Constant Reflection Attenuation Constraint for Incoming Signals on Metasurface in Positional Modulation Design

Metasurface based positional modulation design has been introduced recently, where a given modulation pattern can only be received at certain desired positions by the proposed method. However, the magnitude of weight coefficient of each element on metasurface varies from one other, representing an attenuation of the amplitude of incoming signal in different degrees. In this paper, a constant reflection attenuation constraint for incoming signals is proposed for the first time, and the proposed method can be extended with a post-processing process to the ideal case where there is no reflection attenuation.


I. INTRODUCTION
With the rapid development of wireless network technology, the role of information security becomes more important than ever. In recent years, physical layer security technology has attracted great attention. As one of the technologies, directional modulation (DM) has been studied widely [1]- [18]. However, one of the DM problem is that when eavesdroppers are in the same direction as the desired receiver, their modulation patterns are similar. Then, with a high sensitivity receiver, the private signals transmitted only for the desired receiver can be cracked by eavesdroppers. To solve the problem, DM is extended to positional modulation (PM) where a given modulation pattern can only be received at certain desired positions. In [19], [20], frequency diverse antenna arrays have been used to achieve PM due to its consideration of distance between the transmitter to receivers. For phased antenna array designs, PM can be achieved by exploiting multi-path effect of signals. In [21], a reflecting surface was proposed, followed by multiple antenna arrays design in [22].
Recently, to further increase the degree of freedom in PM design, metasurface has been introduced as a replacement of reflecting surface [23]. As a newly proposed material, The associate editor coordinating the review of this manuscript and approving it for publication was Qilian Liang . metasurface has been applied quickly in wireless communications [24]- [28]. However, to our best knowledge, the magnitude of each element on metasurface varies from one another, representing an attenuation of the amplitude of incoming signal in different degrees. In this paper, a constant reflection attenuation constraint for incoming signals is proposed for the first time, and the proposed method can be extended with a post-processing process to the ideal case where there is no reflection attenuation.
The remaining part of this paper is structured as follows. A review of metasurface based PM design is given in Sec. II. A constant reflection attenuation constraint for incoming signals on metasurface for PM design is presented in Sec. III. Design examples are provided in Sec. IV, followed by conclusions in Sec. V.

II. REVIEW OF METASURFACE BASED PM DESIGN
A Q-element metasurface with an N -element linear antenna array employed for positional modulation design is shown in Fig. 1. The vertical distance between the metasurface and the transmitting array is represented by H . Weight coefficient for the n-th antenna is represented by w n (n = 0, 1, . . . , N − 1), and coefficient for the q-th unit on the metasurface is represented byw q (q = 0, 1, . . . , Q − 1). The distance between antennas is denoted by d n , and the distance between units on the metasurface is denoted by x q . L and E represent the desired receiver and eavesdropper, respectively. In the two-ray model, θ represents the direct angle from the transmitter to the receiver with the distance D 1 , while ζ and φ represent the direct angle from the transmitter to the metasurface, and the direct angle from metasurface to the receiver, respectively. The steering vectors for the three direct paths are functions of θ , ζ and φ, respectively, given by The weight vector for the transmitting array can be represented by and the counterpart for the metasurface is Then, the beam response is where · represents element-wise multiplication.
To have M symbols in the modulation pattern for PM design, the following vector is constructed, representing weight coefficients of the antenna array for the m-th symbol Moreover, without loss of generality, assume r desired locations and R − r eavesdropper locations in the design; then, based on these locations, we have the corresponding transmission angles θ k , ζ k and φ k for k = 0, . . . , r − 1, r, . . . , we can classify beam responses for desired locations p m,L and beam responses for eavesdroppers p m,E , respectively [23].
To keep one set ofw available for all M symbols, the following matrices are formulated whereW = diag(w) and diag represents vector diagonalization.
Note that the metasurface has no amplifying function, then Therefore, the corresponding PM design can be given by

III. CONSTANT REFLECTION ATTENUATION CONSTRAINT FOR INCOMING SIGNALS ON THE METASURFACE
Based on formulation (8), magnitude of the q-th weight coefficient on the metasurface can be any value but no larger than 1 (q = 0, 1, . . . , Q − 1), representing an attenuation of the amplitude of the incoming signal on each element of the metasurface in different degrees. Therefore, in this section, we introduce a constraint below to keep a constant attenuation for incoming signals on the metasurface, and it can be extended to the ideal case where there is no reflection attenuation Then, the PM design with the constant reflection attenuation constraint is given by However, due to the equality constraint (9), formulation (10) is non-convex, resulting in an un-solvable problem by existing optimisation tools. Therefore, in the work, we borrow the MinMax method to keep the same magnitude value for all symbols [29]. Then, constraint (9) can be replaced by where || · || ∞ represents l ∞ norm. Based on it, formulation subject to W H S L + W HŜ LWSL = P L ||w|| ∞ ≤ 1 min ||w|| ∞ .
To our best knowledge, in previous PM designs the formulation has either multiple sets of variables or multiple objective functions, but none of the PM designs consider both scenarios. For the above formulation (12), multiple sets of variables W andw, multiple objective functions min||[Y (1, :), Y(2, :), . . . , Y(M , :)]|| 2 and min ||w|| ∞ are included in the design. Here, we consider a two-phase method to solve the problem, and it is available for all frequencies.
Phase 1: we solve the problem of multiple cost functions. We introduce a parameter µ where µ is a trade-off factor ranging from 0 to 1 to combine multiple cost functions into one. Based on it, the cost function in (12) Phase 2: we solve the problem of multiple sets of variables. Similar to [23], an alternating optimization method can be adapted, with the process below 1) Randomly generate the vectorw with all magnitudes the same as a given value. 2) With the providedw, W in (14) can be optimised. 3) Set W as a given value,w in (14) can be optimised. 4) Repeat 2) and 3) until the cost function in (14) converges. The above problem (14) can be solved by the CVX toolbox in MATLAB [30], [31].
Note that all magnitudes of weight coefficients of the metasurface calculated by the above alternating optimization method cannot be set as a pre-defined value σ . Therefore, we introduce a post-processing process. By keeping the phase while increasing or decreasing the magnitude uniformly, the weight coefficients can be satisfied for all required attenuations. The process is given below: 1) Set the parameter ρ equals to the angle ofw (ρ = w).
2) Setw as the new value with the pre-defined magnitude σ no larger than 1 and phase shift ρ (w = σ e jρ ).
The desired response at the desired location obeys the QPSK modulation mode, while the magnitude is down to 0.2 at eavesdropper locations with a random phase shift. The signal to noise ratio is set at 12 dB at the desired location, with the same level of noise at other locations. µ = 0.2 is selected for the trade-off factor by trial and error. Moreover, all reflection   (14). attenuation of units on the metasurface are required to be the same, and there is no attenuation reflection for the ideal case (σ = 1). The resultant beam and phase patterns for the eavesdroppers based on the PM design without the constant reflection attenuation constraint for incoming singals on the meta-    un-satisfactory of the design requirement. The corresponding weight coefficients are shown in Table 1. For the PM design with the constant reflection attenuation constraint in (14), reflection attenuation for all elements on metasurface are shown in Fig. 4. Here we can see the same reflection attenuation 0.4967 are achieved for all elements, with the corresponding weight coefficients shown in Table 2, demonstrating the effectiveness of the proposed design. Fig. 5 shows the corresponding cost function difference in (14), representing the achievement of the cost function convergence.
For the requirement of no reflection attenuation in the ideal scenario (σ = 1),w calculated by (14) is post-processed, where the angle ofw is kept with the magnitude set at σ = 1, as shown in Fig. 6. Then, with the neww shown in Table 3, the beam pattern and phase pattern are shown in Figs. 7(a) and 7(b). Fig 8 shows the bit error rate (BER) of the proposed design, where the BER of the desired location is the lowest and down to 10 −5 , but it is fluctuated at un-desired locations, demonstrating the achievement of the PM design.

V. CONCLUSION
In this paper, a constant reflection attenuation constraint for incoming signals on the metasurface is proposed for the first time, and the proposed method can be extended to the ideal case with a post-processing process where there is no reflection attenuation. As shown in the given figures, with the proposed constraint, reflection attenuation of all elements on the metasurface can be constrained to a pre-defined value, with a satisfactory of the PM design.