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Frequency diverse array beamforming for physical-layer security with directionally-aligned legitimate user and eavesdropper

The conventional physical-layer (PHY) security approaches, e.g., transmit beamforming and artificial noise (AN)-based design, may fail when the channels of legitimate user (LU) and eavesdropper (Eve) are close correlated. Due to the highly directional transmission feature of millimeter-wave (mmWave), this may occur in mmWave transmissions as the transmitter, Eve and LU are aligned in the same direction exactly. To handle the PHY security problem with directionally-aligned LU and Eve, we propose a novel frequency diverse array (FDA) beamforming approach to differentiating the LU and Eve. By intentionally introducing some frequency offsets across the antennas, the FDA beamforming generates an angle-range dependent beampattern. As a consequence, it can degrade the Eve's reception and thus achieve PHY security. In this paper, we maximize the secrecy rate by jointly optimizing the frequency offsets and the beamformer. This secrecy rate maximization (SRM) problem is hard to solve due to the tightly coupled variables. Nevertheless, we show that it can be reformulated into a form depending only on the frequency offsets. Building upon this reformulation, we identify some cases where the SRM problem can be optimally solved in closed form. Numerical results demonstrate the efficacy of FDA beamforming in achieving PHY security, even for aligned LU and Eve.


I. INTRODUCTION
As an efficient approach to secure wireless communications, the physical-layer (PHY) security technique has been intensively researched recently [1], [2]. One key performance measure of PHY security is secrecy rate, which is characterized by the channel capacity difference of the legitimate user (LU) and the eavesdropper (Eve). Popular PHY security approaches include transmit beamforming [2]- [4] and artificial-noise (AN)based designs [5]- [7], which explore certain channel statistical independence to either enhance the LU's reception or jam the Eve's reception.
However, these approaches may fail when the independence assumption does not hold. For instance, there is an increasing attention recently in using millimeter-wave (mmWave) bands for the next generation wireless networks [8]. One important feature of mmWave communications is the highly directional transmission, especially in line-of-sight (LOS) scenarios [9]. Thus, if Eve is aligned with LU in the same transmit direction, highly correlated channels may result. In the worst case, if the Eve's channel vector is a scaled version of the LU's, then the transmit beamforming and AN-aided methods do not work.
To solve the PHY security problem for directionally-aligned LU and Eve, approaches that are able to discriminate LU and Eve with different range in the same transmit direction should be sought. However, to the best of our knowledge, so far no related research has been reported. Frequency diverse array (FDA) is a potential solution to this problem. Basically, FDA employs some small frequency offsets across the antenna array to generate an angle-range dependent beampattern [10]. Current research on FDA mainly focuses on analyzing the anglerange dependent beampattern characteristics, and exploring its application in radar and/or navigation systems [11]- [13]. An orthogonal frequency division multiplexing (OFDM) secure transmit scheme is designed in [14], based on FDA with some given and fixed frequency offsets.
Departing from existing research, we propose to employ the FDA technique to achieve PHY security for aligned LU and Eve in highly directional transmissions. To handle such a PHY security problem, we apply FDA's range discrimination capability to enhance the LU's reception and degrade the Eve's. In addition, to maximally explore the array's potential, we further integrate beamforming into FDA and finally develop a novel FDA beamforming approach to addressing the directionallyaligned PHY security problem.
Specifically, we maximize the secrecy rate via judiciously designing the FDA beamforming parameters, i.e., the frequency offsets across the array and the transmit beamforming. The secrecy rate maximization (SRM) problem is challenging due to the nonconcave objective and the tightly coupled frequencyoffset variables and the beamforming variables. To handle this problem, we first explore the SRM problem structure in order to gain some insights to the FDA beamforming. After that, we develop a two-stage algorithm for this SRM problem, which optimizes the frequency offsets and the transmit beamforming successively. The main difficulty lies in the frequency offsets optimization, which is also a challenging nonconvex problem. Interestingly, we identify some special cases where the problem can be optimally solved. In particular, linearly increasing the frequency across the array attains the maximal secrecy rate. Moreover, the maximal secrecy rate and the optimal frequency offsets can both be computed in closed form.

A. Frequency Diverse Array
As shown in Fig. 1(a), we consider an LOS mmWave communication system consisting of an M -antenna uniform linear transmit array, a single-antenna LU, and a single-antenna Eve. The array forwards confidential information to the LU, in the presence of the Eve overhearing the transmission exactly in the transmit path. To achieve some confidentiality, the FDA beamforming approach is employed to provide PHY security. As opposed to the conventional phase array, FDA introduces small frequency offsets across the array antennas. As shown in Fig. 1(b), the radiation frequency of the mth antenna is f m = f c +∆f m for m = 1, 2, . . . , M , with f c and ∆f m being the carrier frequency and the frequency offsets, respectively. We assume 0 ≤ ∆f m ≤ ∆F and ∆F f c , where ∆F is the maximal frequency offset. Let d denote the uniform antenna spacing of the transmit array, which is maintained as d = c/[2(f c + ∆F )] to avoid aliasing effects with c being the speed of light.
Without loss of generality, we define the first antenna as the origin of the (range, angle) coordinate system. Furthermore, to concentrate on the range-dependent SRM problem, we ignore the very few multi-path components (MPCs) in LOS mmWave transmissions, since they were found to be attenuated by 20dB compared to the LOS component [8]. For the user at (r, θ), the LOS channel associated with the mth antenna is given by where a(r) denotes the signal attenuation factor at the range of r. Notice that we have assumed far-field model to obtain h m (f m , r, θ), i.e., parallel wavefront and a(r m ) a(r). For mmWave array transmission, the far-field and LOS assumptions can hold simultaneously due to the tiny array size, usually in magnitude of millimeters.
To focus on the FDA characteristics, we temporarily ignore the attenuation factor a(r) and assume an all-one beamformer. Then, the FDA beampattern at (r, θ) is computed as Clearly, the phase terms {Φ 1,m (∆f m , r, θ)} M m=1 in (2) depend on the frequency offsets {∆f m } M m=1 , the range r, and the angle θ. By appropriately choosing {∆f m } M m=1 , FDA can generate an angle-range dependent beampattern. To show this, we plot the typical beampatterns of the conventional phase array and FDA in Fig. 2. For the sake of simplicity, here we have assumed linearly increasing frequency offsets in FDA, i.e., ∆f m = (m − 1)∆f, ∀m. In the case of ∆f = 0,  we have Φ 1,m (∆f m , r, θ) = 0, ∀m, and then FDA reduces to the conventional phase array as shown in Fig. 2(a). As a contrast, we set ∆f = 10 −5 f c in Fig. 2(b) and an angle-range dependent beampattern is thus activated owing to the nonzero phase terms {Φ 1,m (∆f m , r, θ)} M m=1 .

B. Problem Statement
We see from (2) that FDA has a deterministic beampattern for some given f and time t. In general, there is no guarantee that the LU and Eve are located at the beampattern peak and valley simultaneously. Hence, to maximally explore the array's potentials in PHY security, we further integrate beamforming into FDA, yielding a novel FDA beamforming strategy. In particular, let w = [w 1 , w 2 , . . . , w M ] T ∈ C M ×1 denote the array transmit beamformer. The channel vector between the transmit array and the user at The signal-to-noise ratio (SNR) for the user is computed as where σ 2 is the noise power and (·) † denotes the Hermitian transpose. Let (r u , θ u ) and (r e , θ e ) denote the coordinates of the LU and Eve, respectively. As shown in Fig. 1(a), we have θ u = θ e = θ and r u > r e . To simplify the notations, we use h u (f ) and h e (f ) to denote the associated channels, i.e., h u (f ) h(f , r u , θ u ) and h e (f ) h(f , r e , θ e ). Hence, the achievable LU rate and Eve rate are computed by The FDA beamforming based SRM problem is expressed as with P being the transmit power budget.

III. A TWO-STAGE ALGORITHM FOR SRM PROBLEM
The SRM problem (P1) is challenging due to the nonconcave objective and the tightly coupled variables f and w. In this section, we first provide some insights to the solution of (P1), and by leveraging these insights we then develop a lowcomplexity two-stage algorithm to the SRM problem.
A. Some Insights to the Solution of (P1) To obtain some insights into why FDA can provide PHY security, even for directionally-aligned LU and Eve, let us first fix the frequency variable f and briefly review the classical PHY secrecy beamforming design [15]. Obviously, with fixed f in (P1), the SRM problem is the same as the classical multiinput, single-output, single-eavesdropper (MISOSE) secrecy problem in [15], and the optimal beamformer can be computed via generalized eigendecomposition. In particular, Lemma 1. [15] Given f in (P1), let us denote H e (f ) = H e (f ) + 1 P I (10) Then, the optimal secrecy rate in (P1) is given by where [·] + max{·, 0} and λ Σ (f ) is the principle eigenvalue of Σ(f ). Moreover, the optimal beamformer w is given by, where sgn{·} returns 1 if the argument is positive, and 0 otherwise; v Σ (f ) is the principle eigenvector of Σ(f ).
We should mention that the secrecy rate expression in (12) slightly differs from the original form in [15]. One can easily verify the equivalence between them. We adopt (12) here since it severs our purpose better in the following development.
From (12), we see that the optimal secrecy rate depends on the principle eigenvalue of Σ(f ). Since Σ(f ) is indefinite in general, λ Σ (f ) can be either positive or negative. In particular, let us assume ∆f m = 0 and f p = f c 1 with 1 being the M × 1 all-one vector; i.e. the conventional phase array beamforming case. For the aligned LU and Eve as shown in Fig. 1(a), we have h e (f p ) = αh u (f p ) with |α| = a(re) a(ru) > 1. Thus, it can be easily concluded that H u (f p )−H e (f p ) and Σ(f p ) are negative semidefinite matrices, i.e., λ Σ (f p ) ≤ 0. As a consequence, the conventional beamforming approach fails.
However, by introducing frequency offsets ∆f m across the array antennas, in general we have h e (f ) = αh u (f ) due to the nonzero {Φ 1,m (∆f m , r, θ)} in (2). That is, h e (f ) and h u (f ) are no longer linearly dependent and H u (f ) − H e (f ) 0. In consequence, Σ(f ) may have a positive eigenvalue. Intuitively speaking, the frequency offsets ∆f m play a role of rotating the LU and Eve such that in the rotated space their channels are decorrelated. This key observation not only explains why FDA can provide PHY security even for aligned LU and Eve, it also sheds some lights on how to design the frequency offsets.
According to Lemma 1 and the previous discussion, we see that the SRM problem (P1) boils down to finding the optimal frequency offsets ∆f m , ∀ m, such that the principle eigenvalue After finding the optimal f for (P2), the optimal beamformer w (f ) can be computed in closed form based on (13). Thus, in the subsequent development we focus on solving (P2).
Lemma 2 reveals that the optimal frequency offsets should turn the channels of LU and Eve as orthogonal as possible. In the ideal case, we should have h e (f ), h u (f ) = 0. However, due to the limited dynamic range of the frequency offsets, it is in general not possible to make h e (f ) and h u (f ) orthogonal exactly. Next, we will identify some special cases, under which the orthogonality can be guaranteed.

B. Closed-form Solution for (P3) under Some Special Cases
Intuitively, if there is sufficient flexibility to choose f m , i.e., ∆F is large enough, we would expect that h e (f ) and h u (f ) can be turned orthogonal exactly. The following propositions formalize this idea. ], which is the maximum channel capacity of LU (without Eve). Therefore, the presence of Eve does not degrade the rate performance between transmitter and LU.
Remark 2. Given the maximum frequency offset range ∆F , the conditions in Proposition 1 impose a minimum distance between LU and Eve, i.e., |r u − r e | ≥ (M −1)c M ∆F , to achieve the optimality.
Proposition 1 reveals an interesting result. For directionally aligned LU and Eve (with sufficiently large ∆F ), linearly increasing frequency offsets across the array is optimal. Actually, the optimality of linear frequency offset can be generalized to the case of θ u = θ e under some mild conditions. and k is the smallest nonzero integer satisfying ∆f > 0. Then, linearly increasing frequency offset is optimal and the optimal frequency offset is given by Proof : We omit the details of proof due to space limitation. Basically, Proposition 2 can be proved similarly as Proposition 1 except that we ignore the phase term ∆fm(m−1)d(sin θu−sin θe) c in h e (f ), h u (f ) due to the conditions of d (r u − r e ) and ∆f m f c . We remark the conditions of d |r u − r e | and ∆f m f c are mild and usually hold in practice in order to avoid aliasing and decorrelation effects in FDA transmission.
We also remark that (15) is actually a special case of (19) as θ e = θ u . Therefore, the FDA beamforming strategy possesses discrimination capabilities in both direction and range.

C. Two-Stage Algorithm
We summarize the two-stage algorithm for the FDA beamforming based SRM problem in Table 1. In the first stage, we optimize the frequency offsets across the antennas to rotate the channels of LU and Eve as orthogonal as possible. Specifically, for the identified special cases, the optimal frequency offsets can be obtained; for the general case, however, we develop an iterative algorithm utilizing the block successive upper-bound minimization (BSUM) method to find a solution for frequency offsets with stationary convergence guarantee [16]. This is not covered in this paper due to space limitation. The details can be found in the journal version [16] (submitted to IEEE TIFS). In the second stage, the beamformer is optimized based on the given frequency offsets according to (13). k is the smallest integer such that ∆f > 0; (4) if ∆F ≥ (M − 1)∆f , output f m = fc + (m − 1)∆f , ∀m; (5) else, find a stationary solution f m , ∀m [16]; (6) [Stage 2] Optimize the beamformer Output the secrecy rate R s (f ) = log[1 + λ Σ (f )].
We should mention that the two-stage algorithm is done in one shot and there is no alternating between the frequency and beamformer optimization. With each step being solved analytically, the two-stage algorithm can be performed efficiently.

IV. SIMULATION AND CONCLUSION
We consider an LOS mmWave communication system operating at f c = 60GHz. The system consists of an M -antenna transmit array, a single-antenna LU and a single-antenna Eve. We assume −100dBm receive noise power for LU and Eve, i.e., 10 log(σ 2 ) = −100dBm. The signal attenuation factor a(r) is determined by the free-space path loss formula of radio wave propagation, i. where F f c is transmit frequency in megahertz (MHz), and r is the range in kilometer (Km). We fix the LU's coordinate  Table 1); (2) the pure beamforming algorithm, which employs the optimal beamformer of (13) with fixed f = f c 1, i.e., ∆f m = 0, ∀ m; (3) the pure FDA algorithm without beamforming, where the array transmits with the fixed beamformer w = P/M ·1, and the stepped frequency offset ∆f is determined by performing one-dimension search in the range of [0, ∆F ].
The secrecy rate comparison for different antenna numbers M is illustrated in Fig. 3, where we also plot the theoretical secrecy rate upper bound log[1 + a 2 (ru)P M σ 2 ]. When the LU and Eve are directionally aligned, the pure beamforming fails. Performing frequency offsetting yields some positive secrecy rate. However, there is no guarantee that the improved secrecy rate can be achieved without optimizing the beamformer. The proposed FDA beamforming algorithm always achieves the secrecy rate upper bound by jointly optimizing the frequency offsets and the transmit beamformer.
Next, we fix the Eve's range as r e = 500m and change its angle θ e . The secrecy rate comparison is illustrated in Fig. 4, where we used the stepped frequency offset ∆f in Proposition 2 in the FDA beamforming algorithm. As θ e departs from θ u , the channels associated with LU and Eve may span different subspaces. Therefore, the pure beamforming algorithm begins to work gradually. The pure FDA algorithm behaves unstably, depending on the FDA beampattern shape. The proposed FDA beamforming algorithm exhibits a stable performance, which achieves the secrecy rate bound regardless of θ e .
In summary, an innovative FDA beamforming approach is proposed for the directionally-aligned PHY security problem. Specifically, we aim to maximize the secrecy rate by optimizing the frequency offsets and the transmit beamformer jointly. After providing some insights to the FDA beamforming, we design a two-stage algorithm to solve the challenging problem. In our solution development, we further identify some special cases, under which the SRM problem can be optimally solved in closed form.