Secure Hybrid Digital and Analog Precoder for mmWave Systems with low-resolution DACs and finite-quantized phase shifters

# Secure Hybrid Digital and Analog Precoder for mmWave Systems with low-resolution DACs and finite-quantized phase shifters

Ling Xu, Feng Shu, , Guiyang Xia,
Yijin Zhang, Zhihong Zhuang, and Jiangzhou Wang,
This work was supported in part by the National Natural Science Foundation of China (Nos. 61771244, 61472190, 61702258, 61501238, and 61602245)(Corresponding authors: Feng Shu, Ling Xu, and Zhihong Zhuang). Ling Xu, Feng Shu, Guiyang Xia, Yijin Zhang and Zhihong Zhuang  are with School of Electronic and Optical Engineering, Nanjing University of Science and Technology, 210094, CHINA. E-mail:{shufeng, xuling}@njust.edu.cnJiangzhou Wang is with the School of Engineering and Digital Arts, University of Kent, Canterbury CT2 7NT, U.K. Email: {j.z.wang}@kent.ac.uk.
###### Abstract

Millimeter wave (mmWave) communication has been regarded as one of the most promising technologies for the future generation wireless networks because of its advantages of providing a ultra-wide new spectrum and ultra-high data transmission rate. To reduce the power consumption and circuit cost for mmWave systems, hybrid digital and analog (HDA) architecture is preferred in such a scenario. In this paper, an artificial-noise (AN) aided secure HDA beamforming scheme is proposed for mmWave MISO system with low resolution digital-to-analog converters (DACs) and finite-quantized phase shifters on RF. The additive quantization noise model for AN aided HDA system is established to make an analysis of the secrecy performance of such systems. With the partial channel knowledge of eavesdropper available, an approximate expression of secrecy rate (SR) is derived. Then using this approximation formula, we propose a two-layer alternately iterative structure (TLAIS) for optimizing digital precoder (DP) of confidential message (CM), digital AN projection matrix (DANPM) and analog precoder (AP). The inner-layer iteration loop is to design the DP of CMs and DANPM alternatively given a fixed matrix of AP. The outer-layer iteration loop is in between digital baseband part and analog part, where the former refers to DP and DANPM, and the latter is AP. Then for a given digital part, we propose a gradient ascent algorithm to find the vector of AP vector. Given a matrix of AP, we make use of general power iteration (GPI) method to compute DP and DANPM. This process is repeated until the terminal condition is reached. Simulation results show that the proposed TLAIS can achieve a better SR performance compared to existing methods, especially in the high signal-to-noise ratio region.

Hybrid digital and analog, mmWave, security, artificial Noise, low-resolution digital-to-analog converter(DAC)

## I Introduction

With the rapid development of wireless communication technologies as well as explosive access of mobile terminals, the demand of wireless data traffic grows exponentially. However, the spectrum of current exiting wireless systems is highly congested, which brings a bottleneck for increasing wireless access rate ulteriorly. Millimeter wave (mmWave) communication, whose frequency band ranges from 30GHz to 300GHz, emerges as a promising candidate way of addressing the problem of spectrum congestion [1, 2, 3, 4].

Nevertheless, although the mmWave communication becomes more and more popular due to its abundant available and under-utilized spectrum, its realistic applications are under many constraints such as the severe free space path loss and rain attenuation [5]. To overcome these challenges, multiple-input-multiple-output (MIMO) is usually adopted to compensate for these shortages. Meanwhile, hybrid digital and analog (HDA) architecture has been employed to further reduce the energy consumption of mmWave MIMO systems [6, 7, 8, 9, 10, 11, 12, 13]. The design of hybrid precoder was formulated as a problem of sparse signal reconstruction by exploiting the inherent spatial sparse structure of mmWave channel in [6, 7], where orthogonal matching pursuit (OMP) algorithm was used to achieve a comparatively good performance. In [8], the authors proposed three methods of innovative alternative minimization for both fully-connected and partially-connected hybrid combiner in mmWave MIMO systems, which confirms the feasibility and effectiveness of hybrid precoder in mmWave MIMO architecture. In [9], an energy-efficient successive-interference-cancelation-based algorithm with low complexity was proposed for mmWave MIMO systems. Authors in [10] and [11] extended conventional flat-fading mmWave channels to broadband frequency-selective mmWave channels, where the joint analog beamforming for the entire band and respective baseband precoder for each sub-band were required to be designed carefully. Specifically, in [10] the heuristic algorithms in OFDM systems were developed for two scenarios: single-user MIMO and multiple-user multiple-input-single-output scenarios. In [11], the authors developed a novel method by dynamically establishing the structure of sub-arrays. In [12] , the hybrid analog and digital (HAD) architecture was used at receiver to make a measurement of direction of arrival (DOA). Here, three low-complexity DOA estimation methods were proposed and the corresponding HAD Cramer-Rao lower bound was also derived. In [13], a robust beamforming method using HAD receive structure was proposed to achieve interference compression.

However, the aforementioned research works mainly focus on HDA mmWave systems without taking security into consideration. Wyner first proposed a discrete, memoryless wiretap channel model to investigate the secrecy of wireless communication [14]. Based on the framework of Wyner’s wiretap channel, multiple-antenna technique [15] is applied to enhance security and beamforming has been proven to be secrecy-capacity-achieving under the circumstances that the desired receiver has single antenna and full channel knowledge are available for all terminals[16, 17]. If the transmitter can not obtain the simultaneous channel state information (CSI) of eavesdropper or only know the partial CSI of eavesdropper, artificial noise (AN) was shown to be an effective aiding method to strengthen security [18, 19, 20, 21]. Recently, some researchers paid a close attention to security in mmWave MIMO channel. Authors in [22] made a systematical investigation of the security performance in HDA mmWave systems under two different CSIs: full and partial. Here, maximum ratio transmission (MRT) method was extended to such scenarios and minimum secrecy outage probability was adopted to design the corresponding hybrid precoder. Considering the sparse features of millimeter wave channels in [23], a discrete angular domain channel model was proposed to meticulously derive the secure performance for the proposed transmission schemes in slow fading multipath channels. In [24], an AN-aided hybrid precoder is proposed to maximize the lower bound of average secrecy rate (SR).

In practical applications, if low-resolution digital-to-analog converters (DACs) or analog-to-digital converters (ADCs) are adopted, the average power consumption of hybrid transceivers will be greatly reduced. Actually, a high -resolution DACs/ADCs can be power-hungry [25]. In [26], how to strike a trade-off between energy efficiency and spectrum efficiency was investigated for analog, hybrid, and digital receivers with low resolution ADCs, respectively. To further reduce energy consumption, hybrid beamforming and digital beamforming with low resolution ADCs are studied in [27]. Here, the authors find that digital beamforming equipped with finite resolution ADCs can achieve a higher achievable rate and can be more energy efficient than hybrid beamforming in the low signal-to-noise ratio (SNR) region. In [28], taking both low resolution DACs and RF losses in hybrid into account in mmWave MIMO system, a quantized hybrid transmitter with additive quantization noise model (AQNM) has been constructed for both fully-connected and partially-connected hybrid architecture, and a lower bound of achievable rate is presented.

However, to the best of our knowledge, there are few literature of making an investigation on how to achieve a security in hybrid mmWave systems with low resolution DACs. Therefore, in our paper, we propose an AN-aided hybrid mmWave transmitter with low resolution DACs and finite-quantized phase shifters. The transmitted signals with the help of AN are first precoded by digital precoder (DP) in baseband, then passing through low-resolution DACs and RF chains before analog precoder (AP). Here, the AQN model is adopted to approximate the quantized signal as a linear output, which simplifies the analysis of further secrecy rate. Since the optimization problem is non-convex and intractable to tackle, an alternate iteration algorithm is resorted to maximize the approximate expression of secrecy rate. Our main contributions are summarized as follows:

1. An AN-aided secure hybrid precoding system model with low-resolution DACs and finite-quantized phase shifters on RF is established. By taking two kinds of quantization errors (QEs) into consideration, the proposed model is completely distinct from conventional non-secure hybrid precoding system without considering QEs and the AQN quantized model is adopted in our system model to approximate the distortion from QE of low-resolution DACs as a linear output. With partial eavesdropping channel knowledge available, an approximate expression for secrecy rate (SR) is derived. This expression converts the original intractable problem into a more easy-to-handle one. Thus, the approximate SR (ASR) expression will significantly simplify the optimization and design of digital precoding (DP) vector of confidential messages, digital AN projection matrix (ANPM) and analog precoding (AP) vector in what follows.

2. A two-layer alternatively iterative structure (TLAIS) is proposed to maximize the ASR by taking quantization errors from both low-resolution DAC and phase shifters into consideration. Given AP vector, the DP vector of confidential messages and ANPM are alternatively computed within an interior iterative loop by making use of general power iterative (GPI) method with the aim to maximize the approximate SR. In particular, by complex Kronecker product manipulation, the problem of maximizing ASR with respect to the optimization matrix ANPM is converted into one with respect to an optimization vector being the vectorization of the corresponding ANPM. Given the DP vector and digital ANPM, a steepest ascent algorithm is used to attain the updated AP vector. The above process is repeated until the terminal condition is reached. Finally, the phases of AP vector is directly taken out as the inputs of the finite quantized phase shifters on RF. More importantly, to reduce the computation complexity, we abstract the non-zero elements in analog part by taking advantage of the sparsity of analog precoder.

The remainder of this paper is organized as follows. Section II presents the system model of AN-aided HDA mmWave multiple input single output (MISO) system with low resolution DACs and finite-quantized phase shifters, and the approximate expression of SR is given in this section. Based on the approximate formula of SR, a TLAIS among DP of confidential message, digital AN projection matrix (DANPM) and AP is proposed in Section III. Performance analysis and simulation evaluations are presented in Section IV. Finally, we draw our conclusions in Section V.

Notation: throughout the paper, matrices, vectors, and scalars are denoted by letters of bold upper case, bold lower case, and lower case, respectively. , £¬ and denote transpose, conjugate, and conjugate transpose, respectively. and denote the norm of a vector and Frobenius norm of a matrix, respectively. and are matrix trace and matrix vectorization; and indicate the Kronecker products and Hadamard products between two matrices, respectively. returns a diagonal matrix consisting of the corresponding diagonal elements of matrix . returns the block diagonal matrix concatenated of . And returns a vector consisting of the non-zero elements in . denotes the element in row and column, returns a matrix consisting of to row and to column in .

## Ii Channel and System Models

### Ii-a System model

In this section, we consider a system model with HDA transmitter of low resolution DACs as shown in Fig. 1. There are three network nodes: Alice, Bob, and Eve. Working as a transmitter, Alice uses a partially connected hybrid architecture in this paper, where each RF chain is connected to a subset of antennas. Assume the transmit antennas at transmitter is , and the number of RF chains is , then it is clear that with each sub-array having antennas.

The transmit signal can be expressed as follows:

 x=FRFQb(√βPfBBs+√(1−β)PTBBz) (1)

where is the effective transmitted power,  denotes the power allocation (PA) factor of confidential message, and indicates the PA factor of AN. and represent the digital beamforming vector of confidential message and AN projection matrix, respectively. To satisfy the transmit power constraint, we have . is the analog precoding matrix with the following structure:

 FRF =Diagblk(fRF1,⋯,fRFK) =⎡⎢ ⎢ ⎢ ⎢ ⎢⎣fRF10⋯00fRF2⋯0⋮⋮⋱⋮00⋯fRFK⎤⎥ ⎥ ⎥ ⎥ ⎥⎦ (2)

where denotes a vector defined as

 fRFk=[exp(jφk,1),exp(jφk,2),⋯,exp(jφk,M)]T. (3)

where

 φk,i∈FRF={0,2π2bps,⋯,2π2bps−12bps} (4)

where represents the set of -bit-quantized phases with being the number of quantization bits of phase shifters at RF. In this paper, we apply the additive quantization noise (AQN) model in [26, 28] to approximate the DACs quantization as a linear output for the simplicity of analysis, which can be formulated as

 Qb(u)≈(1−η)u+nq (5)

where is defined as the reciprocal of signal-to-quantization-noise ratio. is the additive quantization noise vector and uncorrelated with the input , that is, . Therefore, the transmitted signal in (1) can be rewritten as

 x≈ √βP(1−η)FRFfBBs +√(1−β)P(1−η)FRFTBBz+FRFnq (6)

According to [26], the covariance of the quantization noise can be given by

 Rnqnq=η(1−η)diag{Ruu} (7)

where . Therefore, the received signal at Bob can be represented as

 yb= √βP(1−η)hbFRFfBBs +√(1−β)P(1−η)hbFRFTBBz+hbFRFnq+nb (8)

Similarly, the receive signal at eavesdropper can be given as

 ye =√βP(1−η)heFRFfBBs +√(1−β)P(1−η)heFRFTBBz+heFRFnq+ne (9)

where and are additive white Gaussian noise (AWGN) at Bob and Eve, respectively. For the convenience of analysis, we set .

Therefore, the achievable rate at Bob and Eve can be formulated in (10) and (11), respectively. And we can express the optimization problem of maximizing the SR as

 maxFRF,fBB,TBB,β      Rs=Rb−Re subject~{}to          FRF∈FRF,E{xxH}=PT,0≤β≤1 (12)

Since it is assumed that only the partial knowledge of wiretap channel at Eve is available, we can only obtain the average achievable rate of Eve. Meanwhile, because of partially-connected structure, we have . To obey the power constraint, we should have , and . Therefore, the objective function in optimization problem (II-A) can be lower bounded as:

 Rs =Rb−E[log2{1+SINRe}] (a)≥Rb−log2{1+E[SINRe]} =~Rs (13)

where holds due to

 E[log2(x)]≤log2(1+E[x]). (14)

Therefore, replacing the objective function in (II-A) by forms the following simplified optimization problem

 maxβ,fBB,TBB,FRF      ~Rs=Rb−log2{1+E[SINRe]} subject~{}to   FRF∈FRF,∥fBB∥2=1,∥TBB∥2F=1,0≤β≤1 (15)

### Ii-B Channel model

In what follows, a narrow-band clustered mmWave channel model is used with propagation paths, which can be described as follows:

 h=√NtLL∑l=1glaHt,l(ϕl) (16)

where is the normalized factor, stands for the index of paths, and is the number of channel paths. The path gains depicts the complex gain of the path. is the corresponding response vector of transmit antenna array, with denotes the azimuth angles, respectively. For uniform linear array with elements, the array response vector can be represented as

 a(ϕ)=1√N[1,e−j2πλdsin(ϕ),⋯,e−j2πλ(N−1)dsin(ϕ)]T (17)

where and are the wavelength of the signal and distance spacing between the antenna elements.

Since we consider the mmWave channel with partial channel knowledge of Eve, i.e., the full knowledge of angles of departure (AoD) and the distribution of the gains of mmWave paths . As we can see from (16), the channel can be written more compactly as

 h=√NtLgAt (18)

with and . Since the each element in follows the complex Gaussian distribution with zero mean and unit variance, is the i.i.d. complex Gaussian vector with probability density function as

 f(g)=1πLe−gHg. (19)

Therefore, the expectation can be approximated as (20). The approximation in is similarly adopted in [29, 30, 31].

## Iii Design and Optimization of DP£¬ AP£¬and ANPM

In this section, we propose a TLAIS of maximizing the ASR by taking quantization errors from both low-resolution DAC and phase shifters into account as shown in Fig. 2. For the outer loop, given the initial DP vector of confidential messages and ANPM, a steepest ascent algorithm is used to attain the updated AP vector by maximizing the approximate SR (ASR). After completing this, the optimization process turns to the inner loop. Given the AP matrix, the DP vector of confidential messages and ANPM are alternatively attained within an interior iterative loop by using GPI method with the aim to maximize the ASR. The two loops are repeated until their individual terminal conditions are satisfied.

### Iii-a Design of AP matrix

Observing the structure of the numerator in (20), we can simplify it to the following expression

 Tr[Ate,mFRFfBBfHBBFHRFAHte,m] =Tr[FHRFAHte,mAte,mFRFfBBfHBB] =vec(FRF)H[FBB⊗A]vec(FRF) (21)

where and . In the above equation, holds due to the fact that

 Tr(PHZPW)=vec(P)H(WT⊗Z)vec(P). (22)

Since the analog beamforming matrix (III-A) is block diagonal and most of its elements are zeros, i.e., the dimension of is . In other words, there are non-zero elements and zero elements for matrix , so the matrix can be viewed as a sparse matrix. In order to reduce the computational complexity and dimension in the following, only non-zero elements of are required to be extracted. Because there are non-zero elements in , let us define

 d=N[vec(FRF)]=[fRF1,fRF2,⋯,fRFK]T∈CN×1 (23)

represent the mathematic operation of extracting all non-zero elements from . To match the above operation, now, let us extract extract the corresponding elements from matrix in (III-A). The rule of extracting method is as follows: matrix can be partitioned into or block matrices with each being submatrix,

each block matrix can be expressed as with . Only elements per block matrix are extracted with the following rule:

 Γ((m−1)M+1:mM,(n−1)M+1:nM) = [FBB(m,n)⊗A]((m−1)M+1:mM,(n−1)M+1:nM) (24)

Via the above manipulation, we have a simple form of (III-A) as follows

 vec(FRF)H[FBB⊗A]vec(FRF)=dHΓd (25)

In the same fashion, (20) can be rewritten in a more concise way as follows:

 E{SINRe} ≈dHΓe1ddHΓe2d+dHΓe3d+σ2 =dHΓe1ddH[Γe2+Γe3+σ2KINt]d (26)

Similarly, the achievable rate of Eve is also rewritten as

 log2(1+E[SINRe])≈log2(dHAe,1ddHAe,2d) (27)

with

 Ae,1=Γe1+Γe2+Γe3+σ2KINt, (28)

and

 Ae,2=Γe2+Γe3+σ2KINt. (29)

Similarly, the can be also expressed as (III). Therefore, we can further write the achievable rate of Bob as

 Rb=log2(dHAb,1ddHAb,2d) (31)

with

 Ab,1=Γb1+Γb2+Γb3+σ2KINt

and

 Ab,2=Γb2+Γb3+σ2KINt. (33)

Finally, the optimization problem of maximizing (II-A) can be further recasted as

 ~Rs =Rb−log2(1+E[SINR]) ≈log2(dHAb,1ddHAb,2d⋅dHAe,2ddHAe,1d) (34)

Since the is the non-convex function of . In particular, all elements in have the unit modulus, which satisfies . Therefore, a gradient ascent (GA) method is used to compute the AP matrix. Let us define

 f(d)=dHAb,1ddHAb,2d,g(d)=dHAe,2ddHAe,1d. (35)

The gradient of with respect to in can be given by

 ∇d=(f′(d)g(d)+f(d)g′(d))f(d)g(d)ln2 (36)

where

 f′(d)=AHb,1d(dHAb,2d)−(dHAb,1d)AHb,2d(dHAb,2d)2, (37a) g′(d)=AHe,2d(dHAe,1d)−(dHAe,2d)AHe,1d(dHAe,1d)2. (37b)

After obtaining the , we will renew the value of by with being the searching step. The detailed process of GA algorithm proposed by us is listed in Algorithm 1.

### Iii-B Design of DP vector of confidential message

In this section, we will design the DP vector assuming the other two precoders and are known in advance. To optimize , the achievable rate of Bob is represented as a function of as follows

 Rb =log2(1+ξ1fHBBFHRFhHbhbFRFfBBξ2hbFRFdiag[fBBfHBB]FHRFhHb+γb) (d)=log2(1+ξ1fHBBFHRFhHbhbFRFfBBξ2hbFRF[(fBBfHBB)⊙IK]FHRFhHb+γb) (e)=log2⎛⎝1+ξ1fHBB~hHb~hbfBBξ2fHBB[IK⊙(~hHb~hb)]fBB+γb⎞⎠ =log2(fHBBQbfBBfHBBPbfBB) (38)

where

 ξ1=βP(1−η)2 (39)
 ξ2=η(1−η)βP, (40)
 γb =(1−β)P(1−η)2∥hbFRFTBB∥2 +η(1−η)(1−β)P (41)
 ~hb=hbFRF, (42)
 Qb=ξ2[IK⊙(~hHb~hb)]+γbIK+ξ1~hHb~hb, (43)

and

 Pb=ξ2[IK⊙(~hHb~hb)]+γbIK. (44)

Equality in (III-B) is achieved because and holds due to the fact that .

Now, we can rewrite as a function of from (20) as follows:

 E[SINRe]fBB≈λ1fHBBFHRFAHteAteFRFfBBλ2tr[AteFRFdiag{fBBfHBB}FHRFAHte]+γe =λ1fHBB~AHte~AtefBBλ2tr[~Ate(fBBfHBB)⊙IK~AHte]+γe =λ1fHBB~AHte~AtefBBλ2tr[(fBBfHBB){IK⊙(~AHte~Ate)}]+γe =fHBB(λ1~AHte~Ate)fBBfHBB(λ2[IK⊙~AHte~Ate]+γeIK)fBB (45)

where

 λ1=βP(1−η)2Nt/L, (46)
 λ2=η(1−η)βPNt/L, (47)
 γe=(1−β)P(1−η)2NtLtr[AteFRFTBBTHBBFHRFAHte]+ NtLtr[AteFRFη(1−η)diag{(1−β)PTBBTHBB}FHRFAHte]+σ2, (48)

and . Using the above expression of , we directly have the average achievable rate at Eve as follows:

 log2(1+E[SINRe])≈log2(fHBBQefBBfHBBPefBB) (49)

where

 Qe=λ2[IK⊙~AHte~Ate]+γeIK+λ1~AHte~Ate (50)

and

 Pe=λ2[IK⊙~AHte~Ate]+γeIK. (51)

Combining (III-B)and (49) yields the expression of as

 ~Rs=Rb−log2(1+E[SINRe]) ≈log2(fHBBQbfBBfHBBPbfBB⋅fHBBPefBBfHBBQefBB) (52)

Observing the form of (III-B) is the product of fractional quadratic functions, one efficient way to address this problem is GPI algorithm [32]. Therefore, we can obtain the solution to by resorting to GPI algorithm.

### Iii-C Design of ANPM

In this subsection, we will optimize the digital ANPM by fixing and . Under this condition, we can rewrite the as a function of as follows:

 Rb(TBB) =log2⎛⎝1+κbα1∥~hbTBB∥2+α2~hbdiag{TBBTHBB}~hHb+ωb⎞⎠ =log2⎛⎝1+κbα1∥~hbTBB∥2+α2tr{THBB[I⊙(~hHb~hb)]TBB}+ωb⎞⎠ =log2(1+κbα1∥~hbTBB∥2+α2tr{THBBCbTBB}+ωb)