Optimization of pre-equalized time reversal security transmission systems assisted with artificial noise

Time reversal (TR) transmission technology can focus the power of a signal in both time and space domains and reduce signal energy leakage to unintended receivers, making it suitable for physical layer security systems. By adding a pre-equalizer to the transmitter and optimizing it, the performance of multi-antenna TR transmission system can be improved obviously with an acceptable optimization complexity. In this paper, the pre-equalizer and artificial noise (AN) are jointly optimized to improve the security performance of pre-equalized TR (ETR) systems when eavesdropping channel state information (ECSI) is unknown or known. When ECSI is unknown, null-space AN is adopted, and the pre-equalizer is optimized to minimize the signal power under the constraint of the minimum signal-to-interference-plus-noise ratio (SINR) of the legitimate receiver, so the AN power and the interference with the eavesdropper are maximized. The optimization problem is transformed into the problem of finding the maximum generalized eigenvalue of a matrix pencil. When ECSI is known, under the minimum SINR constraint of the legitimate receiver, the pre-equalizer and AN’s covariance matrix are jointly optimized to minimize the SINR of the eavesdropper. The optimization problem is non-convex and is transformed into a convex problem, which can be solved using the CVX toolbox. The optimization complexity is independent of the number of transmit antennas. Simulation results demonstrate that compared with TR systems, by optimizing the pre-equalizer, the security performance of ETR multi-antenna systems can be significantly improved, and the optimization complexity is acceptable.


Introduction
In wireless communication systems, signals travel along different paths between the transmitter and the receiver. The propagation time of signals on each path is different, resulting in the extension of the symbol period and inter-symbol interference (ISI). RAKE receiving and equalization are two technologies applied by receivers to eliminate or reduce ISI and gather the signal energy from multiple paths to enhance the signal strength. However, the complex equalizer or RAKE receiver is seldom available for the receivers with limited processing capacity. signal processing technology employed at the transmitter to reduce ISI and enhance the signal power at the intended receiver. A TR transmission consists of two steps. In the first step, the receiver sends an impulse signal to the transmitter. The transmitter estimates the multipath channel's channel impulse response (CIR) based on the received signal. The TR pre-filter's impulse response (IR) is the time-reversed and conjugated CIR. In the second step, the signal is filtered by the TR pre-filter and transmitted. Therefore, the multipath channel is the matched filter of the pre-filtered signal. The signals from the different paths are coherently superposed at a specific moment in the intended receiver. So, there is an obvious signal peak while the delay spread is reduced and ISI is alleviated. The channel between the transmitter and any unintended receiver is not the matched filter of the pre-filtered signal, so the power of the signal received at unintended receivers is significantly lower than that of the intended receiver. This feature is called spatial-temporal focusing of TR transmission. Since the signal processing is executed at the transmitter in TR systems, the complexity of the receiver is low.
The early research of TR transmission technology mainly focused on its application in underwater acoustic communications [1]. In recent years, TR transmission technology in wireless communications has been studied by scholars. The spatial-temporal focusing effect of TR has been verified by [2], and the experiment results proved that the leakage of radio frequency energy to unintended receivers is low in TR wireless communication systems. Although ISI is alleviated by TR pre-filtering, ISI still exists and has a severe impact on the received signal when the delay spread of the channel is larger than the symbol period. ISI can be reduced by upsampling the transmitted signal, but the spectrum efficiency will decline [3]. Some research works studied the design of the TR pre-filter to reduce ISI or maximize the signalto-interference-plus-noise ratio (SINR). In [4], a method to reduce ISI by aligning interference is studied. Pre-distortion waveform is designed for each symbol to align the ISI of the signal at the intended receiver, so the signal energy is promoted while ISI is suppressed. In [5], the optimization of the TR pre-filter for cloud access networks is studied, and a content-aware waveform design and an optimal receiving algorithm are proposed. The simulation results show that the performance of the proposed scheme is better than that of the conventional TR pre-filter scheme.
Information security is a crucial issue in communications. Because of the openness of wireless channels, wireless communications meet more severe security threats than wired communications. The security threats include eavesdropping, jamming, fake information, disguising, etc. [6][7][8]. Physical layer security (PLS) exploits the characteristics of wireless channels such as randomness, time-varying, and reciprocity to guarantee the security of information transmission by using physical layer technologies [9,10]. The PLS technologies can be used for anti-eavesdropping, anti-jamming, key generation, identification, etc. Antieavesdropping is the most in-depth research field in PLS. Secrecy capacity is an important metric to evaluate the antieavesdropping performance of PLS. To increase the secrecy capacity, it is necessary to increase the quality advantage of the legitimate channel over the eavesdropping channel. The technologies to increase the secrecy capacity include multiple antenna pre-coding and beamforming [11], artificial noise (AN) [12], cooperative forwarding [13], etc. By beamforming, the signal is radiated to the legitimate receiver and the signal energy leaked to the eavesdropper is suppressed. AN is often employed in multi-antenna systems. By controlling AN's radiation direction, AN can significantly deteriorate the signal quality at the eavesdropper without interfering with the legitimate receiver. However, the number of antennas is limited at small-size and low-cost nodes, which greatly restricts the effect of beamforming and AN. When the channels are multipath channels, the spatial-temporal focusing feature of TR transmission makes the signal energy focus on the target receiver. The received signal power and AN power at the legitimate receiver and the eavesdropper can be controlled by TR pre-filtering, even when there is only one antenna installed at the transmitter. Whether it is a multi-antenna or a single-antenna system, its security performance can be improved by using TR technology. The signal-to-noise ratio (SNR) of the target receiver and the unintended receiver in a distributed TR system is studied by [14]. It demonstrates that TR pre-filtering can improve security performance of the system. In [15], the ergodic achievable secrecy rate of TR systems is derived, and the bit error rates (BERs) at the target receiver and the unintended receiver for binary phase-shift keying (BPSK) are deduced too. The results show that the target receiver can achieve a higher transmission rate and a lower BER than the unintended receiver. In [16], the security performance of multiple-input multiple-output (MIMO) TR systems is studied. The results show that TR pre-processing can significantly improve the security performance of the system.
[17] studies the security performance of a TR multi-user MIMO downlink system. It finds that TR beamforming can promote the secrecy capacity and is an attractive and costeffective solution for security. It also finds that the inter-user correlation may cause a significant loss of achievable secrecy sum-rate. The conventional TR pre-processing, whose prefilter is matched to the channel of the target receiver, can improve the signal power of the target receiver. However, from the perspective of secure transmission, it is not optimal. In [18], for a multi-input single-output (MISO) TR transmission system, the optimal design scheme of TR pre-filter to maximize the secrecy rate is studied. The simulation results show that the achievable secrecy rate of the proposed method is better than that of the conventional TR scheme. In [19], for a multi-user TR secure transmission system, two optimization schemes for the TR filters and AN are proposed respectively for the scenarios when the eavesdropping channel state information (ECSI) is known and unknown. When ECSI is unknown, null-space AN is adopted, and the TR pre-filter is optimized to minimize the signal power and AN power is maximized. When ECSI is known, TR pre-filter and AN are jointly optimized to maximize the sum secrecy rate. In [20], a secure transmission scheme combining AN with TR pre-processing is proposed, and the power allocation scheme between the signal and AN is optimized. A frequency domain TR security precoding and AN injection scheme for orthogonal frequency-division multiplexing (OFDM) systems is proposed by [21]. The TR precoder and the AN power are optimized. The simulation results of [19][20][21] show that AN can effectively improve the secure transmission performance of TR systems.
In multi-antenna TR systems, each antenna assigns a TR pre-filter. In order to obtain a good performance, all TR prefilters need to be optimized jointly. When the number of paths of the channel or the number of antennas is large, the complexity of optimization is very high. Pre-equalized time reversal (ETR) transmission scheme is proposed to reduce the optimization complexity of multi-antenna TR systems [22][23][24][25]. In ETR systems, a pre-equalizer is added ahead of the conventionally TR pre-filters. The pre-equalizer is designed according to a specific criterion, such as zeroforcing (ZF), minimum mean square error (MMSE), etc. The pre-equalizer can also be optimized to improve a specific performance of the system. Because only one pre-equalizer is optimized, the optimization complexity of ETR systems is much lower than that of TR systems. [22] compares the performances of the conventional TR system combined with a ZF pre-equalizer, the pure ZF system and the pure TR system, and demonstrates that the TR system combined with the pre-equalizer has the best BER performance. A theoretical bound on the BER of ZF-ETR systems has been derived in [23]. The numeric results based on the bound show that the ETR system has a better BER performance than the conventional TR system under any SNR. [24] studied the design of the pre-equalizer to mitigate ISI and the multi-stream interference for a spatial multiplexing single-input-multipleoutput ETR system, according to the ZF criterion and MMSE criterion respectively. Simulation results show that the preequalization can considerably improve BER performance of the system. In [25], for a MISO TR transmission system, the pre-equalizer is optimized to minimize ISI under the constraint of the minimum peak power of the received signal. In multi-antenna PLS transmission systems, adopting TR pre-processing and optimizing each pre-filter can significantly improve the security performance. However, because the security performance is related to both the legitimate channel and the eavesdropping channel, the optimization of the TR pre-filters is more complicated than in non-secure TR systems, especially in multi-antenna TR systems. The optimization complexity will be reduced significantly if a pre-equalizer is introduced into a multi-antenna TR PLS system.
For an ETR-MISO system with eavesdroppers, this paper designs an anti-eavesdropping scheme employing AN and TR pre-processing technologies. AN and the pre-equalizer are optimized to enhance the anti-eavesdropping performance when ECSI is known or unknown respectively. When ECSI is unknown, null-space AN is adopted. To maximize AN power, the pre-equalizer is optimized to minimize the signal power under the constraint of the minimum SINR of the legitimate receiver. When ECSI is known, under the constraint of the minimum SINR of the legitimate receiver, the AN covariance matrix and the pre-equalizer are optimized to minimize the SINR of the eavesdropper. To the best of our knowledge, there are not work researching the optimization of ETR PLS systems.

System model
The model of the system studied by this paper is shown in Fig. 1, which is an ETR-MISO system consisting of a multiantenna transmitter, a single-antenna legitimate receiver, and a single-antenna eavesdropper. The number of antennas of the transmitter is N. It is assumed that both the legitimate channel and the eavesdropping channel are frequency selective. For frequency selective fading channels, CIR is a basic function to describe the channels' characteristic. In this paper, CSI is identical to CIR. CIRs from the i-th antenna of the transmitter to the legitimate receiver and to the eavesdropper are expressed as h B,i [l] and h E,i [l], respectively, where l 0, . . . , L − 1, and L is the CIR's length, that is, the number of the paths of the channels. The IR of the TR pre-filter is the time-reversed and conjugated CIR of the legitimate channel, that is, the IR of the TR pre-filter of the i-th antenna is , where (·) * represents the conjugate operation. Before the TR pre-filter, there is a pre-equalizer. The IR of the pre-equalizer is denoted as g pre [l], where l 0, 1, . . . , 2L − 2. The number of taps of the pre-equalizer is 2L − 1. The CIRs and the IRs of the pre-filters and the preequalizer can be expressed in vector form respectively as where the superscript T denotes transpose operation.
The symbol sequence to be transmitted is denoted as {x[n]} M−1 n 0 , where M is the length of the symbol sequence and E |x[n]| 2 1. After filtered by the pre-equalizer and the TR pre-filters, the signals are transmitted by all antennas. The transmitter also emits AN with power P AN to prevent the information from being wiretapped. The AN sequence emitted by each antenna is denoted as The signal sent by the i-th antenna is expressed as where ⊗ represents discrete convolution. The first part on the right side of (2) is the signal carrying information, and the second part is AN. The sequence length of the signal is L s M + 3L − 3, and the sequence length of AN is also L s . The received signal of the legitimate receiver is wherez B [n] is the complex Gaussian channel noise with variance σ 2 B . By substituting (2) into (3), we get Letting h B,TR [n] , the above formula can be rewritten as After the symbol is sent, the symbol peak at the legitimate receiver appears at the (2L − 2)-th sample. So the sampled sequence for detection can be expressed as where . The first part on the right side of the last equal sign is the expected signal, the second part is ISI, and the third part is AN.
We replace the convolution operation in (6) with matrix multiplication for the convenience of description. Firstly, we Furthermore, we denote h B,l as the transpose of the l-th row of H B,TR . We also define a (L s + L − 1) × L sdimensional Toeplitz matrix H B,AN which is composed of h B,AN [n], which is expressed as where z AN [z AN [0], z AN [1], · · · , z AN [L s − 1]] T is the vector form of the AN sequence. The SINR of the symbol where AN E z H AN z AN is the covariance matrix of AN, and (X) H represents the conjugate transpose of matrix or vector X. Similarly where h E,TR [n] . We also denote h E,l as the transpose of the l-th row of H E,TR . Then we define a (L s + L − 1) × L s -dimensional Toeplitz matrix H E,AN as follows: where where z E [n] is the complex Gaussian channel noise with variance σ 2 E . The SINR of the symbol y E [n] is In this paper, we consider the worst-case that the eavesdropper has a strong signal processing ability and can eliminate all ISI, so there is no ISI power in (14).
The achievable secrecy rate is one of the metrics to measure the performance of PLS systems and is widely adopted in the research of PLS, which is the difference between the legitimate channel's capacity and the eavesdropping channel's capacity: where [x] + max{0, x}, γ B and γ E represent the SINR of the legitimate receiver and that of the eavesdropper, respectively. The secrecy rate can be promoted by increasing the SINR of the legitimate receiver or by reducing the SINR of the eavesdropper. From (10) and (14), we can find that to promote the secrecy rate, the IR of the pre-equalizer and the AN's covariance matrix need to be optimized. We will discuss this in the next section.

Scheme and optimization algorithm
In this paper, it is assumed that the CSI of the legitimate channel, that is, h B,i , i ∈ { 1, · · · , N } is known, and the CSI of the eavesdropping channel-ECSI, that is, h E,i , i ∈ { 1, · · · , N } is either known or unknown. The security schemes are designed and optimized for the two scenarios respectively.

Scheme and optimization when ECSI is unknown
AN is widely used in PLS systems. When ECSI is unknown, AN should be radiated in all directions. In order not to interfere with the legitimate receiver, AN is arranged in the null space of the legitimate channel. The expression of the received AN at the legitimate receiver in (6) (N AN ×N AN ) . The covariance matrix of AN is expressed as In this case, the SINR of the legitimate receiver is Because ECSI is unknown, the expression of the secrecy rate cannot be obtained, and the secrecy rate cannot be the objective of the optimization. As an alternative, we optimize the tap coefficients of the pre-equalizer to minimize the signal power under the constraints of the total transmit power limit and the minimum SINR requirement of the legitimate receiver. In this way, the power of AN will be maximized and the eavesdropper is disturbed in the most degree. The optimization problem can be expressed as where γ 0 is the SINR threshold of the legitimate receiver, P max is the maximum transmit power, P s 2 is the signal power, and P s + P AN ≤ P max . The expression of P s can be rewritten in matrix form as where G i is a (3L − 2) × (2L − 1)-dimensional Toeplitz matrix: Letting It is easy to know that A is a symmetric positive definite matrix. The optimization problem (19) can be re-expressed as It is difficult to solve optimization problem (22) directly, so we transform the problem first. Matrix A can be decomposed into A Ã HÃ , whereÃ 1 2 U, U is a matrix composed of the eigenvectors of A, λ 2L−1 ), and λ 1 , λ 2 , · · · , λ 2L−1 are the eigenvalues of A. LettingÃg pre √ P s α, where α is a column vector with 2-norm 1, we can get P s g H pre Ag pre P s α H α and g pre √ P sÃ −1 α. Substituting √ P sÃ −1 α in (18) for g pre , the SINR of the legitimate receiver can be rewritten as Then, substituting (23) into (22) and adding a 2-norm constraint about α, that is, α H α 1, we can get a new form of the optimization problem as It can be found that γ N -AN B (α, P s ) is a generalized Rayleigh quotient, and its maximum value is the maximum generalized eigenvalue of matrix pencil ( 1 , 2 ) and is a monotone increasing function of P s , where We denote the maximum value of γ N -AN B (α, P s ) as γ N -AN B,max (α, P s ). So, the minimum signal power P opt s under the SINR constraint must meet γ N -AN B,max (α opt , P opt s ) γ 0 , where α opt is the generalized eigenvector corresponding to the maximum generalized eigenvalue of the matrix pencil when P s P opt s . The binary search can be applied to find the minimum signal power efficiently. Each search is carried out in two steps. In the first step, P s in matrix pencil ( 1 , 2 ) is substituted with the middle point of the search section, and the generalized eigenvalue decomposition of the matrix pencil is done. In the second step, we find the maximum generalized eigenvalue of matrix pencil and compare it with γ 0 . If the eigenvalue is larger than γ 0 , the upper boundary of the search section is updated with the middle point; otherwise, the lower boundary is updated with the middle point. The above two steps are repeated until the range of the search section is less than ε, which is a small positive number and determines the precision of the search. After the binary search is completed, the optimal solution of the signal power (P opt s ) is the middle point of the search section. The optimal solution α is the generalized eigenvector corresponding to the maximum generalized eigenvalue of the matrix pencil when P s P opt s . The optimal solution to the IR of the preequalizer is g opt pre P opt sÃ −1 α opt , and the power of AN is P AN P max − P opt s . Algorithm is summarized in Table 1

Scheme and optimization when ECSI is known
If the eavesdropper a user of the network, ECSI can be obtained. In this case, it is optimal that the AN's covariance matrix and the pre-equalizer's IR are jointly optimized to maximize the secrecy rate. However, the complexity of Calculate the eigenvalues of matrix pencil ( 1 , 2 )

5)
Get γ N -AN B,max (α, P s ) by finding the maximum generalized eigenvalue of the matrix pencil 6) If the optimization is very high. As a sub-optimal scheme, we adopt null-space AN and jointly optimize the covariance matrix of AN and the IR of the pre-equalizer to minimize the eavesdropper's SINR under the constraints of the minimum requirement of the legitimate receiver's SINR. As that in Sect. 3.1, AN is z AN Wv. AN's covariance matrix is expressed as where v E vv H . Unlike that in Sect. 3.1, v needs to be optimized. The optimization problem is expressed as Constraint v ≥ 0 means that v must be a positive semidefinite matrix.
Problem (27) is non-convex and is difficult to be solved. By introducing a relaxation variable t, problem (27) is transformed into (28) can be further rewritten as The first and second constraints are non-convex, so the problem (29) is still non-convex. The left side of the first constraint can be regarded as the Schur complement of t in [26] It is easy to know that T ≥ 0 is equivalent to t ≥ 0 and pre h E, (2L−1) ≥ 0, so the first constraint can be transformed into Then, by introducing a phase constraint, the second constraint can be changed to another form, and problem (30) is transformed into Since any phase rotation of g pre does not change the value of the objective function and the constraints in problem (27), problem (32) is equivalent to the problem (27). Problem (32) is a convex optimization problem that can be solved using the CVX optimization tool.
The complexity of this convex optimization problem is listed in Table 2. For comparison, the complexities of the algorithms proposed by [19] and [20] are also given. [19] and [20] optimize the TR filers and AN to improve the PLS performance. It can be found that the complexity of our proposed scheme is independent of the number of antennas N, while the complexities of the schemes of [19] and [20] are proportional to N 3 . The more the antennas are, the larger complexity advantage of our scheme is.

Simulation results
In this section, the proposed scheme is evaluated by MAT-LAB simulation. In the simulation, all channels are Rayleigh fading channels, and the parameters of the channels are: the number of paths is L 10; the channel bandwidth is The legitimate receiver's SINR threshold is γ 0 6dB, the number of transmitted symbols is M 3, and the noise power of the channels is 1 × 10 −11 W. The search precision in Algorithm 1 is ε 1 × 10 −6 . The data given in this section are the average values under 10,000 channel realizations. Figure 2 shows the simulation results of the average SINR at the legitimate receiver and the eavesdropper. Figure 3 gives the simulation results of the ergodic capacities of the legitimate channel and that of the eavesdropping channel and the ergodic achievable secrecy rate. The number of transmit antennas N is 2. It can be seen from Fig. 2 that the legitimate receiver's SINR remains unchanged and keeps at 6 dB with the increase of the total transmit power whether the ECSI is known or unknown, which indicates that the solution to the optimization problem meets the constraint of minimum SINR of the legitimate receiver. Because the legitimate receiver's SINR does not change, the capacity of the legitimate channel remains constant, as is shown in Fig. 3. Whether ECSI is known or unknown, the constraint of the optimization is the minimum SINR of the legitimate receiver. Because AN is in the null space of the legitimate channel and the AN power is zero in the legitimate receiver, so the signal power does not change as the total transmit power increases. When ECSI is known, the SINR of the eavesdropper is minimized. When ECSI is unknown, the AN power is maximized. So, in both scenarios, the AN power increases and the SINR of the eavesdropper decreases as the total transmit power increases. AN is radiated in all directions when ECSI is unknown, while it is radiated directly to the eavesdropper by optimizing AN's covariance matrix when ECSI is known. So, when ECSI is known, AN interferes with the eavesdropper more effectively and its SINR is lower and a higher secrecy rate can be achieved, as is shown in Figs. 2 and 3. Figure 4 compares the achievable secrecy rate of the proposed scheme with that of the conventional TR transmission scheme. The number of the transmit antennas is N 4. In the traditional TR scheme, there is no pre-equalizer, and the TR pre-filters for each antenna are the matched filter of the channel from the antenna to the legitimate receiver, that is, , i 1, · · · , N . The null-space AN is employed too, and AN's covariance matrix is obtained in the same way as the proposed scheme in this paper. The signal power meets the legitimate receiver's SINR requirement, and the remaining power is used to transmit AN. It can be seen from Fig. 4 that the secrecy rate of our proposed scheme is higher than that of the conventional TR scheme. The reason is that the pre-equalizer is optimized to minimize the signal power in the proposed scheme under the constraint of the minimum legitimate receiver's SINR, while there is no pre-equalizer and the TR pre-filters are not optimized in the conventional TR scheme, so the signal power in the proposed scheme is lower than conventional TR scheme. As a result, the AN power in the proposed scheme is higher, so the secrecy rate is higher. The comparison indicates that the optimization of the pre-equalizer can improve security performance significantly. Figure 5 and Fig. 6 are the simulation results when the transmitter is equipped with different numbers of antennas. Because the SINR of the legitimate receiver is constrained at 6 dB in the optimization, the legitimate channel capacity will  Figures 5b and 6b show that the more transmit antennas there are, the lower the eavesdropping channel capacity is. This is because the more antennas there are, the larger the gain of the transmit antenna array is, and the smaller the signal power is required to meet the SINR requirement of the legitimate receiver. As a result, the AN power increases and the SINR of the eavesdropper decreases, so the eavesdropping channel capacity decreases. Since the legitimate channel capacity remains unchanged, the achievable secrecy rate increases with the number of antennas, as shown in Figs. 5a and 6a. The performance gain brought by installing more transmit antennas decreases as the transmit power increases. The higher (a) (b) Fig. 6 Security performance with different numbers of antennas when ECSI is known. a Achievable secrecy rate. b Eavesdropping channel capacity the transmit power, the higher the interference power in the eavesdropper and the lower the eavesdropping channel's capacity, so the less decrease of the eavesdropping channel's capacity is due to the increase of the AN power. Because the optimization complexity of the proposed scheme does not increase with the increase of antennas, so the number of the antennas is not limited by the complexity of the optimization.

Conclusion
In this paper, an ETR-MISO system assisted with AN is optimized to improve the secrecy rate either when ECSI is known or unknown. AN is arranged in the null space of the legitimate channel, so it does not interfere with the legitimate receiver. When ECSI is unknown, the pre-equalizer is optimized to minimize the transmit power under the constraint of the minimum SINR of the legitimate receiver and the AN power is maximized. The optimization problem is transformed into the problem of finding the maximum generalized eigenvalue of a matrix pencil and the minimum signal power. When ECSI is known, the covariance matrix of the null-space AN and the pre-equalizer are jointly optimized to minimize the SINR of the eavesdropper under the minimum SINR constraint of the legitimate receiver. The optimization problem is non-convex. By introducing a relaxation variable and transforming the non-convex constraints into convex constraints, the original optimization problem is transformed into a convex optimization problem, which can be solved by using the CVX optimization tool. The proposed scheme is evaluated by simulation. The simulation results show that the secrecy rate of the system can be significantly promoted by optimizing the pre-equalizer and the power allocation of the signal and AN, and the more the transmit antennas, the higher the secrecy rate. In the proposed scheme, only one pre-equalizer needs to be optimized, so the optimization complexity is significantly lower than those of the schemes in which all TR pre-filters need to be jointly optimized. The optimization complexity of the proposed scheme is independent of the number of the transmit antennas, so the configuration of the transmit antennas is not limited by the optimization complexity.