Design of binary sequences with low PSL/ISL

In this paper the long standing major challenge of designing binary sequences with good (aperiodic) autocorrelation properties in terms of Peak Sidelobe Level (PSL) and Integrated Sidelobe Level (ISL) is considered. The problem is formulated as a bi-objective Pareto optimization forcing the binary constraint at the design stage. An iterative novel FFT-based approach exploiting the coordinate descent method is devised to deal with the resulting optimization problem which is non-convex and NP-hard in general. Simulation results illustrate that the proposed algorithm can outperform some counterparts providing sequences with desirable PSL as well as ISL.


I. INTRODUCTION
Binary sequences with low Peak Sidelobe Level (PSL) and Integrated Sidelobe Level (ISL) in aperiodic autocorrelation function have wide applications in active sensing and communication systems [1], [2] being their implementation quite simple. In radar range compression, low PSL binary codes are employed to avoid masking of weak targets in range sidelobes of a strong return [3], [4]. Besides, sequences with low Integrated Sidelobe Level (ISL) are exploited to mitigate the deleterious effects of distributed clutter returns close to the target of interest [5]. Some well-known binary sequences are the Barker codes, Msequences, Gold codes, Kasami codes, and Minimum Peak Sidelobe (MPS) sequences. The Barker sequences shares excelent autocorrelation properties but, unfortunately, are limited to length 13. M -sequences, are renowned for their ideal periodic autocorrelation function but have no constraints/guarantees on the sidelobes of their aperiodic autocorrelation function; hence, they are almost impractical in radar applications (similarly Gold codes and Kasami sequences) [1]. Unlike the case of the periodic correlation, it is not possible to construct sequences with an exact impulsive aperiodic autocorrelation. Therefore, a brute-force approach to obtain good sequences is to perform an exhaustive search, viable especially when the alphabet size is small, i.e., binary case. In this respect, MPS sequences are the best binary codes in terms of PSL (known up to length 105) which are obtained via global search through some supercomputers; a summary of the best known binary sequences is presented in [6]. When the code length increases, it becomes almost impossible to perform the global search. In these situations, derivation of analytical methods to design optimal or nearly-optimal sequences are valuable. To this end, CAN algorithm [7], which minimizes an objective almost equivalent to the ISL, provides high quality continuous phase sequences. However, when phase quantization is implemented, the performance of CAN gets worse especially when the constellation size is small. In the case of discrete phase code design, three important algorithms attempting to minimize the ISL have been proposed in [8]- [10]. The CANARY [8] (which is an extension of the CAN algorithm) proposes an adhoc method for designing complementary sets of sequences. The Iterative Twisted appROXimation (ITROX) [9] method can be used to obtain sequences (or complementary sets of sequences) possessing good periodic (ITROX-P) or aperiodic (ITROX-AP) correlation properties with continuous/discrete phases. The MBIQ&L algorithm [10] provides an effective approach to design continuous/discrete phase-only modulated waveforms sharing a desired range-Doppler response. Notice that the The MBIQ&L and ITROX are highly-computational demanding and useful to build code libraries. In this paper, we propose an attractive iterative method to design binary sequences jointly optimizing PSL and ISL according to a Pareto multi-objective optimization framework 1 . It is worth observing that, the synthesis of optimized binary codes in terms of PSL fills a relevant gap in the open literature and this is indeed the main technical contribution of this study 2 . The problem is formulated as a bi-objective optimization where a binary constraint is forced at the design stage. To tackle the resulting non-convex and, in general NPhard problem, an iterative procedure based on the Coordinate Descent (CD) method is introduced. In each iteration of the proposed algorithm, the solution of a non-convex min-max problem is handled via a DFT-based procedure. Additionally, the selected weighted sum of the ISL and PSL based metrics decreases with iterations until convergence. The rest of this paper is organized as follows. Section II presents the problem formulation. In Section III, the CD-based method is devised together with a technique aimed at solving the optimization problem involved in each iteration. Section IV is devoted to numerical examples. Finally, concluding remarks are given in Section V.

A. Notation
We adopt the notation of using bold lowercase letters for vectors and bold uppercase letters for matrices. The transpose, the conjugate, and the conjugate transpose operators are denoted by the symbols (·) T , (·) * , and (·) H respectively. The l pnorm of a vector x is denoted by x p . The letter  represents the imaginary unit (i.e.,  = √ −1), while the letter i often serves as index. For any x ∈ R, |x| and arg(x) represent the modulus and the argument of x, respectively. The abbreviation "s.t." stands for "subject to".

II. PROBLEM FORMULATION
Let x = [x 1 , x 2 , · · · , x N ] T ∈ R N be the transmitted fasttime radar binary code vector with N being the number of coded sub-pulses (code length). The autocorrelation function associated with x is defined as and represents the output of the matched filter to x when x is the input signal. The PSL and ISL metrics, are commonly used to design waveforms with "good" autocorrelation properties [1] and are formally defined as PSL = max{|r k |} k=N −1 k=1 and ISL = N −1 k=1 |r k | 2 , respectively. This paper is focused on the design of binary sequences considering simultaneously the PSL and the ISL as performance indices. From an analytical point of view the problem can be formulated as the following constrained bi-objective optimization, where In a multi-objective optimization framework, usually a feasible solution that minimizes all the objective functions simultaneously does not exist [12]. Accordingly, the goal is to find the Pareto-optimal solutions to (2) that is in general a formidable task. A viable means to obtain the above solutions is the scalarization technique which exploits as objective a specific weighted sum between f 1 (x) and f 2 (x). Specifically, defining the function f θ (x), parameterized in the Pareto weight θ ∈ [0, 1], scalarization leads to the design problem Remarkably, P θ reduces to pure ISL (PSL) minimization setting θ = 0 (θ = 1). Moreover, for any θ, an optimal solution to (4) is a Pareto-optimal point to Problem (2) (see [13]- [15] and references therein for details).

III. CD-BASED RADAR CODE OPTIMIZATION
This section introduces an iterative algorithm based on the CD minimization procedure [16] (also known as alternate optimization [17]) to sequentially optimize the objective over one variable keeping fixed the others. Otherwise stated, according to the CD approach, the minimization of a multivariable function is pursued optimizing it along one direction at a time [16], [18]. With reference to (4), at each iteration, a code entry is selected as variable to optimize leading to the following problems at step n + 1 indicates the remaining code entries, and Thus, denoting by x d,n+1 the optimal solution to P θ d,x (n) , the optimized radar code at step n + 1 is , · · · of radar codes are obtained iteratively. A summary of the proposed approach can be found in Algorithm 1. Notice that, the monotonic property of the CD technique along with the fact that the objective function is bounded (from below) are sufficient to prove the convergence of the sequence of objective values. It is also worth pointing out that the Maximum Block Improvement (MBI) updating rule 3 [19] can be used in place of the cyclic one (actually involved in Algorithm 1) to ensure the convergence of the algorithm to a stationary point. In practice, a final optimized code can be obtained refining the solution provided Algorithm 1 through the MBI-modification. To proceed further, let us make explicit the functional depen- 1], and required improvement ; Output: Optimal solution x ; 1) Initialization.
, and go to the step 2; 4) Output.
• Set x = x (n) . 3 The MBI method is an iterative algorithm known to achieve excellent performance in the maximization of real polynomial functions subject to spherical constraints [10]. It is proved that any cluster point of the sequence produced by the MBI method is a stationary point for the considered optimization problem [19]. dence of the objective function in P θ d,x (n) , i. e., f θ (x d ; x where 1 A (.) denotes the indicator function of the set A = {1, 2, · · · , N }, i.e., 1 A (α) = 1 if α ∈ A, otherwise 1 A (α) = 0. Defining the real coefficients, a dk = x (n) i+k , the autocorrelation function with the explicit x d -dependence can be written as Thus, the optimization problem P θ d,x (n) can be recast as, which is a non-convex constrained min-max problem. In the next subsection, an efficient algorithm to tackle P θ d,x (n) is derived. This procedure paves the for the design of arbitrary discrete phase codes.

A. Binary Code Design
In this subsection, an efficient procedure to solve P θ d,x (n) is developed exploiting Discrete Fourier Transform (DFT). To this end, notice that in terms of φ d = arg (x d ) ∈ {0, π}, P θ d,x (n) can be recast as, where The following lemma provides a key result to tackle Problem (9).
where DFT(ζ dk ) is the two points DFT of the vector ζ dk and the square modulus is element wise. Now, defining the matrix U ∈ R (N −1)×2 whose kth row is the optimal solution to P θ d,φ d is given by where i = arg min  (11), the optimal phase code entry can be efficiently computed as x d = e φ d using DFT. The proposed approach is reported in Algorithm 2. Remark 1. Algorithm 2 needs the evaluation of N − 1 different two points DFTs. Therefore (computed the parameters) the computational complexity order is O(N ) [20].

Algorithm 2 Binary Code Entry Optimization
Input: Initial code vector x (n) , code entry d and θ; Output: Optimal solution x d ; 1) Set for all k = 1, . . . , N − 1

B. Algorithm Initialization
The solution obtained via the designed method depends evidently on the initial sequence. As a result, the development of a heuristic approach that can be used to provide high quality starting points is valuable. To this end, recall that the minimization of the l p -norm of the autocorrelation vector [r 1 , r 2 , . . . , r N −1 ] allows to trade-off ISL and PSL values of the devised sequence as the value of p increases [21]- [23]. Besides, the PSL coincides with the limit as p → ∞ of the autocorrelation vector l p -norm. According to the above considerations, a heuristi procedure to obtain binary codes with low autocorrelation l p -norm is introduced. In particular, with reference to the PSL metric, a start-stop procedure involving a sequence of l p -norm minimization problems with increasing value of p, p 1 < p 2 < . . . < p e say, is employed similarly to [21]. Specifically, the algorithm is initialized with a binary random sequence and the l p -norm minimization starts with p = 2, i.e., p 1 = 2. Then, p is set to p 2 and the algorithm starts with the solution obtained for p = p 1 , and so on. In general, the l p -norm minimization problem for binary codes can be formulated as To tackle H p the algorithm proposed in [24] is exploited, where each variable block corresponds to one code entry and the surrogate function of [21] is adopted. Specifically, at step n + 1 of the iterative procedure, the following optimization problem is considered, can be cast as Hence, using Lemma III.1 and considering the definition of ν dk in (10), the optimal x d can be efficiently obtained as

IV. PERFORMANCE ANALYSIS
This section is devoted to the performance analysis of the proposed algorithm for Binary Code Design (BCD) exploiting PSL and ISL as figure of merits. For comparison purposes the behavior of the sequences devised via ITROX-AP [9] are reported too. The considered procedures are initialized using the same set of 5 random binary starting codes (drawn from a uniform distribution over the set of the feasible sequences). Hence, the best obtained objective value is reported and the resulting sequence picked up. Finally, the stopping criteria |obj(x (n) ) − obj(x (n−1) )| ≤ 10 −5 is used to terminate all the algorithms.

A. PSL Minimization
In this subsection, the ability of the proposed algorithms to synthesize low PSL sequences is assessed. To this end, the Pareto weight is fixed to θ = 1 and the sequence of p-values for the selection of the initial starting point is 2, 2 2 , 2 3 , · · · , 2 13 , i.e., p i = 2 i , i = 1, . . . , 13. In Fig. 1, the PSL versus the code length N of BCD and ITROX-AP are reported. To highlight the quality of BCD algorithm also the PSL of MPS sequences, obtained via exhaustive search up to the length of 105, is shown in the figure. The plot clearly illustrates the effectiveness of our approach. Indeed, BCD significantly outperforms ITROX-AP. Besides it provides a PSL quite close to the global optimum of MPS sequences but with a much lower computational complexity and without restrictions to the maximum code length. This last feature is particularly appealing since the higher N the higher the pulse compression. Interestingly, BCD provides in some circumstances the global optimal solution (see in Fig. 1 the points where BCD and MPS coincide). In Fig. 2 the autocorrelation function devised via BCD and ITROX-AP for sequence length 126 is displayed. Therein, the PSL of BCD is equal to 8 whereas the PSL of ITROX-AP is 12 which further confirms the effectiveness of the new framework.

B. ISL Minimization
The performance assessment of BCD for ISL minimization is now considered. In this case, θ = 0 and the initialization procedure in Subsection III-B is not performed. Fig. 3 shows the ISL versus the code length N for BCD and ITROX-AP. The result highlights that BCD outperforms ITROX-AP with a maximum ISL gain of 2.25 dB. Remarkably, BCD also provides an ISL close to that of the MPS sequences. In Fig. 4, the autocorrelation function devised via BCD and ITROX-AP for sequence length 126 is reported. In this specific case, BCD provides a ISL-gain over ITROX-AP of 2.1 dB.

C. Pareto-Optimal Solution
In this subsection, the impact of the parameter θ on the designed code is assessed. Table I reports the PSL and the ISL of the solutions obtained via BCD assuming N = 512 and θ ∈ {θ 1 , . . . , θ 4 } with θ 1 = 1, θ 2 = 0.7, θ 3 = 0.3, and θ 4 = 0. The starting sequence used at θ = θ i is the code optimized at θ = θ i−1 ; also, at θ = θ 1 the heuristic approach of Subsection III-B is used. As expected, θ trades-off ISL and PSL values. Specifically, the higher θ the better the PSL and the worst the ISL, that is a classical feature of bi-objective Pareto curves. Otherwise stated, any solution is a "Pareto-optimal" point.

V. CONCLUSION
The synthesis of binary codes exhibiting good aperiodic correlation features has been addressed. Specifically, PSL and ISL have been adopted as performance metrics and the design problem has been formulated as a bi-objective optimization. The non-convex and, in general, NP-hard problem resulting from scalarization is handled via a novel iterative procedure based on the CD method and an efficient DFTbased procedure. Finally, a heuristic procedure based l pnorm minimization has been introduced to suitably initialize the new convergence-ensured algorithm. Simulation results have illustrated the effectiveness of the new BCD algorithm. Specifically the proposed method can outperform ITROX-AP with reference to both PSL and ISL.