Designing Sequence with Minimum PSL Using Chebyshev Distance and Its Application for Chaotic MIMO Radar Waveform Design

Designing Sequence with Minimum PSL Using Chebyshev Distance and its Application for Chaotic MIMO Radar Waveform Design Hamid Esmaeili Najafabadi Student Member, IEEE Mohammad Ataei, and Mohamad F. Sabahi Member, IEEE

Abstract—Controlling peak side-lobe level (PSL) is of great with impulse-like autocorrelation. In general, there exist two importance in high-resolution applications of multiple input major merits to measure the resemblance of a sequence with multiple output (MIMO) radars. In this paper, designing se- impulse: integrated side-lobe level (ISL) and peak side-lobe quences with good autocorrelation properties are studied. The PSL of the autocorrelation is regarded as the main merit level (PSL). Minimizing the first was the topic of several recent and is optimized through newly introduced cyclic algorithms, publications [15], [24]–[26]. However, much less effort is namely; PSL Minimization Quadratic Approach (PMQA), PSL made on minimizing the second. On the other hand, generating Minimization Algorithm, the smallest Rectangular (PMAR) and a set of sequences with minimized PSL is of great importance PSL Optimization Cyclic Algorithm (POCA). It is revealed in high-resolution applications of MIMO radars [14]. In fact, that minimizing PSL results in better sequences in terms of autocorrelation side-lobes when compared with traditional in- for a radar engineer autocorrelation corresponds to the output tegrated side-lobe level (ISL) minimization. In order to improve of the matched filter of the radar system. Generally speaking, the performance of these algorithms, fast-randomized Singular the peak of the side-lobes corresponds to the falsely detected Value Decomposition (SVD) is utilized. To achieve waveform objects (false alarms), while high peak side-lobes result in design for MIMO radars, this algorithm is applied to the masking of the low signature targets next to high signature waveform generated from a modified Bernoulli chaotic system. The numerical experiments confirm the superiority of the newly targets. Hence, in order to have low false alarms, the peak of developed algorithms compared to high-performance algorithms the side-lobes should be lowered as low as possible. In this in mono-static and MIMO radars. regard, the main contribution of this article is to address the Index Terms—MIMO radar waveform design; Peak side-lobe problem of PSL minimization of a sequence and acclimatizing level; Chebyshev Distance. it to MIMO radars through chaotic waveforms as the initial sequences. In [26], authors have solved the problem of ISL mini- I.INTRODUCTION mization for an unimodular sequence (i.e. all elements have AVEFORM design is a traditional problem in radar and unit absolute value). They derived several cyclic algorithms: W communication systems. Classically, in these systems, CAN, CAD, CAP, and WeCAN. In [17], by generalizing some a matched filter is applied to detect the target or message in methods given in [26], several algorithms are developed to the presence of a background white Gaussian noise (WGN). generate waveforms appropriate in MIMO radars applications. In this context, a well-transmitted waveform should have low In addition to autocorrelation, they minimized the cross- autocorrelation side-lobes for preventing false results in detec- correlation between the generated waveforms. In [35], the tion. Study of sequences with good autocorrelation properties, waveform design in presence of clutter and white Gaussian with respect to radar applications in mind, is a classic topic noise is assessed. Accordingly, a cyclic algorithm is developed [1], [3], [4], [9], [10], [12], [28], [33]. In fact, the related to maximize signal-to-clutter-plus-noise ratio (SCNR) under literature covers a wide array ranging from bi-phase and poly- the constant modulus constraint. In [8], through cyclic opti- arXiv:2010.03674v1 [eess.SP] 7 Oct 2020 phase to more recent chaotic and algorithmic methods. For mization, a computationally attractive algorithm is introduced the bi-phase Barker [2], for poly-phase Golomb [12], Frank for the synthesis of constant modulus transmit signals with [11] and Chu [6], for chaotic Lorenz [16] and for algorithmic good auto-correlation properties for MIMO radars. In a recent method cyclic algorithm (CA) [9] and majorization minimiza- work [24], the problem of minimizing lp (for 2 ≤ p < ∞) tion (MM) [25] methods are just a few to be enumerated. In norm on the side-lobes of autocorrelation is addressed. They all the aforementioned references, a good sequence is the one majorized this problem by an l2 problem, and by solving it, they minimized the lp-norm of the autocorrelation side-lobes. Copyright (c) 2015 IEEE. Personal use of this material is permitted. Then, by choosing large values for p, they approached l∞ However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. norm, which is indeed The PSL. However, in the case of peak DOI: 10.1109/TSP.2016.2621728 side-lobe level, their algorithm named “monotonic minimizer Hamid Esmaeili Najafabadi is currently a Ph.D. student at the Department (MM) for lp-metric”, lacks the ability to suppress a specified of Electrical Engineering, University of Isfahan, Hezar jarib St., Isfahan 8174657811, Iran. Phone: +98-3142636343; Fax: +98-3137933071; Email: part of the autocorrelation. Moreover, this minimizer actually [email protected]. does not minimize PSL, but minimizes an lp-metric, then by Mohammad Ataei and Mohamad Farzan Sabahi are with Department choosing large p it approaches PSL minimization. As noted of Electrical Engineering, University of Isfahan, Hezar jarib St., Isfahan 8174657811, Iran. Phone: +98-3137934068; Fax: +98-3137933071; Email: before, like WeCAN and CAP, they lack the ability to suppress [email protected], [email protected]. more than half of the autocorrelation side-lobes to “almost 2 zero”. A. Notations In [18], chaos is introduced to generate phase-coded wave- We use bold lower case for column vectors and bold forms for MIMO radars. In [30], authors introduced quasi- uppercase for matrices. The symbols AT and AH represent orthogonal waveforms for wideband MIMO radars application. transpose and conjugate transpose of the matrix A, respec- They have established that chaotic waveforms possess many tively. ek denotes the k-th canonical vector, a column vector desirable radar properties. Consequently, in the line of previous with zero elements, except for the k-th element, which is one. work, in [31], the selection of parameters for Lorenz system |a| indicates the absolute value of the scalar a. k . k2 denote for wideband radars is studied. In this article, the problem the Frobuinous norm of a vector or matrix. The norm k . k∞ of PSL minimization is formulated and solved. Then, by for a matrix A is defined according to k A k∞= max |Ai,j|, applying chaotic waveforms as the initial sequence to the that is, k A k∞ is the norm defined by Chebyshev distance newly developed algorithms, a set of waveforms proper for over the space of all complex valued N × M matrices. In MIMO radars is constructed. The obtained results can be appendix VIII, it is shown that this operator actually defines a summarized as follows: norm on the vector space of all complex matrices. The symbol d • Problem formulation for PSL minimization and solving ≤ is the lexicographical order or dictionary order on C. That d it is, b ≤ a if and only if, • Suppression of more than half of the autocorrelation side- Im{b} ≤ Im{a} ∨ lobes to “almost zero” (Im{b} ≤ Im{a} ∧ Re{b} = Re{a}), (1) • Better side-lobes both in terms of PSL and ISL • Ability to generate a large set of waveforms with low where, ∨ and ∧ are “logical or” and “logical and”, respectively. d d cross-correlation Accordingly, by max and min , the maximum and minimum d • Ability to deal with long sequences under the dictionary order are meant. Note that, ≤ is a total order on C. Chaotic waveforms have been and are extensively applied to II.PEAK SIDE-LOBE LEVEL MINIMIZATION radar systems [13], [20], [22], [31]. The chaotic systems’ out- Let {x }N represent the sequence to be designed. The puts are bounded, aperiodic and sensitive to initial conditions, n n=1 autocorrelation function of this sequence is: resulting in low peak to average power ratio (PAPR), low auto- N correlation side-lobes and low cross-correlation, respectively. X ∗ ∗ rk = xnx = r , k = 0, ..., N − 1. (2) Moreover, the simplicity of waveform generation in these n−k −k n=k+1 systems enables the design of a set of waveforms with high Commonly, two major merits are considered for sequence cardinality, which is appropriate in MIMO radars [30], [31]. performance in the radar; the ISL and PSL. The first is defined In addition to these properties, chaotic waveforms are noise- by, like deterministic signals; therefore, radars which apply these N−1 kinds of waveforms have a low probability of interception. X 2 ISL = |rk| . (3) Bernoulli map is a typical defining example for chaos [5]. k=1 This system is one of the simplest systems with the capability The PSL is computed by: of generating chaos. It has phase space dimensionality of two, PSL = max |rk|. (4) that is, it can produce high cardinality set of waveforms of any k6=0 N length with very low computational effort [23]. Consequently, The matrix X for the sequence {xn}n=1, is defined according by the juxtaposition of this chaotic system and PSL minimiza- to:   tion algorithm, it is possible to generate an arbitrary number x1 0 ··· 0 of sequences with low cross-correlation, low autocorrelation  . . . .   ......  low-rank and low probability of interception.     The organization of the article is as follows. The prob-  . . ..   . . . 0  lem of PSL minimization is formulated based on Chebyshev    . .  distance and solved through some novel cyclic algorithm in X =  . .  (5) xN . . x1  sectionII. Then, by enhancing the time-consuming parts of    .. . .  the algorithm, the speed performance of this algorithm is  0 . . .    improved in section III. Consequently, in sectionIV, imposing    ......  additional constraints on the problem is studied. In section  . . . .  0 ··· 0 x V, modified Bernoulli map is introduced. It is shown that by N (2N−1)×N applying our algorithm to the sequences generated by a chaotic Having defined X, the autocorrelation of the sequence system, a suitable set of waveforms for MIMO radars can be N {xn}n=1 is represented by, generated. Accordingly, in sectionVI, the proposed methods ∗ ∗  r0 r . . . r  are evaluated. Afterward, the conclusion is given. The last part 1 N−1  .. .  but not the least is the appendix. The presented appendix is to  r1 r0 . .  XH X =   . (6) reveal the fact that the Chebyshev distance indeed defines a  .   . .. .. ∗  complete normed vector space, known to the mathematicians  . . . r1  as Banach space. rN−1 . . . r1 r0 3

Considering the fact that a good autocorrelation is the one Indeed, by considering this configuration, the PSL of se- with r0 > 0 and ri = 0, i 6= 0, a sequence could be designed, lected autocorrelation coefficients rk, k = 1,...Q − 1, can where XH X =∼ NI. Accordingly, let be minimized through solving the following minimization problem, k A k∞= max |Ai,j| (7) √ i,j min k X˜ − NL k A ∞ be the norm defined by Chebyshev distance for matrix H (13) (refer to appendix for the proof that it is actually a norm). s.t. L L = I N Minimizing the PSL of the sequence {xn}n=1 is equivalent where, L is a (N + Q − 1) × Q matrix. Like any other cyclic to minimizing: algorithm, finding the solutions of X˜ and L from each other H in a cyclical manner would suffice. However, because of the k X X − NI k∞ . (8) infinity norm, it is not possible to find the L. To mitigate this It is assumed that coping this quadratic problem is unapproach- problem, here, just an approximate solution of L is sufficient. able, therefore, with an argument to be followed, minimizing In fact, because of the cyclic nature of this algorithm, it is the following equation instead is considered, √ sufficient to have an enhancement at each step. A proper k X − NL k∞, (9) approximate solution of (13) with respect to L can be obtained where the matrix L with the dimensionality of (2N − 1) × N by singular value decomposition, accordingly, first let H satisfies L L = I, and X is of the form defined in (5), H H X˜ = U 1ΣU (14) indicating that the problem of minimizing PSL is equivalent 2 be the economy-sized singular value decomposition (SVD) of to the following minimization problem, the conjugate transpose of the matrix X˜ , then, √ H L = U 2U 1 , (15) min k X − NL k∞ (10) is the solution of the Euclidean version of the problem in s.t. LH L = I (13) (see [17], [19], [26] and the references therein). Fig. 1 N×M Argument: The normed space (C , k . k∞) is a illustrates the conceptual solution of Euclidean and Chebyshev Banach space (for the proof refer to Appendix A) and topo- norm version of the problem in (13), for two dimensions. In N×M ˜ logically equivalent to the Euclidean space (C , k . k2). this figure, the 8 points for X and their projection on the Hence, the norm is a continuous function and space is com- feasible set LH L = I under the Euclidean and Chebyshev plete. Therefore, although (8) and (9) are not equivalent, they distance are illustrated. The length of the dotted and solid are almost equivalent. That is, (9) is zero if and only if (8) is lines, which connect each point and its projection on the zero. Hence, from the continuity, if the global minimum of (9) circle, describes the Chebyshev and Euclidean distance in is sufficiently small, then the sequence where (9) is minimized, this figure, respectively. Almost all other configurations of is arbitrary close to the solution of (8) . the X˜ are symmetrically equivalent to these 8 points. It is Before trying to find the solution of (9), consider the deduced from Fig.1 that the error in this approximation is N−1 less than 1/8 of the circle in all configurations. In this figure, situation where not all {rk}k=0 but some part of them is required to be made small, as considering rk, k = 1,...Q−1, the solutions of the Chebyshev distance problem (i.e. (13)), are where Q ≤ N. That is just Q − 1 first autocorrelation obtained such that by relocating the solution on the circle, the coefficients are of importance. Accordingly, Eq. (5) should triangle side with the maximum length can not be shortened. be altered properly, that is, let Note that, in 4 of the 8 configurations, the approximation   errors are zero. Besides, as the dimensionality of the problem x1 0 ··· 0 increases the approximation error occupies less proportion of  . . . .   ......  the corresponding hyper-ball. The second step in the cycle is  . .    to solve (13) with respect to {x }N for a given L. In order  . . ..  n n=1  . . . 0  to accomplish this let x be an arbitrary element of {x }N   n√ n=1   and {µ } be the corresponding elements of the matrix NL  . . .  k  . . . x1  the positions of which are the same as x in X˜ . Then, the X˜ =   . (11)  . . .  generic form of minimization in (13) becomes, x . . .   N    min max |x − µk| . (16)  .. . .  x k  0 . . .    This problem is a well-acknowledged problem known as  . . . .  the smallest enclosing circle problem, which has matured  ......    algorithms in order to find exact solution (for instance refer 0 ··· 0 x N (N+Q−1)×Q to [34]). First, note that the problem in (16) can be converted Then we have, to,  ∗ ∗  r0 r1 . . . rQ−1 min R xR, xI  .. .  q H  r1 r0 . .  2 2 (17) X˜ X˜ =   . (12) s. t. (xR − Re {µk}) + (xI − Im {µk}) ≤ R,  . . .   . . . ∗  k = 1, ..., Q − 1,  . . . r1  where, xR and xI are the real and imaginary parts of x. rQ−1 . . . r1 r0 4

4 3 4 4 3 3 2 2 2 1 1 1 7 7 7 6 6 5 6 8 5 5 8 8 connecting mind to maxd (): Euclidian distance []: Chebyshev distance Fig. 2: Conceptual illustration for comparing solutions to the smallest circle, smallest rectangular problems and the one introduced in (24) Fig. 1: Conceptual illustration for comparison of the solution of (13) with the solution of Euclidean version of it in 2 dimension in appendix B. Nevertheless, most of these solutions are com- Consider the following problem for a fixed R ≥ 0, putationally very expensive for our application. For instance, the cone optimization approach requires O(Q3.5 | log δ|) arith- min θ metic operations to find the optimal solution of (16), where δ 2 2 2 s.t. (xR − Re {µk}) + (xI − Im {µk}) − θ ≤ R (18) is a user specified parameter for accuracy. Similarly, in the best k = 1, ..., Q − 1. case scenario, the Sub-algorithm 1 requires at least O(Q2) op- Then, the problem in (18) becomes equivalent to the problem erations. Since the solution of (16) is required in “every itera- ∗ ∗ ∗ tion”, the preference is to find an approximate but fast solution. in (17) in a sense that (xR, xI ,R ) is an optimal solution ∗ ∗ Algorithm 1: PMQA (PSL Minimization of (17) if and only if (xR, xI , 0) is an optimal solution of (18) for R∗ = R (see the proof of the theorem 2 in [34]). Quadratic Approach) N Having established this equivalency, the problem in (18) can be 1 Set {xn}n=1 to an initial sequence. (Initialization) converted into a series of constrained quadratic programming 2 Constitute the matrix X˜ and compute L according 2 2 to equation (14) problems, accordingly, define z := xR + xI − θ. Then, (18) Q−1 can be reformulated as 3 For each xn constitute the sequence{µk}k=1 , 2 2 min xR + xI − z solve the smallest circle problem in (16) by fol-

s. t. − 2xRRe {µk} − 2xI Im {µk} + z (19) lowing steps 1,2,3 as in Sub-algorithm 1 2 2 2 4 Go to step 2 till some stop criterion is satisfied ≤ R − (Re {µk} ) − (Im {µk}) , k = 1...Q − 1 (e.g. k X˜ new − X˜ old k∞< ε) This problem is a quadratic programming with linear con- R straints, where, if its optimal value is zero, then is the III.ACCELERATION SCHEMES optimal value for the problem in (17), otherwise, reduce R and solve the newly obtained quadratic problem. Accordingly, In order to accelerate the algorithm, the first step is to avoid one may follow the following steps in order to get the exact the iterative Sub-algorithm 1 by some wise guess. Accord- solution of (17). ingly, instead of solving the problem in (16), the following Sub-algorithm 1: approximated problem is considered min max |x − µk|∞ , (22) 1) Start from an appropriate point e.g., x k maxd {µ } + mind {µ } wherein |a| := max{Re(a), Im(a)}. In fact, given the x = k k . (20) ∞ 2 sequence {µk}, the above mentioned problem is the smallest and compute, enclosing rectangular for this sequence. Fig. 2 depicts the p 2 2 R = max (xR − Re{µk}) + (xI − Im{µk}) (21) smallest rectangular and smallest circle problems for a random k sequence. In fact, in almost all cases the solution to the small- 2) Solve (19) and find z, x and x . R I est rectangular is a good approximation to the smallest circle 3) If x2 + x2 − z < δ then stop. Else, re-compute R in R I problem. Moreover, the smallest rectangular has a closed-form (21) and go to step 2. solution in the form, Consequently, the PSL minimization algorithm based on max{Re{µk}}+min{Re{µk}} x∗ = quadratic approach is introduced through Algorithm 1. In 2 (23) max{Im{µk}}+min{Im{µk}} addition to this approach, other methodologies for solving the +j 2 . smallest circle problem in (16) can be adopted. Two other such This solution consumes O(Q) logic and O(1) arithmetic methodologies for exact solution of the problem are introduced operations (additions and multiplications) in each iteration, 5 hence saving a considerable amount of time. Furthermore, matrix, as a rougher and more fast approach one may consider the ˆ ˆ H B = U 1ΣU 2 . (29) following approximate solution to (16), Finally, form the orthonormal matrix Uˆ , and compute U d d 1 1 max {µk} + min {µk} according to x∗ = . (24) 2 U = QUˆ It is worth mentioning that if one confines problem in (16) to 1 1 (30) real sequences then the exact solution of (16) will be of the It is known that, if the value of S is chosen large enough, then H similar form (see end of appendix B (IX) for proof.), the initial matrix X˜ can be approximated arbitrarily close max {µ } + min {µ } by (see [27], [32] and references therein), x = k k . (25) ∗ H H 2 X˜ ≈ U 1Σˆ U , (31) Consequently, the following two algorithms are obtained for 2 generating sequences with good autocorrelation shapes. In fact, as observed in the aforementioned references even for small values of S this works well. The experiments run here indicate that for S values as small as 4 suffice. Algorithm 2: PMAR (PSL Minimization Algo- By applying this technique, those versions of CA algorithm where the smallest Rectangular) rithm where SVD is involved (see, e.g. [17], [26]) can N 1 Set {xn}n=1 to an initial sequence. (Initialization) be made faster. Regarding the algorithm introduced here, 2 Constitute the matrix X˜ and compute L according i.e. the POCA, this enhancement results in the follow- to equation (14) ing algorithm, where it is named RPOCA for random- Q−1 3 For each xn constitute the sequence{µk}k=1 , ized PSL optimization cyclic algorithm. The other SVD solve the smallest rectangular problem in (22) based cyclic algorithms, like PMQA, PMAR, CA, CAP, and using (23) to achieve the new xn CAD can be dealt with similarly. To avoid unnecessarily 4 Go to step 2 till some stop criterion is met lengthening of the article, they are not developed here. (e.g. k X˜ new − X˜ old k∞< ε) Algorithm 4: RPOCA (Randomized PSL Opti- mization Cyclic Algorithm) N 1 Set {xn}n=1 to an initial sequence. Choose S Algorithm 3: POCA (PSL Optimization Cyclic Q ≤ N and set Ω to a Gaussian random matrix. Algorithm) (Initialization) N ˜ 1 Set {xn}n=1 to an initial sequence (Initialization) 2 Constitute the matrix X (equation(11)) and com- 2 Constitute the matrix X˜ and compute L according pute the low rank SVD of it using the equations to equation (14) (31) then form L according to equation (14) Q−1 x {µ }Q−1 3 For each xn constitute the sequence{µk}k=1 , find 3 For each n constitute the sequence k k=1 , the maximum and minimum of this sequence under find lexicographic maximum and minimum of this the dictionary order and set xn to arithmetic mean sequence according to (24) and set xn to their of them arithmetic mean. 4 Go to step 2 till some stop criterion is satisfied 4 Go to step 2 till some stop criterion is satisfied. (e.g. k X˜ − X˜ k < ε) new old ∞ Note that the computational cost of low-rank SVD is in Alongside the enclosed iteration, the most computationally the order of O((N + Q − 1)QS). Besides, RPOCA needs intensive part of our algorithms and those that minimize no random access to matrix X˜ , that is, it can be implemented autocorrelation related factors like CA, CAP, CAD, is SVD. simpler (see [27], [32]). In fact, in computing a low-rank SVD This is just the problem when large values of N is of version of X˜ (equation (31)), just the matrices of the size concern. Reminding aforementioned argument that even an S×Q should be fitted in the random access memory (RAM). approximate value of L works very well, fast low rank On the contrary, the conventional SVD needs all the X˜ , with SVD algorithms can be applied ( [32] and [27]) to improve dimensions of (N + Q − 1)×Q, to be fitted in RAM. the complexity costs of the algorithms. Specifically, let IV. SOLVINGTHEPROBLEMWITHCONSTRAINTS S Q ≤ N and Ω be a Gaussian random matrix of the size No additional constraint was assigned to the develope- (N + Q − 1) × S. Constitute, ment of the above algorithms. However, based on practical H Y = X˜ Ω. (26) requirements some constraints can be added to this kind where Y is of the size Q × S. Next, let of development. Such constraints have been and are being adopted by many authors, like [15], [24]–[26]. One of such Y = QR (27) requirements is the unimodularity constraint. The sequence Y Q Q×S N be the QR factorization of the matrix , such that be {xn}n=1 is called unimodular if and only if, unitary matrix. Then, form matrix, |xn| = 1, n = 1, ..., N. (32) H ˜ H B = Q X . (28) Unimodular sequences have the lowest possible peak to av- Note that B has much lower dimensions (S × (N + Q − 1)) erage power ratio (PAPR). Alternatively, one might consider compared to original X˜ , that is, SVD of this matrix is restricting PAPR itself. In order to enforce such a restriction T H ∼ computationally cheaper. Compute the SVD of this thiner define x = [x1, ..., xn] . Since in our developed X X = NI 6

x 1 H ∼ n+1 the average power is N x x = 1. Hence, in order to restrict PAPR it is sufficient to restrict peak power or equivalently xn+1 A adding the following constraints to each problem in the 1. development of previous sections. k|x|k ≤ a (33) A ∞ B wherein |.| denotes the element-wise absolute value of the complex vector x and k.k denotes the Chebyshev norm −1. A 0 x 1. ∞ − B n of x. In order to preserve brevity, we avoid rewriting the whole problems again. Nevertheless, such constraints should be added to the equations (8), (10), (13), (16), (17), (18), 0 1 − λ 1. xn −A (19),(21). This alteration, illustrates itself in the sub-algorithm 1, Eq. (21), where the equation becomes, p 2 2 R = max (xR − Re{µk}) + (xI − Im{µk}) (34) k x2 + x2 = 1 Fig. 3: Comparison of the phase space of modified Bernoulli system I R (right), with phase space of Bernoulli System (left) for unimodularity constraint and Therefore, the correlation scales up by a factor of N. Accord- p 2 2 R = max (xR − Re{µk}) + (xI − Im{µk}) (35) k ingly, the lower-bound in this case is of the form, s x2 + x2 ≤ a M − 1 I R c ≥ N . (39) for constraining the PAPR. It is interesting to mention that the max M(2N − 1) − 1 development of acceleration schemes (Sec. III) subject to the Nevertheless, all the above algorithms generate sequences above-mentioned constraints are also possible. Accordingly, with good autocorrelation. In the next section, by applying (22) should be solved subject to the these constraints. the modified Bernoulli system, an “arbitrarily large” set of min max |x − µk|∞ , sequences with low cross-correlation is generated. By com- x k bining these two approaches, one can design large set of |x| = 1 or |x| ≤ a (36) sequences with low cross-correlation and arbitrary shape of where the first constraint is for unimodularity and the second autocorrelation. is for constraining PAPR. Regarding unimodularity constraint, V. MODIFIED BERNOULLI MAP finding an approximate and fast solution can be achieved by solving it without constraint and then project it to unit Bernoulli shift map belongs to a family called piecewise circle. Accordingly, the alteration on the algorithms PMAR, linear maps, where its elements consist of a number of POCA and RPOCA can be achieved by introducing following piecewise linear segments [5], [22]. Due to its chaotic nature, intermediate step between steps 3 and 4. sequences generated from this map are highly sensitive to 3.5 Project xn to unit circle. initial conditions. Therefore, by changing its initial condition, Similarly, in the case of restricted PAPR, it is sufficient to very different sequences can be produced (see [30]). Bernoulli solve (22) without any constraints and if the solution satisfies map is renowned for its simple structure consisting of two the constraint (i.e. |x| ≤ a), mission accomplished, else the linear segments as follows: projection onto |x| = a circle will restrict the PAPR. There- ( xn fore, the alteration for algorithms PMAR, POCA and RPOCA λ 0 < xn < 1 − λ xn+1 = (40) can be made by introducing the following intermediate step xn−(1−λ) λ 1 − λ < xn < 1 between steps 3 and 4. The system defined above is commonly acknowledged as 3.5 If xn > a then project xn to the |x| = a circle. Bernoulli system in its special case of λ = 1/2, represented Remark. By introducing unimodularity constraint, a trade- in the form, off involving cross-correlation and autocorrelation are derived m xn+1 = 2 xn mod 1, x0 ∈ (0, 1) . (41) [29]. Consider the uni-power set of sequences {xn }, n = 1, ..., N, m = 1, ..., M, where The classical form of Bernoulli map given in (40) is not N appropriate to be applied here, since it is observed that, X m 2 the signals generated from this map are all positive which |xn | = 1, m = 1, ..., M. (37) n=1 result in a non-uniform high cross-correlation. To alleviate this It is known that for such a sequence the following lower-bound phenomenon, the origin of the phase space of this dynamical 1 exists. system to is changed 2 , 1 − λ , hence the following dynam- s M − 1 ical system, a modified version of Bernoulli system. cmax ≥ , (38) ( B M(2N − 1) − 1 Bxn + A, − A < xn < 0 xn+1 = B (42) where, cmax is the maximum of the correlation side-lobes Bxn − A, 0 < xn < A (consisting of all cross-correlation and all autocorrelation lags Fig. 3 illustrates the phase space of the modified Bernoulli except zero). When sequences are unimodular, each sequence system along with original Bernoulli system. Note that in PN m 2 will be of power N, (i.e. n=1 |xn | = N, m = 1, ..., M). addition to the change in the center of the phase space, the 7

 B xn POCA N1 u() t x,, B A B x A,   x  0 zn 0  n n u()() t z g t  nT  A or  n xn1   n0  B Bxn-A , 0  x n  RPOCA  A

Fig. 4: Baseband processing block diagram of MIMO radar at each antenna slopes of the lines are altered as well. In Theorem V.1, it This metric resembles the correlation level given in [24], [26]. is revealed that the modified Bernoulli system can indeed The name “Normalized Autocorrelation” is preferred here to produce chaos. avoid confusion with cross-correlation. Afterwards, the peak Theorem V.1. The modified Bernoulli system is a chaotic correlation level (PCL) is, system for B > 1. max rk PCL = 20 log . (48) 10 r Proof. In order to maintain that the one-dimensional system 0 When suppressing a specified part of the autocorrelation is xn+1 = f (xn) , x ∈ I (43) of concern, in [26] a useful metric named the modified merit is chaotic, its being of positive Lyapunov exponent, defined factor (MMF) is developed, by (see [7] and references therein) must be proved, N 2 MMF = . (49) 1 d n PQ−1 2 λ = lim log | f (x)|, (44) 2 |ri| n→∞ n dx i=1 n times The equivalent merit factor of the suppressed part of the z }| { autocorrelation is measured through this metric. Similarly, a where f n(x) f(f...(f(x))). In case of modified Bernoulli , modified peak correlation level (MPCL) is defined according system by successive substitution, to n n f (x) = B x + K, (45) max{ri|i = 1, ..., Q − 1} MPCL = . (50) is yield, where, K is a constant depending solely on B,A and r0 n. Therefore, Note that MPCL is similar to PCL when suppressing all 1 d autocorrelation side-lobes is of concern. The merits defined λ = lim log | f n(x)| = log |B|. (46) n→∞ n dx in (47) and (49) are chosen from [17], [24], [26] in order Now, by selecting B > 1, the system’s Lyapunov exponent to compare newly developed algorithms with theirs on their becomes positive. footings. The merits in (48) and (50) are defined here to minimize peak side-lobe level. Hereafter, we compare POCA Chaotic waveforms possess many appropriate radar proper- and RPOCA with other methods. The CAN and WeCAN are ties. In fact, the autocorrelation and cross-correlation, due to selected from [26], the “Monotonic minimizer for Weighted their aperiodicity and sensitivity to the initial condition, are ISL” (MWISL) and the “monotonic minimizer (MM) for usually low. Besides, they can be generated with a very low l ” are chosen from [24]. The CAN, WeCAN and MWISL computational burden for any length and quantity. p minimize ISL, and “MM for lp” minimizes lp-norm on side- By choosing different initial conditions, modified Bernoulli lobes. Like [24], very large p is considered for “MM for lp” in system can produce waveforms with very low cross- order to approximate the Chebyshev norm, with p set at 10000 correlation. By applying the algorithms introduced in the in specific. For both the “MM for lp” and MWISL the iteration previous section, the autocorrelation side-lobes levels of the counter is set to 106 (which is way beyond the suggestions in waveforms can be enhanced. In order to have a proper set of [24]) to ensure the convergence. Unless specified otherwise, sequences, it is sufficient to generate random sequences from all the algorithms are initialized by Golomb sequence, defined the modified Bernoulli system beginning different initial con- by, ditions and then applying one of the algorithms introduced in G(n) = e(j(n−1)nπ/N), n = 1, ..., N. previous sections (i.e. PMQA, PMAR, POCA or RPOCA). A (51) block diagram is proposed for baseband waveform generation Finally, it should be mentioned that all the following simu- at the transmitter side in each antenna in Fig. 4. lations are implemented through MatLab on an i7 3.2 GHz machine with 6 GBytes of RAM. VI.SIMULATION RESULTS 1) Comparison with Barker A. Autocorrelation Performance One of the widely embraced sequences in the context of the good autocorrelation sequences is Barker. Barker is the most In this section, the proposed methods in previous sections acknowledged member of the bi-phase sequences. Although are evaluated. Subsequently, in order to be able to compare it is not fair to compare a bi-phase sequence to a polyphase the autocorrelation, some metrics are defined herein. The sequence, due to its prevalent application, comparison with normalized autocorrelation is defined according to, Barker is illustrative. The longest Barker sequence is of the

rk length 13 and is as follows, Normalized Autocorrelation = 20 log10 , r0 x = [1 1 1 1 1 − 1 − 1 1 1 − 1 1 − 1 1]. (52) k = 1, ..., N. (47) The comparison between the autocorrelation of Barker 13, 8

0 0 RPOCA PMQA Chu -50 -20 PMAR MM-Corr Barker 13

) -100 -40 B d ( n o i t

a -60 l -150 e r r o c o t

u -80

A -200 Autocorrelation (dB)

-100 -250

-120 -300 -100 -80 -60 -40 -20 0 20 40 60 80 100 -10 -5 0 5 10 lag k lag k Fig. 8: Comparison between the Normalized Autorotation of mono- Fig. 5: Comparison for PMQA, PMAR and Barker, all at length 13 tonic minimizer (MM) for PSL minimization, RPOCA, and Chu sequence, for N = 100 0 POCA CAN 2) Suppressing all side-lobes −20 First, consider a scenario where suppressing all the side- lobes of the sequence is of equal importance. For instance, −40 consider a case where designing a sequence with length N = 100 is required, thus, −60 ( Q = 100 , (53) −80 N = 100

Normalized autocorrelation in dB In this scenario, T = IN×N . The comparison between −100 POCA and CAN algorithms for N = 100 is depicted in Fig. 6. The CAN algorithm is the best performing algorithm −120 −100 −80 −60 −40 −20 0 20 40 60 80 100 amongst several algorithms developed in [26]: CAP, CAN, and lag k WeCAN, while, it lacks the ability to suppress some specified part of the autocorrelation, where, WeCAN is developed there Fig. 6: Normalized autocorrelation of POCA vs. CAN [17] to accomplish this task. Note that WeCAN and CAN have PMQA and PMAR is depicted in Fig. 5 for parameters the same functionality when suppression all side-lobes is ε = 10−12, N = 13, Q = 12. The fact that both algorithms contemplated. can suppress the autocorrelation side-lobes to almost zero in The CAN, POCA and MWISL are compared in terms of wide range of correlation lags is revealed by this figure. Here, the PCL in Fig. 7 for different lengths for the same scenario the difference is in the consumed time, since 2.26 sec and as in (53). It is observed that POCA easily beats MWISL and 1.87 sec are consumed by PMQA and PMAR to generate this CAN with respect to PCL, where, the supremacy of POCA figure. In the experimentations hereafter the focus is on POCA over CAN is of order 5 dB. This figure is generated by setting and RPOCA, the reason is being the degree of approximations equal terminating points for both CAN and POCA. applied in deriving them. Accordingly, their applicability is Golomb sequence belongs to the family of polyphase se- asserted through various scenarios. quences. Another member of this family is the so-called Chu sequence, which is considered to be the best amongst them

Peak Correlation Level ISL (refer to [21] and references therein). This sequence is

−10 deﬁned according to ( 2 e(j(n−1) π/N) if n is even −15 C(n) = , n = 1, ..., N. (j(n−1)nπ/N) −20 e if n is odd

−25 (54) The comparison between “MM for l ”, RPOCA, and Chu −30 p sequence is demonstrated in Fig. 8 for the same scenario −35 as above (i.e. equation (53)), indicating that, in terms of −40 autocorrelation side-lobe levels RPOCA defeats state-of-the- −45 Peak Correaltion Level in dB art methods. Like in [24], the p is set at 10000 in order to Peak Correlation Level of POCA −50 Peak Correlation Level of CAN approximate the PSL minimization. −55 Peak Correlation Level of MWISL 3) Suppressing Less Than Half of the Side-lobes −60 1 2 3 10 10 10 In the second scenario, a situation is considered, where, sup- Length of Sequence pressing a speciﬁed part of the autocorrelation is of concern. Fig. 7: Comparison of PCL of CAN, POCA and MWISL algorithms 9

TABLE I: Comparison of MPCL and MMF for the scenario given 0 in (55) -50 POCA RPOCA WeCAN +Modiﬁed +Modiﬁed CAP -100 +CAP Bernoulli Bernoulli 1.12 × 2.96 × 3.096 × 2.23 × -150 MPCL 10−14 10−15 10−15 10−13 -200 5.20 × 4.54 × 1.08 × MMF 2.37 × 1026 1032 1032 1023 -250 Consumed

Autocorrelation (dB) > 140 2.36 2.607 140.06 -300 Time (s)

-350 POCA 341 iterations 0 MWISL 106 iterations -400 -100 -50 0 50 100 -50 lag k Fig. 9: Normalized autocorrelation of POCA for N = 100,Q = -100 39(341 iterations), and MWISL [24] the same scenario (106 itera- -150 tions) -200 Consider the example given in equation (66) of [26], where -250 suppression of r1, ..., r39 is required when N = 100, that is, ( Q = 39 -300Autocorrelation (dB) . (55) N = 100 -350 POCA random initialization POCA initialized by modified Bernoulli In [26], the CAP+WeCAN is revealed to be the best- -400 performing configuration (Refer to Fig. 4.b of [26]. Re- -20 -15 -10 -5 0 5 10 15 20 picturing is avoided to preserve brevity). The POCA’s out- lag k put initialized by modified Bernoulli is illustrated in Fig. 9 Fig. 10: Comparison between POCA with random initializa- in companion with MWISL’s output for the same scenario, tion and POCA initialized with modified Bernoulli where, the smallest correlation level and the peak correlation rithms are able to provide ”almost zero” autocorrelation side- level in the suppressing region for POCA is -340 dB and -308 lobes ”just” when suppressing less than half of the side-lobes dB, respectively. In comparison, the corresponding figures for are of concern. And if suppression of more than half is taken CAP+WeCAN are -320 and -280 dB, respectively, indicating into account, then the side-lobes for either CAP or WeCAN an improvement of more than 20 dB. In this figure the number becomes higher. According to the assessment made here, this 6 of iterations for MWISL is set to 10 to ensure convergence. holds for MWISL as well. The reason behind this phenomenon In comparison, the required number of iterations for POCA is that the problem is formulated with unimodularity (a desir- −14 algorithm in Fig. 9 is 341(ε = 10 ). It is obvious that the able but limiting) constraint. This additional constraint results supremacy of POCA to MWISL is more than 40 dB. in the elimination of the half of the degrees of freedom in The comparison of the MMF and MPCL figures for We- nullifying autocorrelation values. Luckily, this limitation does CAN initialized by CAP (CAP+WeCAN), POCA and RPOCA not hold for PMQA, PMAR, POCA, and RPOCA, except initialized by Modified Bernoulli (POCA+Modified Bernoulli when suppressing all the side-lobes is of concern, where the and RPOCA+Modified Bernoulli) as well as the CAP algo- newly developed algorithms cannot provide almost zero side- rithm are tabulated in TableI. In this table MPCL and MMF lobes as well. Note that the scenario in Fig. 5 affirms this fact (Eq. (49),(50)) numbers are in magnitude. From numbers for the first two algorithms. To illustrate this fact for POCA in TableI it is deduced that, although POCA and RPOCA and RPOCA, several scenarios are contemplated here. In the minimize PSL they show superior merit factors or equivalently first, suppression of r1, ..., r64 at N = 100 is envisioned and ISL, that is, by decreasing peak of the side-lobes (or PSL), all bearing in mind that typical surveillance pulse Doppler radars the side-lobes and therefore their power integration (or ISL) commonly utilize short or medium-size sequences for pulse would be decreased. This phenomenon is proved by authors compression, in the additional scenarios, the suppression of for asymptotic case (i.e. for very large sequences) in [13]. r1, ..., r32 at N = 40 and suppression of r1, ..., r17 at N = 20 The table provides a comparison between consumed time in are examined. Fig. 10 illustrates these scenarios. seconds for each one of the aforementioned methods, as well. The question at hand now is that how much is the effect The RPOCA and POCA both consumes less than 3 sec, in of chaotic sequence initialization on the autocorrelation side- comparison with more than 140 sec by the other methods. Note lobes. To answer this question, a ’suppressing more than half that the number for RPOCA is not less than POCA since the of the side-lobes’ scenario is contemplated due to the fact sequence is not long enough. In fact, the RPOCA is superior that it has more practical value. Accordingly, the comparison to POCA regarding time for large values of matrix dimension for N = 20,Q = 18 is illustrated in Fig. 10. From this where the speedup of the fast SVD dominates time added by comparison, it is observed that the POCA initialized by the additional steps in RPOCA. modified Bernoulli performs slightly better (about 4 dB) for 4) Suppressing More Than Half of the Side-lobes most of the suppression lags. As mentioned in [26], the CAP and CAP+WeCAN algo- 5) Dealing With Large N 10

0 TABLE II: Comparison of Computational burden when considering CCP -50 Consumed Time in seconds N=20,Q=18 Length 10 100 300 -100 Algorithm POCA initialized by modified Bernoulli 0.1 1.3 20.3 -150 MIMO-CAN initialized by modified Bernoulli 0.1 3.2 83 MIMO-CAN random sequence initialization 0.02 2.54 67.1 -200 of the cross-correlation, the cross-correlation peak (CCP) is Autocorrelation (dB) N=40,Q=33 defined according to, -250 CCP = max (rxy (k)) , (56) -300 where, rxy (k) is the cross-correlation between the sequences N=100,Q=65 {x }N and {y }N , -350 n n=1 n n=1 -100 -80 -60 -40 -20 0 20 40 60 80 100 N lag k X ∗ ∗ rxy(k) = xnyn−k = rxy(−k) , k = 0, ..., N − 1. Fig. 11: Normalized Autocorrelation of POCA for N = 100,Q = n=k+1 65; N = 40,Q = 33 and N = 20,Q = 18 (57) By this definition, the CCP goes to zero whenever all the Another drawback of the CAP and WeCAN is that they cross-correlation values tends to zero. The comparison of cannot easily cope with large sequences (i.e. the lengths more cross-correlation peaks for Bernoulli system before and after than N ∼ 1000). In these situations, POCA and its fast modification is depicted in Fig. 14a. This figure is generated by version, RPOCA, can be applied. It should be noted that due to averaging CCP over 100 instances of the sequences generated their iterative nature, the POCA and RPOCA have restrictions by equations (40) and (42) initialized by different initial as well. In fact, the superiority of these algorithms compared values. The parameters of the systems in (40) and (42) are to their predecessors is of order one, two or three decades. λ = 1.9 and B = 1/λ. Hence, the modification of the That is, POCA is restricted to lengths up to N ∼ 104 and Bernoulli system results in the generation of a set of sequences RPOCA cannot be applied for sequences more than N ∼ 106. with low cross-correlation. The comparison between the CCP The application of this algorithm on Golomb sequence with of POCA + Modified Bernoulli is presented in Fig. 14b, lengths N = 103 and N = 104 are illustrated in Fig 12a and when initialized by different initial values and MIMO-CAN 12b, when suppression of r , ..., r is required (Q = 65). 1 64 algorithm [17], where the number of antenna’s in MIMO-CAN The experiments here indicate that for RPOCA S as low as 4 is set to M = 40. The MIMO-CAN is the generalization of would suffice. CAN approach to MIMO radars by considering the cross- 6) Imposing PAPR restriction and Unimodularity constraint correlation in addition to autocorrelation. In Fig. 14b, the Generally speaking, imposing additional restrictions like the superiority of the POCA + Modified Bernoulli compared to ones introduced in sectionIV projects time burden on the MIMO-CAN is at least 12 dB. Furthermore, by comparing algorithm for total convergence. In another perspective, for a 14a and 14b it is inferred that, when using POCA algorithm certain amount of iteration, PAPR restriction results in higher in order to suppress the autocorrelation of the modified side-lobe levels. In order to show this fact, iteration num- Bernoulli, the cross-correlation degenerates (i.e. increases). bers for POCA algorithm implemented with unimodularity This fact indicates the existence of a trade-off of obtaining and PAPR constraints are restricted. In order to assess that good autocorrelation and reduced cross-correlation. the constrained version of the algorithm can suppress the side-lobes down to almost zero, the POCA constrained to C. Time performance unimodularity for N = 100,Q = 20 is simulated, where, One of the advantages of the POCA algorithm over the the number of iterations is set to 10000 in order to assure monotonic minimizer is its low time consumption. Therefore, convergence. The result is depicted in Fig. 13 along with other the MWISL [24], which is able to suppress the desired part of scenarios. Furthermore, in order to study the effect of PAPR the autocorrelation is chosen as a benchmark and the scenario bound, the POCA with constrained PAPR is examined, and in (55) is contemplated to compare consumed time and the to distinguish this case from others, the simulation parameters MPCL. These two parameters are depicted in Fig. 15 in an are N = 100,Q = 30. Nevertheless, for a certain amount of x-y plot, where it is observed that for an equal amount of iterations (1000 iterations) the POCA with constrained PAPR consumed time, the suppression level for POCA, measured by is simulated in three cases: unimodularity, a = 1.02, and MPCL, is considerably more. a = 1.2, Fig. 13, where it is observed that less suppression (higher side-lobe levels) is reached when restriction on PAPR VII.CONCLUSION tightens. B. Cross-Correlation Performance The waveform generation for MIMO radars has been ex- From the above arguments, it is clear that the autocorrelation amined. Our idea was that by juxtaposition of the chaotic related metrics are improved. However, the cross-correlation systems and cyclic algorithms, it is possible to design large between the transmitted waveforms should be as low as set of sequences which, possible in MIMO radars, hence, to quantify the goodness • have very low autocorrelation side-lobes. 11

0 0

-50 -50

-100 -100

-150 -150

-200 -200 Autocorrelation (dB) Autocorrelation (dB) -250 -250

-300 -300

-350 -350 -1000 -500 0 500 1000 -1 -0.5 0 0.5 1 lag k lag k ×104 (a) (b) Fig. 12: Autocorrelation of POCA for N = 1000,Q = 65 and N = 10000,Q = 65

0 • Positive deﬁniteness: k A k∞> 0 for all nonzero vectors and -50 (58) k A k∞= 0 if and only if A = 0 -100 ¯ ¯ • Linearity: -150 Unimodular k αA k∞= α k A k∞ (59) -200 • Triangle inequity: a=1.02 -250 k A + B k∞ ≤ k A k∞ + k B k∞ . (60)

Autocorrelation (dB) -300 The two first properties, (58) and (59) are straightforward. a=1.2 -350 To prove that last note that POCA Constrained to Unimodularity after 1000 iterations POCA with Constrained PAPR a = 1.02 after 1000 iterations k A + B k∞ = max |Ai,j + Bi,j| -400 POCA with Constrained PAPR a = 1.2 after 1000 iterations POCA Constrained to Unimodualirity after 10000 iterations ≤ max(|Ai,j| + |Bi,j| )= max |Ai,j| + max |Bi,j| (61) -100 -80 -60 -40 -20 0 20 40 60 80 100 lag k = k A k∞ + k B k∞, Fig. 13: Effect of imposing PAPR restriction on the suppres- where, triangle inequity for complex numbers is used in sion level for certain amount iterations forming (61). For a normed space to be Banach space an additional condition known as “completeness” is essential. In • can have “almost zero” side-lobes in specified regions of specific terms, given any Cauchy series in the space, it should autocorrelation. N×M converge. The completeness of (C , k . k∞) is revealed • maintain low cross-correlation between the elements of here through Theorem VIII.1. Before that, the term “Cauchy the sets. series” is defined, In the way of obtaining the aforementioned set, the problem of Definition: Consider a normed vector space (X, k . k), a ∞ minimizing peak side-lobe level is formulated. Consequently, series {xn}n=1 with its elements in X is a Cauchy series if, several algorithms are developed namely, PMQA, PMAR, ∀ε > 0 ∃N0 ; ∀m, n > N0 |xn − xm| < ε. (62) POCA, and RPOCA. The self-applied algorithms to mono- static pulse Doppler (PD) radars outperform their predecessors Theorem VIII.1. The metric space defined by the k . k∞ both in the terms of implementation time and their generated norm over CN×M is complete. side-lobe level of the sequences. Moreover, the Bernoulli Proof. Let, {An}∞ be a Cauchy series. Note that in this chaotic system is modified to have low cross-correlations. n=0 notation n is just a superscript and does not denote the power. By juxtaposition of the previously mentioned algorithms and For any ε > 0, there exists an integer positive N such that modified Bernoulli, appropriate waveforms for MIMO radar 0 for every n, m > N0, applications can be generated. m n k A − A k∞ ≤ ε. (63) VIII.APPENDIX A In order to show that such a sequence converges, it is sufficient N×M Throughout the paper, the notion that (C , k . k∞) is a to prove that there exists a matrix like A such that for any Banach space has been used extensively. In order to maintain ε > 0, there is an integer N0 > 0 such that the entirety of the article, the credibility of this fact should n k A − A k∞ ≤ ε. (64) be verified. Consider the vector space of all N × M matrices From (63), it is revealed that over the field of complex numbers, or N×M . Let A and B C n m be two arbitrary elements of this vector space. Moreover, let max Ai,j − Ai,j < ε (65) α be an arbitrary complex number. In order to maintain that Therefore, for any ε > 0, there exists a positive integer, N×M (C , k . k∞) is a normed space, it is sufficient to prove N such that for any n, m > N and for every i and j the n m that, inequality Ai,j − Ai,j < ε is guaranteed. Alternatively, for 12 CCP of Modified Bernouli vs CCP of Bernouli Cross Correlation Peak Comparison

Averaged CCP of Modified Bernouli Sequences -5 -10 Averaged CCP of Bernouli Sequences -10 -20 -15

-30 -20

-40 -25

-30 -50 CCP of MIMO-CAN Sequences -35

Cross Correlation Peak in dB CCP of POCA initilized by modified Bernoulli Cross Correlation Peak in dB -60 CCP of MIMO-CAN initialized by modified Bernoulli -40

-70 1 2 3 4 5 -45 10 10 10 10 10 101 102 103 Length of Sequence Length of Sequence (a) (b) Fig. 14: Comparison of cross-correlation peak (CCP) between (a) Bernoulli and modified Bernoulli (b) MIMO-CAN with random sequences initialization vs MIMO-CAN + Modified Bernoulli initialized by different initial values vs POCA + Modified Bernoulli initialized by different initial values. IX.APPENDIX B -20 In this appendix alternative approaches to exact solution -40 of the smallest circle problem in (16) is developed. For any Q−1 -60 complex sequence {µk}k=1 consider the following problem, min max |x − µk| -80 x k q 2 2 = min max (xR − Re {µk}) + (xI − Im {µk}) . -100 xR, xI k MPCL in dB (70) -120 First, notice that this unconstrained problem is equivalent to the following constrained problem, -140 POCA MWISL min R xR, xI -160 (71) 0 5 10 15 q 2 2 Time (sec) s. t. (xR − Re {µk}) + (xI − Im {µk}) ≤ R, Fig. 15: Comparison of the consumed time between POCA where, xR and xI are real and imaginary parts of x. Then, and MWISL for the scenario given in (55). by defining yk := xI − Im {µk}, wk := xR − Re {µk}, k = 1,...,Q − 1 this problem can be restated as, min R n o∞ x , x ,{w },{y } i j An R I k k every and the complex sequence generated by i,j n=0 s. t. wk + xR = Re {µk} , k = 1,...,Q − 1 is a Cauchy series in (C, |.|) normed vector space, where |.| (72) y + x = Im {µ } , k = 1,...,Q − 1 denotes the absolute value. Note that, ( C, |.|) is known to k I k q be a complete normed vector space, indicating that, every 2 2 ∞ R ≥ w + y , k = 1,...,Q − 1. Cauchy series in it converges. For the sequences An k k i,j n=0 The problem in (72) is a cone optimization problem and can assume, they converge to f for every i and j. Mathematically i,j be solved through interior point method with an arithmetic speaking, 3.5 n cost of O(n | log δ|), where δ is the user defined accuracy ∀ε > 0 ∃N0 ; ∀n > N0 ∀i,j Ai,j − fi,j < ε. (66) parameter. Note that, the inequality holds for every i and j. Therefore, Another approach to solving (70) is obtained through notic- it also holds when “every” is changed to “max”. That is, the ing that this problem is also an unconstrained nondifferentiable statement in (66) is equivalent to, convex programming, solvable through subgradient method. n ∀ε > 0 ∃N0 ; ∀n > N0 max Ai,j − fi,j < ε. (67) Accordingly, one may follow the following steps in order to Hence, find the exact solution. n ∀ε > 0 ∃N0 ; ∀n > N0 k A − F k∞ ≤ ε. (68) 1) Start form an appropriate point, for instance maxd {µ } + mind {µ } where, F is defined according to the following equation. x0 = k k . (73) F = [f ] , i = 1, ..., N, j = 1, ..., M. (69) 2 i,j 2) Compute the subgradient of n ∞ The statement in (68) states that the sequence {A }n=0 f(x , x ) = converges to F in the Chebyshev distance sense, resulting in R I q 2 2 (74) the completeness of the proof. max (xR − Re {µk}) + (xI − Im {µk}) , k i i at (xR, xI ) 13

3) Determine the step size by using line search. To this end, 2) The elements of feasible set that are less than x∗, that is, let max µk + min µk x ≤ . (83) K = {k : f(xR, xI ) = 2 (75) p 2 2 Similarly, here we have: (xR − Re{µk}) + (xI − Im{µk}) }, be the active functions index set. Then, constitute the set, max µk − x ≥ |x − µk| , ∀k (84) ∂f(xR, xI ) = or,     xR − Re {µk} max |x − µk| = max µk − x. (85)     k      xI − Im {µk}  (76) Thus, in this case the minimization problem in (16) is co √ |k ∈ K , (x −Re{µ })2+(x −Im{µ })2 equal to the following problem:  R k I k    min(max µk − x)   x . (86) s.t. x ≤ max µk+min µk where ”co” and ∂f(xR, xI ) denote the convex hull and 2 With the change of variable z = −x, this problem can the sub-differential of f(xR, xI ), respectively. Therefore, any member of this set is a sub-gradient of f. 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H. E. Najafabadi received B.S. and M.S. in electronics, and communication in 2007 and 2009 from Isfahan University of Technology and Amirkabir University of Technology, respectively. From 2009 to 2010, he was with the ICTI group (icti.ir), where he was working on the national airborne radar system as a team member. Cur- rently, He works toward Ph.D. in University of Isfahan.

M. Ataei received the B.Sc. degree from the Isfahan University of Technology, Iran, in 1994, the M. Sc. degree from the Iran University of Science & Technology, Iran, in 1997, and PhD degree from K. N. University of Tech- nology, Iran, in 2004 (joint project with the University of Bremen, Germany) all in control Engineering. He is an associate professor at the Department of Electrical Eng. at University of Isfahan, Iran. His main areas of research interest are control theory and applications, nonlinear control, and chaotic systems’ analysis and control.