A method for signal modulation and low complexity detection of multi-reflection devices

Abstract

The present application belongs to the technical field of wireless communication, and particularly relates to a multi-reflection device signal modulation and beamforming cooperation optimization method. The method of the present application mainly gives the modulation mode of the main system signal and the modulation order of the multi-reflection device, optimizes the signal constellation point of the reflection device and the beamforming to maximize the minimum Euclidean distance of the noise-free signal at the receiving end, thereby improving the symbol error rate performance of the main system and the Internet of Things communication. The scheme of the present application can be applied to the signal modulation of the main system and the Internet of Things system of any order, the scheme is simple to implement, and has strong application value.

Description

Technical Field

[0001] This invention belongs to the field of wireless communication technology, specifically relating to a method for signal modulation and low-complexity detection of multi-reflection devices. Background Technology

[0002] Passive Internet of Things (Passive IoT) is one of the key technologies of 6G, in which thousands of IoT devices are interconnected, providing people with an immersive and ultimate experience. The international standards organization 3GPP has begun discussions on Passive IoT, covering standards for its use cases, key supporting technologies, and service requirements.

[0003] In recent years, symbiotic wireless communication technology has emerged as a potential technology to support passive Internet of Things (IoT) due to its high spectral efficiency and high energy efficiency. In symbiotic communication, the reflective device acts as the IoT transmitter, using the main system signal as the modulation carrier to send IoT monitoring information to the receiver via reflective communication. The receiver then uses joint demodulation to recover the main system signal and the IoT device signal. Furthermore, the reflective device can provide a reflective multipath to improve the transmission performance of the main system.

[0004] Current coexisting communication modulation techniques primarily consider single-reflector scenarios. However, for future applications such as smart homes and smart cities, simultaneous access by multiple reflectors is a crucial scenario. Therefore, there is an urgent need to research cooperative modulation of signal constellation points for multiple reflectors in coexisting communication. In particular, passive reflectors are constrained by hardware limitations such as power consumption, and their reflection coefficients must meet peak amplitude constraints. Under these constraints, traditional symmetrical regular constellation point structures are not optimal for reflectors. Therefore, new methods are needed to find the optimal constellation point modulation design. Furthermore, the signals from multiple reflectors are coupled with the main system signal, making signal detection very complex. Therefore, there is an urgent need to design low-complexity signal detection methods. Summary of the Invention

[0005] The main content of this invention is to propose a signal modulation and low-complexity detection method for multi-reflection devices in passive Internet of Things (IoT). Symbiotic wireless communication is one of the potential technologies supporting passive IoT. By integrating environmental monitoring IoT devices into the reflective device, the reflective device can transmit IoT monitoring information to the receiver using the principle of reflective communication. Simultaneously, the reflective device can provide a multipath link for the main system transmission to improve its performance. Due to the application of reflective modulation, the transmission performance of the main system and the IoT communication system are coupled. Designing a good signal constellation point for the reflective device can simultaneously improve the transmission performance of both the main system and the IoT communication.

[0006] The technical solution adopted in this invention is:

[0007] like Figure 1As shown, the present invention considers a coexisting wireless communication system, comprising a single-antenna master system transmitter and a configuration N r A receiver with one antenna and K multi-antenna reflectors, each device configured with N antennas. k One reflective unit, remember This indicates the channel response from the transmitter to the receiver in the main system. This represents the channel response from the main system transmitter to the k-th reflecting device. This represents the channel response from the k-th reflecting device to the receiver. During the n-th symbol period, the symbols for the main system and the k-th reflecting device are represented as follows: and in and Let represent the signal constellation points of the main system and the k-th reflecting device, respectively. The received signal at the receiver can be represented as...

[0008]

[0009] Where P t This indicates the power of the main system transmitter; This represents additive white Gaussian noise at the receiver, where each element follows a mean of 0 and a variance of σ. 2 The complex Gaussian distribution; Let c represent the common reflection coefficient matrix of the k-th reflecting device. k (n)Φ k This forms the time-varying reflection coefficient matrix of the k-th reflecting device.

[0010] The receiver recovers the main system signal and the signals from all reflecting devices using a joint demodulation method. Assuming that both the main system signal and the reflecting device signals are transmitted with equal probability, the maximum likelihood detection method at the receiver is:

[0011]

[0012] The maximum likelihood detector described above reveals that the main system signal and the IoT device signal are coupled at the receiving signal in a multiplicative manner. This occurs when a direct link is absent, i.e., h... d =0, maximum likelihood detection will encounter ambiguity problems. Specifically, assuming If it is a set of optimal estimates, then under the condition that... conditions, This is also a set of optimal estimates. Therefore, the problem of detecting ambiguity arises in this case. A typical example of the problem of detecting ambiguity is... This means that the traditional symmetrical constellation points are not the optimal design for reflective devices. Therefore, it is urgent to study the cooperative optimization of signal constellation points of reflective devices in coexisting communication scenarios with multiple reflective devices to solve the detection ambiguity problem and improve detection bit error rate performance. Simultaneously, multiple antennas of reflective devices can be used for joint beamforming to improve the equivalent channel strength, thereby enhancing transmission performance.

[0013] Furthermore, it can be seen from formula (2) that although using a maximum likelihood detector can guarantee optimal demodulation performance, its complexity is very high because it requires exhaustively searching every possible combination of constellation points. Its complexity is comparable to... Proportional, where M s and M k Let represent the modulation order of the main signal and the k-th reflecting device, respectively. This phenomenon is more pronounced when the number of reflecting devices is large. Therefore, designing a low-complexity signal detection method is crucial.

[0014] The method of the present invention specifically includes:

[0015] The optimization problem, with the objective of maximizing the minimum Euclidean distance of the received signal, is as follows:

[0016]

[0017] Among them, the constraint condition |c k [m]|≤1 indicates the signal peak amplitude constraint for each reflecting device, M k This represents the modulation order of the k-th reflecting device. Constraint Φ k [n,n]=1 indicates the constant modulus constraint of the common reflection coefficient. Denotes the diagonal matrix of common reflection coefficients of the k-th reflecting device; constraint condition D min This represents the minimum Euclidean distance at which there is no noise signal at the receiving end. This represents the channel from the k-th reflecting device to the receiver. This represents the channel from the main system transmitter to the k-th reflecting device. This represents the channel from the transmitter to the receiver in the main system, s i and s j Let c represent the i-th and j-th elements in the set of modulation constellation points of the main system transmitter. k [p k ] and c k [q k ] represents the p-th point in the set of modulation constellation points of the k-th reflecting device. k The qth element and the qth element k One element;

[0018] Solving the optimization problem involves transforming it into two sub-problems related to the signal constellation points and beamforming variables:

[0019]

[0020]

[0021] First, solve the subproblem concerning the signal constellation points. Specifically, stack the constellation point sets of all reflecting devices into a column vector, denoted as . Thus making Let m be the m-th element of an M×M identity matrix. k The Euclidean distance between any two constellation points receiving the signal is expressed as:

[0022]

[0023] in,

[0024] Optimization variables Converting it to the format of vector c, and introducing a non-negative auxiliary variable t, we get:

[0025]

[0026] t≥0,

[0027] By scaling the quadratic function, we further obtain:

[0028]

[0029] t≥0,

[0030] Where c (l) This represents the set of signal modulation constellation points of the reflective device obtained through optimization during the l-th iteration, and the optimized design of the signal modulation constellation points of the reflective device is obtained by solving the optimization tool CVX.

[0031] After obtaining the optimized design of the modulation signal constellation points of the reflection device, the sub-problem of beamforming variables is then solved, specifically:

[0032] Define variables:

[0033]

[0034] The Euclidean distance between the two constellation points receiving the signal is re-expressed as:

[0035]

[0036] Where a = h d (si -s j ), F = Hdiag(g), φ k The vector form of the reflection coefficient matrix of the k-th reflecting device. This represents the total number of antennas for all reflecting devices;

[0037] Transform the matrix form of the optimization variable Φ into the vector form v, and let... And introduce a positive semi-definite matrix Satisfying the rank-one constraint rank(V) = 1 and the matrix positive semi-definite condition Introducing a non-negative auxiliary variable t≥0 transforms the subproblem of beamforming variables into:

[0038]

[0039] rank(V) = 1,

[0040]

[0041] in, The transformed problem is a semidefinite programming problem. After relaxing the rank-one constraint, it is solved using the standard optimization tool CVX, and then a rank-one solution is recovered using the Gaussian random method.

[0042] Finally, by iteratively solving two sub-problems related to signal constellation points and beamforming variables, the original optimization problem is solved, and the optimized design of signal constellation points and beamforming variables is obtained.

[0043] The receiver detects signals using the following method:

[0044] Given a master signal s, reconstruct the direct link signal as follows: P t This represents the transmit power of the main system transmitter. Then, the reconstructed direct link signal is subtracted from the received signal to obtain:

[0045]

[0046] Using beamforming matrix From the received signal To obtain estimates of the signals from all reflecting devices:

[0047]

[0048] The linear receiver beamforming matrix is ​​represented as:

[0049]

[0050] in h k =H k Φ k g k , σ 2 It is the variance of the additive white Gaussian noise at the receiver;

[0051] make Given a master signal s, the estimated set of signals from all reflecting devices is used to... The following demodulation method is used to recover the signal of each reflecting device:

[0052]

[0053] in, Representing vectors The kth element;

[0054] Using a method to obtain signal estimation of the reflecting device from a given master signal, M is obtained respectively. s The estimated values ​​of the main signal and the reflected device signal will be used to determine the M values. s Substitute the estimated values ​​into the maximum likelihood detector and select the set of estimated values ​​that minimizes the mean squared error as the final estimated value:

[0055]

[0056] The final signal estimate of the reflecting device is denoted as:

[0057] The beneficial effects of this invention are that, given the modulation method of the main system signal and the modulation order of the multiple reflection devices, the minimum Euclidean distance of the noise-free signal at the receiving end is maximized by optimizing the signal constellation points and beamforming of the reflection devices, thereby improving the symbol error rate performance of the main system and IoT communication. The solution of this invention can be applied to the signal modulation of any order of the main system and IoT system. The solution is simple to implement and has strong application value. Attached Figure Description

[0058] Figure 1 A schematic diagram of the system composition of the present invention is shown;

[0059] Figure 2 A design diagram of the signal constellation points of the reflecting device in this invention is shown;

[0060] Figure 3 This is a graph showing the symbol error rate performance of the main system signal in this invention;

[0061] Figure 4 This is a diagram showing the symbol error rate performance of the signal from the reflecting device in this invention.

[0062] Figure 5This is a graph showing the performance of low-complexity signal detection in this invention. Detailed Implementation

[0063] The present invention will now be described in detail with reference to the accompanying drawings:

[0064] Take, for example, a wireless communication system with multiple antennas and multiple reflection devices coexisting. Figure 1 As shown, the system includes a single-antenna main system transmitter and a configuration N r A receiver with one antenna and K reflecting devices, where the k-th reflecting device is configured with N... k One reflective unit, remember and This represents the symbols transmitted by the main system and the k-th reflecting device in the n-th symbol period, where and Let represent the constellation sets of the transmitted signals of the main system and the k-th reflecting device, respectively. The received signal of the receiver can be represented as...

[0065]

[0066] in This represents the channel from the transmitter to the receiver in the main system. This represents the channel from the main system transmitter to the k-th reflecting device. This represents the channel from the k-th reflecting device to the receiver. P represents the diagonal matrix of common reflection coefficients of the k-th reflecting device. t z(n) represents the transmit power of the main system transmitter, and z(n) represents the additive complex Gaussian noise at the receiver.

[0067] Using the maximum likelihood detection scheme, the detection formula at the receiver is as follows:

[0068]

[0069] Existing reflective devices employ traditional phase modulation schemes, but detection ambiguity arises when the direct link is blocked. Therefore, the modulation method for reflective devices needs to be redesigned and optimized. This invention proposes a collaborative optimization method for signal constellation points and beamforming of multiple reflective devices. For simplicity, the time index (n) of the transmitted signal is omitted hereafter.

[0070] Since ambiguity can cause partial overlap of constellation points in the received signal, this invention aims to solve the ambiguity detection problem by maximizing the minimum Euclidean distance of the received signal. The specific problem model is as follows:

[0071]

[0072] The first constraint represents the peak signal amplitude constraint for each reflecting device, and M k The first constraint represents the modulation order of the k-th reflecting device; the second constraint represents the constant mode constraint of the common reflection coefficient; in the third constraint, D... min This represents the minimum Euclidean distance at which there is no noise signal at the receiving end.

[0073] By solving problem P1, the signal constellation point design and beamforming design for each reflecting device can be obtained. Preliminary analysis shows that this problem is non-convex, and the optimization variables constellation points and beamforming are coupled. The solution approach utilizes the idea of ​​alternating optimization to transform the problem into two sub-problems related to the signal constellation points and beamforming variables, expressed as follows:

[0074]

[0075] and

[0076]

[0077] (1) Algorithm for solving problem P2

[0078] Stack the constellation points of all reflecting devices into a column vector, denoted as . in and Let represent the set of modulation signal constellation points for the k-th reflecting device. Therefore, the p-th point in the set of modulation constellation points for the k-th reflecting device... k Each element can be represented as in Let m be the m-th element of an M×M identity matrix. k List.

[0079] Therefore, the Euclidean distance between any two constellation points receiving the signal can be expressed as:

[0080]

[0081] in a = h d (s i -s j ),

[0082] Optimization variables Converting to the format of vector c and introducing a non-negative auxiliary variable t, problem P2 can be transformed into:

[0083]

[0084] Problem P2-A is a non-convex quadratic constrained quadratic programming problem. For non-convex quadratic constraints, the continuous convex approximation method can obtain a linear constraint on the optimization variables through a first-order Taylor expansion. By scaling the quadratic function in equation (6), problem P2-A can be transformed into the following form:

[0085]

[0086] Where c (l) Let represent the set of signal modulation constellation points of the reflecting device obtained through optimization during the l-th iteration. Thus, problem P2-B is a standard convex problem with optimization variables c and t, which can be solved efficiently using existing optimization tools, such as the CVX optimization toolkit built into Matlab.

[0087] (2) Algorithm for solving problem P3

[0088] After solving subproblem P2-B, the optimized design of the modulation constellation points of the reflecting device can be obtained. Then, given the optimized result of the modulation constellation points of the reflecting device, the design of the reflection coefficient matrix can be obtained by solving subproblem P3. To facilitate optimization, variables are first defined...

[0089]

[0090] Therefore, the Euclidean distance between the two constellation points receiving the signal can be rewritten as:

[0091]

[0092] Where a = h d (s i -s j ), F = Hdiag(g), φ k The vector form of the reflection coefficient matrix of the k-th reflecting device. This represents the total number of antennas for all reflecting devices.

[0093] Transform the matrix form of the optimization variable Φ into the vector form v. Let... And introduce a positive semi-definite matrix Satisfying the rank-one constraint rank(V) = 1 and the matrix positive semi-definite condition Then, by introducing a non-negative auxiliary variable t≥0, problem P3 can be transformed into the following form.

[0094]

[0095] in Problem P3-A is a semidefinite programming problem. After relaxing the rank-one constraint, it can be solved using the standard optimization tool CVX. Then, a rank-one solution is recovered using the Gaussian stochastic method described in the literature "Sidiropoulos ND, Davidson TN, Luo Z Q. Transmit beamforming for physical-layer multicasting[J].IEEE Transactions on Signal Processing,2006,54(6):2239-2251". The original problem P1 can be solved by iteratively solving the two subproblems P2 and P3.

[0096] After solving problem P1, a low-complexity receiver design is proposed to recover the main signal and the signals from all reflecting devices. From the received signal model, it can be observed that the modulated carrier of all reflecting device signals is the main signal. Based on this phenomenon, a linear receiver design is then proposed, consisting of two steps.

[0097] Step 1: Given the main signal s, the direct link signal can be reconstructed as follows: Then, the reconstructed direct link signal is subtracted from the received signal to obtain...

[0098]

[0099] After eliminating the direct link signal, the remaining reflected link signal can be viewed as a classic MIMO (Multiple-Input-Multiple-Output) channel model. Therefore, classic MIMO signal detection methods, such as linear receive beamforming, can be used to recover the signal from the reflecting device. First, a beamforming matrix is ​​used... From the received signal The estimates of the signals from all reflecting devices are obtained as follows:

[0100]

[0101] The linear receiver beamforming matrix is ​​expressed as follows:

[0102]

[0103] in h k =H k Φ k g k .

[0104] remember This is the set of estimated signals for all reflecting devices given the main signal s. Then, the estimated set of reflecting device signals is used... The following demodulation methods can be used to recover the signal from each reflecting device:

[0105]

[0106] in Representing vectors The k-th element. Since the signals of all reflecting devices are recovered independently, the search complexity of formula (14) is .

[0107] Step 2: Following Step 1, given a master signal constellation point, a set of signal estimates for all reflecting devices can be obtained. Therefore, there are a total of M s Estimate the values ​​of the main signal and the signal from the reflecting device. Then, use this M... s Substituting the estimated values ​​into the maximum likelihood detector, a set of estimates that minimizes the mean square error is selected as the final estimate. The above process is given by the following formula:

[0108]

[0109] Finally, the signal estimate of the reflecting device is denoted as: Since the search complexity of the proposed low-complexity receiver is This is far less complex than the search complexity of directly using a maximum likelihood detector. Therefore, the proposed low-complexity receiver design can be extended to scenarios with multiple reflective devices, and it has more advantages when there are a large number of reflective devices.

[0110] Below, this invention presents simulation results to verify the effectiveness of the proposed multi-reflection device signal constellation point and beamforming cooperative optimization scheme. Specifically, this invention considers three benchmark schemes.

[0111] Reference Scheme 1: The multi-reflection device adopts a cooperative beamforming design, and the constellation points adopt a phase shift keying design.

[0112] Reference Scheme 2: The multi-reflection device adopts a random beamforming design, and the constellation points adopt a phase shift keying design.

[0113] Benchmark Scheme 3: The multi-reflection device adopts a random beamforming design, and the constellation points adopt a cooperative optimization design.

[0114] For the channel model, this invention considers the Ricean channel model, where the channel includes both large-scale and small-scale fading. Large-scale fading is modeled as PL(d,ξ) = 10. -3 d -ξWhere d represents the distance between the two nodes, and ξ represents the path loss exponent. Small-scale fading includes direct path components and indirect path components. This invention considers a scenario with two reflecting devices, where the position of the main system transmitter is x. T = (0,0), the positions of the two reflecting devices are x and x respectively. R1 = (100,5) and x R2 = (95, -5), the receiver's position is x R = (105, 5). Without loss of generality, this invention considers the path loss index from the main system transmitter to the receiver to be ξ1 = 3.8, the path loss index from the main system transmitter to the reflecting device to be ξ2 = 2.2, and the path loss index from the reflecting device to the receiver to be ξ3 = 2.1. The noise power is set to σ. 2 = -100dBm. Each reflector has 32 reflector elements, and the signal modulation order is 4. The main system transmitter signal uses binary phase shift keying modulation. The channel implementation count is 10. 3 Second-rate.

[0115] Figure 2 This paper demonstrates the optimal constellation design for two reflecting devices using the proposed cooperative optimization scheme when the receiver antenna is a single antenna and a direct link is absent. For comparison, the invention also illustrates a constellation design using traditional phase-shift keying (PSK) for the reflecting devices. A special case is considered where the reflection link strengths of the two reflecting devices are consistent. In this case, it can be seen that if a traditional PPSK constellation design is used, the constellation points of the noise-free signal at the receiver will partially overlap, leading to demodulation ambiguity during joint demodulation. However, by employing the proposed constellation optimization scheme, this ambiguity problem can be solved. It can be observed that the constellation points of the noise-free signal at the receiver will separate, thus resolving the ambiguity problem. After cooperative optimization, the signal constellation points of the reflecting devices are no longer the traditional zero-mean symmetrical regular constellation structure, but rather a non-zero-mean constellation structure.

[0116] Figure 3 and Figure 4 The symbol error rates (BER) of the master system signal and the reflector signal are shown when the direct link is absent. From the BER performance perspective, the proposed constellation point and cooperative beamforming optimization scheme for the reflector signal outperforms the three benchmark schemes. It can be observed that benchmark schemes 1 and 2 lead to demodulation ambiguity issues, resulting in a BER of 0.5 for both the master system signal and the reflector signal. Although benchmark scheme 3 can resolve the ambiguity issue through cooperative signal constellation point design, its use of a random beamforming strategy leads to a very weak equivalent channel strength for the reflector link. Therefore, benchmark scheme 3 performs worse than the proposed constellation point and beamforming cooperative optimization scheme, validating the effectiveness of the proposed cooperative optimization scheme.

[0117] Figure 5 The graphs show the symbol error rate (BER) performance of the reflecting device signal as a function of transmitted power under low-complexity receiver schemes. Three receiver beamforming schemes are considered: MRC, ZF, and MMS receivers. The case of a weak reflecting link is also considered, where the path loss exponent from the main system transmitter to the reflecting device is set to 2.9. Clearly, the maximum likelihood detector outperforms the other three linear receiver schemes. For the other three low-complexity receivers, the MMSE receiver outperforms the ZF and MRC receivers. This is because the MRC receiver maximizes the SNR of only one reflecting link, treating signals from other reflecting links as interference. The ZF receiver eliminates signal interference from other reflecting devices through matrix zero-forcing. Although the ZF method performs well in the high SNR range, it may amplify noise power in the low SNR range, resulting in poor BER performance. The MMSE receiver achieves a balance between eliminating reflected signal interference and amplifying noise power.

Claims

1. A method of signal modulation and low complexity detection for a multi-reflecting device system, the system comprising a main system transmitter configured with a single antenna, a receiver configured with N r antennas, and K multi-antenna reflecting devices, each reflecting device configured with N k reflecting units, the method comprising the steps of: The method is: An optimization problem is established to maximize the minimum Euclidean distance of the received signal: where the constraint |c k [m]|≤1 represents the signal peak amplitude constraint of each reflecting device, M k represents the modulation order of the kth reflecting device, the constraint Φ k [n,n]=1 represents the constant modulus constraint of the common reflection coefficient, represents the common reflection coefficient diagonal matrix of the kth reflecting device; the constraint D min represents the minimum Euclidean distance of the noiseless signal at the receiving end, represents the channel from the kth reflecting device to the receiver, represents the channel from the main system transmitter to the kth reflecting device, represents the channel from the main system transmitter to the receiver, s i and s j represent the i-th element and the j-th element in the main system transmitter modulation constellation point set, c k [p k ] and c k [q k ] represent the p k -th element and the q k -th element in the kth reflecting device modulation constellation point set; Solving the optimization problem, specifically converting the optimization problem into two sub-problems related to the signal constellation point and the beamforming variable, respectively: First, the sub-problem about the constellation points of signals is solved, specifically, the constellation points of all reflecting devices are stacked into a column vector, denoted as Thus, let The mthcolumn of a M × M unit matrix is denoted as k The Euclidean distance between any two constellation points of the received signal is denoted as: wherein a = h d (s i -s j ), The optimization variable is converted to the format of the vector c and an auxiliary non-negative variable t is introduced, resulting in: Scaling the quadratic function further obtains: where c (l) represents the set of reflection device signal modulation constellation points obtained by optimization in the lth iteration process, so as to obtain the optimization design of the reflection device modulation signal constellation points by solving the optimization tool CVX. After obtaining the optimization design of the reflection device modulation signal constellation point, solve the sub-problems of the beamforming variable, specifically: Define the variable: The Euclidean distance between the two constellation points of the received signal is re-expressed as: where a = h d (s i -s j ), F = Hdiag(g), φ k denotes the vector form of the reflection coefficient matrix of the k-th reflecting device, denotes the sum of the number of antennas of all reflecting devices; The matrix form of the optimization variable Φ is converted into the vector form v, let and a semi-positive definite matrix satisfy the rank-one constraint condition rank(V) = 1 and the matrix semi-positive definite condition A non-negative auxiliary variable t ≥ 0 is introduced, and the sub-problem of the beamforming variable is converted into: rank(V) = 1, where, The transformed problem is a semidefinite programming problem, which is solved by CVX after relaxing the rank-one constraint, and a rank-one solution is recovered by the Gaussian random method. Finally, by iteratively solving the two sub-problems related to the signal constellation point and the beamforming variable, the original optimization problem is solved, and the optimization design of the signal constellation point and the beamforming variable is obtained; The detection method of the receiver for the signal is: Given a host signal s, the reconstructed direct link signal is P t denotes the transmit power of the host system transmitter, and the reconstructed direct link signal is subtracted from the received signal to obtain Using a beamforming matrix From the received signal Obtain an estimate about all reflected device signals: The linear receive beamforming matrix is represented as: wherein h k = H k Φ k g k , σ 2 is the variance of the additive white Gaussian noise at the receiver; Let For the set of signal estimates of all reflecting devices given the main signal s, use the estimated set of reflecting device signals The signal of each reflecting device is recovered using the following demodulation: wherein denotes the kth element of the vector denotes the kth element of the vector The M s group of main signal and reflection device signal estimates are obtained by a method of obtaining reflection device signal estimates from a given main signal, and the M s group of estimates are substituted into a maximum likelihood detector, and a set of estimates that can minimize the mean square error is selected as the final estimates. The signal estimate of the reflecting device is finally obtained and denoted as

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