Directional Modulation Design Under a Constant Magnitude Constraint for Weight Coefficients

Directional modulation (DM) as a physical layer security technique has been studied from many different aspects recently. Normally all existing designs based on antenna arrays lead to varying weight coefficients for different antennas and for different signal symbols, which poses a particular challenge for feed circuits design in analogue implementation. In this paper, to reduce the implementation complexity, a constant magnitude constraint is proposed for the first time, and the resultant non-convex constraint can be modified to a convex form so that the problem can be solved conveniently by existing convex optimisation toolboxes. Design examples are provided to show the effectiveness of the proposed design.


I. INTRODUCTION
With the fast development of the Fifth Generation (5G) network, communication has been more important than ever [1], [2]. As physical layer security technique to keep known constellation mappings in a desired direction or directions, while scrambling them for the remaining ones, directional modulation (DM) has been studied widely [3]. In [4], a four-element reconfigurable antenna array was designed, where the DM design was achieved by changing elements for each symbol. In [5], phased antenna array was employed for DM, with an individual tailor-made feed circuit (including phase shift and amplitude change) prepared for each antenna. Compared with a given antenna array design, to further reduce the number of antennas, DM design was extended to sparse antenna arrays [6]. To overcome the inherent limitation of DM, where eavesdroppers and the desired users share the same received signal when they are in the same spatial direction of the antenna array, two positional modulation (PD) designs were proposed, with one based on a reflecting surface [7], and the other employing multiple antenna arrays [8]. Both designs are based on the idea that if the received signal is a combination of signals from different The associate editor coordinating the review of this manuscript and approving it for publication was Qilian Liang . paths with different directions, then signals at these locations can be distinguished. To increase the channel capacity of DM, two antenna array structures were proposed recently. One uses a crossed-dipole antenna array [9], where two DM signals with orthogonal polarisations are transmitted in the same direction. The second one employs multiple frequencies, leading to an orthogonal frequency division multiplexing (OFDM) type structure based on the inverse Discrete Fourier Transform (IDFT) [10], [11]. A method named dual beam DM was introduced in [12]. Different from the traditional design where inphase and quadrature (IQ) components of signals are transmitted by the same antenna, dual beam DM design transmits these two components by different antennas. In [13], directional antennas were used in the design instead of isotropic antennas, and a narrower low BER range was achieved. In [14], the BER was employed for DM transmitter synthesis by linking the BER performance to the settings of phase shifters. A pattern synthesis approach was presented in [15], [16], where information pattern and interference patterns are created together to achieve DM, followed by an eight-element time-modulated antenna array in [17], an artificial-noise-aided zero-forcing synthesis approach in [18], and a multi-relay design in [19].
Recently, the introduction of artificial noise (AN) has further advanced the directional modulation technology. AN can VOLUME 7, 2019 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ be categorised into 'static' AN and 'dynamic' AN. Static AN results in scrambled but fixed constellation points in undesired directions, while dynamic AN keeps the scrambled points change with time. For the construction of AN, two methods were introduced. One is the orthogonal vector method [20], [21], where the added AN vector is orthogonal to the steering vector of the desired direction. The other one is the AN projection matrix method [22], [23], where by designing an artificial noise projection matrix, the AN vector is projected into the zero space of the derivative of the desired direction. However, the resultant weight coefficients for existing DM designs vary with different antennas, in particular in their magnitudes. When implemented by analogue circuits, both magnitude and phase responses of the feed circuits will be different for different antennas and different symbols, which will increase the implementation complexity of the whole system. To reduce this complexity, in this paper, a constant magnitude constraint is proposed for the DM design, and the resultant non-convex optimization problem is then transformed into a convex one to facilitate its solution. In general, the resultant weight coefficient magnitude is the same for all antennas for the same symbol. When all the symbols have the same magnitude, such as in phase shift keying (PSK) modulation, the weight coefficient magnitude will be the same for all antennas and symbols, further reducing the implementation complexity of the whole system.
The remaining part of this paper is structured as follows. A review of DM design based on narrowband linear antenna arrays is given in Sec. II. The constant magnitude constraint for weight coefficients is introduced and a solution to the constrained design problem is derived in Sec. III. In Sec. IV, design examples are provided, with conclusions drawn in Sec. V.

II. REVIEW OF DM DESIGN BASED ON NARROWBAND LINEAR ANTENNA ARRAYS A. NARROWBAND BEAMFORMING
A narrowband linear antenna array for transmit beamforming is shown in Fig. 1, which has N omni-directional antennas with the corresponding weight coefficient w n (n = 0, 1, . . . , N −1) and spacing d n for n = 1, . . . , N −1 between the first antenna to its subsequent antennas. The spatial angle θ ∈ [0 • , 180 • ]. The steering vector of the array as a function of angular frequency ω and spatial angle θ , is given by where {·} T is the transpose operation, and c is the speed of propagation. For a uniform linear array (ULA) with a half-wavelength spacing (d n −d n−1 = λ/2) between adjacent antennas, the steering vector can be simplified to s(ω, θ) = [1, e jπ cos θ , . . . , e jπ(N −1) cos θ ] T .
The corresponding weight coefficients can be put together as a vector w, Then, the beam response of the array can be given by where {·} H represents the Hermitian transpose.

B. DM DESIGN BASED ON THE ABOVE STRUCTURE
The aim of DM design is to keep known constellation mappings in a desired direction or directions, while scrambling them for the remaining ones. To achieve it, we need to find the corresponding sets of weight coefficients for all symbols.
Here, for M -ary signaling, such as multiple phase shift keying (MPSK), there are M sets of desired array responses p m (θ ), with the corresponding weight vector Each desired response p m (ω, θ) can be classified into one of the two regions: the mainlobe response p m,ML , and the sidelobe response p m,SL . Here we assume r points θ 0 , θ 1 , . . . , θ r−1 are sampled in the mainlobe, and R−r points θ r , θ r+1 , . . . , θ R−1 in the sidelobe. Then, the desired beam responses in the mainlobe and sidelobe regions for the m-th symbol can be represented by Similarly, the steering matrix for mainlobe and sidelobe ranges can be expressed as Note that all symbols for a fixed θ share the same steering vector. Based on the above parameters, for the m-th symbol, its corresponding weight coefficients for DM design can be obtained by solving the following problem where || · || 2 denotes the l 2 norm. The cost function is to minimise the difference between desired and designed sidelobe responses, and the equality constraint is to make sure the main lobe direction has the desired symbol response.

III. CONSTANT MAGNITUDE CONSTRAINT FOR WEIGHT COEFFICIENT
However, the magnitudes of weight coefficients in (8) are not the same for different antennas and different symbols. In other words, we need an individual tailor-made feed circuit (including phase shift and amplitude change) for each antenna for analogue implementation. To simplify implementation of a DM design, in this section, we introduce a constant magnitude constraint for weight coefficients, i.e., where δ m represents the given magnitude for each weight coefficient for the m-th symbol, so that we only need to change the phase response of the feed circuit for different antennas. The mainlobe and sidelobe regions are shown in Table 1. The DM design under the constant magnitude constraint for weight coefficient can be formulated as follows However, δ m is not arbitrary, and there is a valid range for it. To derive it, we first have forr = 0, 1, . . . , R − 1 including mainlobe and sidelobe regions.
Due to the proposed constant magnitude constraint, where the magnitudes of all weight coefficients have to be the same, i.e., (11) can be further simplified to Therefore, we have Since the maximum value of |p m (ω,θr )| where θ 0 represents the mainlobe direction with the maximum magnitude response value, we have Then, with the equality constraint |w m,n | = δ m for n = 0, 1, . . . , N − 1 from (10), the inequality (16) can be replaced by Then, the formulation in (10) For the purpose of saving the transmission power for each antenna, i.e., setting the minimum magnitude of weight coefficient, we select the minimum value of the constant magnitude Note that in practice if we need to increase the transmission power of the array to increase its communication range, we can uniformly increase the magnitude of all coefficients accordingly. Therefore, using (19) to set the minimum transmission power will not cause any additional issues in practice. Then, the formulation (18)  Note that for the above formulation to work, the maximum magnitude response in the mainlobe can only be located at one single direction (θ 0 in our case), i.e., a flat top main beam can not be achieved. We will prove this later at the end of Section III. Here, with the single maximum mainlobe direction represented by θ 0 , the formulation (20) can be simplified to Moreover, the equality constraint |w m,n | = |p m (ω,θ 0 )| N in (21) is non-convex. To solve the problem using the CVX toolbox in Matlab, we propose a new constraint to replace the equality constraint |w m,n | = |p m (ω,θ 0 )| N in (21), where || · || ∞ represents the l ∞ norm (the maximum magnitude of the entries in the vector). This inequality constraint is to set the entry with the maximum magnitude of the vector w m no greater than |p m (ω,θ 0 )| N . According to (16), if the maximum value or any other entries of the vector is less than |p m (ω,θ 0 )| N , then the constraint w H m s(ω, θ 0 ) = p m (ω, θ 0 ) cannot be satisfied. Therefore, the proposed constraint (22) The above problem (23) can be solved by the CVX toolbox in MATLAB [24], [25]. Note that if for m = 0, . . . , M −1, representing different symbols with the same magnitude, such as in PSK-type modulation schemes, then i.e. the magnitudes of coefficients for different antennas and different symbols will become the same. This will further reduce the implementation complexity of the whole DM system. Now we prove the formulation (20) only works for one desired direction with maximum magnitude response. Without loss of generality, the maximum beam response in the mainlobe direction for one symbol (m = 0) is represented by p 0 (ω, θ 0 ) = √ 2 2 + √ 2 2 i. If there are two desired directions θ 0 and θ 1 receiving the same symbol, then the equality constraints in (20) can be represented by We add up the first two constraints and have the following results 2w * 0,0 + w * 0,1 (e j2πd 1 cos θ 0 + e j2πd 1 cos θ 1 ) + . . .
However, according to (27), it should be larger or equal to 2, which contradicts the above result. As a result, the number of directions with the same maximum magnitude value cannot be larger than one. For eavesdroppers with a known direction, we can apply a corresponding constraint to make the beam response at this direction as low as possible. This constraint will not affect the proposed constant magnitude constraint.
Another note is that the proposed constraint cannot only be applied to a linear antenna array, but also a planar antenna array or a circular antenna array by changing the corresponding steering vectors.

IV. DESIGN EXAMPLES
In this section, we consider an N = 20 ULA with a half wavelength spacing between adjacent antennas. Note that the proposed constraint works well irrespective of the number of antennas. Both broadside and off-broadside designs are provided. For the broadside design, the desired direction is for symbols '00', '01', '11', '10', and a value of 0.1 (magnitude) with random phase shifts over the sidelobe regions. The constant magnitude of weight coefficient To verify the performance of the proposed design, the beam and phase patterns for the designs with and without constant magnitude constraint for weight coefficients are given. Bit error rate (BER) is calculated based on which quadrant the received complex-valued signal falls into. Here the signal to noise ratio (SNR) is set at 12 dB in the mainlobe direction, 10 6 bits are transmitted, and the same additive white Gaussian noise (AWGN) power levels for all directions are also assumed.
For the broadside design without constant magnitude constraint in (8)   with a low sidelobe level, and the phases in the desired direction follow the standard QPSK constellation mappings, but random for the rest of the angles. However, as shown in Tables 2, 3, 4     are not the same; in other words, constant magnitude for weight coefficients are not achieved.
In contrast, for the design with constant magnitude constraint in (23), Figs. 4 and 5 show the corresponding beam and phase patterns, which satisfies the DM design. The magnitudes of all weight coefficients are shown in Table 6. It can be seen that all magnitudes are equal to 0.05, the same as the given magnitude |p m (ω,θ 0 )| N = 0.5, demonstrating the effectiveness of the proposed design. The BER performance of the proposed design is shown in Fig. 6, where we can see that in 90 • (the desired direction) the value is down to 10 −5 , while at other directions it fluctuates around 0.5, illustrating the practicality of the proposed design.
For the off-broadside design with constant magnitude constraint in (23)    and phase patterns, where we can see that the main beam points to the desired direction 60 • , with a standard QPSK. The magnitudes of all weight coefficients are the same as 0.05, as shown in Table 6. The corresponding BER performance is shown in Fig. 9, again with a satisfactory result.

V. CONCLUSION
To reduce the implementation complexity, a constant magnitude constraint for weight coefficients has been introduced into the directional modulation design for the first time, and by employing the absolute value inequalities, the resultant non-convex constant magnitude constraint is transformed into a convex form so that its solution can be found conveniently by existing convex optimisation toolboxes. As shown in the provided design examples, with the proposed constraint, a constant magnitude of weight coefficients can be achieved for different antennas with a given symbol, and when all the magnitudes of the symbols are the same, we will have a constant weight coefficient magnitude for all antennas and symbols, with an even lower implementation complexity.