Precoding and Transmit Antenna Subarray Selection for Secure Hybrid Spatial Modulation

As an essential key technique for future generation wireless communications like 6G and WiFi 7, multiple-input-multiple-output (MIMO) has obtained a continuous extensive and intensive huge amount of research activities from the end of the last century, which can greatly improve the system performance in wireless communications. Two typical forms of MIMO are bell laboratories layer space-time (BLAST) in[1] and space time coding (STC) in [2]. They make their effort to strike a good balance between spatial multiplexing and diversities in MIMO systems. However, as the third form of MIMO, spatial modulation (SM) is attracting more and more research attention from industry and academia due to its advantages of no inter-channel interference (ICI) and inter-antenna synchronization (IAS) [3]. It works by mapping a block of information bits into two information-carrying units. The first information carrying unit is a amplitude/phase modulation (APM) symbol chosen from the signal-constellation diagram. The second information-carrying unit is a transmit antenna index chosen, while other transmit antennas are not activated[4, 3, 5].

There are several ways to improve the performance of SM including transmit antenna selection[6, 7, 8], linear precoding [9, 10], power allocation[11, 12] and so on. The physical layer security of conventional SM is an urgent problem. The authors in [13] focused their attention on the secrecy behavior of SSK and SM, and derived the expression for secrecy rate (SR). In [14], the secrecy performance of SM was improved by making a combination of useful signal and jamming against unknown eavesdropping, and the corresponding SR performance is analyzed. The authors in [15] redefined the transmit antenna indices from the viewpoint of channel state information (CSI) to prevent eavesdropping by assuming that the imperfect legal CSI is available for the eavesdropper. Then, a full-duplex receiver assisted secure spatial modulation scheme was proposed in [16]. Here, jamming signals are designed to protect legitimate receivers from self-interference (SI) and to interfere eavesdroppers. Besides, in [17], the authors proposed two secure transmit antenna selection procedures, which can achieve a better secrecy performance. The problem of power allocation (PA) between confidential message and artificial noise (AN)[18, 19] was addressed in [11]. And the authors in[11, 12] have proposed two PA strategies with performance approaching that of the optimal PA factor.

It is very important to study the security of multiple-antenna communication systems[20, 21, 22, 23, 24]. Furthermore, the security of SM systems is also very important. The precoding schemes of traditional SM can also be generalized to secure SM. For most conventional PSM schemes, the antenna indices at receiver is utilized to carry bits rather than the antenna indices of the transmitter as spatial bits. The optimization of the precoding matrix at transmitter is to address the issues of preprocessing and detection of PSM signals at receiver in order to improve bit-error-rate (BER) performance [10]. However, the authors in [25] have proposed a time-varying precoder for the secret PSM (SPSM), which generated a time-varying interference to the eavesdropper and retains all advantages of PSM at the legal receiver. Then they also derived the upper bounds for BERs at legal receiver and eavesdropper in the massive MIMO systems in [26], and designed a precoder by jointly minimizing the receive power at eavesdropper and maximizing the receive power at legal receiver in [27]. The kind of PSM is also extended to secure multiuser MIMO downlink scenario by introducing a scrambling matrix to disturb the eavesdropper[28].

Until now, we make a literature review about fully-digital (FD) SM. However, as the number of transmit antennas tends to medium-scale or large-scale, the circuit cost and complexity will become a burden on SM. To deal with this problem, hybrid SM, combining hybrid MIMO structure in [29, 30] and SM, emerges as the times require[31, 32, 33, 34, 35, 36, 37]. It could make a good balance among cost, complexity and performance.

In the case of hybrid partially-connected architecture, the total antenna array is divided into multiple transmit antenna subarrays (TASs) with each being connected to single RF chain. Thus, the spatial bits can be carried by selecting the indices of TASs rather than those of transmit antennas. However, there were several papers focusing on hybrid SM. The hybrid SM was first proposed in [31]. But in [31], it actually used analog precoding, and its digital precoder was replaced by the process of TAS selection (TASS) in SM. Besides, hybrid SM was also extended to the multi-user scenario in [32], which showed that the hybrid SM system with hybrid beamforming at transmitter and digital combining at the receiver can achieve an excellent BER performance.

The authors in [33] made an investigation of the macro SM, in which the indexes of the small base stations were used to carry information bits and they also proposed a low-complexity detection method. A multimode hybrid precoder is designed for proposed analog precoding-aided virtual SM and obtain BER performance gain compared with the conventional precoding-aided MIMO systems[34].

For hybrid generalized SM (GSM), in [35], an analog precoder of maximizing spectrum efficiency (SE) was designed with the verified superiority in SE performance. In [36], the digital and analog precoders by turbo optimization was proposed for hybrid GSM systems to improve its SE performance. Furthermore, in [37], the closed-form expression of the achievable SE lower bound of was derived firstly. Then, via exploiting the concavity of the SE expression in digital precoding vectors, the digital precoding vectors were computed, the convex relaxation was invoked to handle the non-convex constraints of analog precoding, and the corresponding problem of analog precoding was converted into a convex optimization problem. Finally, the designed digital and analog precoding schemes harvested higher SE gains over existing methods.

In this paper, several techniques in this paper will be developed to improve the secrecy performance of hybrid SM as follows: TASS, digital precoding for confidential message (CM), AN projection, and analog precoding. Our main contributions are summarized as follows:

1. 1.

   A hybrid secure SM system model is established, where the transmitter is equipped with the partially-connected architecture, and the desired receiver and eavesdropping receiver are equipped with fully digital architecture. Since each RF chain is connected to a TAS in the partially-connected architecture, the spatial bits are carried by activating the TAS rather than a single transmit antenna, while the APM symbols are transmitted by the active TAS. Each TAS carries single bit stream by using multiple antennas. This will create spatial diversity and will be exploited to improve the secure performance of SM. In addition, with the help of AN interfering with the eavesdropper, the useful signal is sent along the desired subspace to further improve the security of this system.

2. 2.

   To improve the secrecy performance, two hybrid precoding methods, maximizing the approximate SR (ASR) based on the gradient ascent (Max-ASR-GA) and maximizing the approximate SR (Max-ASR) based on alternating direction method of multipliers (Max-ASR-ADMM), are proposed. Due to the fact that there is no closed-form expression of SR, an ASR expression is presented for hybrid precoding design. The proposed Max-ASR-ADMM method can be expressed as a general form consensus problem, which can be addressed by exiting ADMM. Simulation results show that the SR performance of the proposed Max-ASR-GA is better than conventional hybrid precoding method: SDR-AltMin and the proposed Max-ASR-ADMM. In particular, the proposed Max-ASR-ADMM has a better SR performance than SDR-AltMin in the medium and high regions.

3. 3.

   Generally, the number of TASs is not a power of two, so TASS is required. In order to improve the secrecy performance, three secure TASS schemes are proposed. Maximizing approximate SR (Max-ASR) TASS method is proposed to consider the effect of the above two factors on secrecy performance. To reduce the complexity of Max-ASR, the maximizing the eigenvalue (Max-EV) TASS method and the maximizing the product of signal-to-interference-plus-noise ratio and artificial noise-to-signal-plus-noise ratio (Max-P-SINR-ANSNR) are proposed as two low complexity methods. Since each TAS corresponds to a sub-channel, it can achieve a good rate performance by choosing the TASs corresponding to the sub-channels with large channel gains. The eigenvalues of each sub-channel are sorted in order and the corresponding TASs of the sub-channels with large eigenvalues are used. The proposed Max-EV performs slightly better than the proposed Max-P-SINR-ANSNR in the medium and high SNR regions with the same complexity as the latter in terms of SR. The proposed Max-ASR harvests a substantial SR performance gain over Max-EV and Max-P-SINR-ANSNR.

Consider a system of hybrid SM as shown in Fig. 1, where the transmitter (Alice) is equipped with $N_{\rm{RF}}$ RF chains with each having $N_{\rm{AA}}$ antennas to transmit one data stream to the legitimate receiver (Bob) equipped with $N_{b}$ antennas and $N_{b}^{\rm{RF}}$ RF chains. There is an eavesdropper (Eve) with $N_{e}$ antennas and $N_{e}^{\rm{RF}}$ RF chains trying to eavesdrop on the data stream from Alice to Bob. Alice uses the hybrid sub-connected structure, while Bob and Eve use fully digital structure, namely $N_{b}\!=\!N_{b}^{\rm{RF}}$ and $N_{e}\!=\!N_{e}^{\rm{RF}}$.

Now, we extend conventional SM to hybrid structure transmitters in MIMO systems. Therefore, in Fig. 1 system, the first part information bits are used to choose one of TASs and the second part information bits are mapped to the modulation symbol. In what follows, it is assumed that $N_{t}=\lfloor N_{\rm{RF}}\rfloor_{2}$ due to the reason that the number of TASs is confined to a power of two. Let $I_{\rm{ASS}}=\{l_{i}\}_{i=1}^{N_{t}}$ be the legitimate TAS subset, where the element $l_{i}$ is selected from set $\left\{1,2,\cdots,N_{\rm{RF}}\right\}$. A proper TASS strategy can clearly improve system performance. $\mathbb{I}$ is the set of enumerations of all possible $Q=\begin{pmatrix}N_{\rm{RF}}\\ N_{t}\end{pmatrix}$ combinations of TAS subsets $I_{\rm{ASS}}$. The modulation symbol first experiences the digital precoder and then passes through the RF chain connected by the active TAS. Considering the security of communication, AN is needed to disturb the eavesdropper as an efficient secure tool. AN is also projected in digital precoding stage. After the precoded CM plus the projected AN passing the corresponding RF chains,and the phase shifters (PSs) of analog precoding and AN. Finally, the transmit signal is in the following form

where $b_{j}$ is the digital symbol chosen from the $M$-ary constellation for $j\in\mathbb{M}=\left\{1,2,\cdots,M\right\}$, and satisfies $\mathbb{E}|b_{j}|^{2}=1$. $\textbf{E}_{i}\!=\!\textrm{diag}[\textbf{0},\cdots\!,\textbf{0},\textbf{e}_{i},\textbf{0},\cdots\!,\textbf{0}]\!\in\!\mathbb{R}^{N_{\rm{AA}}N_{\rm{RF}}\times N_{\rm{AA}}N_{\rm{RF}}}$ is a block diagonal matrix, and its $i$-th sub-block is the identity matrix, that is, $\textbf{e}_{i}=\textbf{I}_{N_{\rm{AA}}}$, which means that the $i$-th TAS is activated with $i\in I_{\rm{ASS}}$. $\textbf{T}=\textrm{diag}[\textbf{t}_{1},\textbf{t}_{2},\cdots,\textbf{t}_{N_{\rm{RF}}}]$ is a block diagonal TASS matrix similar to $\textbf{E}_{i}$, where $\textbf{t}_{i}=\textbf{I}_{N_{\rm{AA}}}$ for all $i\in I_{\rm{ASS}}$, and the other sub-blocks are all-zero matrix, which means that $N_{t}$ TASs are selected. Since there are Q combinations for $I_{\rm{ASS}}\in\mathbb{I}$, i.e., there are Q possibilities of T, which can be expressed as $\textbf{T}_{1},\textbf{T}_{2},\ldots,\textbf{T}_{Q}$. $\textbf{n}\sim\mathcal{CN}(0,\textbf{I}_{N_{t}})$ is the AN vector, and $\beta$ is the power allocation factor between CM and AN. $P$ is the transmit power. And the digital precoding matrix of CM is given by

and $\textbf{T}_{\rm{BB}}\!\in\!\mathbb{C}^{N_{\rm{RF}}\times N_{t}}$ is the digital projection matrix of AN, also called the AN precoding matrix. Since only $N_{t}$ TASs are selected from $N_{\rm{RF}}$ TASs to work, and the AN n is only projected to the selected $N_{t}$ TASs, so $\textbf{T}_{\rm{BB}}$ only has the corresponding $N_{t}$ rows with non-zero vectors and the remaining rows are all 0. After passing through the corresponding RF chain, the precoded CM plus AN from each RF chain is delivered to the corresponding $N_{\rm{AA}}$ PSs to perform the corresponding analog precoding. This process can be denoted by the $N_{\rm{AA}}N_{\rm{RF}}\times N_{\rm{RF}}$ analog precoding matrix $\textbf{F}_{\rm{RF}}$ as follows

where $\textbf{f}_{i}\in\mathbb{C}^{N_{\rm{AA}}\times 1}$ is the $i$th analog weighting vector for $i=1,2,\cdots,N_{\rm{RF}}$, and all its elements have the same amplitude $1/\sqrt{N_{\rm{AA}}}$ but distinct phases. Therefore, $\textbf{P}=\textbf{F}_{\rm{RF}}\textbf{F}_{\rm{BB}}$ can be expressed as

where $\textbf{p}_{i}\!\in\!\mathbb{C}^{N_{\rm{AA}}\times 1},i\!=\!1,2,\cdots,N_{\rm{RF}}$, and $\textbf{P}_{\rm{AN}}\!=\!\textbf{F}_{\rm{RF}}\textbf{T}_{\rm{BB}}$. The associated transmit vector after precoding and TASS can be expressed as

where $\textbf{b}_{j}\in\mathbb{C}^{1\times N_{\rm{AA}}}$ is the modulation symbol $b_{j}$ after hybrid precoding, and $i\in I_{\rm{ASS}}$ denotes the index of the activated TAS. The received signals at Bob and Eve can be represented as

where $\textbf{H}\in\mathbb{C}^{N_{b}\times N_{\rm{AA}}N_{\rm{RF}}}$ is the channel matrix of Alice-to-Bob which can be thought as $\textbf{H}=[\textbf{h}_{1},\textbf{h}_{2},\cdots,\textbf{h}_{N_{\rm{RF}}}]$ and $\textbf{h}_{i}\in\mathbb{C}^{N_{b}\times N_{\rm{AA}}}$ is the channel matrix from TAS $i$ to Bob. $\textbf{G}\in\mathbb{C}^{N_{e}\times N_{\rm{AA}}N_{\rm{RF}}}$ is the channel matrix from Alice to Eve and G can be given by $\textbf{G}=[\textbf{g}_{1},\textbf{g}_{2},\cdots,\textbf{g}_{N_{\rm{RF}}}]$ and $\textbf{g}_{i}\in\mathbb{C}^{N_{e}\times N_{\rm{AA}}}$ is the channel matrix from the $i$th TAS to Eve. $\textbf{n}_{b}\sim\mathcal{CN}(0,\sigma_{b}^{2}\textbf{I}_{N_{b}})$ and $\textbf{n}_{e}\sim\mathcal{CN}(0,\sigma_{e}^{2}\textbf{I}_{N_{e}})$ denote the complex additive white Gaussian noise (AWGN) vectors at Bob and Eve, respectively. Thus, the received signals observed at Bob and Eve after decoding processing can be respectively formulated as follows

where $\textbf{W}_{b}\!\in\!\mathbb{C}^{N_{b}\times N_{\rm{RF}}}$ is the combiner of Bob. It can be expanded as $\textbf{W}_{b}\!=\![\textbf{w}_{1}^{b},\textbf{w}_{2}^{b},\ldots,\textbf{w}_{N_{\rm{RF}}}^{b}]$, and $\textbf{w}_{i}^{b}\in\mathbb{C}^{N_{b}\times 1},i\!=\!1,2,\ldots,N_{\rm{RF}}$. $\textbf{W}_{e}\!\in\!\mathbb{C}^{N_{e}\times N_{\rm{RF}}}$ is the combiner of Eve, and $\textbf{W}_{e}\!=\![\textbf{w}_{1}^{e},\textbf{w}_{2}^{e},\ldots,\textbf{w}_{N_{\rm{RF}}}^{e}]$, where $\textbf{w}_{i}^{e}\!\in\!\mathbb{C}^{N_{e}\times 1},i\!=\!1,2,\ldots,N_{\rm{RF}}$.

The maximum-likelihood (ML) detector at Bob is given by

The average SR is given as

where

where $\textbf{y}^{\prime}_{b}\!=\!\bm{\Omega}_{\rm{B}}^{-1/2}\textbf{y}_{b}$, $\textbf{y}^{\prime}_{e}\!=\!\bm{\Omega}_{\rm{E}}^{-1/2}\textbf{y}_{e}$, $\textbf{n}^{\prime}_{b}\!=\!\bm{\Omega}_{\rm{B}}^{-1/2}\textbf{W}_{b}^{H}\textbf{n}_{b}$, and $\textbf{n}^{\prime}_{e}\!=\!\bm{\Omega}_{\rm{E}}^{-1/2}\textbf{W}_{e}^{H}\textbf{n}_{e}$,

$\textbf{d}_{ij}\!=\!\textbf{x}_{i}\!-\!\textbf{x}_{j}$ and $\textbf{d}_{mk}\!=\!\textbf{x}_{m}\!-\!\textbf{x}_{k}$, where $\textbf{x}_{i}$, $\textbf{x}_{j}$, $\textbf{x}_{m}$, and $\textbf{x}_{k}$ are the possible transmit vectors. $\bm{\Omega}_{\rm{B}}$ and $\bm{\Omega}_{\rm{E}}$ are the covariance matrices of interference plus noise of Bob and Eve respectively, where

with

respectively. Since $\bm{\Omega}_{\rm{B}}$ and $\bm{\Omega}_{\rm{E}}$ are adopted to whiten colored noise into an white noise, they don’t change the mutual information, that is, $I(\textbf{x};\textbf{y}_{b})\!=\!I(\textbf{x};\textbf{y}^{\prime}_{b})$ and $I(\textbf{x};\textbf{y}_{e})\!=\!I(\textbf{x};\textbf{y}^{\prime}_{e})$.

In addition, to facilitate the hybrid precoding design, $\textbf{P}_{\rm{AN}}$ is chosen to be the projection matrix in the null space of channel H, satisfying $\textbf{W}_{b}^{H}\textbf{HT}\textbf{F}_{\rm{RF}}\textbf{T}_{\rm{BB}}=\textbf{W}_{b}^{H}\textbf{HTP}_{\rm{AN}}=\textbf{0}$. Therefore,

where $\textbf{H}_{\textbf{T}}\!=\![\textbf{h}_{l_{1}},\textbf{h}_{l_{2}},\cdots,\textbf{h}_{l_{N_{t}}}]\!\in\!\mathbb{C}^{N_{b}\times N_{\rm{AA}}N_{t}}$ is the channel matrix corresponding to the selected subchannels $I_{\rm{ASS}}\!=\!\{l_{i}\}_{i=1}^{N_{t}}$. Due to the relation $\textbf{P}_{\rm{AN}}\!=\!\textbf{F}_{\rm{RF}}\textbf{T}_{\rm{BB}}$, once $\textbf{F}_{\rm{RF}}$ is determined, so $\textbf{T}_{\rm{BB}}$ can be obtained readily.

The analog precoding corresponding to the selected channel is expressed as

And the digital precoding matrix corresponding to the selected channel is $\textbf{T}_{\rm{BB},\textbf{T}}\!\in\!\mathbb{C}^{N_{t}\times N_{t}}$ and satisfies $\textbf{T}_{\rm{BB},\textbf{T}}\!=\!\textbf{ST}_{\rm{BB}}$, where S is the selection matrix, and the $l_{i}$-th column of S is the $i$-th column of $\textbf{I}_{N_{t}}$.

Therefore, $\textbf{W}_{b}^{H}\textbf{HT}\textbf{F}_{\rm{RF}}\textbf{T}_{\rm{BB}}\!=\!\textbf{W}_{b}^{H}\textbf{H}_{\textbf{T}}\textbf{F}_{\rm{RF},\textbf{T}}\textbf{T}_{\rm{BB},\textbf{T}}\!=\!\textbf{0}$. Let us define $\textbf{H}^{\prime}\!=\!\textbf{W}_{b}^{H}\textbf{H}_{\textbf{T}}\textbf{F}_{\rm{RF},\textbf{T}}$, $\textbf{T}_{\rm{BB},\textbf{T}}$ can be given by

where $\mu$ is the normalized factor. According to the power constraint at transmit side, and considering $\|\textbf{T}\textbf{F}_{\rm{RF}}\textbf{T}_{\rm{BB}}\|_{F}^{2}\!=\!\|\textbf{F}_{\rm{RF},\textbf{T}}\textbf{T}_{\rm{BB},\textbf{T}}\|_{F}^{2}\!=\!\|\textbf{T}_{\rm{BB},\textbf{T}}\|_{F}^{2}\!=\!1$, then we have $\|\textbf{T}_{\rm{BB},\textbf{T}}\|_{F}^{2}\!=\!1$ and $\mu\!=\!\|\textbf{I}\!-\!\textbf{H}^{\prime H}(\textbf{H}^{\prime}\textbf{H}^{\prime H})^{\dagger}\textbf{H}^{\prime}\|_{\textrm{F}}$. Therefore, we obtain the digital precoder $\textbf{T}_{\rm{BB}}\!=\!\textbf{S}^{T}\textbf{T}_{\rm{BB},\textbf{T}}$ of AN. $\textbf{A}^{\dagger}$ is the Moore-Penrose pseudo inverse of A.

For hybrid SM systems, the design of hybrid precoder at Alice and combiner at Bob is very important to improve the system performance. In this section, the approximate expression of SR is derived by using the definition of cut-off rate in [38]. Then, two hybrid precoders, called Max-ASR-ADMM and Max-ASR-GA, are proposed to enhance the security of SM systems. Additionally, the SDR-AltMin Algorithm in [39] is also extended to be suitable for hybrid SM and used as a performance reference.