SVD-Aided Multi-Beam Directional Modulation Scheme Based on Frequency Diverse Array
Qian Cheng Vincent Fusco Jiang Zhu Shilian Wang Chao Gu
eess.SP










SVD-Aided Multi-Beam Directional Modulation Scheme Based on Frequency Diverse Array
    Qian Cheng, Student Member, IEEE,
        Vincent Fusco, Fellow, IEEE,
        Jiang Zhu, Member, IEEE,
        Shilian Wang, Member, IEEE,
        and Chao Gu
This work was supported by a scholarship (No. 201803170247) from China Scholarship Council (CSC). (Corresponding author: Shilian Wang.)
Q. Cheng is with the College of Electronic Science, National University of Defense Technology, Changsha 410073, China, and also with the ECIT Institute, Queen's University Belfast, Belfast BT3 9DT, UK. (e-mail: chengqian14a@nudt.edu.cn)
J. Zhu and S. Wang are with the College of Electronic Science, National University of Defense Technology, Changsha 410073, China (e-mail: {jiangzhu, wangsl}@nudt.edu.cn).
V. Fusco and C. Gu are with the ECIT Institute, Queen's University Belfast, Belfast BT3 9DT, UK (e-mail: v.fusco@ecit.qub.ac.uk; chao.gu@qub.ac.uk).

    December 30, 2023
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With the assistance of singular value decomposition (SVD), a multi-beam directional modulation (DM) scheme based on symmetrical multi-carrier frequency diverse array (FDA) is proposed. The proposed DM scheme is capable of achieving range-angle dependent physical layer secure (PLS) transmissions in free space with much lower complexity than the conventional zero-forcing (ZF) method. Theoretical and simulated results about secrecy rate and complexity verify the improved computational efficiency and considerable memory savings despite a very small penalty of secrecy rate.




Directional modulation; frequency diverse array; physical layer security; singular value decomposition.






§ INTRODUCTION


Directional modulation (DM) has increasingly become a promising technique capable of achieving physical layer secure (PLS) communications.


The work in <cit.> proposed an angle-dependent DM scheme based on phased arrays (PA) by optimizing the phase shifters at the radio frequency (RF) frontend. A low-complexity DM synthesis method was proposed in <cit.>, which transferred the DM synthesis from RF frontend to baseband. Afterwards, multi-beam angle-dependent DM synthesis methods were researched in <cit.>-<cit.>. Specifically, the orthogonal vector synthesis approach was implemented in <cit.>, while <cit.> proposed a robust synthesis method for multi-beam DM scheme with imperfect direction knowledge. The multipath nature of the channel was exploited in <cit.> to create a multi-user DM transmissions. Artificial noise (AN) aided multi-beam DM system was studied in <cit.> using zero-forcing (ZF) criterion.

The above-mentioned single- or multi-beam DM schemes can only achieve angle-dependent secure transmissions, the security of which will fail when the eavesdropper is located along the same direction as the desired receiver. To address this problem, the frequency diverse array (FDA) with non-linear frequency increments was introduced to synthesize both range-angle dependent DM in <cit.>. The work in <cit.> proposed a random FDA-based range-angle dependent DM synthesis method, while the AN-aided FDA-DM communication over Nakagami-m fading channels was studied in <cit.>. 

These FDA-DM schemes <cit.>-<cit.>   only considered single-beam DM transmissions for a single desired receiver. To achieve range-angle dependent multi-beam FDA-based DM transmissions, one possible approach <cit.> is to jointly optimizing the frequency increments, the beamforming vector, and the AN orthogonal matrix, which is too complicated to realize practically. The other approach is to extend the ZF-aided multi-beam PA-DM  synthesis <cit.> to multi-beam FDA-DM synthesis <cit.>. The ZF method, however, consumes a large amount of memory for the orthogonal matrix and the AN vector.

This paper aims to reduce the complexity of the range-angle dependent multi-beam DM system in free space by re-designing the AN vector and orthogonal matrix with the aid of singular value decomposition (SVD). Compared with the conventional ZF method, the proposed SVD method can improve computational efficiency and save much memory with a small loss of secrecy rate. This makes it possible to achieve secure range-angle dependent multi-beam DM transmissions with lower complexity and lower DC power consumption. 







Notations: The operators ( · )^T, ( · )^H, ( · )^-1, and ( · )^† represent the transpose, Hermitian transpose, inverse, and Moore-Penrose inverse operations of a matrix, respectively.







§ SYSTEM MODEL


As shown in Fig. 1, we consider a (2N+1)-element symmetrical linear FDA with multiple carriers <cit.>. The spacing d between adjacent elements is designed as half central wavelength, and L carriers are transmitted via each element. The frequency of the l-th (l=0,1,⋯,L-1) carrier for the n-th (n=-N,⋯,0,⋯,N) element of FDA is designed as

    f_n,l=f_0+Δ f_n,l=f_0+Δ fln[(|n|+1)(l+1)]

where f_0 denotes the central carrier frequency, and Δ f is a fixed frequency offset satisfying |Δ f|≪ f_0.





Let r and θ represent the range between an arbitrary spatial position and the central FDA element, and the azimuth angle, respectively. The normalized steering vector radiated from the transmit antenna array at (r,θ) is a (2N+1)L× 1 vector for such a system, which can be expressed as

    𝐡(r,θ)
       =1/√((2N+1)L) [𝐚_-N^T(r,θ) ⋯𝐚_n^T(r,θ) ⋯𝐚_N^T(r,θ) ]^T
 
where 𝐚_n(r,θ)=
[ 
e^jϕ_n,0⋯  e^jϕ_n,l⋯  e^jϕ_n,L-1 ]^T is an L × 1 sub-steering vector caused by the L carriers of the n-th antenna element <cit.>. The phase of the l-th entry of 𝐚_n(r,θ) is 

    ϕ_n,l=2π[ Δ fln[(|n|+1)(l+1)] (t-r/c)+ f_0ndsinθ/c]

where c is light speed, d=.λ/ 2 . denotes the spacing between adjacent elements, t refers to the time variable, and λ=.c / f_0. represents the wavelength of the central carrier.

The multi-beam DM model consists of a transmitter with a (2N+1)-element FDA, K stationary desired receivers, and U passive eavesdroppers in different locations. The spatial coordinate of the k-th (k=1,2,⋯,K) desired receiver is assumed to be (r_d^k,θ_d^k) and the combined set of the K desired receivers' spatial coordinates is expressed as

    (Υ_d,Θ_d)={(r_d^1,θ_d^1),(r_d^2,θ_d^2),⋯,(r_d^K,θ_d^K)}
 




A steering matrix 𝐇(Υ_d,Θ_d) with size (2N+1)L× K can be obtained by combining the steering vectors 𝐡(r_d^k,θ_d^k) of the K desired spatial positions (Υ_d,Θ_d), i.e.,

    𝐇(Υ_d,Θ_d)=[𝐡(r_d^1,θ_d^1) 𝐡(r_d^2,θ_d^2) ⋯ 𝐡(r_d^K,θ_d^K)]
 

Let 𝐱_d=[x_d^1 x_d^2 ⋯ x_d^K]^T denote the confidential baseband symbol vector for the K desired receivers. The weighted N-tuple transmitted signal vector of the FDA is designed as

    𝐬=β_1√(P_s)𝐏_1𝐱_d+αβ_2√(P_s)𝐏_2𝐳
 
where P_s is the total transmit power constraint, β_1 and β_2 are the power splitting factors satisfying β_1^2+β_2^2=1, α denotes the normalization factor for the inserted AN, and 𝐳 is the inserted complex AN vector with each entry having zero mean and variance σ_z^2. In order to obtain a desired standard modulation constellation at the desired receivers while distorting the received signals at other undesired receivers, the normalization matrix 𝐏_1 and the orthogonal matrix 𝐏_2 should satisfy the following criteria:

    𝐇^H(Υ_d,Θ_d)𝐏_1=𝐈,  𝐇^H(Υ_d,Θ_d)𝐏_2=0
 

In this paper, the normalization matrix is directly designed as the Moore-Penrose inverse matrix of the steering matrix 𝐇^H(Υ_d,Θ_d), i.e.,

    𝐏_1=𝐇(Υ_d,Θ_d)[𝐇^H(Υ_d,Θ_d)𝐇(Υ_d,Θ_d)]^-1
 
which satisfies 𝐇^H(Υ_d,Θ_d)𝐏_1=𝐈_K, and the size of which is (2N+1)L× K.

It is known that the null space of a matrix can be derived from its SVD. Therefore, in order to design the orthogonal matrix 𝐏_2, we perform the SVD operation on the steering matrix 𝐇^H(Υ_d,Θ_d), which is now expressed as

    𝐇^H(Υ_d,Θ_d)=𝐔[𝐃 0][𝐕_1 𝐕_0]^H
 

Next, we can obtain a null space, that is, 𝐕_0∈ℂ^(2N+1)L× (2N+1-K) for 𝐇^H(Υ_d,Θ_d) from (<ref>), which means 𝐇^H(Υ_d,Θ_d)𝐕_0=0_K× (2N+1-K). Therefore, the orthogonal matrix of the SVD method can be designed as

    𝐏_2=𝐕_0
 









The normalized line-of-sight (LoS) channel is considered in this paper. In fact, the proposed method can also hold over fading channels like  Nakagami-m fading <cit.>, as long as the channel state information (CSI) is perfectly estimated. After passing through an LoS channel, the combined vector of the received signals at the desired receivers can be expressed as

    𝐲(Υ_d,Θ_d)   =𝐇^H(Υ_d,Θ_d)𝐬+𝐰_d
    = β_1√(P_s)𝐇^H(Υ_d,Θ_d)𝐏_1𝐱_d_Useful Signal
          +αβ_2√(P_s)𝐇^H(Υ_d,Θ_d)𝐏_2𝐳_Artificial Noise+𝐰_d_AWGN
       =β_1√(P_s)𝐱_d+𝐰_d
 
where 𝐰_d∼ CN(0_K× 1,σ_w_d ^2𝐈_K) is the circularly symmetric complex additive white Gaussian noise (AWGN) vector with each entry having zero mean and variance σ_w_d ^2.


Similarly, the received signal of an arbitrary eavesdropper at (r_e,θ_e), (r_e,θ_e)≠ (r_d^k,θ_d^k), can be expressed as

    y(r_e,θ_e)   =𝐡^H(r_e,θ_e)𝐬+w_e 
    = β_1√(P_s)𝐡^H(r_e,θ_e)𝐏_1𝐱_d_Distortion Signal
          +αβ_2√(P_s)𝐡^H(r_e,θ_e)𝐏_2𝐳_Artificial Noise+w_e_AWGN
 
where w_e∼ CN(0,σ_w_e ^2)  is the AWGN with zero mean and variance σ_w_e ^2. It is worth emphasizing that the first term of (<ref>) denotes distortion signal for the undesired receiver, and the second term is the inserted AN which cannot be eliminated at the undesired receiver due to the fact that the designed 𝐏_2 is non-orthogonal to its steering vector 𝐡^H(r_e,θ_e).



§ PERFORMANCE ANALYSIS







 §.§ Secrecy Rate


We assume the U eavesdroppers are located at  (r_e^u,θ_e^u), 1≤ u≤ U, respectively, which  satisfy  (r_e^u,θ_e^u)≠ (r_d^k,θ_d^k) for ∀ u∈{1,2,⋯,U} and ∀ k∈{1,2,⋯,K}.

The achievable rate of the link from the transmitter to the k-th desired receiver can be calculated by <cit.>

    R(r_d^k,θ_d^k)=log_2[1+γ(r_d^k,θ_d^k)]
 
where γ(r_d^k,θ_d^k) is the ratio of signal to interference plus noise (SINR) at the k-th desired receiver with the expression of 

    γ(r_d^k,θ_d^k)   =β_1^2 P_s𝐡^H(r_d^k,θ_d^k) 𝐏_1𝐏_1^H𝐡(r_d^k,θ_d^k)/σ_w_d^2 + α ^2β_2^2 P_s𝐡^H(r_d^k,θ_d^k) 𝐏_2𝐏_2^H𝐡(r_d^k,θ_d^k)


Similarly, the achievable rate of the link from the transmitter to the u-th eavesdropper is expressed as

    R(r_e^u,θ_e^u)=log_2[1+γ(r_e^u,θ_e^u)]
 
where

    γ(r_e^u,θ_e^u)   =β_1^2 P_s𝐡^H(r_e^u,θ_e^u) 𝐏_1𝐏_1^H𝐡(r_e^u,θ_e^u)/σ_w_e^2 + α ^2β_2^2 P_s𝐡^H(r_e^u,θ_e^u) 𝐏_2𝐏_2^H𝐡(r_e^u,θ_e^u)


Therefore, the secrecy rate of the proposed multi-beam DM system can be defined as

    R_s=max_k∈{1,⋯,K }[min_u∈{1,⋯,U }(R(r_d^k,θ_d^k)-R(r_e^u,θ_e^u))  ]^+

where [·]^+=max{0,·}.



 §.§ Time Complexity


Both the normalization matrices of the ZF method in <cit.><cit.> and the proposed SVD method are designed as [ 𝐇^H(Υ_d,Θ_d)]^†, so the complexity depends on the orthogonal matrix 𝐏_2 and the inserted AN 𝐳, as shown in Table I.


Here, we consider the time complexity of calculating the orthogonal matrix 𝐏_2. For the ZF method,  the orthogonal matrix is designed as  𝐏_2^ZF=𝐈_(2N+1)L - [𝐇^H(Υ_d,Θ_d)]^†𝐇^H(Υ_d,Θ_d), the time complexity of which is 𝒪((2N+1)^2L^2K) <cit.><cit.>. For the SVD method, the orthogonal matrix can be directly obtained from the SVD of 𝐇^H(Υ_d,Θ_d), which means the time complexity of calculating 𝐏_2^SVD lies in the time complexity of calculating the SVD of 𝐇^H(Υ_d,Θ_d). Since the size of 𝐇^H(Υ_d,Θ_d) is K× (2N+1)L, (K<(2N+1)L), the time complexity of calculating 𝐏_2^SVD can be obtained by 𝒪(K^2(2N+1)L) <cit.>.

Additionally, the Moore-Penrose inverse operation of a matrix can be acquired using SVD algorithm <cit.>, which means the normalization matrix 𝐏^SVD_1 can be directly calculated using the output of SVD. But for the ZF method, the normalization matrix 𝐏^ZF_1 has to be calculated independently. 












 §.§ Space Complexity


The sizes of the orthogonal matrix 𝐏_2 and the AN 𝐳 can impact the the memory consumption significantly. As shown in Table I, the size of the orthogonal matrix 𝐏_2^ZF of the ZF method is (2N+1)L× (2N+1)L, and the size of the inserted AN 𝐳^ZF is (2N+1)L× 1. By contrast, the size of the orthogonal matrix 𝐏_2^SVD of the proposed SVD method is (2N+1)L× (2N+1-K), and the size of the inserted AN 𝐳^SVD is (2N+1-K)× 1.

We define a metric ζ as the ratio of the total memory consumed by 𝐏_2^SVD and 𝐳^SVD to that of the ZF method, i.e.,

    ζ   =(2N+1)L× (2N+1-K)+(2N+1-K)/(2N+1)L× (2N+1)L + (2N+1)L× 100%
       =(2N+1)^2L- (2N+1)(LK-1)-K/(2N+1)^2L^2+(2N+1)L× 100%








It is worth noting that ζ→.1/L. when K is determined and N→∞, which means the proposed SVD method can consume at most .1/L. of the memory required by the ZF method, thereby reducing the amount of memory and lowering DC power consumption requirements.



§ SIMULATION RESULTS


In our simulations, the parameters are set as f_0=10 GHz, Δ f=2 kHz, β_1=0.9, P_s=1, N=8, L=7, K=3, and Gray-coded .π/4.-QPSK, respectively. The signal-to-noise ratio (SNR) is set as 10 dB. Without loss of generality, we further assume σ_w_d^2=σ_w_e^2, and the locations of the K desired receivers are (r_d^1,θ_d^1)=(150km,50^∘), (r_d^2,θ_d^2)=(180km,-40^∘), and  (r_d^3,θ_d^3)=(260km,0^∘), respectively.  



Fig. 2(a) shows the secrecy rate versus SNR (dB) for ZF- and SVD-aided multi-beam DM systems, where the eavesdroppers' locations are randomly selected in the simulations. It is observed that the SVD method requires slightly higher SNR (dB) than the ZF method in order to achieve the same secrecy rate. For example, only 0.5 dB of additional SNR is required for the SVD method to match the ZF method when R_s = 8 bits/s/Hz and U=2. The secrecy loss is due to that the size of the inserted AN of the SVD method is smaller than that of the ZF method, which makes the SINRs of eavesdroppers a little higher than that of the ZF method.



Fig. 2(b) and (c) illustrate the simulated BER performances for ZF and SVD methods versus angle and range, respectively. In the angle dimension, the BER lobes of the ZF method are slightly narrower than the proposed SVD method around the desired receivers. In the range dimension, both ZF and SVD methods can achieve almost the same BER performances.



In order to illustrate the time complexity, we conducted 10^4 Monte Carlo experiments to record the average time consumption of calculating 𝐏_2 using MATLAB[1][1]Computer configurations: Intel(R) Xeon(R) CPU E5-1620 v2 @ 3.7 GHz; 8.0 GB RAM; 64-bit operating system. MATLAB version: R2016a.. The result shows that the average time consumptions of ZF and SVD methods are 0.2258 and 0.1933 ms, respectively, which verifies that the computational efficiency of the proposed SVD method is better than that of the ZF method. 


Moreover, Fig. 3 shows the total amount of memory required by the orthogonal matrix 𝐏_2 and the AN 𝐳, and the ratio ζ of SVD to ZF versus N, L and K, respectively. From Fig. 3(a), the SVD method only consumes up to 14.29% of the memory required by the ZF method with L=7 and K=3, which is due to ζ→.1/L. when N→∞ as shown in (<ref>). From Fig. 3(b) and Fig. 2(c), the ratio of SVD to ZF decreases as L and K increases. Therefore, the SVD method is more efficient than the ZF method with respect to memory required.





§ CONCLUSION

A new SVD-aided low-complexity range-angle dependent multi-beam DM scheme based on symmetrical multi-carrier FDA was proposed. The proposed SVD method outperforms the ZF method in regard to the computational complexity and the amount of memory required for processing, while introducing only a small performance loss of secrecy rate. The SVD method opens a way to reduce the implementing complexity and lower DC power consumption requirements for range-angle dependent multi-beam DM systems.









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