 1, 11, 2L and 31 the subcarriers for a first STSO-implemented MOPRO exchange. Alice and / or Bob can then simultaneously use 2, l 2, 22, and 32 as a second STSO implemented MOPRO exchange in parallel to the first SISO-implemented MOPRO. Alice and / or Bob can then use 3, 13, 23, and 33, etc.

[0082] Furthennore, in the above examples, both Alice and Bob split their transmissions into two separate 4-component symbols: doing so may not always be necessary. ln some instances, subcarrier spacing can be chosen such that neighboring subcarriers have the same channel coefficient. In that case, all 8 components {e.g. from r1 and t~) could be sent simultaneously. Adding to the above example, instead of sending two consecutive 4 component transmissions in subcarriers l, ll, 2 l, and 3 l, 25. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 Alice (and Bob, when responding) can pul symbol E1 in subcarriers ] , l L 2 L and 31, and put s11nbol f2 into 2, 12, 22, and 32. roo83] In pe1fom1ing a MOPRO operation usmg a SiSO system, one difference compared to performing an enha.nced-MOPRO operation using a SISO system would have been that the matrix transmitted by Bob wouldn't have had the factor of (Gt A) t in the matrix. Therefore, the term (GT A) t could be simply replaced by an identity tenn. As aresull, Bob transmils matrix B'F_71b back to Alice. Alice then receives matrix: A* DF;lb. Alice can remove the b, and the SVD \Vill immediately give her Fn. roo84] In any OFDM-like or UBDM-like system, assuming appropriate circularization by a cyclic prefix, the action of the channel is a single complex coefficient on each subcarrier value. In other words, a symbol of (b1 . b2, b3 .. b4 , .,. ) will become (h1bi, h2b2 .. h3b3, h4 b4 , .,. ). How much h1 differs from hi+1 and hi-l depends on the subcarrier spacing, which can be selected or modified. In more mathematical language, the channel will take the transmitted symbol b = (b1, b2, ... ) and perform a 'Hadamard' product with lhe channel vector h = (h,, h2 , ... ). The Hadamard product is denoted h 0 b, and is defined as: [00851 Tn some instances, a method of performing l\iiOPRO and / or enhanced-MOPRO using a SISO system includes starting with some collection of vectors bi (for i in some indexing set) and applying some linear transformations to get a set of vectors vi = Mi bi (where Mi represents a set of linear operators). The method can farther include arranging the components of the vectors i\ into a new set of vectors Vi so that the Hadamard action of the channel li. on Vi produces output that the recipient can rearrange back into the fonn Hi M / hi, ,vhere Hi are matrices whose components depend in some vvay on the channel vector h. roo86] FIG. 4 is a schematic representation of a PLS communication system using UBDM or OFDM with SISO-implemented physical layer security, capable of perfonning the SlSO-implernented-1\iOPRO and enhanced-MOPRO processes described above, according to an embodiment. As shown in FIG. 4, lhe communication svstem 400 includes a first set of communication devices 40 l and a second set of 26. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 communication devices 411, communicably coupled to one another via a communication mediwn 440 (e.g., free space, a multipath wireless environment, etc.). The first set of communication devices 40 l is communicably coupled to a first processor 404, and the second set of communication devices 411 is communicably coupled to a second processor 414. The first processor 404 is operably coupled to a memory 405 and the second processor 414 is operably coupled to a memory 415. Each of the first processor 404 and the second processor 414 is operably coupled to a storage repository storing a codebook of unitary matrices 450, vvhich may be publiclyaccessible. During operation of the PLS communication system 400, the processor 404 produces a first encoded vector and a second encoded vector and transmits the first encoded vector and second encoded veclOr to the second set of communication devices 411 via a communication channel of the communication medium 440. The communication channel applies a channel transfomiation to the first encoded vector and the second encoded vector durmg transmission, thereby producing a first transfom1ed signal and a second transformed signal. l11e second processor 4 l 4 receives the first transformed signal and the second transfom1ed signal, constructs a matrix from the first transfom1ed signal and the second transformed signal, detem1ines an effoctive channel representation / matrix thereof, and identifies left and right singular vectors of the effective channel. The second processor 414 selects a precoding matrix from the codebook of unitary matrices 450 based on a message, and produces a third encoded vector and a fourth encoded vector based on a second knovvn vector, the precoding matrix, and a complex conjugate of the singular vectors. l11e second processor 414 then sends the second encoded vector to the first set of communication devices 401 for identification of the message. The first set of communication devices 401 can then recerve, from the second set of communication devices 411 the third encoded vector and lhe fourth encoded vector. The processor 404 delects a represenlation of an effective channel based on the third encoded vector and the fourth encoded vector, and performs a singular value decomposition of the representation of the effective channel to identify a singular veclOr of the representation of the effective channel. l11e processor 404 then perfrmns a query on the codebook of unitary matrices 450 to identify a message associated with the third encoded vector and the fourth encoded vector.

[0087] T11e methods and apparatus presented here are representative of many other possible methods and apparatus covering other temporal and / or speclral dimensions than can perform MOPRO and / or enhanced-MOPRO using a SISO system. In some 27. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 embodiments, methods and apparatus to perfonn MOPRO and / or enhanced-MOPRO using a SISO system can cover temporal coherence and frequency / spectral coherence approach. [00881 FIG. 5 is a flowchart illuslrating a first method 500 of perfom1ing UBDM or OFDM with SISO-implemented PLS, according to some embodiments. The method 500 can be implemented, for example, by the system 400 of FIG. 4, to perfom1 the SISO-implemented-MOPRO and enhanced-MOPRO processes described above. As shown in FIG. 5, the method 500 includes receiving, at 502, via a first communication device ( e.g., a communication device from the first set of communication devices 401 of FIG. 4) and at a first processor (e.g., processor 404 of FIG. 4), a first signal representing a first symbol of a first encoded vector and a channel transfom1ation. The method 500 also includes receiving, at 504, via the first communication device and at the first processor, a second signal representing a second symbol of the first encoded vector and a channel transfonnation. roo89] The first processor detecls, at 506, a representation of an effective channel based on the first signal and the second signal, and performs, at 508, a singular value decomposition of the representation of the effective channel to identi{y a left singular vector of the representation of the effective channel and a right smgular vector of the representation of the effective channel. The first processor selects, at 510, a precoding matrix from a codebook of unitary matrices (optionally, a publicly-accessible codebook). The precoding matrix is associated vvith an index for a message for transmission. 111e first processor produces, at 512, a second encoded vector based on a second known vector, the precoding matrix, a complex conjugate of the left singular vector, and the right singular vector of the representation of the effective channel. 111e method 500 also includes transmitting, at 514, (l) a signal representing a first symbol of the second encoded vector and (2) a signal represenling a second symbol of the second encoded vector, through a communication channel, to a second communication device {e.g., communication device 41 l of FIG. 4), for identification of the message at a second processor associated with lhe second communication device. The method 500 optionally also includes transmitting signals representing a plurality of additional encoded vectors through the communication channel to the second communication device until a predetenn ined number of messages have been sent. [00901 Tn some implementations, producing the second encoded vector includes multiplying the complex cor~jugate of the left singular vector by the precoding matrix 28. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 to produce an intem1ediate matrix, and multiplying the intennediate matrix by the right singular vector of the representation of the effective channel to produce the second encoded vector. Alternatively or in addition, the precoding matrix can be a first precoding matrix, the message can be a first message, and the index can be a first index, the method 500 further including selecting a second precoding matrix from the codebook of unitary matrices (the second precoding matrix associated with a second index for a second message for transmission), and producing a third encoded vector based on a third knovm vector, the second precoding matrix, a complex conjugate of the left singular vector, and the right singular vector of the representation of the effective channel. A signal representing the third encoded vector is then transmitted, through the communication channel, to the second communication device fbr identification of the second message. [00911 FlG. 6 is a flowcha1t illustrating a second method of perfom1ing UBDM or OFDM with STSO-implemented PLS, according to some embodiments. T11e method 600 can be implemented, for example, by the system 400 of FIG. 4, to perfimn the SISO-implemented Iv10PRO and enhanced-MOPRO processes described above. As shown in FlG. 6, the method 600 includes generating, at 620, at a processor of a first communication device (e.g., a communication device from the first set of communication devices 40 l of FIG. 4), a first encoded vector using a known vector and a unitary matrix. The method 600 also includes transmitting, at 622, to a second communication device (e.g., a communication device from the second set of communication devices 4 l l of FIG. 4) and through a communication channel, a first signal representing a first symbol of the first encoded vector. l11e communication channel applies a channel transfonnation to the first signal during transmission. 111e method 600 also mcludes transmitting, at 624, to the second communication device and through the communication channel, a second signal representing a second symbol of the first encoded vector. 111e communication channel also applies a channel transfomiation to the second signal during transmission. The processor receives, from the second communication device: (l) a third signal (at 626) representing a first symbol of a second encoded vector and the channel transfom1ation, and (2) a fourth signal (at 628) representing a second symbol of the second encoded vector and the channel transfonnation. The processor detects, at 630, a representation of an effective channel based on the third signal and the fi.mrth signal, and perfonns, at 632, a singular value decomposition of the representation of the effective channel to identify a singular vector 29. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 of the representation of the effective channel. The method 600 also includes querying a codebook of unitary matrices (the codebook optionally publicly-accessible), at 634, to identify a message associated with the tlmd signal and the fomth signal based on the singular vector of the representation of the effective channel and the unitary matrix. [00921 In some implementations, the method 600 also includes receiving, from the second communication device and at the processor, a plurality of additional signals representing a plurality of additional encoded vectors via the communication channel from the second communication device until a predetennined number of messages have been received. Alternatively or in addition, the method 600 also includes detecting, via the processor, a precoding matnx associated with an index for the message, with the querying of the codebook of unitary matrices being based on the precoding matrix. [00931 FIG. 7A is a flowchart of a method 700A of preparing, producing, and transmitting a signal using a first communication device (such as the signal transmitter 20 l shown and described with respect to FTG. 2) in a secure and efficient manner using a UBDM system, according to an embodiment. As shown in FIG. 7 A, the method 700A includes, at step 701A, producing, via at least one processor operatively to a first communication device, a first encoded vector, using a first known vector and a unitary matrix. The method 700A fmther includes, at step 702A, transmitting a signal representing a first symbol of the first encoded vector to a second communication device, through a communication channel that applies a channel transfonnation to the first symbol during transmission to produce a first transfom1ed symbol. The method 700A further includes, at step 703A, transmitting a signal representing a second symbol of the first encoded vector to the second communication device, through a communication channel that applies the channel transfom1ation to the second symbol durmg transmission to produce a second transformed symbol. [00941 Tn some embodiments, the first communication device recerves data that represent an original signal to be transmitted in a secure and efficient manner. The data can also represent attributes associated with the signal such as, for example, information related to the nature of the signal, the nature of the input bits, the size, sensitivity of the infomrntion contained, security requirements, etc. In some instances, the signal transmitter can generate a plurality of symbols, with a symbol being described as a pulse ma digital complex baseband signal. In some implementations, a symbol can be a wavefom1, or a state that, when transmitted through a communication channel defined in a connmmications net\vork, can change / alter and / or maintain a state or a significant SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 condition of lhe communication channel such that the slate or condition persists, for a fixed period of time. In some instances, the first communication device can break up the data mto a plurality of symbols that can be modified and / or transmitted in parallel using a SISO system of transmission and a MIMO system of transmission as described further below. In some instances, a signal transmitter can use a converter (e.g., converter 214) to convert a parallel data to a serial data. In some other instances, a signal transmitter can use the conveiter to convert a serial data into a parallel data. In some implemenlations, the generating a plurality of symbols based on the data can be via using a bit-to-symbol map.

[0095] In some embodiments, the first communication device produces a serial plurality of symbols associated with a serial signal and breaks the serial plurality of symbols up into pluralities of blocks, each plurality of blocks representing a vector from a plurality of vectors, the vectors being configured to be encoded and / or transmitted in series using a SISO system of transmission as described herein. In some mstances, a signal transmitlercan use a convener (e.g., converter 214 ofFTG. 2) to convert a parallel plurality of symbols into a serial plurality of blocks.

[0096] In some implementations, the first communication device selects, based at least partially on the plurality of vectors, an arbitrary transfonnation configured to be applied to the vectors to generate a plurality of encoded vectors. For example, the signal transmitter can have access to a library of known arbitrary transfonnations including unitary transformations, equiangular tight frame (ETF) transfomiations, and a nearly equiangular tight frame (NETF) transformations. l11e signal transmitter can use an arbitrary transformation selector (e.g., arbitrary transfom1ation selector 215 shown and describe with respect to FlG. 2) to select the arbitrary transformation, for example a unitary transformation, to be applied on the plurality of vectors to generate a plurality of encoded vectors. In some instances, the arbilrary transfom1ation can selecl an equiangular tight frame (ETF) transformation, or in some other instances the arbitrary transfom1ation selector can select a nearly equiangular tight frame (NETF) transfom1ation. Tn some implementations, the arbilrary transfom1ation selector can be configured such that the arbitrary transformation selected is based on a matrix that is not an identity matrix or a discrete Fourier matrix. ln some implementations, the arbitrary transfom1ation selector can be configured such that the arbitrary transfonnation selected is based on a matrix that is nol a direct sum of discrete Fourier matrices. l11e first communication device applies the arbitrary transformation to each 31. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 vector of the plurality of vectors lO produce the plurality of encoded veclOrs. In some instances, applying the arbitrary transformation can be such that the plurality of encoded vectors have a total magnitude that subst1.ntially equals a total magnitude of the plurality of vectors. [00971 In some implementations, the first comrrnmication device transmits (e.g., at 702.A) a signal representing a first symbol of a first encoded vector from the plurality of encoded vectors to the second communication device (such as the signal recerver 30 l shown and described with respect to HG. 3), through a communication channel that applies a dumnel transformation to the first symbol during tra.nsmission to produce a first transfonned symbol. In some instances, the first communication device sends a signal representing the first sy1nbol of the first encoded vector lO at least one transmitter antenna for transmission of a signal representing the first encoded vector from the at least one antenna to the second communication device. In some instances, the plurality of encoded vectors can be configured to be sent in series via at least one transmitter antenna associated vvith the first communication device (e.g., signal transmitter antenna 217 associated with the signal transmitter 20 l shown and described with respect to FIG. 2) and through at least one communication channel using a such that the first encoded vector is sent in series can be received bv at least one receiver associated with the 1:JBDl\!1 system being used. For example, the at least one receiver can include at least one antenna, and the at least one receiver be associated with the second communication device (e.g., the signal receiver 301) and the at least one transmitter antenna can be associated vvith the first communication device (e.g., the signal transmitter 201 ), where in the first communication device and the second communication device are configured to perfonn Single Input Single Output (MIMO) operations.

[0098] In some implementations, the first communication device transmits {e.g., at 703A) a signal representing a second symbol of the first encoded vector to the second communication device (such as the signal receiver 301 shown and described with respect to FIG. 3), through the commumcation channel that applies the channel transfom1ation to the second symbol during transmission to produce the second transformed symbol. ln some instances, the first communication device sends a signal representing the second symbol of the first encoded vector to at least one transmitter antenna for transmission of a signal representing the first encoded vector from the at least one antenna to the second communication device. In some instances, the plurality of encoded vectors can be configured to be sent in series via at least one transmitter 32. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 antenna associated vvith the first communication device (e.g., signal transmitter antenna 217 associated with the signal transmitter 201 shown and described with respect to FIG. 2) and through at least one communication channel using a such that the first encoded vector sent in series can be received by at least one receiver associated with the UBDM system being used. for example, the at least one receiver can include at least one antenna, and the at least one receiver be associated with the second communication device (e.g., the signal receiver 301) and the at least one transmitter antenna can be associated vvith the first communication device (e.g., the signal transmitter 201), where in the first communication device and the second communication device are configured to perfonn SlSO operations. [00991 Tn some implementations, the signal includes a set of transfom1ed symbols associated with the first encoded vector and the first communication device (e.g., signal transmitter 201) can place a set of transfonned symbols on the communication channel(s) (e.g., via a signal transmitter antenna 217) at a fixed and known symbol rate. l11e second communication device (e.g., signal receiver 30 l) can perfonn the task of detecting the sequence of transfom1ed symbols to reconstmct the first encoded vector. lOlOO] In some implementations, the first commw1ication device can be configured to send the signal representing the first encoded vectors to the plurality of transmitters via a physical layer associated with an open system interconnection model (OSI). l11e OSI model is a conceptual model that characterizes and standardizes the communication functions of a telecommunication or computing system without regard to its underlying internal structure and technology with the goal of achieving interoperability of diverse communication systems using standard communication protocols. TI1e OSI model uses partitioning of information exchanged via communication channels of a communication network into abstraction Javers (e.g., seven layers) with each layer including infom1ation of a specific type. [OHH I For example, a layer can include a physical layer used for the transmission and reception of unstructured rmv data between a signal transmitter and a physical transmission medium (e.g., a wireless communication channel in a communication network such as communication network 106 as shown and describe with respect to FIG. 1). It is configured to convert data included in the signals transmitted into electrical, radio, or optical signals. Layer specifications define characteristics such as voltage levels, the timing of voltage changes, physical data rates, maximum transmission distances, modulation scheme, channel access method and physical 33. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 connectors. 111is includes the layout of pms, voltages, line impedance, cable specifications, signal timing and frequency for wireless devices. Bit rate control is done at the physical layer and may define transmission mode as simplex, half duplex, and full duplex. The components of a physical layer can be described in terms of a network topology. The communications channel used to transmit the signal can have specifications for a physical layer.

[0102] In some instances, a first arbitrary transformation is used to produce the first encoded vector and a second arbitrary transfom1ation is used to produce the second encoded vector. Providing a signal representing the first arbitrary transformation and / or the second arbitrary transfomiation can include providing a first signal representing the first arbitrary transfrmnation and providing a second signal representing the second arbitrary transformation. In some implementations, the transmitting the first transfom1ed signal and the providing the first signal representing the first arbitrary transfom1ation can be to a first receiver associated with a first receiver, and the transmitting the second transfonned signal produced using the second arbitrary transformation and the providing the second signal representing the second arbitrary transformation can be to a second receiver antenna associated with a second receiver different from the first receiver. In some instances, the first and second signals representing the first arbitrary transformation and the second arbitrary transformations can be broadcast together to a wide audience including the first and second signal receivers. In some instances the first signal representing the arbitrary transformation can be broadcast widely but not the second signal representing the arbitrary transformation, such that the first signal receiver is able to recover the first encoded vector but the second receiver is unable to recover the second encoded vector until the second signal representing the second arbitrary transfomiation is provided or broadcast. [01031 To generate a maximal set of mutually orthogonal spreading codes, a unitary matrix A E U(N) is chosen. Tfthe nth column is denoted (or row, doesn't matter which as long as there is consistency) of A as in, then the N codes are {(iln) for n E [ l, ... , N]. if one device is to transmit data on all N codes, then it will be able to take the N symbols bn, multiply each symbol by every component of its spreading code, and then add the resulting vectors together. So the transmitted vectors is: N ,5 = L bnc(.4n), n=l 34. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 where bn are the symbols. [01041 The transmitter multiplies a symbol bn EC which is typically a complex number (a float double, elc), times all M::::: N componenls of c(An) This is repeated for all N symbols hn. So, there are N symbols, each being multiplied by N components of the code. 111is makes the complexity O(N 2), which is prohibitive for wide-band applications and is larger compared to complexity of OFDM, ,vhich has a complexity of O(N logN).

[0105] Notably for multiple access applications, where each user is given a subset of the codes, multiple access applications only have to do O(N) work, which is belterthan OFDM. That makes the DSSS implementation a suitable option fi.)r multiple access applications. [01061 To obtain a lJBDIV1 that has a complexity in the order of O(N log N), to match OFDM reinterpret. 111e transmitted baud is [s]m L, .m.. N: :::l bn [ C~ cA-n)· 1. m . . , (rn 1) L 'N . b~ e --- 2rnR k· -----. k=l k 'JvI 2 • This can be interpreted (up to nonnalization) as a discrete Fourier transfom1 of the svmbols: N bk I bnA,nk· n=l

[0107] FTG. 7B illustrates a flowchart describing a method 700B, continued from method 700A of FIG. 7 A, of receiving a first set of signals, retrieving infonnation from the first set of signals, and transmitting a second set of signal using a second communication device (such as the signal receiver 301 shown and describe with respect to FIG. 3) in a secure and efficient manner usmg a lJBDM system, according to an embodiment. The method 700B can be implemented by a processor associated with the second communication device (e.g., signal receiver 301). As shown in FIG. 7B, the 35. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 method 700B includes, at step 70 l B, receiving, al a second communication device, a first transformed signal including a first transformed symbol. The method 700B further includes, at step 702R receiving a second transfonned signal mcluding a second transfonned symbol. 1l1e method 700B further includes, at step 703B, constructing a matrix based on the first transfom1ed signal and the second transformed signal. TI1e method 700B further includes, at step 704B, detecting a representation of an effective channel based on the matrix, the effective channel associated with the communication channel. 111e method 700B further includes, at step 705B, perfonning a singular value decomposition of the representation of the effective channel to identify a singular vector of the representation of the effective channel. 111e method 700B fuither includes, at step 706B, selecting a precoding matrix from a codebook of unitary matrices based on a message for transmission, the precoding matrix associated with an index for the message for transmission. The method 700B further includes, at step 707R producing a second encoded vector based on a second known vector, the precoding matrix, a complex conjugate of the singular vector. 1l1e melhod 700B funher includes, at step 708B, transmitting ( 1) a signal representing a first symbol of the second encoded vector, and (2) a signal representing a second symbol of the second encoded vector, through the communication channel, to the first communication device for identification of the message. TI1e method 700B is explained in greater detail below. [01081 ln some implementations, the second communication device receives (e.g., at 701B) the first transfonned signal including a first transfom1ed symbol. The second communication device can include at least one receiver anlenna (e.g., the signal receiver antenna 317 as shown and described with respect to FIG. 3.) In some instances, the second communication device receives the first transformed signal representing the first symbol of the first encoded vector from the at least one receiver antenna for reception of a signal representing lhe first encoded vector from the at least one antenna from the first communication device. In some instances, a plurality of encoded vectors can be configured to be received in series via the at least one receiver antenna associated with the second communication device (e.g., signal receiver antenna 317 associated with the signal receiver 301 shown and described with respect to FIG. 3) and through at least one communication channel. For example, the at least one receiver can include at least one antenna, and the at least one receiver can be associated vvith the second communication device (e.g., the signal receiver 301) and the at least one lrnnsmitter antenna can be associated with the first communication device (e.g., the signal 36. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 transmitter 20 l ), vvhere in the first communication device and the second communication device are configured to perform SISO operations.

[0109] In some embodiments, the second communication device receives (e.g., at 702B) the second transfom1ed signal including a second transfom1ed symbol. in some instances, the second communication device receives the second transformed signal representing the second symbol of the first encoded vector from the at least one receiver antenna for reception of a signal representing the first encoded vector from the at least one antenna from the first communication device. in some instances, a plurality of encoded vectors can be configured to be received in series via the at least one receiver antenna associated with the second communication device and through at least one communication channel. For example, the at leasl one receiver can include at least one antenna, and the at least one receiver can be associated with the second communication device (e.g., the signal receiver 301) and the at least one transmitter antenna can be associated with the first communication device {e.g., the signal transmitter 201), vvhere in the first communication device and the second communication device are configured to perform SISO operations.

[0110] In some implementations, the second communication device constmcts {e.g., at 703B) a matrix based on the first transformed symbol of the first transformed signal and the second transfom1ed symbol of the second transfom1ed signal. In some instances, the second communication device constructs the matrix by arranging elements of the first transfom1ed symbol and the second transfom1ed symbol to a matrix with at leasl two rovvs and two columns. The second communication device decomposes the matrix to at least a symbol matrix and a communication channel matrix. At 404B, the second communication device detects a representation of an effective channel based on the matrix and the effective channel associated with the communication channel. [01111 ln some implementations, the second communication device performs a singular value decomposition (e.g., at 705B) of the representation of the effective channel lO idenli-fy a singular vector of the representation of the effective channel. In one example, the singular value decomposition can be a factorization of a real or complex matrix, such as, for example, a factorization of the representation of the effective channel. At 706B, the second communication device selects a precoding matrix from a codebook of unitary matrices based on a message for transmission, the precoding matrix associated with an index for the message for transmission. TI1e 37. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 precoding matrix can be selected from a codebook of unitary matrices that may or may not be publicly available.

[0112] In some implementations, the second communication device produces a second encoded vector (e.g., at 707B) based on a second known vector, the precoding matrix, a complex conjugate of the singular vector. At 70813, the second communication device transmits (l) a signal representing a first symbol of the second encoded vector, and (2) a signal representing a second symbol of the second encoded vector, through the communication channeL to the first communication device for identification of the message. The communication channel can have a channel vector h that transforms the encoded vector to a transformed symbol of the second encoded vector.

[0113] In some embodiments, the UBDM system (e.g., UBDM system 100) can be in some aspects partly similar in structure and / or function to an Oithogonal Frequency Divisional Multiplexing (OFDM) syslem. For example, an exan1ple pipeline for an OFDM system 800A can include a set of operations as presented in FlG. 8A, where vector b can be a set of symbols bn. ro114] Compared lO the above described OFDM syslem 800A, operations carried out by a UBDM system 800B described herein (e.g., UBDM system 100) are illustrated in FIG. 8B. The Ul3DM 800B can include an extra operator (e.g., a linear operator) "A' between the S / P block 80213 and the iFFT block, as shown in FIG. 8B. In use, according to the example embodiment associated with FIG.SB, the UBDM 800B operates such that symbols bn are received by the signal transmitter and are first put through a serialto- parallel block (e.g., conve1ter similar to converter 214 of the signal transmitter 201) to generate a converted set of vectors. The converted set of vectors then undergo the linear transfom1ation A to generate a set of transfonned vectors. For example, the transformation can be carried out by an arbitrary transfomrntion applier 80313 similar to arbitrary transformation applier 216 and the linear transformation A being selected by arbitrary transfomiation selector similar to the arbitrary transfomiation selector 215. Tn some embodiments, the transfonned vectors are then put through an iFFT block to generate a second transfonned vectors and the resulting second transfonned vectors can be transmitted to one or more receivers in the UBDM system.

[0115] In some other embodiments, the iFFT block can be skipped and the transfom1ed vectors generated by the arbitrary transformation applier can be transmitted to one or more receivers in lhe UBDM system. Expressed in another vvay, 38. SUBSTITUTE SHEET (RULE 26) 39. 𝑏 􀴤 → 𝐴𝑏􀴤 → 𝑠̅ 􀵌 ℱ𝐴𝑏 􀴤 (where ℱ is the discrete Fourier matrix). In some embodiments, 𝐴 can be unitary by design, as described herein, and ℱ is known to be unitary. By property of unitary matrices as a group, the product ℱ𝐴 will also be unitary. Therefore, because 𝐴 can be any unitary, including the iFFT matrix is unnecessary, and according to some embodiments a UBDM system can be configured by replacing the iFFT block with an arbitrary unitary 𝐴, as illustrated in FIG. 8C showing the operations in a UBDM system 800C, including an arbitrary transform applier 803C, according to an embodiment.

[0116] Following the above description a signal transmitter and a signal receiver operable with an OFDM system (e.g., OFDM system 800A of FIG. 8A) can be easily adapted to be used with a UBDM system described herein with the replacement of an iFFT operation with an arbitrary transformation operation using 𝐴 at the transmitter and the FFT with a linear operation 𝐴′ at the signal receiver to reverse the transformation. Other details of an OFDM system can remain.

[0117] Some embodiments described herein relate to methods. It should be understood that such methods can be computer implemented methods (e.g., instructions stored in memory and executed on processors). Where methods described above indicate certain events occurring in certain order, the ordering of certain events can be modified. Additionally, certain of the events can be performed repeatedly, concurrently in a parallel process when possible, as well as performed sequentially as described above. Furthermore, certain embodiments can omit one or more described events.

[0118] All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents, and / or ordinary meanings of the defined terms.

[0119] Examples of computer code include, but are not limited to, micro-code or microinstructions, machine instructions, such as produced by a compiler, code used to produce a web service, and files containing higher-level instructions that are executed by a computer using an interpreter. For example, embodiments can be implemented using Python, Java, JavaScript, C++, and / or other programming languages and development tools. Additional examples of computer code include, but are not limited to, control signals, encrypted code, and compressed code.

[0120] The drawings primarily are for illustrative purposes and are not intended to limit the scope of the subject matter described herein. The drawings are not necessarily to scale; in some instances, various aspects of the subject matter disclosed herein can be CA 3164606 Date reçue / Received date 2025-04-03 CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 shown exaggerated or enlarged in the dravvings to facilitate an understanding of different features. In the drawings, like reference characters generally refer to like features (e.g., functionally similar and / or structurally similar elements). [0121 l 111e acts perfonned as part of a disclosed method(s) can be ordered in any suitable way. Accordingly, embodiments can be constrncted in which processes or steps are executed in an order different than illustrated, which can include performing some steps or processes simultaneously, even though shown as sequential acts in illustrative embodiments. Put differently, it is to be understood that such features may not necessarily be limited to a particular order of execution, but rather, any number of threads, processes, services, servers, and / or the like that may execute serially, asynchronously, concurrently, in parallel, simultaneously, synchronously, and / or the like in a manner consistent with the disclosure. As such, some of these features may be mutually contradictory, in that they cannot be simultaneously present in a single embodiment Similarly, some features are applicable to one aspect of the innovations, and inapplicable to others. !0122] Where a range of values is provided, it is understood that each intervening value, to the tenth of the unit of the lower limit unless the context dearly dictates otherwise, between the upper and lower limit of that range and any other stated or intervenmg value in that stated range is encompassed within the disclosure. That the upper and lower limits of these smaller ranges can independently be included in the smaller ranges is also encompassed within the disclosure, subject to any specifically excluded limit in the stated range. ·where the stated range includes one or both of the limits, ranges excluding either or both of those included limits are also included in the disclosure. !0123] "11le phrase ''and / or," as used herein in the specification and in the embodiments, should be understood to mean "either or both" of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with "and / or" should be construed in the same fashion, 1.e., "one or more" of the elements so conjoined. Other elements can optionally be present other than the elements specifically identified by the "and / or" clause, whether related or mu-elated to those elements specifically identified. Tims, as a non-limiting example, a reference to "A and / or B", when used in conjunction with open-ended language such as "comprising" can refer, in one embodiment, to A only {optionally including elements other than B); in another embodiment to B only (optionally 40. SUBSTITUTE SHEET (RULE 26) CA 03164606 2022-06-13 WO 2021 / 162974 PCT / 0S2021 / 017043 including elements olher than A); in yet another embodiment, lO bolh A and B (optionally including other elements); etc.

[0124] As used herein in the specification and in the embodiments, "or" should be understood to have lhe same meaning as "and / or" as defined above. For example, when separating items in a list, "or" or "and / or'' shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only tenns clearly indicated to the contrary, such as "only one of' or "exactly one of," or, when used in the embodimenls, ''consisting oC' will refer to the inclusion of exactly one element of a number or list of elements. In general, the term ''or"' as used herem shall only be interpreted as indicating exclusive allematives (i.e., "one orthe other but not both") when preceded by lenns of exclusivity, such as "either,'' "one of," "'only one of," or "exactly one of" "Consisting essentially o±:" when used in the embodiments, shall have its ordinary meaning as used in the field of patent lavv. ro125] As used herein in the specification and in the embodiments, the phrase "at least one," in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements can optionally be present other than the elements specifically identified within the list of elements to which the phrase "at least one" refers, whelher related or unrelated to those elements specifically identified. Thus, as a non-limiting example, "at least one of A and B" (or, equivalently, "at least one of A or B, ,, or, equivalently "at least one of A and / or B") can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B): in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A): in yet another embodiment, to at least one, optionally including more than one, A, and at leasl one, optionally including more than one, B (and optionally including other elements); etc.

[0126] In the embodiments, as well as in the specification above, all transitional phrases such as "comprismg," "including," "carrying," ''having," "containing," "involvmg," "holding," ''composed of," and lhe like are to be understood to be open-ended, i e , to mean including but not limited to. Only the transitional phrases "consisting of' and 41. SUBSTITUTE SHEET (RULE 26) 42. “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively. CA 3164606 Date reçue / Received date 2025-04-03

Claims

43 CLAIMS 1. A system, comprising: a first communication device; a second communication device; at least one processor operatively coupled to the first communication device, the at least one processor for the first communication device configured to: produce a first encoded vector; transmit a signal representing a first symbol of the first encoded vector to the second communication device, through a communication channel that applies a channel transformation to the first symbol during transmission to produce a first transformed symbol; transmit a signal representing a second symbol of the first encoded vector to the second communication device, through a communication channel that applies a channel transformation to the second symbol during transmission to produce a second transformed symbol; and at least one processor operatively coupled to the second communication device, the at least one processor for the second communication device configured to: receive a first transformed signal including the first transformed symbol; receive a second transformed signal including the second transformed symbol, construct a matrix based on the first transformed signal and the second transformed signal; detect a representation of an effective channel based on the matrix, the effective channel associated with the communication channel; perform a singular value decomposition of the representation of the effective channel to identify a singular vector of the representation of the effective channel; select a precoding matrix based on a message for transmission, the precoding matrix associated with an index for the message for transmission; CA 3164606 Date reçue / Received date 2024-08-27 44 produce a second encoded vector based on the precoding matrix and the singular vector; and transmit (1) a signal representing a first symbol of the second encoded vector, and (2) a signal representing a second symbol of the second encoded vector, through the communication channel, to the first communication device for identification of the message.

2. The system of claim 1, wherein the at least one processor operatively coupled to the second communication device is configured to produce the second encoded vector by: multiplying the complex conjugate of the singular vector by the precoding matrix to produce an intermediate matrix; and multiplying the intermediate matrix by training values to produce the second encoded vector.

3. The system of claim 1, wherein the at least one processor for the second communication device is configured to select the precoding matrix from a codebook of unitary matrices.

4. The system of claim 1, wherein the precoding matrix is a first precoding matrix, the message is a first message, and the index is a first index, the at least one processor operatively coupled to the second communication device is further configured to: select a second precoding matrix associated with a second index for a second message for transmission; produce a third encoded vector; and transmit (1) a signal representing a first symbol of the third encoded vector, and (2) a signal representing a second symbol of the third encoded vector, through the communication channel, to the first communication device for identification of the second message. CA 3164606 Date reçue / Received date 2024-08-27 45 5. The system of claim 1, wherein the at least one processor operatively coupled to the second communication device is further configured to transmit signals representing a plurality of additional encoded vectors through the communication channel to the first communication device until a predetermined number of messages have been sent.

6. A non-transitory, processor-readable medium storing instructions that, when executed by a processor, cause the processor to: receive, via a communication channel. a first transformed signal including a version of a first symbol that has been transformed by a channel transformation associated with the communication channel; receive a second transformed signal including a version of a second symbol that has been transformed by the channel transformation; construct a matrix based on the first transformed signal and the second transformed signal; detect a representation of an effective channel based on the matrix, the effective channel associated with the communication channel; perform a singular value decomposition of the representation of the effective channel to identify a left singular vector of the representation of the effective channel and a right singular vector of the representation of the effective channel; produce a second encoded vector based on the left singular vector of the representation of the effective channel, and the right singular vector of the representation of the effective channel; and transmit (1) a signal representing a first symbol of the second encoded vector, and (2) a signal representing a second symbol of the second encoded vector, through the communication channel, to a communication device for identification of a message. CA 3164606 Date reçue / Received date 2024-08-27 46 7. The non-transitory, processor-readable medium of claim 6, wherein the instructions to cause the processor to produce the second encoded vector include instructions to: multiply a complex conjugate of the left singular vector of the representation of the effective channel by a precoding matrix to produce an intermediate matrix; and multiply the intermediate matrix by the right singular vector of the representation of the effective channel to produce the second encoded vector.

8. The non-transitory, processor-readable medium of claim 6, further storing instructions to cause the processor to select a precoding matrix from a publicly-accessible codebook of unitary matrices based on the message.

9. The non-transitory, processor-readable medium of claim 6, wherein the message is a first message, the non-transitory, processor-readable medium further storing instructions to cause the processor to: select a first precoding matrix from a publicly-accessible codebook of unitary matrices; select a second precoding matrix associated with a second index for a second message for transmission; produce a third encoded vector based on the second precoding matrix; and cause transmission of (1) a signal representing a first symbol of the third encoded vector, and (2) a signal representing a second symbol of the third encoded vector, through the communication channel, to the communication device for identification of the second message.

10. The non-transitory, processor-readable medium of claim 6, further storing instructions to cause the processor to cause transmission of signals representing a plurality of additional encoded vectors through the communication channel to the communication device until a predetermined number of messages have been sent. CA 3164606 Date reçue / Received date 2024-08-27 47 11. The non-transitory, processor-readable medium of claim 6, further storing instructions to cause the processor to: receive a third transformed signal including a version of the first symbol of the second encoded vector that has been transformed by the channel transformation; and receive a fourth transformed signal including a version of the second symbol of the second encoded vector that has been transformed by the channel transformation, wherein identification of the message includes removing a representation of the right singular vector of the representation of the effective channel from each of the third transformed signal and the fourth transformed signal.

12. A method, comprising: receiving, via a first communication device and at a first processor, a first signal representing a first symbol of a first encoded vector and a channel transformation; receiving, via the first communication device and at the first processor, a second signal representing a second symbol of the first encoded vector and a channel transformation; detecting, via the first processor, a representation of an effective channel based on the first signal and the second signal; performing, via the first processor, a singular value decomposition of the representation of the effective channel to identify a left singular vector of the representation of the effective channel and a right singular vector of the representation of the effective channel; selecting, via the first processor, a precoding matrix associated with an index for a message for transmission; producing, via the first processor, a second encoded vector based on a complex conjugate of the left singular vector, and the right singular vector of the representation of the effective channel; and transmitting (1) a signal representing a first symbol of the second encoded vector and CA 3164606 Date reçue / Received date 2024-08-27 48 (2) a signal representing a second symbol of the second encoded vector, through a communication channel, to a second communication device, for identification of the message at a second processor associated with the second communication device.

13. The method of claim 12, wherein producing the second encoded vector includes: multiplying the complex conjugate of the left singular vector by the precoding matrix to produce an intermediate matrix; and multiplying the intermediate matrix by the right singular vector of the representation of the effective channel to produce the second encoded vector.

14. The method of claim 12, wherein the precoding matrix is selected from a codebook of unitary matrices.

15. The method of claim 12, wherein the precoding matrix is a first precoding matrix, the message is a first message, and the index is a first index, the method further comprising: selecting a second precoding matrix associated with a second index for a second message for transmission; producing a third encoded vector; and transmitting a signal representing the third encoded vector, through the communication channel, to the second communication device for identification of the second message.

16. The method of claim 12, further comprising transmitting signals representing a plurality of additional encoded vectors through the communication channel to the second communication device until a predetermined number of messages have been sent.

17. A method, comprising: generating, at a processor of a first communication device, a first encoded vector; transmitting, to a second communication device and through a communication channel, CA 3164606 Date reçue / Received date 2024-08-27 49 a first signal representing a first symbol of the first encoded vector, the communication channel applying a channel transformation to the first signal during transmission; transmitting, to the second communication device and through the communication channel, a second signal representing a second symbol of the first encoded vector, the communication channel applying a channel transformation to the second signal during transmission; receiving, from the second communication device and at the processor, a third signal representing a first symbol of a second encoded vector and the channel transformation; receiving, from the second communication device and at the processor, a fourth signal representing a second symbol of the second encoded vector and the channel transformation; detecting, via the processor, a representation of an effective channel based on the third signal and the fourth signal; performing, via the processor, a singular value decomposition of the representation of the effective channel to identify a singular vector of the representation of the effective channel; and identifying a message associated with the third signal and the fourth signal based on the singular vector of the representation of the effective channel.

18. The method of claim 17, wherein the message is identified based on a codebook of unitary matrices.

19. The method of claim 17, further comprising receiving, from the second communication device and at the processor, a plurality of additional signals representing a plurality of additional encoded vectors via the communication channel from the second communication device until a predetermined number of messages have been received.

20. The method of claim 17, further comprising: detecting, via the processor, a precoding matrix associated with an index for the message, the identifying the message being based on the precoding matrix. CA 3164606 Date reçue / Received date 2024-08-27