# Secrecy Capacity of Time Reversal Beamforming for Multi-User MIMO System under Spatial Correlation

Authors: Bingyan He; Tao Sun; Chuanmu Li; Xingwang Huang
Journal: Security and Communication Networks
 (2021)

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## Abstract

In this paper, for strengthening the security of wireless transmission system, the time reversal (TR) beamforming method is proposed for the downlink of multi-user MIMO system with multiple users who potentially act as eavesdroppers. We develop a multi-input, single-output, multi-eavesdropper (MISOME) wiretap channel model in which Rayleigh fading and spatial correlation are taken into account. Using the proposed model, we further analyze the confidentiality provided by TR beamforming and we use achievable secrecy rates as our performance metrics. In particular, we derive novel closed-formed expressions for the average secrecy-SINR and the mean secrecy sum-rate in order to characterize the influences of propagation conditions on network secrecy metrics. These expressions provide deeper insights into the impact of network interference on communication confidentiality. We find that TR beamforming can deliver the maximum secrecy capacity potential in uncorrelated Rayleigh channels and achieve perfect confidential communication without any extra secrecy cost. On the other hand, even weak inter-user correlation may cause a significant loss of achievable secrecy sum-rate and therefore result in high secrecy cost. But benefiting more from larger signal bandwidth and rich-scattering environment, the TR beamforming technique is still an attractive and cost-effective solution for low-power indoor applications.


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## Body

## 1. Introduction

As the diversity and the number of users in wireless networks keep growing, wireless security appears to be a crucial matter in today’s communication systems. Due to the broadcast nature of wireless channels, wireless multi-user communications are very susceptible to eavesdropping. Traditionally, wireless security is ensured by data cryptography techniques, which mainly depend on secret keys and also rely on the limited computational power of eavesdroppers [1–4]. However, future wireless systems demand ubiquitous coverage and large-scale deployment of wireless radio devices. The market forecasts that in the era of the Internet of Things (IoT), billions of connected devices built with more than a trillion sensors will be installed worldwide [5]. The emergence of large-scale and dynamic networks imposes new challenges on conventional cryptographic techniques, due to the complexity of distributing and maintaining secret keys. To this end, physical-layer security has reemerged as an alternative to achieve perfect secrecy without the need for an encryption key and complex encryption/decryption algorithms [1–11].The concept of communication secrecy is built on information-theoretic notion of perfect secrecy and was first postulated by Shannon in his ground-breaking treatise [12]. Based on this concept, Wyner in his work [13] introduced the discrete memory-less wiretap channel and showed that the secrecy capacity is determined by the difference between the capacity of the legitimate link and that of the eavesdropping link. In wireless environments, radio links are degraded by path loss, shadowing, and multi-path fading. The secrecy capacity in fading channels was investigated in [14–18]. Later on, secrecy capacity topic has been further studied in the context of multi-access channels, multiple-input multiple-output (MIMO) communications [1, 19, 20], cooperative networks [21, 22], cellular networks [14, 23, 24], and IoT networks [5, 25, 26].Recently, physical-layer security has also been extended to multi-user MIMO channels [22, 27, 28]. But most existing works only consider noise to evaluate secrecy capacity of multi-user networks and there is still a lack of fine-grained investigation of interference impact on network secrecy [19, 22, 29]. In fact, in multi-user networks, more users cause more severe inter-user interference (IUI) imposed on the receiver, which is considered deleterious for communications. Moreover, as signal bandwidth increases, more multi-path components can be resolved in a rich-scattering environment, such as indoors or in metropolitan areas [30]. Essentially, multi-path components are naturally existing degrees of freedom in the environment and channel capacity grows linearly with the available degrees of freedom [30–32]. But in practical communication systems, strong multi-path propagation causes severe inter-symbol interference (ISI) and then causes crucial system performance degradation, especially when the symbol rate is very high. Therefore, it is of critical importance to characterize the impact of interference (i.e., ISI and IUI) on secrecy capacity achievable in multi-user MIMO system over multi-path channels. We advocate the exploitation of wireless channel intrinsic properties (e.g., multi-path propagation) to strengthen communication secrecy.As a linear precoding technique, TR beamforming can fully harvest energy from a rich-scattering environment and achieve superior focusing effect in both time and spatial domains, which can effectively suppress interference and improve the energy efficiency of wireless transmission [33–38]. In addition, thanks to its high-resolution spatial focusing effect, the TR-based system can achieve low probability of intercept. In other words, spatial focusing effect can also improve the system security [39–43].In this paper, the secrecy performance of the multi-user MIMO TR-based system is presented and mainly studied in terms of achievable secrecy rates and secrecy cost. We take into account a multi-user MIMO TR-based system model that is generalized by considering (i) the frequency selective channel that has arbitrary power in each tap, (ii) the correlation at the transmitter side, and (iii) the correlation at the user side. Moreover, the average effective secrecy-SINR is derived using exact power expressions of desired signal, inter-symbol interference, and inter-user interference terms at both legitimate and illegitimate nodes. Finally, the validity of our analysis is verified by means of Monte Carlo simulation. The key contributions of the paper can be summarized as follows:(1)
We develop a framework for design and analysis of a multi-user MIMO TR-based system with intrinsic secrecy that accounts for propagation conditions and aggregates network interference.(2)
We derive approximations for the average secrecy-SINR and the mean per-user secrecy rate achievable by TR beamforming, the accuracies of which are verified by Monte Carlo simulations.(3)
The upper and lower bounds on mean per-user secrecy rate are determined. We find that the secrecy rate in a multi-user network does not grow unbounded with the transmit power.(4)
We find that the secrecy performance of TR beamforming depends highly on the degrees of freedom in the environment, i.e., multi-path components in a Rayleigh channel. The larger the number of multi-path components, the higher the TR focusing gain and thus the larger the secrecy capacity of TR-based system.(5)
We examine quantitatively the impact of spatial correlation on secrecy performance of the multi-user MIMO TR-based system.The remainder of the paper is organized as follows. In Section2, the system model is formulated. The secrecy rate achievable by the TR-based system is characterized in Section 3. In Section 4, numerical simulation results and corresponding discussions are provided. Finally, concluding remarks are drawn in Section 5.Notation. E, ∗, and ⊗ denote expectation, discrete-time convolution, and the Kronecker product, respectively. The boldface lowercase a and uppercase A indicate vectors and matrices, respectively. In addition, the superscripts .T and .H represent the transpose and transpose conjugate, respectively. For a complex value, we denote Re . as the real part. The notation ℂm×n denotes m×n complex matrix.


## 2. System Model

### 2.1. Preliminaries

The TR-based system under consideration is comprised of a base station (BS) withM antennas which simultaneously transmits N independent confidential messages to N single-antenna users. Our model is based on the assumption of a static (i.e., block fading) channel with perfect channel state information (CSI) available at the BS. This assumption is particularly appropriate for indoor wireless communications.In order to develop an analytical expression, we only consider discrete-time signals. Tapped delay line (TDL) is used for modeling multi-path fading channel in a rich-scattering environment. Every tap in a tapped delay line represents a resolved path in a multi-path fading channel. In the multi-path fading channel, we assume that the length of channel impulse response (CIR) isL. Thus, the CIR between the i-th transmit antenna and the n-th user is(1)hnik=∑l=1Lαni,lδk−l,where αni,l is the complex path gain of the l-th tap in the CIR while l represents the corresponding path delay. Additionally, we assume the following: (i) each hnil is independent circular symmetric complex Gaussian (CSCG) random variable with zero mean Ehnil=0,Ehnil2=σni,l2; (ii) the average power of each tap decays exponentially described as Ehnil2=e−lT/σT, where T is the sampling period of the system such that 1/T equals the channel bandwidth B and σT is the delay spread of the channel. In practice, the CIR can be expressed as a vector hni∈ℂL×1. Then, the channel matrix of the whole system can be written as(2)H=H1H2⋯HN=h11h12⋯h1Mh21h22⋯h2M⋯⋯⋱⋯hN1hN2⋯hNM,where Hi is the channel matrix of dimension L×M from the BS to the i-th user. As mentioned above, when signal bandwidth increases, L will increase correspondingly. Let the transmitted signal xj for the j-th user be complex random variable with zero mean and variance of θ. This signal is obtained at the BS by performing a linear processing on the confidential messages un,n=1,…,N. Thus, the received signal vector at user n is given by(3)yn=∑i=1M∑j=1Nxj∗hni+zn,where zn represents additive white Gaussian noise with variance σ2zn∼CN0,σ2.It is required that the BS securely transmits each confidential messageun, ensuring that the unintended users receive no information for other users. This is performed at the secrecy rate Rn,s, defined as follows. The secrecy capacity is the largest secrecy rate of communication between the source and the destination nodes with the eavesdropper knowing no information of the messages. In general, the behavior of the users cannot be known by the BS. Considering a worst-case scenario in our system, we assume that for each user n, the remaining N−1 users can cooperate to jointly eavesdrop on the message un. For each user n, the alliance of the N−1 eavesdroppers is equivalent to a single eavesdropper with N−1 receiving antennas, which is denoted by n¯ [29].


### 2.2. TR Beamforming

The TR-based system model is depicted in Figure1. We assume that the CIRs are stationary for at least one probing-and-transmitting cycle. During channel probing phase, the BS records the CIR of each link, also known as CSI. After the channel probing phase, the system starts its transmission phase and the BS applies the complex-conjugated time-reversed version of the CIR to prefilter the transmitted signal at the transmitter. Such prefiltering acting as a beamformer in the spatial domain focuses the RF power on the receiver and therefore can control the amount of crosstalk between the users [30, 40]. TR beamforming is particularly interesting because it can allow low-complexity implementation. If hni is supposed to be perfectly known at the BS and the channel is static, we define the prefiltering vector fni∈ℂL×1 for the link hni. Each tap of fni can be given by(4)fnik=hniHL+1−kPh,(5)Ph=∑i=1M∑l=1Lhnil2,where hniHL+1−k is the time-reversed conjugate version of hnik and Ph is a power normalization factor introduced to make sure that the total transmitted power remains constant in every realization. Let F denote the global prefiltering matrix for the channel matrix H, which is expressed as(6)F=F1F2⋯FN=f11f12⋯f1Mf21f22⋯f2M⋯⋯⋱⋯fN1fN2⋯fNM,where Fi∈ℂL×M is the prefiltering matrix matched for the channel matrix Hi shown in (2). Let confidential messages u=u1,u2,…,uN go through TR precoder, and un is assumed to have unit power (i.e., Eun2=1). After that, the transmitted signal from the i-th antenna to the n-th user is(7)xn=θun∗fni,where θ is the transmit power and xn∈ℂL×1.Figure 1
The block diagram of TR-based system.By employing linear prefiltering in (4), the received signals at users n and n¯ are, respectively,(8)yn=∑j=1N∑i=1Mxj∗hni+zn=θ∑j=1N∑i=1Muj∗fji∗hni+zn=θ∑i=1Mun∗fni∗hni+θ∑j=1,j≠nN∑i=1Muj∗fji∗hni+zn=θ∑i=1Mun∗fni∗hniL︸desired signal+θ∑k=1,k≠L2L−1∑i=1Mun∗fni∗hnik︸ISI+θ∑j=1,j≠nN∑i=1Muj∗fji∗hni︸IUI+zn︸Noise,(9)yn¯=θ∑j=1,j≠nN∑i=1Mun∗fni∗hji+∑j=1,j≠nN∑i=1Mxj∗hji+∑j=1j≠nNzj=θ∑j=1,j≠nN∑i=1Mun∗fni∗hji+uj∗fji∗hji+∑j=1,j≠nNzj=θ∑j=1,j≠nN∑i=1Mun∗fni∗hjiL︸desired signal+θ∑j=1,j≠nN∑k=1,k≠L2L−1∑i=1Mun∗fni∗hjik︸ISI+θ∑j=1,j≠nN∑i=1Muj∗fji∗hji︸IUI+∑j=1,j≠nNzj︸Noise,where zn,zj∼CN0,σ2 and yn,yn¯∈ℂ2L−1×1. Note that each user n, along with its own eavesdropper n¯ and the transmitter at the BS, forms an equivalent multi-input, single-output, multi-eavesdropper (MISOME) wiretap channel [44]. The transmitter, the intended receiver, and the eavesdropper of this MISOME wiretap channel are equipped with M, 1, and N−1 virtual antennas, respectively.Due to the simultaneous transmission of theN messages at the BS, the signal from the message un is desired for user n, and user n experiences inter-user interference from all uj,j≠n as seen in (8). In addition, because the eavesdropper n¯ wants to intercept the message un, the signal from all the uj is regarded as IUI for the message un as seen in (9).We have the equivalent time-reversed CIR (TR-CIR) formula from thei-th transmit antenna to the n-th user as(10)fni∗hnik=∑l=1khniHL+1−lhink+1−lPh,with k=1,2,…,2L−1 (i.e., fni∗hni∈ℂ2L−1×1).So, the peak amplitude of the equivalent TR-CIR is(11)fni∗hniL=∑l=1Lhnil2Ph.The energy is concentrated at the center tap (i.e.,k=L of equivalent TR-CIR and is zero (or very low power) elsewhere (i.e., k∈1,2⋯,2L−1,k≠L). Thus, only one-tap detection is needed at the receiver and the sampling point is just at the center tap. This will effectively reduce computational complexity at the terminal devices. Because the CIRs associated with different users are uncorrelated or weakly correlated, the power of fji∗hnikk≠L is also very small compared to the power of fni∗hniL. Thanks to TR prefiltering, ISI and IUI can be effectively mitigated.


### 2.3. Channel Correlation Model

In a rich-scattering environment, we usually assume a model of independent channels because the channels of different users are often spatially uncorrelated. However, channels may become correlated when the environment is less scattering and users are very close to each other. In our system, we assume that the distances between the BS and users are large and only antennas from the same equipment (either the transmitter or the receiver) are spatially correlated due to scattering and electromagnetic coupling. The correlation can be included into the channel model by introducing fixed transmit antenna and inter-user correlation matrices following the well-known Kronecker model [45]. The M×N spatially correlated channels are therefore given by(12)Hk=Rr1/2T⊗Rt1/2Hwk,where Hwk∈ℂMN×L is the channel matrix containing the independent CIRs for a given k. Hkni=hnik, i∈1⋯M, n∈1⋯N, and k∈1⋯L. Rt is the transmit antenna correlation matrix with dimension M×M and Rr is the inter-user correlation matrix with dimension N×N. The correlation matrix follows the general model with arbitrary positive coefficients (i.e., ρt,ii′≥0). We give an example of transmit antenna correlation matrix Rt:(13)Rt=1ρt,12⋯ρt,1Mρt,211⋯ρt,2M⋯⋯⋱⋯ρt,M1ρt,M2⋯1.


## 2.1. Preliminaries

The TR-based system under consideration is comprised of a base station (BS) withM antennas which simultaneously transmits N independent confidential messages to N single-antenna users. Our model is based on the assumption of a static (i.e., block fading) channel with perfect channel state information (CSI) available at the BS. This assumption is particularly appropriate for indoor wireless communications.In order to develop an analytical expression, we only consider discrete-time signals. Tapped delay line (TDL) is used for modeling multi-path fading channel in a rich-scattering environment. Every tap in a tapped delay line represents a resolved path in a multi-path fading channel. In the multi-path fading channel, we assume that the length of channel impulse response (CIR) isL. Thus, the CIR between the i-th transmit antenna and the n-th user is(1)hnik=∑l=1Lαni,lδk−l,where αni,l is the complex path gain of the l-th tap in the CIR while l represents the corresponding path delay. Additionally, we assume the following: (i) each hnil is independent circular symmetric complex Gaussian (CSCG) random variable with zero mean Ehnil=0,Ehnil2=σni,l2; (ii) the average power of each tap decays exponentially described as Ehnil2=e−lT/σT, where T is the sampling period of the system such that 1/T equals the channel bandwidth B and σT is the delay spread of the channel. In practice, the CIR can be expressed as a vector hni∈ℂL×1. Then, the channel matrix of the whole system can be written as(2)H=H1H2⋯HN=h11h12⋯h1Mh21h22⋯h2M⋯⋯⋱⋯hN1hN2⋯hNM,where Hi is the channel matrix of dimension L×M from the BS to the i-th user. As mentioned above, when signal bandwidth increases, L will increase correspondingly. Let the transmitted signal xj for the j-th user be complex random variable with zero mean and variance of θ. This signal is obtained at the BS by performing a linear processing on the confidential messages un,n=1,…,N. Thus, the received signal vector at user n is given by(3)yn=∑i=1M∑j=1Nxj∗hni+zn,where zn represents additive white Gaussian noise with variance σ2zn∼CN0,σ2.It is required that the BS securely transmits each confidential messageun, ensuring that the unintended users receive no information for other users. This is performed at the secrecy rate Rn,s, defined as follows. The secrecy capacity is the largest secrecy rate of communication between the source and the destination nodes with the eavesdropper knowing no information of the messages. In general, the behavior of the users cannot be known by the BS. Considering a worst-case scenario in our system, we assume that for each user n, the remaining N−1 users can cooperate to jointly eavesdrop on the message un. For each user n, the alliance of the N−1 eavesdroppers is equivalent to a single eavesdropper with N−1 receiving antennas, which is denoted by n¯ [29].


## 2.2. TR Beamforming

The TR-based system model is depicted in Figure1. We assume that the CIRs are stationary for at least one probing-and-transmitting cycle. During channel probing phase, the BS records the CIR of each link, also known as CSI. After the channel probing phase, the system starts its transmission phase and the BS applies the complex-conjugated time-reversed version of the CIR to prefilter the transmitted signal at the transmitter. Such prefiltering acting as a beamformer in the spatial domain focuses the RF power on the receiver and therefore can control the amount of crosstalk between the users [30, 40]. TR beamforming is particularly interesting because it can allow low-complexity implementation. If hni is supposed to be perfectly known at the BS and the channel is static, we define the prefiltering vector fni∈ℂL×1 for the link hni. Each tap of fni can be given by(4)fnik=hniHL+1−kPh,(5)Ph=∑i=1M∑l=1Lhnil2,where hniHL+1−k is the time-reversed conjugate version of hnik and Ph is a power normalization factor introduced to make sure that the total transmitted power remains constant in every realization. Let F denote the global prefiltering matrix for the channel matrix H, which is expressed as(6)F=F1F2⋯FN=f11f12⋯f1Mf21f22⋯f2M⋯⋯⋱⋯fN1fN2⋯fNM,where Fi∈ℂL×M is the prefiltering matrix matched for the channel matrix Hi shown in (2). Let confidential messages u=u1,u2,…,uN go through TR precoder, and un is assumed to have unit power (i.e., Eun2=1). After that, the transmitted signal from the i-th antenna to the n-th user is(7)xn=θun∗fni,where θ is the transmit power and xn∈ℂL×1.Figure 1
The block diagram of TR-based system.By employing linear prefiltering in (4), the received signals at users n and n¯ are, respectively,(8)yn=∑j=1N∑i=1Mxj∗hni+zn=θ∑j=1N∑i=1Muj∗fji∗hni+zn=θ∑i=1Mun∗fni∗hni+θ∑j=1,j≠nN∑i=1Muj∗fji∗hni+zn=θ∑i=1Mun∗fni∗hniL︸desired signal+θ∑k=1,k≠L2L−1∑i=1Mun∗fni∗hnik︸ISI+θ∑j=1,j≠nN∑i=1Muj∗fji∗hni︸IUI+zn︸Noise,(9)yn¯=θ∑j=1,j≠nN∑i=1Mun∗fni∗hji+∑j=1,j≠nN∑i=1Mxj∗hji+∑j=1j≠nNzj=θ∑j=1,j≠nN∑i=1Mun∗fni∗hji+uj∗fji∗hji+∑j=1,j≠nNzj=θ∑j=1,j≠nN∑i=1Mun∗fni∗hjiL︸desired signal+θ∑j=1,j≠nN∑k=1,k≠L2L−1∑i=1Mun∗fni∗hjik︸ISI+θ∑j=1,j≠nN∑i=1Muj∗fji∗hji︸IUI+∑j=1,j≠nNzj︸Noise,where zn,zj∼CN0,σ2 and yn,yn¯∈ℂ2L−1×1. Note that each user n, along with its own eavesdropper n¯ and the transmitter at the BS, forms an equivalent multi-input, single-output, multi-eavesdropper (MISOME) wiretap channel [44]. The transmitter, the intended receiver, and the eavesdropper of this MISOME wiretap channel are equipped with M, 1, and N−1 virtual antennas, respectively.Due to the simultaneous transmission of theN messages at the BS, the signal from the message un is desired for user n, and user n experiences inter-user interference from all uj,j≠n as seen in (8). In addition, because the eavesdropper n¯ wants to intercept the message un, the signal from all the uj is regarded as IUI for the message un as seen in (9).We have the equivalent time-reversed CIR (TR-CIR) formula from thei-th transmit antenna to the n-th user as(10)fni∗hnik=∑l=1khniHL+1−lhink+1−lPh,with k=1,2,…,2L−1 (i.e., fni∗hni∈ℂ2L−1×1).So, the peak amplitude of the equivalent TR-CIR is(11)fni∗hniL=∑l=1Lhnil2Ph.The energy is concentrated at the center tap (i.e.,k=L of equivalent TR-CIR and is zero (or very low power) elsewhere (i.e., k∈1,2⋯,2L−1,k≠L). Thus, only one-tap detection is needed at the receiver and the sampling point is just at the center tap. This will effectively reduce computational complexity at the terminal devices. Because the CIRs associated with different users are uncorrelated or weakly correlated, the power of fji∗hnikk≠L is also very small compared to the power of fni∗hniL. Thanks to TR prefiltering, ISI and IUI can be effectively mitigated.


## 2.3. Channel Correlation Model

In a rich-scattering environment, we usually assume a model of independent channels because the channels of different users are often spatially uncorrelated. However, channels may become correlated when the environment is less scattering and users are very close to each other. In our system, we assume that the distances between the BS and users are large and only antennas from the same equipment (either the transmitter or the receiver) are spatially correlated due to scattering and electromagnetic coupling. The correlation can be included into the channel model by introducing fixed transmit antenna and inter-user correlation matrices following the well-known Kronecker model [45]. The M×N spatially correlated channels are therefore given by(12)Hk=Rr1/2T⊗Rt1/2Hwk,where Hwk∈ℂMN×L is the channel matrix containing the independent CIRs for a given k. Hkni=hnik, i∈1⋯M, n∈1⋯N, and k∈1⋯L. Rt is the transmit antenna correlation matrix with dimension M×M and Rr is the inter-user correlation matrix with dimension N×N. The correlation matrix follows the general model with arbitrary positive coefficients (i.e., ρt,ii′≥0). We give an example of transmit antenna correlation matrix Rt:(13)Rt=1ρt,12⋯ρt,1Mρt,211⋯ρt,2M⋯⋯⋱⋯ρt,M1ρt,M2⋯1.


## 3. Achievable Secrecy Rates

In this section, we derive secrecy sum-rate for the multi-user MIMO TR-based system.


### 3.1. SINR at a Typical User

For then-th equivalent MISOME wiretap channel in (8) and (9), we denote Psn,Pisin, and Piuin as the powers of the desired signal and the ISI and the IUI terms, respectively. Using these notations, we propose the following closed-form expression of the SINR for the message un at user n:(14)γn=EPsnPisin+Piuin+σ2,Psn=θ∑i=1Mfni∗hniL2,Pisin=θ∑k=1,k≠L2L−1∑i=1Mfni∗hnik2,Piuin=θ∑j=1,j≠nN∑k=12L−1∑i=1Mfji∗hnik2.For simplicity of calculation, we ignore the difference betweenfji∗hnik and fj′i∗hnik and let ∑k=12L−1∑i=1Mfji∗hnik2 represent the IUI power caused by a single eavesdropper to user n. So, Piuin can be rewritten as(15)Piuin=θN−1∑k=12L−1∑i=1Mfji∗hnik2,j≠n.Note thatPiuin denotes maximum cumulative interference power imposed on user n from the alliance of the N−1 cooperating eavesdroppers.


### 3.2. SINR at the Malicious Users

Here, eavesdroppern¯ attempts to intercept the message un. Based on (9), the instantaneous effective SINR for the message un at eavesdropper n¯ is usually formulated as(16)γn¯=Psn¯Pisin¯+Piuin¯+σ2,(17)Psn¯=θ∑j=1,j≠nN∑i=1Mfni∗hjiL2=θN−1∑i=1Mfni∗hjiL2,j≠n,(18)Pisin¯=θ∑j=1,j≠nN∑k=1,k≠L2L−1∑i=1Mfni∗hjik2,(19)Piuin¯=θ∑j=1,j≠nN∑k=12L−1∑i=1Mfji∗hjik2.Considering the worst-case scenario where all the malicious users can cooperate to eavesdrop on the message intended for the typical user, each malicious user is likely to decode his own message and can indirectly pass this information to all the other malicious users. This will lead to the worst results that all the malicious users can therefore subtract the IUI generated by all the messagesuj,j≠n [29]. In this case, the term Piuin¯ in (16) may be omitted. A simplified expression for (16) is obtained:(20)γn¯=Psn¯Pisin¯+σ2.


### 3.3. Achievable Secrecy Rates

Lemma 1.
An instantaneous achievable secrecy sum-rateRs for the multi-user MIMO system with malicious users is given by(21)Rs=∑n=1NRn,s,where Rn,s is achievable per-user secrecy rate for the n-th MISOME wiretap channel in (8) and (9), n=1,…,N.Lemma 2.
An instantaneous per-user secrecy rateRn,s for the MISOME wiretap channel in (8) and (9) is given by(22)Rn,s=log21+γn−log21+γn¯+=log21+γn1+γn¯+=log21+γn−γn¯1+γn¯+=log21+γn,s+,where γn,s=γn−γn¯/1+γn¯ is the secrecy-SINR. Rn,s is the secrecy rate achievable by user n in the presence of eavesdropper n¯. From (22), it is clearly observed that for secrecy rate maximization, maximizing γn and minimizing γn¯ are required.Then, the secrecy-SINR expression is given by(23)γn,s=γn−γn¯1+γn¯,γn>γn¯,0,otherwise.In more detail,γn,s can be expressed as(24)γn,s=Psn/Pisin+Piuin+σ2−Psn¯/Pisin¯+σ21+Psn¯/Pisin¯+σ2.According to (24), the expectation of secrecy-SINR can be formulated as(25)γ¯n,s=Eγn,s=EPsn/Pisin+Piuin+σ2−Psn¯/Pisin¯+σ21+Psn¯/Pisin¯+σ2.The closed-form expression for (25) is hard to obtain. With the help of [46, 47], we define an approximation for average secrecy-SINR as(26)γ¯^n,s=EPsn/EPisin+EPiuin+σ2−EPsn¯/EPisin¯+σ21+EPsn¯/EPisin¯+σ2.More details are described in Appendix A. Averaging (22) over all realizations of the random ergodic channels, the expected value of the instantaneous achievable secrecy rate is a good reference of the long-term performance.Theorem 1.
The mean per-user secrecy rate can be approximated as(27)R¯n,s=Elog21+γn,s+≈R¯^n,s=log21+γ¯^n,s+.Then, the mean secrecy sum-rate can be approximated as(28)R¯s=E∑n=1Nlog21+γn,s+≈N×R¯^n,s.


## 3.1. SINR at a Typical User

For then-th equivalent MISOME wiretap channel in (8) and (9), we denote Psn,Pisin, and Piuin as the powers of the desired signal and the ISI and the IUI terms, respectively. Using these notations, we propose the following closed-form expression of the SINR for the message un at user n:(14)γn=EPsnPisin+Piuin+σ2,Psn=θ∑i=1Mfni∗hniL2,Pisin=θ∑k=1,k≠L2L−1∑i=1Mfni∗hnik2,Piuin=θ∑j=1,j≠nN∑k=12L−1∑i=1Mfji∗hnik2.For simplicity of calculation, we ignore the difference betweenfji∗hnik and fj′i∗hnik and let ∑k=12L−1∑i=1Mfji∗hnik2 represent the IUI power caused by a single eavesdropper to user n. So, Piuin can be rewritten as(15)Piuin=θN−1∑k=12L−1∑i=1Mfji∗hnik2,j≠n.Note thatPiuin denotes maximum cumulative interference power imposed on user n from the alliance of the N−1 cooperating eavesdroppers.


## 3.2. SINR at the Malicious Users

Here, eavesdroppern¯ attempts to intercept the message un. Based on (9), the instantaneous effective SINR for the message un at eavesdropper n¯ is usually formulated as(16)γn¯=Psn¯Pisin¯+Piuin¯+σ2,(17)Psn¯=θ∑j=1,j≠nN∑i=1Mfni∗hjiL2=θN−1∑i=1Mfni∗hjiL2,j≠n,(18)Pisin¯=θ∑j=1,j≠nN∑k=1,k≠L2L−1∑i=1Mfni∗hjik2,(19)Piuin¯=θ∑j=1,j≠nN∑k=12L−1∑i=1Mfji∗hjik2.Considering the worst-case scenario where all the malicious users can cooperate to eavesdrop on the message intended for the typical user, each malicious user is likely to decode his own message and can indirectly pass this information to all the other malicious users. This will lead to the worst results that all the malicious users can therefore subtract the IUI generated by all the messagesuj,j≠n [29]. In this case, the term Piuin¯ in (16) may be omitted. A simplified expression for (16) is obtained:(20)γn¯=Psn¯Pisin¯+σ2.


## 3.3. Achievable Secrecy Rates

Lemma 1.
An instantaneous achievable secrecy sum-rateRs for the multi-user MIMO system with malicious users is given by(21)Rs=∑n=1NRn,s,where Rn,s is achievable per-user secrecy rate for the n-th MISOME wiretap channel in (8) and (9), n=1,…,N.Lemma 2.
An instantaneous per-user secrecy rateRn,s for the MISOME wiretap channel in (8) and (9) is given by(22)Rn,s=log21+γn−log21+γn¯+=log21+γn1+γn¯+=log21+γn−γn¯1+γn¯+=log21+γn,s+,where γn,s=γn−γn¯/1+γn¯ is the secrecy-SINR. Rn,s is the secrecy rate achievable by user n in the presence of eavesdropper n¯. From (22), it is clearly observed that for secrecy rate maximization, maximizing γn and minimizing γn¯ are required.Then, the secrecy-SINR expression is given by(23)γn,s=γn−γn¯1+γn¯,γn>γn¯,0,otherwise.In more detail,γn,s can be expressed as(24)γn,s=Psn/Pisin+Piuin+σ2−Psn¯/Pisin¯+σ21+Psn¯/Pisin¯+σ2.According to (24), the expectation of secrecy-SINR can be formulated as(25)γ¯n,s=Eγn,s=EPsn/Pisin+Piuin+σ2−Psn¯/Pisin¯+σ21+Psn¯/Pisin¯+σ2.The closed-form expression for (25) is hard to obtain. With the help of [46, 47], we define an approximation for average secrecy-SINR as(26)γ¯^n,s=EPsn/EPisin+EPiuin+σ2−EPsn¯/EPisin¯+σ21+EPsn¯/EPisin¯+σ2.More details are described in Appendix A. Averaging (22) over all realizations of the random ergodic channels, the expected value of the instantaneous achievable secrecy rate is a good reference of the long-term performance.Theorem 1.
The mean per-user secrecy rate can be approximated as(27)R¯n,s=Elog21+γn,s+≈R¯^n,s=log21+γ¯^n,s+.Then, the mean secrecy sum-rate can be approximated as(28)R¯s=E∑n=1Nlog21+γn,s+≈N×R¯^n,s.


## 4. Simulation Experiments and Numerical Analysis

In this section, we present the secrecy performance of the multi-user MIMO TR-based system. First, the focusing property of TR beamforming is assessed by analyzing the CIR under different propagation conditions. Then, the evaluation of network secrecy metric defined in Section3 is provided and fully discussed. The channel model is used with bandwidth (B)∼20 MHz to 125 MHz and the number of taps L∼5to33. All the experimental results were obtained by averaging 10000 random channel realizations.


### 4.1. Time Compression Validation Test

The temporal focusing property of TR beamforming is presented in Figure2 by comparing the amplitudes of both original CIRs and the equivalent TR-CIRs at the receiver, respectively. The power of each CIR has been normalized to 1 (i.e., ∑l=1Lhnil2=1). The original CIRs before TR demonstrate dispersive effects of multi-path channels under different bandwidth conditions. The more the multi-path components resolved in the channel are, the more quickly the signal strength fades as shown in Figures 2(a) and 2(b). Multi-path effects are particularly relevant in high data rate communication systems. Typically, this problem can be solved with equalization method at the receiver. But this would increase computational complexity at the receiver [48]. In Figure 2(c), the equivalent CIR after TR shows that most of signal energy is aggregated at the center tap (i.e., k=L=11) while ISI components at the other taps (i.e., k∈1,2⋯,2L−1,k≠L) have been suppressed as much as possible. This indicates that there is temporal focusing and therefore ISI can be mitigated. Thanks to the superior focusing effect in time domain, the receiver can perform a simple one-tap detection at the center tap and achieve good energy efficiency.Figure 2
An illustration of CIR vs. multi-path channel. (a) The original CIR forB=20MHz. (b) The original CIR for B=40MHz. (c) The equivalent TR-CIR at the intended user for ρt=0 and ρr=0. (d) The equivalent TR-CIR at the intended user for ρt=0.9 and ρr=0. (e) The equivalent TR-CIR at the eavesdropper for ρt=0 and ρr=0. (f) The equivalent TR-CIR at the eavesdropper for ρt=0 and ρr=0.9.
(a)(b)(c)(d)(e)(f)From Figure2(d), one can see that for the multi-path channels with strong transmit antenna correlation ρt=0.9, there is only minor degradation in focusing effect of TR beamforming at the intended user. Comparing Figure 2(e) with Figure 2(f), we can observe that for the multi-path channels with inter-user correlation, as the inter-user correlation increases, the signal for the message un at the eavesdropper is markedly enhanced at the center tap while suppressed at other taps. Obvious, due to the presence of inter-user correlation, the focusing effect of TR beamforming also occurs at the eavesdropper. Unfortunately, this will lead to more interception of confidential information of legitimate user by the eavesdropper. Unlike inter-user correlation, transmit antenna correlation does not improve the CIR of the eavesdropper.


### 4.2. Mean Secrecy Capacity

In this part, we investigate the secrecy performance of the multi-user MIMO TR-based system. We evaluate quantitatively and discuss the influence of propagation conditions, such as the number of taps, the number of transmit antennas, and the number of eavesdroppers on the mean per-user secrecy rate achieved by TR beamforming. To facilitate the discussion, we presume the ratioβ between the number of users N and the number of transmit antennas M, i.e., β=N/M.All the results from Sections4.2 to 4.4 assume that the channels of different users are spatially uncorrelated.Figure3 depicts the simulation results obtained with the mean per-user secrecy rate R¯n,s and the analytical results achieved with the approximation R¯^n,s given in Theorem 1. One can see that the analytical results approximate well the simulation results, which proves the reasonableness and effectiveness of the approximation R¯^n,s.Figure 3
(a) Mean per-user secrecy rate vs. SNR forL=33andN=2. (b) The impact of IUI on mean per-user secrecy rate for L=33andN=2.
(a)(b)In addition, Figure3(a) gives the upper bound of mean per-user secrecy rate R¯n,s under different system configurations, ignoring the effects of IUI and spatial correlation. Note that for a given N, smaller β implies more transmit antennas deployed at the BS. More transmit antennas deployed at the BS also indicate higher per-user secrecy rate achieved, especially notable at high SNR. When the SNR is up to 20 dB, R¯n,s reaches its peak. Thereafter, even if further increasing SNR, R¯n,s still remains unchanged. The reason for this situation lies in the fact that as the SNR increases, the noise power gradually becomes weaker. When the SNR reaches 20 dB, the ISI power far exceeds the noise power so that the noise power can be ignored. The expression for γn,s shown in (24) can be rewritten as(29)γn,s≈Psn/Pisin−Psn¯/Pisin¯1+Psn¯/Pisin¯.Eventually, both the desired signal power and the ISI power determine an upper bound onR¯n,s. Unlike Figure 3(a), Figure 3(b) considers the effects of the IUI on the mean per-user secrecy rate and gives the lower bound of mean per-user secrecy rate R¯n,s. Compared with Figure 3(a), it can be observed that IUI causes R¯n,s to fall by about 0.3∼1 bps/Hz.In Figure4, we have fixed SNR = 20 dB. Figure 4 shows the mean per-user secrecy rat as a function of β for different values of L. As expected, the mean per-user secrecy rate R¯n,s increases as the length of CIR L increases (larger L implies more multi-path components in the channel and stronger scattering in the channel). This figure clearly shows the remarkable benefit of richer multi-path environment on network secrecy capacity. For example, for L=5, the network goes into secrecy outage (i.e., R¯n,s=0) at β=4, whereas for L=11, secrecy outage occurs at β=16. This behavior can be attributed to the fact that by fully exploiting degrees of freedom provided by the environment, i.e., the abundant multi-path components in the channel, the TR-based system can support more coexisting users with non-zero secrecy rate and the secrecy performance of the system is significantly improved.Figure 4
Per-user secrecy rate as a function ofβ, for different values of L. Circles denote secrecy outage.Table1 displays the distributions of the mean per-user secrecy rate and the mean secrecy sum-rate achieved by TR beamforming, for different combinations of M and N, respectively. All the results in Table 1 were obtained at the SNR of 20 dB, for L=33. As indicated in Table 1, this implies a trade-off between the network capacity (in terms of the number of serviced users) and the secrecy rate at each user. When N takes 2, 4, and 8 in turn and M is successively 8, 16, and 32, the mean per-user secrecy rates are 2.25 bps/Hz, 2.24 bps/Hz, and 2.24 bps/Hz, respectively. In the cases, the number of N as well as M is different each time, but the ratios of N to M remain the same (i.e., β=N/M=2/8=4/16=8/32=0.25). These results demonstrate that in the absence of spatial correlation, the mean per-user secrecy rate is just a function of the SNR, L, and β, and thus the average secrecy sum-rate is a function of the SNR, L, β, and N.Table 1
Secrecy rate comparison.
R¯s/R¯n,s (bps/Hz)MN=2N=4N=8N=16N=3221.94/0.972.20/0.552.22/0.281.92/0.121.28/0.0443.08/1.543.84/0.964.32/0.544.48/0.283.84/0.1284.50/2.256.12/1.537.60/0.958.64/0.548.64/0.27166.10/3.058.96/2.2412.16/1.5215.2/0.9517.28/0.54327.74/3.8712.16/3.0417.92/2.2424.32/1.5230.4/0.95


### 4.3. Power Control Strategy

In Figure3(b), it can be seen that for β≤1, R¯n,s is always positive and monotonically increasing with the SNR until the SNR goes up to 20 dB, whereas for β>1, the per-user secrecy rate is irrespective of the SNR. This indicates that the mean per-user secrecy rate does not grow unbounded with the increase of transmit power. Moreover, TR beamforming performs poorly in the high-SNR regime. When increasing the SNR from 10 dB up to 20 dB, only a minor increase in Rn,s is achieved. Therefore, a power control strategy is introduced by reducing the transmit power properly to control the SNR to the value that can maximize the mean secrecy sum-rate (e.g., SNR at 10 dB) and works well for most of the cases. This power control strategy can save transmit power, thus effectively improving energy efficiency. For 1<β<32, the TR-based system with power reduction strategy would achieve an optimal non-negative secrecy sum-rate. However, for β≥31, the secrecy sum-rate is always zero irrespective of the SNR and this strategy would not be effective.Figure3 also shows that the mean per-user secrecy rate R¯n,s is always a decreasing function of β. The main reason for this phenomenon lies in the fact that when β>1, the intended user n suffered from a larger amount of IUI caused by more coexisting eavesdroppers while the alliance of cooperating eavesdroppers n¯ is always free from IUI, as shown in (20). Therefore, the secrecy-SINR and the per-user secrecy rate are both interference-limited, especially limited by the IUI caused by the alliance of cooperating eavesdroppers.


### 4.4. Secrecy Cost

Guaranteeing secrecy and serving multiple (potentially malicious) users at the same time may come at a cost in terms of the per-user transmission rate. We discuss the cost of achieving physical-layer security in the multi-user MIMO TR-based system, which is the sum-rate loss due to the secrecy requirements. It can be obtained by comparing the mean secrecy sum-rateR¯s achievable by TR beamforming to the mean sum-rate R¯ without secrecy requirements. The gap between R¯s and R¯ represents how much guaranteeing secrecy costs in terms of the achievable sum-rate.(30)R¯=NR¯n=Nlog21+γ¯n.Figure5 compares R¯s and R¯ of TR beamforming with and without secrecy requirements, respectively, for different values of β. For the uncorrelated multi-user Rayleigh channels, the difference between R¯s and R¯ (i.e., R¯-R¯s is negligible, which demonstrates that TR beamforming can achieve secure communication without sacrificing achievable sum-rate.Figure 5
Comparison between the per-user secrecy rate (solid) and the per-user rate without secrecy requirements (dashed).


### 4.5. Secrecy Rate Loss due to Spatial Correlation

In the preceding sections, we assume a model of independent channels which is suitable for the rich-scattering multi-path environment. For the case when users are near to each other and the environment is less scattering, the channels of different users may become spatially correlated and the effect of spatial correlation on the system performance cannot be ignored. To gain a more comprehensive understanding of TR beamforming for secrecy transmission, it is important to develop a quantitative assessment of its performance degradation due to transmit antenna correlation or inter-user correlation, respectively. Figure6 displays the mean per-user secrecy rate loss due to transmit antenna correlation. It is noticeable that low-to-moderate correlation among transmit antennas (i.e., ρt<0.4 already causes a remarkable loss in the mean per-user secrecy rate. When ρt rises from 0 to 0.2, the mean per-user secrecy rate decreases to 30% of the original. Transmit antenna correlation causes poor performance of TR beamforming, particularly at high SNR. In such case, the power control strategy can be adapted to prevent the secrecy rate from decreasing fast.Figure 6
Per-user secrecy rate vs.ρt, for SNR = 20 dB and N=16.Figure7 displays the mean per-user secrecy rate loss due to inter-user correlation. From Figure 7, it can be observed that when ρr reaches 0.2, R¯n,s all drop to zero, which means that the BS cannot transmit to any typical user at a non-zero secrecy rate and the network goes into secrecy outage. Obviously, inter-user correlation causes a severe degradation in the secrecy performance of the TR-based system. The worst part is that the inter-user correlation is much more destructive to R¯n,s than transmit antenna correlation.Figure 7
Per-user secrecy rate vs.ρr, for SNR = 20 dB and N=16.In fact, inter-user correlation not only causes much stronger IUI imposed on the intended usern but also helps the eavesdropper n¯ gain the focusing effect of TR beamforming that significantly improves γn¯ (i.e., the SINR at the eavesdropper n¯ in (16). These are major reasons for a drastic degradation in R¯n,s in the presence of inter-user correlation. Moreover, the improved SINR at the eavesdropper n¯ enables him to intercept more confidential information of user n. On the other hand, Figures 6 and 7 also demonstrate that smaller β causes a faster secrecy rate loss. This implies that when channel correlation is present, more transmit antennas cannot improve the system performance any more.More details are listed in Table2, in which R¯n,sρt stands for R¯n,s in the presence of transmit antenna correlation and R¯n,sρr stands for R¯n,s in the presence of inter-user correlation.Table 2
Secrecy rate loss due to channel correlation.
CoefficientR¯n,sρt/R¯n,sρr (bps/Hz), β=0.25ρt/ρr00.10.20.30.40.50.60.70.80.9N=22.25/2.252.09/2.111.75/1.721.38/1.181.08/0.60.84/0.020.66/00.53/00.42/00.35/0N=82.24/2.241.95/1.581.42/0.310.99/00.69/00.5/00.37/00.28/00.21/00.16/0N=162.23/2.231.72/1.051.04/00.63/00.40/00.26/00.18/00.12/00.09/00.06/0Figures8 and 9 reveal the secrecy costs of R¯n,s in the presence of transmit antenna correlation and R¯n,s in the presence of inter-user correlation, respectively. Note that in Figures 8 and 9, the red solid lines represent R¯n,s and the red dashed lines represent R¯n, which plot the cases of uncorrelated Rayleigh channels (i.e., ρt=0 and ρr=0. Similarly, the blue lines plot the cases of correlated multi-user channels ρt=0.1orρr=0.1. It can be seen that in the presence of transmit antenna correlation, the gap between R¯n,s and R¯n is very small in all cases, and thus the secrecy cost (i.e., R¯n-R¯n,s is close to zero, as shown in Figure 8. In Figure 9, when ρr slightly rises from 0 to 0.1, the secrecy cost increases from 0 to 0.7 bps/Hz and the secrecy cost is very high. Worst of all, when ρr reaches 0.2, the network goes into secrecy outage (i.e., R¯n,s=0), whereas R¯n is always greater than 0.6 bps/Hz.Figure 8
Comparison between the per-user secrecy rate (solid) and the per-user rate without secrecy requirements (dashed) under transmit antenna correlation, forN=16, β=0.25, and ρr=0.Figure 9
Comparison between the per-user secrecy rate (solid) and the per-user rate without secrecy requirements (dashed) under inter-user correlation, forN=16, β=0.25, and ρt=0.Furthermore, Figure10 depicts the effects of the number of users on the R¯n,s in the presence of transmit antenna correlation. In Figure 10, M takes 8, 16, 32, and 64 in turn and N takes 2, 4, 8, and 16 in turn such that β=N/M always remain constant. When no spatial correlation is present (i.e., ρt=ρr=0, the original per-user secrecy rates are equal to each other and at its peak, no matter how big N is. This indicates that for uncorrelated Rayleigh channels, the number of users N has no effect on R¯n,s, and hence R¯n,s is only related to L, β, and the SNR. In fact, a larger N means more users and stronger cumulative IUI. Why the original per-user rates for different number of users are the same? This is because more transmit antennas compensate for the adverse effects of substantial increasing of cumulative IUI caused by more users. This also proves that when no spatial correlation is present, TR beamforming has good focusing performance and can effectively eliminate the interference.Figure 10
The impact of the number of users on the per-user secrecy rate, for SNR = 20 dB andβ=0.25.Unfortunately, whenρt is greater than zero but still in the lower range (e.g., from 0 to 0.2), there is a significant reduction in R¯n,s. Both transmit antenna correlation and more users jointly give rise to the sharp increase of the cumulative IUI interference imposed on user n, which eventually leads to severe degradation in R¯n,s. When spatial correlation is present, the per-user secrecy rate is the function of L, β, ρt, ρr, N, and the SNR.


## 4.1. Time Compression Validation Test

The temporal focusing property of TR beamforming is presented in Figure2 by comparing the amplitudes of both original CIRs and the equivalent TR-CIRs at the receiver, respectively. The power of each CIR has been normalized to 1 (i.e., ∑l=1Lhnil2=1). The original CIRs before TR demonstrate dispersive effects of multi-path channels under different bandwidth conditions. The more the multi-path components resolved in the channel are, the more quickly the signal strength fades as shown in Figures 2(a) and 2(b). Multi-path effects are particularly relevant in high data rate communication systems. Typically, this problem can be solved with equalization method at the receiver. But this would increase computational complexity at the receiver [48]. In Figure 2(c), the equivalent CIR after TR shows that most of signal energy is aggregated at the center tap (i.e., k=L=11) while ISI components at the other taps (i.e., k∈1,2⋯,2L−1,k≠L) have been suppressed as much as possible. This indicates that there is temporal focusing and therefore ISI can be mitigated. Thanks to the superior focusing effect in time domain, the receiver can perform a simple one-tap detection at the center tap and achieve good energy efficiency.Figure 2
An illustration of CIR vs. multi-path channel. (a) The original CIR forB=20MHz. (b) The original CIR for B=40MHz. (c) The equivalent TR-CIR at the intended user for ρt=0 and ρr=0. (d) The equivalent TR-CIR at the intended user for ρt=0.9 and ρr=0. (e) The equivalent TR-CIR at the eavesdropper for ρt=0 and ρr=0. (f) The equivalent TR-CIR at the eavesdropper for ρt=0 and ρr=0.9.
(a)(b)(c)(d)(e)(f)From Figure2(d), one can see that for the multi-path channels with strong transmit antenna correlation ρt=0.9, there is only minor degradation in focusing effect of TR beamforming at the intended user. Comparing Figure 2(e) with Figure 2(f), we can observe that for the multi-path channels with inter-user correlation, as the inter-user correlation increases, the signal for the message un at the eavesdropper is markedly enhanced at the center tap while suppressed at other taps. Obvious, due to the presence of inter-user correlation, the focusing effect of TR beamforming also occurs at the eavesdropper. Unfortunately, this will lead to more interception of confidential information of legitimate user by the eavesdropper. Unlike inter-user correlation, transmit antenna correlation does not improve the CIR of the eavesdropper.


## 4.2. Mean Secrecy Capacity

In this part, we investigate the secrecy performance of the multi-user MIMO TR-based system. We evaluate quantitatively and discuss the influence of propagation conditions, such as the number of taps, the number of transmit antennas, and the number of eavesdroppers on the mean per-user secrecy rate achieved by TR beamforming. To facilitate the discussion, we presume the ratioβ between the number of users N and the number of transmit antennas M, i.e., β=N/M.All the results from Sections4.2 to 4.4 assume that the channels of different users are spatially uncorrelated.Figure3 depicts the simulation results obtained with the mean per-user secrecy rate R¯n,s and the analytical results achieved with the approximation R¯^n,s given in Theorem 1. One can see that the analytical results approximate well the simulation results, which proves the reasonableness and effectiveness of the approximation R¯^n,s.Figure 3
(a) Mean per-user secrecy rate vs. SNR forL=33andN=2. (b) The impact of IUI on mean per-user secrecy rate for L=33andN=2.
(a)(b)In addition, Figure3(a) gives the upper bound of mean per-user secrecy rate R¯n,s under different system configurations, ignoring the effects of IUI and spatial correlation. Note that for a given N, smaller β implies more transmit antennas deployed at the BS. More transmit antennas deployed at the BS also indicate higher per-user secrecy rate achieved, especially notable at high SNR. When the SNR is up to 20 dB, R¯n,s reaches its peak. Thereafter, even if further increasing SNR, R¯n,s still remains unchanged. The reason for this situation lies in the fact that as the SNR increases, the noise power gradually becomes weaker. When the SNR reaches 20 dB, the ISI power far exceeds the noise power so that the noise power can be ignored. The expression for γn,s shown in (24) can be rewritten as(29)γn,s≈Psn/Pisin−Psn¯/Pisin¯1+Psn¯/Pisin¯.Eventually, both the desired signal power and the ISI power determine an upper bound onR¯n,s. Unlike Figure 3(a), Figure 3(b) considers the effects of the IUI on the mean per-user secrecy rate and gives the lower bound of mean per-user secrecy rate R¯n,s. Compared with Figure 3(a), it can be observed that IUI causes R¯n,s to fall by about 0.3∼1 bps/Hz.In Figure4, we have fixed SNR = 20 dB. Figure 4 shows the mean per-user secrecy rat as a function of β for different values of L. As expected, the mean per-user secrecy rate R¯n,s increases as the length of CIR L increases (larger L implies more multi-path components in the channel and stronger scattering in the channel). This figure clearly shows the remarkable benefit of richer multi-path environment on network secrecy capacity. For example, for L=5, the network goes into secrecy outage (i.e., R¯n,s=0) at β=4, whereas for L=11, secrecy outage occurs at β=16. This behavior can be attributed to the fact that by fully exploiting degrees of freedom provided by the environment, i.e., the abundant multi-path components in the channel, the TR-based system can support more coexisting users with non-zero secrecy rate and the secrecy performance of the system is significantly improved.Figure 4
Per-user secrecy rate as a function ofβ, for different values of L. Circles denote secrecy outage.Table1 displays the distributions of the mean per-user secrecy rate and the mean secrecy sum-rate achieved by TR beamforming, for different combinations of M and N, respectively. All the results in Table 1 were obtained at the SNR of 20 dB, for L=33. As indicated in Table 1, this implies a trade-off between the network capacity (in terms of the number of serviced users) and the secrecy rate at each user. When N takes 2, 4, and 8 in turn and M is successively 8, 16, and 32, the mean per-user secrecy rates are 2.25 bps/Hz, 2.24 bps/Hz, and 2.24 bps/Hz, respectively. In the cases, the number of N as well as M is different each time, but the ratios of N to M remain the same (i.e., β=N/M=2/8=4/16=8/32=0.25). These results demonstrate that in the absence of spatial correlation, the mean per-user secrecy rate is just a function of the SNR, L, and β, and thus the average secrecy sum-rate is a function of the SNR, L, β, and N.Table 1
Secrecy rate comparison.
R¯s/R¯n,s (bps/Hz)MN=2N=4N=8N=16N=3221.94/0.972.20/0.552.22/0.281.92/0.121.28/0.0443.08/1.543.84/0.964.32/0.544.48/0.283.84/0.1284.50/2.256.12/1.537.60/0.958.64/0.548.64/0.27166.10/3.058.96/2.2412.16/1.5215.2/0.9517.28/0.54327.74/3.8712.16/3.0417.92/2.2424.32/1.5230.4/0.95


## 4.3. Power Control Strategy

In Figure3(b), it can be seen that for β≤1, R¯n,s is always positive and monotonically increasing with the SNR until the SNR goes up to 20 dB, whereas for β>1, the per-user secrecy rate is irrespective of the SNR. This indicates that the mean per-user secrecy rate does not grow unbounded with the increase of transmit power. Moreover, TR beamforming performs poorly in the high-SNR regime. When increasing the SNR from 10 dB up to 20 dB, only a minor increase in Rn,s is achieved. Therefore, a power control strategy is introduced by reducing the transmit power properly to control the SNR to the value that can maximize the mean secrecy sum-rate (e.g., SNR at 10 dB) and works well for most of the cases. This power control strategy can save transmit power, thus effectively improving energy efficiency. For 1<β<32, the TR-based system with power reduction strategy would achieve an optimal non-negative secrecy sum-rate. However, for β≥31, the secrecy sum-rate is always zero irrespective of the SNR and this strategy would not be effective.Figure3 also shows that the mean per-user secrecy rate R¯n,s is always a decreasing function of β. The main reason for this phenomenon lies in the fact that when β>1, the intended user n suffered from a larger amount of IUI caused by more coexisting eavesdroppers while the alliance of cooperating eavesdroppers n¯ is always free from IUI, as shown in (20). Therefore, the secrecy-SINR and the per-user secrecy rate are both interference-limited, especially limited by the IUI caused by the alliance of cooperating eavesdroppers.


## 4.4. Secrecy Cost

Guaranteeing secrecy and serving multiple (potentially malicious) users at the same time may come at a cost in terms of the per-user transmission rate. We discuss the cost of achieving physical-layer security in the multi-user MIMO TR-based system, which is the sum-rate loss due to the secrecy requirements. It can be obtained by comparing the mean secrecy sum-rateR¯s achievable by TR beamforming to the mean sum-rate R¯ without secrecy requirements. The gap between R¯s and R¯ represents how much guaranteeing secrecy costs in terms of the achievable sum-rate.(30)R¯=NR¯n=Nlog21+γ¯n.Figure5 compares R¯s and R¯ of TR beamforming with and without secrecy requirements, respectively, for different values of β. For the uncorrelated multi-user Rayleigh channels, the difference between R¯s and R¯ (i.e., R¯-R¯s is negligible, which demonstrates that TR beamforming can achieve secure communication without sacrificing achievable sum-rate.Figure 5
Comparison between the per-user secrecy rate (solid) and the per-user rate without secrecy requirements (dashed).


## 4.5. Secrecy Rate Loss due to Spatial Correlation

In the preceding sections, we assume a model of independent channels which is suitable for the rich-scattering multi-path environment. For the case when users are near to each other and the environment is less scattering, the channels of different users may become spatially correlated and the effect of spatial correlation on the system performance cannot be ignored. To gain a more comprehensive understanding of TR beamforming for secrecy transmission, it is important to develop a quantitative assessment of its performance degradation due to transmit antenna correlation or inter-user correlation, respectively. Figure6 displays the mean per-user secrecy rate loss due to transmit antenna correlation. It is noticeable that low-to-moderate correlation among transmit antennas (i.e., ρt<0.4 already causes a remarkable loss in the mean per-user secrecy rate. When ρt rises from 0 to 0.2, the mean per-user secrecy rate decreases to 30% of the original. Transmit antenna correlation causes poor performance of TR beamforming, particularly at high SNR. In such case, the power control strategy can be adapted to prevent the secrecy rate from decreasing fast.Figure 6
Per-user secrecy rate vs.ρt, for SNR = 20 dB and N=16.Figure7 displays the mean per-user secrecy rate loss due to inter-user correlation. From Figure 7, it can be observed that when ρr reaches 0.2, R¯n,s all drop to zero, which means that the BS cannot transmit to any typical user at a non-zero secrecy rate and the network goes into secrecy outage. Obviously, inter-user correlation causes a severe degradation in the secrecy performance of the TR-based system. The worst part is that the inter-user correlation is much more destructive to R¯n,s than transmit antenna correlation.Figure 7
Per-user secrecy rate vs.ρr, for SNR = 20 dB and N=16.In fact, inter-user correlation not only causes much stronger IUI imposed on the intended usern but also helps the eavesdropper n¯ gain the focusing effect of TR beamforming that significantly improves γn¯ (i.e., the SINR at the eavesdropper n¯ in (16). These are major reasons for a drastic degradation in R¯n,s in the presence of inter-user correlation. Moreover, the improved SINR at the eavesdropper n¯ enables him to intercept more confidential information of user n. On the other hand, Figures 6 and 7 also demonstrate that smaller β causes a faster secrecy rate loss. This implies that when channel correlation is present, more transmit antennas cannot improve the system performance any more.More details are listed in Table2, in which R¯n,sρt stands for R¯n,s in the presence of transmit antenna correlation and R¯n,sρr stands for R¯n,s in the presence of inter-user correlation.Table 2
Secrecy rate loss due to channel correlation.
CoefficientR¯n,sρt/R¯n,sρr (bps/Hz), β=0.25ρt/ρr00.10.20.30.40.50.60.70.80.9N=22.25/2.252.09/2.111.75/1.721.38/1.181.08/0.60.84/0.020.66/00.53/00.42/00.35/0N=82.24/2.241.95/1.581.42/0.310.99/00.69/00.5/00.37/00.28/00.21/00.16/0N=162.23/2.231.72/1.051.04/00.63/00.40/00.26/00.18/00.12/00.09/00.06/0Figures8 and 9 reveal the secrecy costs of R¯n,s in the presence of transmit antenna correlation and R¯n,s in the presence of inter-user correlation, respectively. Note that in Figures 8 and 9, the red solid lines represent R¯n,s and the red dashed lines represent R¯n, which plot the cases of uncorrelated Rayleigh channels (i.e., ρt=0 and ρr=0. Similarly, the blue lines plot the cases of correlated multi-user channels ρt=0.1orρr=0.1. It can be seen that in the presence of transmit antenna correlation, the gap between R¯n,s and R¯n is very small in all cases, and thus the secrecy cost (i.e., R¯n-R¯n,s is close to zero, as shown in Figure 8. In Figure 9, when ρr slightly rises from 0 to 0.1, the secrecy cost increases from 0 to 0.7 bps/Hz and the secrecy cost is very high. Worst of all, when ρr reaches 0.2, the network goes into secrecy outage (i.e., R¯n,s=0), whereas R¯n is always greater than 0.6 bps/Hz.Figure 8
Comparison between the per-user secrecy rate (solid) and the per-user rate without secrecy requirements (dashed) under transmit antenna correlation, forN=16, β=0.25, and ρr=0.Figure 9
Comparison between the per-user secrecy rate (solid) and the per-user rate without secrecy requirements (dashed) under inter-user correlation, forN=16, β=0.25, and ρt=0.Furthermore, Figure10 depicts the effects of the number of users on the R¯n,s in the presence of transmit antenna correlation. In Figure 10, M takes 8, 16, 32, and 64 in turn and N takes 2, 4, 8, and 16 in turn such that β=N/M always remain constant. When no spatial correlation is present (i.e., ρt=ρr=0, the original per-user secrecy rates are equal to each other and at its peak, no matter how big N is. This indicates that for uncorrelated Rayleigh channels, the number of users N has no effect on R¯n,s, and hence R¯n,s is only related to L, β, and the SNR. In fact, a larger N means more users and stronger cumulative IUI. Why the original per-user rates for different number of users are the same? This is because more transmit antennas compensate for the adverse effects of substantial increasing of cumulative IUI caused by more users. This also proves that when no spatial correlation is present, TR beamforming has good focusing performance and can effectively eliminate the interference.Figure 10
The impact of the number of users on the per-user secrecy rate, for SNR = 20 dB andβ=0.25.Unfortunately, whenρt is greater than zero but still in the lower range (e.g., from 0 to 0.2), there is a significant reduction in R¯n,s. Both transmit antenna correlation and more users jointly give rise to the sharp increase of the cumulative IUI interference imposed on user n, which eventually leads to severe degradation in R¯n,s. When spatial correlation is present, the per-user secrecy rate is the function of L, β, ρt, ρr, N, and the SNR.


## 5. Conclusions

This paper has examined TR beamforming strategy for providing confidentiality at the physical layer in the multi-user MIMO system. This work enables a deeper understanding of how intrinsic properties of wireless networks can be exploited to enhance the network secrecy. Our study demonstrates that when no spatial correlation is present, TR beamforming can achieve perfect zero-cost secure communication and rich-scattering environment can be beneficial for network secrecy. We also found that in the presence of inter-user correlation, achievable mean secrecy sum-rate decreases so quickly that there is significant increase in secrecy cost in terms of the achievable sum-rate.Fortunately, as 5G communication and future 6G communication are expected to adopt high-frequency band and larger signal bandwidth, these will greatly improve the spatial resolution and time resolution of the channels and therefore more abundant degrees of freedom can be resolved. In addition, with the increase of frequency band and the reduction of signal wavelength, the channels’ difference of users will be more intense so that spatial correlation may be minimized almost to zero. By virtue of its good focusing property in the richer scattering environment and its low complexity, TR beamforming is still an ideal solution for enhancing the system security in the future indoor wireless communications.



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*Source: 3670101-2021-12-18.xml*