A Uniﬁed Framework for Low Autocorrelation Sequence Design ... · Abstract—In this paper, we consider the low autocorrelation sequence design problem. We optimize a uniﬁed metric

438 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 65, NO. 2, JANUARY 15, 2017

A Unified Framework for Low AutocorrelationSequence Design via Majorization–Minimization

Licheng Zhao, Junxiao Song, Prabhu Babu, and Daniel P. Palomar, Fellow, IEEE

Abstract—In this paper, we consider the low autocorrelationsequence design problem. We optimize a unified metric over a gen-eral constraint set. The unified metric includes the integrated side-lobe level (ISL) and the peak sidelobe level (PSL) as special cases,and the general constraint set contains the unimodular constraint,Peak-to-Average Ratio (PAR) constraint, and similarity constraint,to name a few. The optimization technique we employ is themajorization–minimization (MM) method, which is iterative andenjoys guaranteed convergence to a stationary solution. We carryout the MM method in two stages: in the majorization stage, wepropose three majorizing functions: two for the unified metric andone for the ISL metric; in the minimization stage, we give closed-form solutions for algorithmic updates under different constraints.The update step can be implemented with a few Fast Fourier Trans-formations (FFTs) and/or Inverse FFTs (IFFTs). We also show theconnections between the MM and gradient projection method un-der our algorithmic scheme. Numerical simulations have shownthat the proposed MM-based algorithms can produce sequenceswith low autocorrelation and converge faster than the traditionalgradient projection method and the state-of-the-art algorithms.

Index Terms—Sequence design, low autocorrelation, unifiedframework, majorization minimization.

I. INTRODUCTION

S EQUENCES with low autocorrelation sidelobes enjoy awide range of applications in wireless communications and

signal processing. Some important engineering applications in-clude Code-Division Multiple Access (CDMA) cellular systems[1], radar systems, security systems, and even cryptographysystems [2]. We define x � {xn}N

n=1 ∈ CN as a complex-valued sequence of length N . Our objective is to design a se-quence whose autocorrelation sidelobes are jointly at a low level.The aperiodic and periodic autocorrelations of the sequence xare given as

rk =N −k∑

n=1

xnx∗n+k = r∗−k (aperiodic) (1)

and

rk =N∑

n=1

xnx∗(n+k)(mod N ) = r∗−k (periodic) (2)

Manuscript received May 24, 2016; revised August 10, 2016 and September12, 2016; accepted October 4, 2016. Date of publication October 21, 2016;date of current version November 17, 2016. The associate editor coordinatingthe review of this manuscript and approving it for publication was Prof. TaoJiang. This work was supported by the Hong Kong RGC 16206315 researchgrant and by the Hong Kong RGC Theme-based Research Scheme (TRS) GrantT21-602/15R.

The authors are with the Hong Kong University of Science and Technology,Hong Kong (e-mail: [email protected]; [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2016.2620113

for k = 0, 1, · · · , N − 1. Usually, the sequence to be designedhas a limited energy budget, so we fix the sequence energywithout loss of generality.

A. Related Work

In order to measure the goodness of the autocorrelation side-lobes of a sequence, researchers have put forward various met-rics. One example is the integrated sidelobe level (ISL)

ISL =N −1∑

k=1

|rk |2 , (3)

or equivalently the merit factor (MF) [3]

MF =r20

2∑N −1

k=1 |rk |2=

‖x‖42

2ISL. (4)

With the sequence energy fixed, maximizing the MF is equiva-lent to minimizing the ISL. As an extension to the ISL, Stoicaet al. [4] also proposed the Weighted ISL (WISL) metric

WISL =N −1∑

k=1

wk |rk |2 , (5)

where {wk}N −1k=1 are the nonnegative weights. This metric is

useful when the rk ’s are not equally important and we want tosuppress some rk ’s in particular. Another example of the popularmetrics is the peak sidelobe level (PSL)

PSL = maxk=1,2,··· ,N −1

{|rk |} . (6)

Both the ISL and PSL can be viewed as the p-norm of |rk |’swith (6) as the limit of large p, and thus these two metrics canbe unified.

In practice, there are many constraints to be considered apartfrom the aforementioned energy budget constraint. In terms ofmodulus, [4]–[6] have mentioned the constant modulus (uni-modular, to be accurate) constraint, and [7]–[9] handled PARconstraint. In terms of phase, [10] focused on the polyphaseconstraint, and [11], [12] studied the similarity constraint. Phaseconstraints are imposed together with the constant modulus con-straint.

There is a rich literature on finding low autocorrelation se-quences. The existing construction methods are either analyt-ical or computational. Using analytical methods, we are ableto construct sequences with closed-form expressions, such asthe Frank sequences [13], the Chu sequences [14], and theGolomb sequences [15]. Traditional computational methodsare either based on search algorithms (exhaustive search orstochastic search) or evolutionary algorithms. They becomecomputationally expensive when the sequence length goes upto a few thousand. In addition, the evolutionary algorithms are

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ZHAO et al.: UNIFIED FRAMEWORK FOR LOW AUTOCORRELATION SEQUENCE DESIGN VIA MAJORIZATION–MINIMIZATION 439

heuristic methods in nature whose convergence is not guaran-teed. To summarize, the existing sequence construction methodscannot effectively design extremely long sequences (up to 104)with very low autocorrelation, and that is why modern optimiza-tion techniques are desperately needed.

Recently, a number of optimization-oriented approaches [4],[16]–[18] have been put forward to design long sequences withlow autocorrelation sidelobes. Among them, the Cyclic Algo-rithm New (CAN) algorithm [4] enjoys a satisfactory perfor-mance in producing unimodular sequences of length up to sev-eral million. Rather than minimizing the ISL, the CAN algorithmminimizes a simpler but “almost equivalent” metric, changingthe quartic objective function to a quadratic one. When it comesto the WISL metric, there is no simple “almost equivalent” re-formulation available. Hence, as is shown in [4], the WeightedCAN (WeCAN) algorithm performs much worse than CAN.

B. Contribution

The major contributions of this paper (with respect to [5], [6])lie in the following three aspects.

1) The first contribution is that we propose a unified frame-work to design low autocorrelation sequences, whichprovides a global perspective for the sequence designproblem. In terms of the objective, we unify the existingISL, WISL, and PSL metrics; in terms of the constraint,we consider the constant modulus constraint, PAR con-straint, polyphase constraint, similarity constraint, and soon. We can solve the sequence design problem in a uni-fied manner instead of dealing with each metric and eachconstraint separately in a specific way.

2) The second contribution is that we systematically applythe Majorization Minimization (MM) method [19]. Wecarry out the MM method in two separate stages: ma-jorization function construction and minimization solu-tion derivation. In the majorization stage, we construct atotal of three majorizing functions. Two of them apply tothe unified metric, and the remaining one, which is notmentioned in [5], [6], applies to the more specific ISLmetric. Its achieved ISL level is as good as the other twomajorizing functions, and the convergence speed is thefastest, 2–4 times as fast as the second fastest, accordingto the simulation results. In the minimization stage, weprovide efficient closed-form solutions to the minimiza-tion problems depending on the constraints. The mini-mization stage is by nature solving a projection problem,which is not pointed out in [5], [6]. We maintain that thewhole MM process is more than a simple combination ofexisting works. Furthermore, the MM update step can beimplemented with a few FFT (IFFT) operations which arehighly efficient.

3) The third contribution is that we establish the connec-tions between the MM and gradient projection methodunder our algorithmic scheme, which is very insightful.The update step of the MM method has exactly the samestructure as that of the gradient projection method.However, the MM method is still superior because of the-oretical stationarity convergence guarantee. Apart fromthat, the MM method has faster convergence speed thanthe gradient projection method, as can be seen in thesimulation results.

C. Organization and Notation

The rest of the paper is organized as follows. In Section II, weintroduce the problem formulation. In Section III, a brief pre-liminary description of the MM method is given. In Sections IVand V, we construct majorization functions and derive mini-mization solutions, respectively. In Section VI, we show theconnections between the MM and gradient projection methodunder our implementation scheme. In Section VII, we elabo-rate on the algorithmic implementation. Finally, Section VIIIpresents numerical simulations, and the conclusions are givenin Section IX.

The following notation is adopted. Boldface upper-case let-ters represent matrices, boldface lower-case letters denote col-umn vectors, and standard lower-case letters stand for scalars.R (C) denotes the real (complex) field. |·| denotes the absolutevalue. ‖·‖p denotes the p-norm of a vector. ∇ (·) represents thegradient of a vector function (the way to derive the complex-valued gradient follows [20]). I stands for the identity matrix.Xij denotes the (i, j)th element of the matrix X. XT , X∗, XH ,Tr (X), λmax (X), λmin (X), and vec (X) denote the transpose,complex conjugate, conjugate transpose, trace, the largest eigen-value, the smallest eigenvalue and stacking vectorization of X,respectively. Diag (x) is a diagonal matrix with x filling itsprincipal diagonal and diag (X) is the vector consisting of allthe diagonal elements of matrix X. Finally, � stands for theHadamard product.

II. PROBLEM STATEMENT

A. Objective Function

We propose a unified metric named Weighted Peak or Inte-grated Sidelobe Level (WPISL) as follows:

WPISL =N −1∑

k=1

wk |rk |p , (7)

where 2 ≤ p < +∞, and {wk}N −1k=1 are nonnegative weights.

This metric readily includes the ISL (or WISL) and the PSL asspecial cases: let p = 2, and we get the WISL metric; furtherlet wk = 1, ∀k, and we get the ISL metric; let p → +∞ and

wk = 1, ∀k, and we get(∑N −1

k=1 |rk |p)1/p

→ maxk {|rk |}, i.e.,

the PSL metric.

B. Constraints of Interest

We have already introduced the energy budget constraint,i.e., ‖x‖2

2 = c2e (constant). Besides that, some additional con-

straints may also be of interest, as are described in the following.1) Strict Constant Modulus Constraint: In [4]–[6], strict

constant modulus constraint or, in particular, strict unimodu-lar constraint is of interest due to the limitations of hardwarecomponents and/or efficiency of amplifiers [21]. This constraintis expressed as: for n = 1, 2, · · · , N ,

|xn | = cm =ce√N

(constant, specially cm = 1) . (8)

This constraint is denoted as C1 for short.2) ε-Uncertainty Constant Modulus Constraint: Sometimes

the strict constant modulus constraint is too harsh and we maywant to relax the constant modulus within a small ε-uncertainty


TABLE IMISCELLANEOUS CONSTRAINTS AND THEIR NOTATIONS

region, which is: for n = 1, 2, · · · , N ,

cm − ε1 ≤ |xn | ≤ cm + ε2 (0 ≤ ε1 ≤ cm , 0 ≤ ε2) . (9)

This constraint is a modest relaxation of the previous constraintC1 , and it is denoted as C2 for short.

3) PAR Constraint: The signal PAR measures the ratio ofthe largest signal magnitude to its average power [7]–[9]:

PAR (x) =maxn |xn |2

‖x‖22 /N

, (10)

and 1 ≤ PAR (x) ≤ N . We require PAR (x) ≤ ρ (a particu-lar threshold), so that the analog-to-digital converters and thedigital-to-analog converters in the system can have lower dy-namic range, and fewer linear power amplifiers are needed.Since we have already assume ‖x‖2 = ce , the PAR constraintis equivalent to: for n = 1, 2, · · · , N ,

|xn | ≤ cp (constant) , (11)

where ce/√

N ≤ cp ≤ ce . This constraint is more relaxed thanthe previous two in that the modulus is not lower bounded.When cp = ce/

√N , the PAR constraint degenerates into strict

constant modulus constraint. When cp > ce/√

N (= cm ), thePAR constraint becomes a large uncertainty set around cm (ε1 =cm and ε2 = cp − cm ). This constraint is denoted as C3 for short.

4) Discrete Phase Constraint: Besides modulus constraints,phase constraints are also desired. One commonly usedconstraint is the discrete phase constraint, also known asthe polyphase constraint [10]. It is expressed as: for n =1, 2, · · · , N ,

arg (xn ) ∈ {φ1 , φ2 , · · · , φI } , (12)

where φ1 , φ2 , · · · , φI are I fixed angles. This constraint is de-noted as C4 for short, and is often accompanied by the strictconstant modulus constraint C1 .

5) Similarity Constraint: Sometimes we want the designedsequence to lie in the neighborhood of a reference onewhich already has good properties [11], [12]. It is writtenas ‖x − xr‖∞ ≤ δ (0 ≤ δ ≤ 2) with xr being the referencesequence, which is equivalent to |xn − xr,n | ≤ δ, ∀n. Simi-larly, this constraint is often accompanied by the strict constantmodulus constraint C1 , and can be further simplified as: forn = 1, 2, · · · , N ,

arg (xn ) ∈ [γn , γn + ϑ] , (13)

where γn = arg (xr,n ) − arccos(1 − δ2/2

), and ϑ = 2arccos(

1 − δ2/2). Note that when δ = 0, we get ϑ = 0, indicating

that x = xr ; when δ = 2, we get ϑ = 2π, and this constraint isalways satisfied. This constraint is denoted as C5 for short.

For clarity, we summarize all the aforementioned constraintsand their notations in Table I.

C. Problem Formulation

The problem formulation consists of the minimization of theWPISL metric in (7) subject to the constraints mentioned inTable I, and it reads

minimizex

∑N −1k=1 wk |rk (x)|p

subject to x ∈ X ,(14)

where rk (x) = xH Ukx, for the aperiodic case, i.e., (1), Uk ∈RN ×N is a Toeplitz matrix with only the kth diagonal entriesbeing 1 and 0 elsewhere; while for the periodic case, i.e., (2), Uk

is a Toeplitz matrix with only the kth and (k − N)th diagonalentries being 1 and 0 elsewhere, and

X ={x ∈ CN

∣∣∣ ‖x‖22 = c2

e

}∩ (∩iCi) , (15)

which means only a few Ci’s are activated.Remark 1: In the following, we focus on the aperiodic case

and the results can be easily extended to the periodic case withminor modifications.

III. PRELIMINARIES: MAJORIZATION MINIMIZATION METHOD

In this section, we are going to briefly introduce the MMmethod, and its algorithmic framework [19].

A. The MM Method

The MM method can be applied to solve the following generaloptimization problem:

minimizex

f (x)subject to x ∈ X ,

(16)

where f is differentiable on the whole C space and X is someconstraint set. Rather than minimizing f (x) directly, we con-sider successively solving a series of simple optimization prob-lems. The algorithm initializes at some feasible starting pointx(0) , and then iterates as x(1) , x(2) , . . . until some convergencecriterion is met. For any iteration, say, the lth iteration, the up-date rule is

x(l+1) ∈ arg minx∈X f(x;x(l)

), (17)

where f(x;x(l)

)is the majorizing function of f (x) at x(l) ,

which satisfies1) f

(x;x(l)

)≥ f (x), ∀x ∈ X ;

2) f(x(l) ;x(l)

)= f

(x(l)).

In words, f(x;x(l)

)is a tight global upper bound of f (x) in

the constraint set and also coincides with f (x) at x(l) .

B. Stationarity Convergence

The MM method can guarantee convergence to a stationarysolution of (16) as long as the following two additional condi-tions are satisfied:


3) ∇f(x(l) ;x(l)

)= ∇f

(x(l));

4) f(x;x(l)

)is continuous in both x and x(l) .

Interested readers may refer to [22] for detailed proofand here we only briefly mention its idea. Firstnote that f

(x(l))

= f(x(l) ;x(l)

)≥ minx∈X f

(x;x(l)

)=

f(x(l+1);x(l)

)≥ f

(x(l+1)

), from which we get f

(x(0))≥

f(x(1))≥ f

(x(2))≥ · · · , implying the monotonicity of the

sequence{f(x(l))}

. Then assume a subsequence of{x(l)}

,denoted by

{x(lj )

}, converges to a limit point z. Thus, for

all x ∈ X , f(x;x(lj )

)≥ f

(x(lj +1);x(lj )

)≥ f

(x(lj +1)

)≥

f(x(lj + 1 )

)= f

(x(lj + 1 ) ;x(lj + 1 )

). Letting j → +∞, we obtain

for all x ∈ X , f(x;x(∞)

)≥ f

(x(∞) ;x(∞)

), which indicates

f ′(x(∞) ;x(∞) ,d

)≥ 0, ∀d ∈ TX

(x(∞)

), (18)

where f ′ stands for directional derivative and TX(x(∞)

)is the

Boulingand tangent cone of X at x(∞) . The definition of direc-tional derivative of f in the direction d is

f ′ (x;d) = lim infλ↓0

f (x + λd) − f (x)λ

. (19)

If f is differentiable, f ′ (x;d) can be computed ef-fectively as dT ∇f (x). Interested readers may refer to[23], [24] for more knowledge of tangent cone. Since∇f(x(∞) ;x(∞)

)= ∇f

(x(∞)

), we have f ′ (x(∞) ;x(∞) ,d

)=

dT ∇f(x(∞) ;x(∞)

)= dT ∇f

(x(∞)

)= f ′ (x(∞) ;d

). There-

fore, the limit point x(∞) should satisfy

f ′(x(∞) ;d

)≥ 0, ∀d ∈ TX

(x(∞)

), (20)

and thus x(∞) is a (Boulingand) stationary point.

C. Acceleration Techniques

The convergence speed of the MM method is mainly de-termined by the majorizing function. In some cases where themajorizing function is not well designed or cannot be betterdesigned, we should adopt some techniques to accelerate theconvergence speed.

1) Acceleration via SQUAREM: SQUAREM stands forsquared iterative method, which can be viewed as an “off-the-shelf” accelerator of the MM method. It was proposed by Varad-han and Roland [25] and is claimed to have the following twoadvantages: 1) requiring nothing more than the MM variableupdating scheme, and 2) enjoying global convergence. The de-tailed implementation of SQUAREM is shown in Algorithm 1.To make SQUAREM better fit in the MM method, we make thefollowing modifications. To guarantee feasibility, we projectthe infeasible points back to the constraint set by the opera-tion PX (·). Moreover, to preserve the monotonicity of the MMmethod, we adopt the strategy of backtracking, which repeatedlyhalves the distance between α and −1 until the monotonicityis achieved. To see why it works, interested readers may referto [6].

2) Acceleration via Local Majorization: This idea has pre-viously shown up in [5]. As is mentioned above, the slow con-vergence speed of the MM method is due to an ill-designedmajorizing function. An alternative acceleration technique isto modify the ill-designed global majorizing function to a lo-cal one. This technique can be readily applied to an objective

Algorithm 1: MM Acceleration via SQUAREM.

Require: l = 0, the initial feasible point x(0) .1: repeat2: x(l)

1 = arg minx∈X f(x;x(l)

);

3: x(l)2 = arg minx∈X f

(x;x(l)

1

);

4: r = x(l)1 − x(l) ;

5: v = x(l)2 − x(l)

1 − r;6: α = −‖r‖2 / ‖v‖2 ;7: x(l) = PX

(x(l) − 2αr + α2v

);

8: while f(x(l))

> f(x(l))

do9: α = (α − 1) /2;

10: x(l) = PX(x(l) − 2αr + α2v

);

11: end while12: x(l+1) = x(l) ;13: l = l + 1;14: until convergence

Algorithm 2: MM Acceleration via Local Majorization.

Require: l = 0, the initial feasible point x(0) ,θ =

[ 1T , 2

T , · · · , 1]T

with T being the grid size.1: repeat2: i = 1;3: x(l) = arg minx∈X fθ

(x;x(l) , θ (i)

);

4: while fθ

(x(l) ;x(l) , θ (i)

)< f

(x(l))

do5: i = i + 1;6: x(l) = arg minx∈X fθ

(x;x(l) , θ (i)

);

7: end while8: x(l+1) = x(l) ;9: l = l + 1;

10: until convergence

function f (x) that is Lipschitz continuous on the constraintset X . In this case, the majorizing function is naturally con-structed as f

(x;x(l)

)= f

(x(l))

+ ∇T f(x(l)) (

x − x(l))

+L2

∥∥x − x(l)∥∥2

2 where L is a constant with L ≥ Lf ≥ 0 (Lf

is the Lipschitz constant for the function f ). In most situations,Lf is very difficult to calculate and we can only derive an up-per bound, possibly quite loose. Let fθ

(x;x(l) , θ

)� f

(x(l))

+∇T f

(x(l)) (

x − x(l))

+ θ · L2

∥∥x − x(l)∥∥2

2 with 0 ≤ θ ≤ 1.It can be seen that fθ

(x;x(l) , 1

)= f

(x;x(l)

), and for

0 ≤ θ < 1, fθ

(x;x(l) , θ

)is a local majorizing func-

tion. Following the argument of the monotonicity of theMM method, we see that f

(x(l))

= fθ

(x(l) ;x(l) , θ

)≥

minx∈X fθ

(x;x(l) , θ

)= fθ

(x(l) ;x(l) , θ

)still holds, but

fθ

(x(l) ;x(l) , θ

)≥ f

(x(l))

may not. To achieve monotonicity,we start from a small-valued θ and gradually increase it (upto 1) until fθ

(x(l) ;x(l) , θ

)≥ f

(x(l))

is satisfied. Then we setx(l+1) = x(l) . The detailed implementation of this accelerationtechnique is elaborated in Algorithm 2.

IV. MAJORIZING FUNCTION CONSTRUCTION

In this section, we are going to put forward a total of threemajorizing functions. Two of them apply to the unified WPISLmetric and the remaining one applies to the more specific ISL


metric. It is generally preferred that the minimization of themajorizing function has a closed-form solution. Driven by thismotivation, we construct quadratic majorizing functions so thata closed-form solution is easily obtained.

A. Majorizing Functions for the WPISL Metric

We first study a scalar function xp . It is generally known thatan arbitrary power function xp does not have a global quadraticmajorizing function for p > 2. However, there does exist a localone, which comes from the lemma below.

Lemma 2 ([6, Lemma 10]): Let g (x) = xp with p ≥ 2. Thefunction g (x;x0) = ax2 + bx + ax2

0 − (p − 1) xp0 is a local

majorizing function of g (x) at x0 ∈ [0, x) on the interval [0, x]where a = [xp − xp

0 − pxp−10 (x − x0)]/(x − x0)2 ≥ 0 and b =

pxp−10 − 2ax0 ≤ 0. In particular, when p = 2, we get a = 1,

b = 0, and g (x;x0) = x2 = g (x) (no majorization).Remark 3: Using such a local majorizing function maintains

monotonicity and will not cause infeasibility. The global min-imizer of g (x;x0) with respect to x is − b

2a . If p > 2, thena > 0 and b ≤ 0, making − b

2a nonnegative. In addition, we

observe− b2a = − pxp −1

0 −2ax0

2a = x0 − p2a xp−1

0 < x0 < x. Sincethe global minimizer still falls within [0, x], infeasibility will notoccur.

Suppose x(l) is the designed sequence at the lth MM iteration,and we want to construct a majorizing function around x(l)

for the WPISL metric. We denote f (x) �∑N −1

k=1 wk |rk (x)|pwith rk (x) = xH Ukx and handle the summation term by termusing Lemma 2. We regard |rk (x)| as x and

∣∣rk

(x(l))∣∣ as

x0 , obtaining (constant terms are represented with const forsimplicity)

f (x) ≤N −1∑

k=1

wk

(ak |rk (x)|2 + bk |rk (x)| + const

), (21)

where ak (≥ 0) and bk (≤ 0) follow the structure of a andb in Lemma 2, and the corresponding interval upper limitx (just as that mentioned in Lemma 2) is different for each

k: xk =

{(1

wk

∑N −1j=1 wj

∣∣rj

(x(l))∣∣p)1/p

wk �= 00 wk = 0

. Next, we

observe bk ≤ 0 and |rk (x)||r−k (x(l))| ≥ Re[rk (x) · r−k (x(l))],so (recall bk ≤ 0)

wkbk |rk (x)| ≤ wkbkRe[

r−k (x( l ) )|r−k (x( l ) )|rk (x)

]=

12xH

(w−k b−k

rk (x( l ) )|rk (x( l ) )|U−k + wkbk

r−k (x( l ) )|r−k (x( l ) )|Uk

)x, (22)

with U−k = UHk , w−k = wk , w0 = 0 and b−k = bk , b0 = 0.

Combining (21) and (22), we can further majorize f (x) in thefollowing manner:

f (x) ≤N −1∑

k=1

wkak |rk (x)|2 +12xH Bx + const

=N −1∑

k=1

wkak

∣∣xH Ukx∣∣2 +

12xH Bx + const, (23)

where

B =N −1∑

k=1−N

wkbkr−k (x( l ) )|r−k (x( l ) )|Uk . (24)

Now we already have a quadratic term 12 x

H Bx, but still have

an intractable quartic term∑N −1

k=1 wkak

∣∣xH Ukx∣∣2 , which is

the focus of the following majorization step. We are going tomajorize the quartic term in two different ways, obtaining twomajorizing functions eventually.

1) Majorizing via the Largest Eigenvalue: First we intro-duce a lemma addressing how to majorize a quadratic functionby another quadratic function.

Lemma 4 ([5, Lemma 1]): Given M � M0 and x0 , thequadratic function xH M0x is majorized by xH Mx +2Re

[xH (M0 − M)x0

]+ xH

0 (M − M0)x0 at x0 .With some simple transformations, the quartic term∑N −1k=1 wkak

∣∣xH Ukx∣∣2 can be rewritten as

12vecH (X) · L · vec (X) , (25)

where

L =N −1∑

k=1−N

wkakvec (U−k ) vecH (U−k ) , (26)

X = xxH , a−k = ak , and a0 = 0. It is easy to see that (25) isquadratic in X and by applying Lemma 4 with M0 = L andM = λmax (L) I, we can get the following result.

Lemma 5: The expression∑N −1

k=1 wkak |xH Ukx|2+12 x

H

Bx is majorized by

12λmax (L) ‖x‖4

2 + xH(R − λmax (L)x(l)x(l)H

)x + const,

(27)where

R =N −1∑

k=1−N

p

2wk

∣∣∣rk

(x(l))∣∣∣

p−2r−k

(x(l))Uk . (28)

Proof: See Appendix A for the detailed proof. �We apply Lemma 4 again onxH

(R − λmax (L)x(l)x(l)H

)x

with M = λuI where λu is chosen such that λu ≥ λmax (R)and finally get

xH(R − λmax (L)x(l)x(l)H

)x

≤ λuxH x + 2Re[xH(R − λmax (L)x(l)x(l)H

− λuI)x(l)]

+ const (29)

= λu ‖x‖22 − 2λuRe

[yH

1 x]+ const,

where

y1 =(1 + λm a x (L)

λu

∥∥x(l)∥∥2

2

)x(l) − 1

λuRx(l) . (30)

To conclude, the first majorizing function we propose is

f1

(x;x(l)

)=


2 + λu ‖x‖22

− 2λuRe[yH

1 x]+ const. (31)


Remark 6: Recall that we have an energy budget constraint‖x‖2

2 = c2e , which means 1

2 λmax (L) ‖x‖42 + λu ‖x‖2

2 = const.Constructing a majorizing function like this will bring conve-nience to the minimization step.

2) Majorizing via a Diagonal Matrix: Here we propose analternative way of majorization, which is based on the fact that Lis both positive semidefinite and nonnegative. We can majorize∑N −1

k=1 wkak |rk (x)|2 + 12 x

H Bx in another way by choosing adifferent M matrix, cf. Lemma 7 below.

Lemma 7 ([6, Lemma 5], [26, Lemma 2]): Let L be a realsymmetric nonnegative matrix. Then, Diag (L1) � L.

With the help of Lemma 7, we can majorize∑N −1

k=1 wk

ak |xH Ukx|2 + 12 x

H Bx in another way (continuing from (23)).The key idea is to apply Lemma 4 with M0 = L and M =Diag (L1).

Lemma 8: The expression∑N −1

k=1 wkak |xH Ukx|2 + 12 x

H

Bx is majorized by

12xH(E �

(xxH

))x + xH

(R − E �

(x(l)x(l)H

))x+const,

(32)with R given in (28) and

E =N −1∑

k=1−N

wkak (N − |k|)U−k . (33)

Proof: See Appendix B for the detailed proof. �We apply Lemma 4 again on xH

(R − E �

(x(l)x(l)H

))x

withM = λuI − λlIwhere λu and λl are chosen such that λu ≥λmax (R) and λl ≤ λmin

(E �

(x(l)x(l)H

)), respectively:

xH(R − E �

(x(l)x(l)H

))x

≤ (λu − λl)xH x

+ 2Re[xH(R − E �

(x(l)x(l)H

)(34)

− λuI + λlI)x(l)]

+ const

= (λu − λl)(‖x‖2

2 − 2Re[yH

2 x])

+ const.

where

y2 =(I +

E�(x( l ) x( l )H )λu −λl

)x(l) − 1

λu −λlRx(l) . (35)

To conclude, the second majorizing function we propose is

f2

(x;x(l)

)=

12xH(E �

(xxH

))x

+ (λu − λl)(‖x‖2

2 − 2Re[yH

2 x])

+ const. (36)

Remark 9: When the strict constant modulus constraintis imposed, i.e., |xn | = cm , we have i) 1

2 xH (E � (xxH ))x

=12 Tr((E � (xxH )) · (xxH ))

(a)= 1

2 Tr(((xxH ) � (x∗xT )E)=c4

m

2 1T E1=const where (a) Tr(AT (B � C))=Tr((AT � BT )C), cf. [27, Ch. 3, Section VI, Theorem 7(a)], ii) ‖x‖2

2 = c2e =

Nc2m and iii) λmin(E � (x(l)x(l)H )) = c2

m λmin(Diag(x(l)/cm ) · E · DiagH

(x(l)/cm

))= c2

m λmin (E). It will also bringconvenience to the minimization step.

B. A Majorizing Function for the ISL Metric

In particular, when p = 2, and wk = 1 for all k, f (x) =∑N −1k=1 |rk (x)|2 . We are going to propose a majorizing function

for this specific metric. We can easily obtain that∑N −1

k=1

∣∣xH Ukx∣∣2 X=xxH

= 12 vecH (X)

[∑N −1k=1−N vec(Uk ) vecH

(Uk ) − vec (I) vecH (I)]vec (X). For majorization, we also

need the following lemma.Lemma 10: The following equation holds:

N −1∑

k=1−N

vec (Uk ) vecH (Uk )=1

2N

2N∑

i=1

vec(fifH

i

)vecH

(fifH

i

)

(37)where fi = [1, exp (jωi (2 − 1)) , · · · , exp (jωi (N − 1))]T ∈CN ×1 and ωi = 2π

2N (i − 1) for i = 1, 2, · · · , 2N .From Lemma 10, we can get

f (x) =1

4N

2N∑

i=1

∣∣fHi x∣∣4 − 1

2‖x‖4

2 . (38)

We observe that the first term is the summation of 4th power(p = 4), and we can follow the trick as is mentioned inSection IV-A. First we apply Lemma 2:

14N

2N∑

i=1

∣∣fHi x∣∣4 ≤ 1

4N

2N∑

i=1

(ai

∣∣fHi x∣∣2 + bi

∣∣fHi x∣∣+ const

),

(39)where ai (≥ 0) and bi (≤ 0) follow the structure of a and b in

Lemma 2, and the corresponding x is(∑2N

i=1

∣∣fHi x(l)

∣∣4)1/4

.

With bi

∣∣fHi x∣∣ ≤ biRe

[x( l )H fi

|f Hi x( l ) | f

Hi x]

, we further have (leaving

out the constant and scaling factor in (39))

2N∑

i=1

(ai

∣∣fHi x∣∣2 + bi

∣∣fHi x∣∣)

≤2N∑

i=1

(aixH fifH

i x + biRe

[x(l)H fi∣∣fHi x(l)

∣∣ fHi x

])(40)

= xH Gx + Re[x(l)H Hx

]

where G =∑2N

i=1 aififHi and H =

∑2Ni=1

bi

|f Hi x( l ) | fif

Hi . A

second majorization is still needed (using Lemma 4):

xH Gx + Re[x(l)H Hx

] (a)≤ λvxH x

− 2Re[xH

(λv I − G − 1

2H)

x(l)]

+ const (41)

(b)= λvxH x − 2Re

[xH (λv I − S)x(l)

]+ const

(c)= λv ‖x‖2

2 − 2λvRe[yH

3 x]+ const


TABLE IIPROPOSED MAJORIZING FUNCTIONS

where (a) λv ≥ λmax (G), (b)

G +12H =

2N∑

i=1

(ai +

bi

2∣∣fH

i x(l)∣∣

)fifH

i

=2N∑

i=1

2∣∣∣fH

i x(l)∣∣∣2fifH

i = S, (42)

and (c)

y3 = x(l) − 1λv

Sx(l) . (43)

To conclude, the third majorizing function we propose is

f3

(x;x(l)

)=

14N

(λv ‖x‖2

2 − 2λvRe[yH

3 x])

− 12‖x‖4

2 + const. (44)

Remark 11: The energy budget constraint ‖x‖22 = c2

e makestwo terms of f3

(x;x(l)

)constant.

For clarity, we summarize all the aforementioned majorizingfunctions and their intended metrics in Table II.

Remark 12: We would like to provide more insight into thequartic term of f1 and f2 . Let’s start from that of f1 :


2

=12xH(λmax (L)xH x

)x

=12xH(λmax (L) · 11T � xH x

)x (45)

=12xH(mat (λmax (L)1) � xH x

)x,

where mat (L1) reshapes a length-N 2 vector into an N × Nmatrix (reverse operation of vec (·)); now f2 : from (73),we know 1

2 xH(E �

(xxH

))x = 1

2 xH(mat (L1) �

(xxH

))

x. Recall that when constructing f1 , we choose M =λmax(L)I = Diag(λmax(L)1) and f2 , M = Diag (L1). Thus,the quartic term of f1 and f2 can be unified as12 x

H(Z �

(xxH

))x where Z = mat (diag (M)).

V. MINIMIZATION SOLUTION DERIVATION

Now we move on to the minimization step. We shouldcarefully choose majorizing functions so that the minimiza-tion solution can be easily obtained. In the following, wewould like to specify the general constraint set X in (15)and discuss four sets of combinations of the constraints men-tioned in Table I. They are:X1 =

{x ∈ CN

∣∣∣ ‖x‖22 = c2

e

},X2 =

{x ∈ CN

∣∣∣ ‖x‖22 = c2

e

}∩ C3 , X3 =

{x ∈ CN

∣∣∣ ‖x‖22 = c2

e

}∩

C2 , and X4 ={x∈CN

∣∣∣ ‖x‖22 = c2

e

}∩ C1∩ (C4 , C5 optional).

Let’s look into the four cases one by one.

A. Fixed Energy Constraint Only (X = X1)

In this case, we adopt majorizing function f1(x;x(l)

)for

the WPISL metric, and f3(x;x(l)

)for the ISL metric. Since

λu , λv > 0 and ‖x‖22 = c2

e , the minimizing problems in bothscenarios reduce to:

minimizex

− Re[yH

j x]

subject to ‖x‖22 = c2

e (46)

where j = 1, 3. The (update) solution is

x(l+1) =ce

‖yj‖2yj , (47)

following the Cauchy-Schwartz inequality.

B. PAR Constraint With Fixed Energy (X = X2)


)for


)for the ISL metric. The

minimizing problem is:

minimizex

− Re[yH

j x]


e (48)

|xn | ≤ cp , ∀n

where j = 1, 3. The (update) solution is already given in [7,Algorithm 2] by using the Karush–Kuhn–Tucker (KKT) condi-tion. The phases of x(l+1) follow those of yj . Denote the numberof nonzero elements of yj as M (≤ N), and the set containingall the corresponding indexes as M. With ce/

√N ≤ cp , this

problem always has a feasible solution.� If Mc2

p < c2e ≤ Nc2

p , the solution is:

∣∣∣x(l+1)n

∣∣∣ ={

cp ∀n ∈ M,√c2

e −M c2p

N −M ∀n /∈ M;(49)

� If c2e ≤ Mc2

p , the solution is:∣∣∣x(l+1)

∣∣∣ = [β |yj |]cp

0 (50)

where β satisfies∥∥[β |yj |]cp

0

∥∥2 = ce (|·| denotes ele-

mentwise absolute value, [x]vv means projecting x el-ementwisely onto the interval [v, v]) . Observing thath1 (β) =

∥∥[β |yj |]cp

0

∥∥2 is a strictly increasing function

on[0,

cp

minn ∈M{|yj , n |}

], there is a unique β satisfying

h1(β)=ce .


C. ε-Uncertainty Constant Modulus Constraint With FixedEnergy (X = X3)


)for


)for the ISL metric. The

minimizing problem is:

minimizex

− Re[yH

j x]


e (51)

cm − ε1 ≤ |xn | ≤ cm + ε2 , ∀n

where j = 1, 3. The (update) solution is analogous with the pre-vious case. The phases of x(l+1) follow those of yj . M and Mfollow the same definitions as above. The problem has a feasi-ble solution if and only if N (cm − ε1)

2 ≤ c2e ≤ N (cm + ε2)

2 .The solution of

∣∣x(l+1)∣∣ is:

� If M(cm + ε2)2 + (N − M)(cm − ε1)2 < c2e≤N(cm +

ε2)2 , The solution is:

∣∣∣x(l+1)n

∣∣∣ ={

cm + ε2 ∀n ∈ M,√c2

e −M (cm +ε2 )2

N −M ∀n /∈ M;(52)

� If N(cm − ε1)2 ≤ c2e ≤ M(cm + ε2)2 + (N − M)(cm −

ε1)2 , the solution is:∣∣∣x(l+1)

∣∣∣ = [β |yj |]cm +ε2cm −ε1

(53)

where β satisfies ‖[β|yj |]cm +ε2cm −ε1

‖2 = ce . Observing thath2(β) = ‖[β|yj |]cm +ε2

cm −ε1‖2 is a strictly increasing function

on [ cm −ε1maxn ∈M{|yj , n |} ,

cm +ε2minn ∈M{|yj , n |} ], there is a unique β sat-

isfying h2 (β) = ce .

D. Strict Constant Modulus Constraint With MiscellaneousPhase Constraints (X = X4)


)and

f2(x;x(l)

)for the WPISL metric, and f3

(x;x(l)

)for the ISL

metric. Some of the nontrivial issues in minimizing f2(x;x(l)

)

are 1) λu −λl ≥ λmax (R) −λmin(E �

(x(l)x(l)H

)) (a)= λmax

(R) − c2m λmin (E)

(b)> 0 where (a) is due to Remark 9, and (b)

R [cf. (28)] and E [cf. (33)] are nonzero Hermitian matriceswith their diagonal entries being all zero (because w0 = 0); 2)12 x

H(E �

(xxH

))x = const, cf. Remark 9. The minimizing

problem is:minimize

x− Re

[yH

j x]

subject to |xn | = cm , ∀n (54)

arg (xn ) ∈ Φn , ∀n

where j = 1, 2, 3, and Φn is a phase constraint set for xn whichwill be specified later. The (update) solution is expressed ele-mentwisely: ∀n,

x(l+1)n = cm exp (jϕn ) (55)

with� C4 , C5 neither included, Φn = [0, 2π), ϕn = arg (yj,n );� C4 included, Φn = {φ1 , φ2 , · · · , φI }, ϕn = arg min{φi }

(|φi − arg (yj,n )|);� C5 included, Φn = [γn , γn + ϑ], ϕn = [arg (yj,n )]γn +ϑ

γn.

VI. CONNECTIONS WITH GRADIENT PROJECTION METHOD

In this section, we are going to show the connections with thegradient projection method. A gradient projection step takes thefollowing format:

x(l+1) = PX(x(l) − [step size] · ∇f (x)

), (56)

where the notation PX denotes projection onto X . The gradientof the WPISL metric f (x) =

∑N −1k=1 wk

∣∣xH Ukx∣∣p is given by

∇f (x) = Rx, (57)

with R given in (28), and that of the ISL metric f (x) =∑N −1k=1

∣∣xH Ukx∣∣2 is

∇f (x) =1

4NSx(l) − c2

ex(l) , (58)

with S given in (42). We will relate the gradient projection stepto our MM methods. Let’s start from the following theorem.

Theorem 13: The solution of the following problem can beexpressed as PX (y):

minimizex

− Re[yH x

]

subject to x ∈ X , (59)

where X is given in (15). For all c > 0, PX (cy) = PX (y)holds. In particular, whenX = X4 =

{x ∈ CN

∣∣∣ ‖x‖22 = c2

e

}∩

C1 ∩ (C4 , C5 optional), PX (c � y) = PX (y) holds with c =[c1 , c2 , · · · , cN ]T elementwise positive.

Proof: See Appendix C for the detailed proof. �When we minimize the majorizing function f1

(x;x(l)

), the

minimization solution is given as

x(l+1) = PX (y1)

= PX((

1 + λm a x (L)λu

∥∥x(l)∥∥2

2

)x(l) − 1

λuRx(l)

)(60)

= PX(x(l) − 1

λu +λm a x (L)c2e

∇f(x(l)))

.

When we minimize the majorizing function f2(x;x(l)

)over

X = X4 , the minimization solution is

x(l+1) = PX (y2)

= PX

((I − R−E�(x( l ) x( l )H )

λu −λl

)x(l))

= PX

(x(l) − 1

λu −λl∇f(x(l))

+E�(x( l ) x( l )H )

λu −λlx(l))

= PX((

1 + c2m

λu −λlE1)� x(l) − 1

λu −λl∇f(x(l)))

(a)= PX

(x(l) − 1

λu −λl

(1 + c2

m

λu −λlE1)−1

�∇f(x(l)))

,

(61)

where (a) λu − λl > 0, E is a nonnegative matrix, and (·)−1 isan elementwise inverse operator. When we minimize majorizing


function f3(x;x(l)

), the minimization solution is

x(l+1) = PX (y3)

= PX((

I − 1λv

S)x(l))

= PX(x(l) − 4N

λv

(∇f(x(l))

+ c2ex

(l)))

(62)

(a)= PX

(x(l) − 4N

λv −4N c2e

∇f(x(l)))

,

where (a) λv > 4Nc2e and the proof is given in Appendix D.

It is very explicit that PX (y1) and PX (y3) have the samestructure as the update step of the gradient projection method,butPX (y2) is different because a diagonal matrix is used duringthe construction of the majorizing function f2 . Basically, it isa tradeoff between two concerns. The first is the simplicity ofthe minimization problem after majorization (we want closed-form solutions) and the second is the goodness of the majorizingfunction when it is used to approximate the original objectivefunction. Depending on the constraint set, sometimes we canonly use majorizing functions of the forms of f1 and f3 to geta closed-form solution in each iteration, and for some otherconstraint sets we may choose the majorizing function f2 whichadmits a closed-form solution in each iteration and at same timeapproximates the original objective better.

In addition, when we adopt the MM method, the update solu-tion of projection on a nonconvex set has convergence guaran-tee, which results from the nature of the MM method. Moreover,the MM method proves to be superior in terms of convergencespeed judging from the simulation results in the simulation sec-tion. This is because the MM step size is automatically computedfrom the majorizing function and it is adaptively changing, whilefor the traditional gradient projection method, some backtrack-ing line search strategy needs to be used.

VII. ALGORITHMIC IMPLEMENTATION

In this section, we will look into the algorithmic implementa-tion. We observe that all the minimization solutions are closelyrelated to yj ’s (j = 1, 2, 3), and thus we need to compute themefficiently to reduce complexity and save CPU time.

A. Computation of y1

The expression of y1 is(1 + λm a x (L)

λu

∥∥x(l)∥∥2

2

)x(l) −

1λu

Rx(l) , and we need to compute λmax (L), λu , and

Rx(l) . Firstly, according to [6, Lemma 2], λmax (L) =maxk {wkak (N − k) |k = 1, 2, · · · , N − 1}, which is of com-plexity O (N).1 Then, we compute λu ≥ λmax (R) where R[cf. (28)] is a Toeplitz matrix. The value λu follows that in [6,Lemma 3]. The computation process needs 3 FFT(IFFT) oper-ations. In particular, for the ISL metric, only 1 FFT operationis needed. Lastly, we compute Rx(l) . This process addition-ally requires 1 IFFT. All the relevant details are elaborated inAppendix E.

1The computation of ak ’s is not counted for the moment. The ak ’s arereadily computed with (78) and (79) since they are functions of autocorrelationsrk

(x( l))

.

Claim 14: The overall process of computing y1 takes4 FFT(IFFT) for WPISL metric and 2 FFT(IFFT) for the ISLmetric, which is of complexity O (N log N).

B. Computation of y2

The expression of y2 is

(I +

E�(x( l ) x( l )H )λu −λl

)x(l) −

1λu −λl

Rx(l) , and we need to compute λu , λl , Rx(l) , and(E �

(x(l)x(l)H

))x(l) . With the previous subsection, we still

have to additionally compute λl and(E �

(x(l)x(l)H

))x(l) .

Because f2(x;x(l)

)is only used when each element of x(l) has

strict constant modulus cm , we can take advantage of this ex-tra property. Firstly, we compute λl ≤ λmin

(E �

(x(l)x(l)H

))

where E [cf. (33)] is a Toeplitz matrix. The value λl follows thatin [6, Lemma 3]. The computation process needs 1 FFT(IFFT)operations. Then, we compute

(E �

(x(l)x(l)H

))x(l) . Only 1

additional IFFT is needed per iteration. In particular, for theISL metric, no additional FFT(IFFT) operation is needed. Thedetails are elaborated in Appendix F.

Claim 15: The overall process of computing y2 takes 6FFT(IFFT) for WPISL metric and 2 FFT(IFFT) for the ISLmetric, which is of complexity O (N log N).

C. Computation of y3

The expression of y3 is x(l) − 1λv

Sx(l) , and we need to com-

pute λv and Sx(l) . Firstly, we compute λv ≥ λmax (G) whereG =

∑2Ni=1 aififH

i is also a Toeplitz matrix. The value λv canalso be derived from [6, Lemma 3]. The computation complex-ity of λv is O (N). Then, we compute Sx(l) where S is in (42).This process requires 2 FFT(IFFT). The details are elaboratedin Appendix G.

Claim 16: The overall process of computing y3 takes2 FFT(IFFT) for the ISL metric, which is of complexityO (N log N).

VIII. NUMERICAL SIMULATIONS

We give some numerical results in this section. All exper-iments were performed on a PC with a 3.20 GHz i5-4570CPU and 8GB RAM. We would like to specify the unifiedWPISL metric to be the ISL, WISL, and PSL to comparethe performance with various benchmarks and the gradientprojection method (if applicable). The Armijo step size ruleis used to implement the gradient projection method. Weinitialize the algorithms with either a random sequence inthe constraint set or some known sequence (Frank, Golomb,etc.). Since we are working on a nonconvex problem, initial-ization does affect the performance. Empirically, initiatingfrom an existing sequence endowed with low autocorrelationsidelobes could reach good performance. However, randominitialization may not do as well. The stopping criterion is:∣∣WPISL

(x(l+1)

)− WPISL

(x(l))∣∣ /max

(1,WPISL

(x(l)))

≤ Tol, where Tol is the tolerant precision. In case the stoppingcriterion is too harsh, we set the maximum number of iterationsto be MaxIter.

A. ISL Minimization

Set p = 2, wk = 1, ∀k, and we get the ISL metric. The firstexperiment is to present the ISL values or the merit factors(MF = N 2

2ISL ) of different algorithms under different sequence


Fig. 1. ISL value versus iterations.

Fig. 2. Merit factor versus sequence length.

lengths: N = 25 , 26 , · · · , 213 . The constraint set is (strict) uni-

modular constraint: X ={x ∈ CN

∣∣∣ |xn | = 1, ∀n}

. For the

proposed MM-based algorithm, we have 6 choices: majoriz-ing functions f1 , f2 , f3 with 2 acceleration techniques. Thebenchmark is the CAN algorithm, and the gradient projectionmethod is applicable here. The initial sequence is the Golombsequence, Tol = 10−8 , and MaxIter = 5 × 104 . In Fig. 1, weshow the convergence property of different algorithms. All thealgorithms display a monotonic decreasing property, and theSQUAREM-accelerated MM algorithms converge the fastest.Next, we would present the results of merit factor and weadditionally add a benchmark from a recent radar paper [28]. Theauthors provide an algorithm for designing space-time trans-mit code and receive filter. The algorithm alternately optimizestransmit code and receive filter, but it lacks theoretical guaran-tee for monotonic property and stationarity convergence thoughempirical convergence is observed. In Fig. 2, we see a significantincrease at all sequence lengths in merit factor except the radarbenchmark if the algorithms are initialized by some known se-quence. Both the MM-based algorithms and CAN can achieve

Fig. 3. Correlation level of two sequences: N = 256 and N = 4096.Majorizing function: f1 , accelerated by SQUAREM.

Fig. 4. Average running time versus sequence length.

high merit factor, and thus they are equally good. To see thesequence correlation level, we plot two examples from the firstexperiment in Fig. 3 (correlation level = 20 log10 |rk/r0 | , k =1 − N, · · · , N − 1.). The radar benchmark only has four datapoints due to its prohibitive computational cost, as will be shownlater.

The second experiment is to present the average runningtime of different algorithms under different sequence lengths:N = 25 , 26 , · · · , 213 . The initial sequence is a random unimod-ular sequence and we repeat 100 times to compute the averagerunning time. Tol = 10−8 , and MaxIter = 3 × 104 . The gradi-ent projection method is implemented with the Armijo step sizerule. The radar benchmark is required to initialize with some spe-cific sequence, so only one realization is carried out. In Fig. 4,we see that the radar benchmark always takes the most CPUtime and the consumption goes up to 103 when the sequencelength is merely 256, which reflects a prohibitive computationalcost in this algorithm. Also, the time consumption may notincrease with the sequence length, and this is because it takes


Fig. 5. ISL under different modulus constraints versus iterations.

many more iterations for the N = 25 case to converge than theN = 26 case. The gradient projection method takes the secondmost CPU time, which is 1–2 orders of magnitude slower thanthe CAN and MM methods. Some of the MM-based algorithms,i.e., those accelerated by SQUAREM, are faster than the existingCAN algorithm.

Remark 17: Up to this point, we have done the comparisonof our proposed algorithms and the methods in [5], [6]. It can bechecked that the first two proposed majorizing functions (f1 andf2) are variants of [5], [6], with [5] proposing f1 only and [6]proposing both. However, the third majorizing function (f3) isnewly proposed and not mentioned in [5], [6]. The comparisonis elaborated in Figs. 1, 2, and 4. As can be seen in Figs. 1 and2, the ISL level achieved by f3 is as good as that from f1 andf2 , which indicates that we can get as good results as [5], [6].Moreover, in Fig. 4, we observe that the convergence speed off3 is the fastest, 2–4 times as fast as [5] (proposing only f1)and 2–6 times as fast as [6] (proposing f1 and f2). In termsof time consumption, we can achieve even better results than[5], [6].

The third experiment is to show the ISL under miscella-neous constraints, i.e., modulus constraints and phase con-straints. In terms of modulus constraints, we have threedifferent types: strict constant modulus constraint (|xn | =cm ), ε-uncertainty constant modulus constraint (cm − ε1 ≤|xn | ≤ cm + ε2), and PAR constraint (|xn | ≤ cp ). We setN = 256, cm = 1, ε1 = ε2 = ε, and cp = cm + ε2 = 1 + ε.We use MM-based algorithm to compute the ISL, using ma-jorizing function f1 and SQUAREM acceleration. The initialsequence is the Golomb sequence, Tol = −10−8 (stoppingcriterion is deactivated), and MaxIter = 5 × 103 . In Fig. 5,we relax the unimodular constraint gradually as shown, andthere is remarkable ISL decrease even if the constraint setis relaxed a little bit. In terms of phase constraints, wealso have three different types: no phase constraint, dis-crete phase constraint (arg (xn ) ∈ {φ1 , φ2 , · · · , φI }), simi-larity constraint (arg (xn ) ∈ [γn , γn + ϑ], γn = arg (xr,n ) −arccos

(1 − δ2/2

), and ϑ = 2arccos

(1 − δ2/2

)). Note that all

the phase constraints are accompanied by the strict constantmodulus constraint. We set N = 256, cm = 1, φi = 2π

I (i − 1),i = 1, · · · , I , and the reference sequence xr is the Golomb

Fig. 6. ISL under different phase constraints versus iterations.

sequence. The MM-based algorithm is based on majorizingfunction f1 and local majorization acceleration. The initial se-quence is the Golomb sequence, Tol = −10−8 , and MaxIter =1.5 × 103 . In Fig. 6, we tightened the unimodular constraintgradually as shown, and we observe an increase in ISL. Anal-ogous result can be achieved when we optimize under othermetrics (WISL and PSL), and thus we do not replicate.

B. WISL Minimization

Set p = 2 only, and we get the WISL metric. We do one exper-iment is to present the convergence speed of different algorithmsunder N = 100. The weight is

wk ={

1 k ∈ {1, · · · , 20} ∪ {51, · · · , 70}0 otherwise, (63)

such that only r1 ∼ r20 and r51 ∼ r70 have small correlations.The constraint set is unimodular constraint. For the proposedMM-based algorithm, we have 4 choices: majorizing functionsf1 , f2 with 2 acceleration techniques. The benchmark is theWeCAN algorithm, and the gradient projection method is ap-plicable here. The initial sequence is a random unimodular se-quence, Tol = 10−10 , and MaxIter = 106 . In Fig. 7, we showthe monotonic decreasing property plotting WISL versus itera-tions. In Fig. 8, we see that all the algorithms can achieve WISLdown to 10−10 , but all the MM-based algorithms are much fasterthan the gradient projection method and the existing WeCANalgorithm. To see the sequence correlation level, we refer toFig. 9 to see one example of the converged sequence. We seethat the designed sequences have low autocorrelation sidelobesat the required lags.

C. PSL Minimization

Set p to be large, wk = 1, ∀k, and we approximately getthe PSL metric. The first experiment is to present the PSLof different algorithms under different sequence lengths: N =102 , 202 , 402 , 602 , 802 , 1002 . We set p = 100. The constraintset is unimodular constraint. For the proposed MM-based al-gorithm, we have 4 choices: majorizing functions f1 , f2 with 2acceleration techniques. We do not have benchmarks here and


Fig. 7. Weighted ISL versus iterations.

Fig. 8. Weighted ISL versus CPU time.

Fig. 9. Correlation level of the designed sequence. Majorizing function: f1 ,accelerated by SQUAREM.

Fig. 10. PSL value versus iterations.

Fig. 11. PSL versus sequence length. (G): initialized by the Golomb sequence,and (F): initialized by the Frank sequence.

the gradient projection method is not applicable because of thenumerical issue caused by large value of p. The initial sequencesare the Golomb sequence and the Frank sequence, Tol = 10−10 ,and MaxIter = 2 × 105 . In Fig. 10, we show the monotonic de-creasing property plotting PSL versus iterations when N = 400and the initial sequence is the Frank sequence. In Fig. 11, wesee a significant decrease in PSL if the algorithm is initializedby some known sequence.

The second experiment is to present the convergence speedof different values of p: p = 10, 102 , 103 , 104 . The initial se-quence is the Frank sequence, N = 400, Tol = −10−10 , andMaxIter = 5 × 104 . In Fig. 12, we see that when p is small(p = 10), the convergence speed is fast, but the converged PSLis high; when p is large (p = 10000), the situation is right theopposite. Therefore, we can adopt an increasing scheme of pto get low PSL and fast convergence speed. We increase p as2, 22 , · · · , 213 . The initial sequence is the Frank sequence withN = 1002 , Tol = 10−5/p for each p, and MaxIter = 5 × 103 .


Fig. 12. PSL versus iterations (majorizing function: f1 , accelerated bySQUAREM).

Fig. 13. Correlation level of the Frank and designed sequence (majorizingfunction: f1 , accelerated by SQUAREM).

We plot the sequence correlation level in Fig. 13 compared withthe initial Frank sequence. The autocorrelation sidelobes of theFrank sequence become larger for k close to 0 and N − 1, whilethose of the designed sequence are much more uniform acrossall lags.

D. Metric Comparison: ISL and PSL

In this subsection, we compare the results of different met-rics, and we focus on ISL and PSL. The designed sequenceis unimodular. Note that it is not a good idea to optimize theweighted sum of ISL and PSL because in this case the pth orderterm in PSL will dominate the objective, implicitly suppress-ing the effect of the ISL term. If we want to achieve a tradeoffbetween ISL and PSL, we may as well tune the order parame-ter p. In Fig. 14, we plot the optimized ISL and PSL value withrespect to different p, ranging from 2 to 128. The initial sequenceis chosen as the Frank sequence, N = 256, Tol = 10−10 , andMaxIter = 3 × 104 . The majorizing function is chosen as f1 ,accelerated by SQUAREM. The best tradeoff is achieved when

Fig. 14. (Optimized) ISL and PSL levels versus order parameter p.

p lies between 8 and 16, where the level of PSL is already lowenough and that of ISL is still not too high.

IX. CONCLUSION

We have proposed a unified framework to design low auto-correlation sequences. We have optimized a unified metric overa general constraint set. We have carried out the MM methodin two stages. In the majorizing function construction stage,we have constructed three majorizing functions. Two of themapply to the unified WPISL metric, and the remaining one ap-plies to the specific ISL metric. In the minimization solutionderivation stage, we have provided closed-form solutions to dif-ferent minimization problems. Additionally, we have shown theconnections between the MM and gradient projection methodunder our algorithmic scheme. Thereafter, we efficiently im-plement the MM step with FFT (IFFT) operations. Numericalsimulations have shown that the proposed MM-based algorithmsconverge faster than the traditional gradient projection methodand the state-of-the-art algorithms.

APPENDIX APROOF OF LEMMA 5

Proof: We apply Lemma 4 with M0 = L and M =λmax (L) I, obtaining (constant terms are represented withconst for simplicity)

12vecH (X) · L · vec (X)

≤ 12λmax (L) vecH (X) vec (X) + Re

[vecH

(X(l)

)· (64)

(L − λmax (L) I) vec (X)]

+ const.

Now we recover X and X(l) as xxH and x(l)x(l)H , respectively:

N −1∑

k=1

wkak

∣∣xH Ukx∣∣2 ≤ 1

2λmax (L) ‖x‖4

2

+ xH(A − λmax (L)x(l)x(l)H

)x + const,

(65)


where A is defined as

A =N −1∑

k=1−N

wkakr−k

(x(l))Uk = AH . (66)

We add 12 x

H Bx to (65):

N −1∑

k=1

wkak |rk (x)|2 +12xH Bx ≤ 1

2λmax (L) ‖x‖4

2

+ xH

(A +

12B − λmax (L)x(l)x(l)H

)x + const, (67)

where A + 12 B can be further simplified [cf. (66) and (24)]:

A +12B =

N −1∑

k=1−N

wk

(ak + b k

2 |r −k (x ( l ) )|)r−k

(x(l))Uk

=N −1∑

k=1−N

wk

(ak +

p |r k (x ( l ) )|p −1−2 a k |r k (x ( l ) )|

2 |r −k (x ( l ) )|)r−k

(x(l))Uk

=N −1∑

k=1−N

p

2wk

∣∣∣rk

(x(l))∣∣∣

p−2r−k

(x(l))· Uk = R. (68)

�

APPENDIX BPROOF OF LEMMA 8

Proof: We start from the following:

12vecH (X) · L · vec (X)

≤ 12vecH (X) Diag (L1) vec (X) + Re

[vecH

(X(l)

)· (69)

(L − Diag (L1)) vec (X)]

+ const.

Then we recover X and X(l) as xxH and x(l)x(l)H , respectively.We do this term by term:

12vecH (X) Diag (L1) vec (X)

=12vecH (X) ((L1) � vec (X))

(a)=

12Tr(xxH mat ((L1) � vec (X))

)

=12xH(mat (L1) �

(xxH

))x

(b)=

12xH(E �

(xxH

))x,

(70)

Re[vecH

(X(l)

)Lvec (X)

]= xH Ax, (71)

Re[vecH

(X(l)

)Diag (L1) vec (X)

]

(c)= xH

(E �

(x(l)x(l)H

))x (72)

where (a) mat(·) is the inverse operation of vec(·), (b)

E = mat (L1) = mat

(N −1∑

k=1−N

wkakvec (U−k ) vecH (U−k )1

)

= mat

(N −1∑

k=1−N

wkak (N − |k|) vec (U−k )

)

=N −1∑

k=1−N

wkak (N − |k|)U−k = EH , (73)

and (c) follow (70). Therefore,

N −1∑

k=1

wkak

∣∣xH Ukx∣∣2 ≤ 1

2xH(E �

(xxH

))x

+ xH(A − E �

(x(l)x(l)H

))x + const. (74)

We add 12 x

H Bx to (74):

N −1∑

k=1

wkak |rk (x)|2 +12xH Bx ≤ 1

2xH(E �

(xxH

))x

+ xH(R − E �

(x(l)x(l)H

))x + const. (75)

�

APPENDIX CPROOF OF THEOREM 13

Proof: The problem (59) is equivalent to the following onedue to the blanket constraint ‖x‖2

2 = c2e :

minimizex

12‖x − y‖2

2

subject to x ∈ X , (76)

which can be interpreted as a projection problem.Judging from (59), it is obvious that PX (cy) = PX (y)

for c > 0. When X = X4 , we simplify it as X4 ={x ∈ CN

∣∣∣ |xn | = cm , arg (xn ) ∈ Φn ,∀n}

where Φn is a

phase constraint set for xn . We express PX (y) elementwiselyas [PX (y)]n = cm exp (jϕn (yn )), ∀n.

� C4 , C5 neither included, Φn =[0, 2π), ϕn (yn )=arg(yn )= arg(cnyn ) = ϕn (cnyn ), thus [PX (y]n = [PX (Diag(c)y)]n ;

� C4 included, Φn ={φ1, φ2 , · · ·, φI }, ϕn (yn )=argmin{φi }(|φi − arg(yn )|) = arg min{φi }(|φi − arg(cnyn )|)= ϕn

(cnyn ), thus [PX (y)]n = [PX (Diag(c)y)]n ;� C5 included, Φn = [γn , γn + ϑ], ϕn (yn ) = [arg(yn )

]γn +ϑγn

= [arg(cnyn )]γn +ϑγn

= ϕn (cnyn ), thus [PX (y)]n =[PX (Diag(c)y)]n .

Therefore, PX (c � y) = PX (y) for any c elementwisepositive. �

APPENDIX DPROOF OF THE CLAIM λv > 4Nc2

e

Proof: The proof is given as follows: λv = 12

(max1≤i≤N 2Na2i+max1≤i≤N 2Na2i−1) ≥∑2N

i=1 ai =∑2N

i=1


x4 −|f Hi x( l ) |4 −4|f H

i x( l ) |3 (x−|f Hi x( l ) |)

(x−|f Hi x( l ) |)2 with x=(

∑2Ni=1 |fH

i x(l) |4)1/4 .

We know that x4 −|f Hi x( l ) |4 −4|f H

i x( l ) |3 (x−|f Hi x( l ) |)

(x−|f Hi x( l ) |)2 = 3|fH

i x(l) |2 +

2|fHi x(l) |x + x2 ≥ 3|fH

i x(l) |2 + x2 . Also, x = (∑2N

i=1 |fHi

x(l) |4)1/4 ≥ (2N)1/4( 12N

∑2Ni=1 |fH

i x(l) |2)1/2 = (2N)1/4ce .

Then, λv ≥ 3∑2N

i=1 |fHi x(l) |2 + 2N

√2Nc2

e=(6N+2N√

2N)c2e > 4Nc2

e . �

APPENDIX ECOMPUTATION PROCESS OF λu AND Rx(l)

From [29] and [6, Lemma 3], we see that in order to sat-isfy λu ≥ λmax(R), λu can be chosen as 1

2 (max1≤i≤N μ2i +max1≤i≤N μ2i−1) with μ= Fc where F∈C2N ×2N is a2N−DFT matrix with Fmn = exp(−j 2π (m−1)(n−1)

2N ), 1 ≤ m,

n ≤ 2N , and c = [c0 , c1 , · · · , c2N −1 ]T with

ck =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

0 k = 0, Np2 wk

∣∣x(l)H Ukx(l)∣∣p−2 ·

(x(l)H Ukx(l)

) k = 1, · · · , N − 1

c∗2N −k k = N + 1, · · · , 2N − 1.(77)

In short, μ can be computed as μ = FFT (c). Moreover,the vector c can also be computed with FFT. We define r =[r0(x(l)), r1(x(l)), · · · , rN −1

(x(l)), 0, r∗N −1

(x(l)), · · · ,

r∗1(x(l)) ]T

. Since r is comprised of the autocorrelations ofx(l) , it can be efficiently computed via FFT:

r = IFFT(|t|2)

, (78)

where

t = FFT([

x(l)T ,0TN ×1

]T )(79)

and |·|p denotes the elementwise absolute value to the pth power.Here we need 1 FFT and 1 IFFT. Knowing r, we can computec with simple Hadamard product:

c =p

2w � |r|p−2 � r, (80)

where w = [0, w1 , · · · , wN −1 , 0, wN −1 , · · · , w1 ]T , of com-

plexity O (N). To sum up, the computation process is:

t = FFT([

x(l)T ,0TN ×1

]T )

r = IFFT(|t|2)

μ = FFT(p

2w � |r|p−2 � r

)(81)

λu =12

(max

1≤i≤Nμ2i + max

1≤i≤Nμ2i−1

),

3 FFT(IFFT) operations. When p = 2 and wk = 1, ∀k, i.e., forthe ISL metric, we have μ = |t|2 and only 1 FFT operation isneeded.

The Toeplitz matrix can be expressed as R =1

2N FH:,1:N Diag (μ)F:,1:N (F:,1:N stands for the first N columns

of F, the 2N−DFT matrix). Then,

Rx(l) =[

12N

FH Diag (μ)F[

x(l)

0N ×1

]]

1:N(82)

= [IFFT (μ � t)]1:N ,

where [·]1:N means taking the first N elements of a vector, andt follows (79). Here we additionally need 1 IFFT.

APPENDIX FCOMPUTATION PROCESS OF λl AND (E � (x(l)x(l)H ))x(l)

We have λmin(E � (x(l)x(l)H )) = c2m λmin(E), cf. Remark

9. From [29] and [6, Lemma 3], we see that in order to sat-isfy λl ≤ c2

m λmin(E), λl is set to be c2m · 1

2 (min1≤i≤N ν2i +min1≤i≤N ν2i−1) where ν = Fu, F is the 2N−DFT matrix,and u = [u0 , u1 , · · · , u2N −1 ]T with

uk =

⎧⎪⎨

⎪⎩

0 k = 0, N

wkak (N − |k|) k = 1, · · · , N − 1

u2N −k k = N + 1, · · · , 2N − 1.

(83)

In short, ν can be computed as ν = FFT (u).Next, we observe

(E �

(x(l)x(l)H

))x(l) = Diag

(x(l))· E ·

DiagH(x(l))x(l) = c2

m (E1) � x(l) , where E1 can also be im-plemented with FFT:

E1 =

[1

2NFH Diag (ν)F

[1N ×1

0N ×1

]]

1:N

=[IFFT

(ν � FFT

([1T

N ×1 ,0TN ×1]T ))]

1:N. (84)

Since FFT([

1TN ×1 ,0

TN ×1

]T )only needs to be computed once,

so only 1 additional IFFT is needed per iteration. When p = 2and wk = 1, ∀k, i.e., the ISL metric, u is a constant vector(ak = 1, ∀k), and thus ν and E1 are both constant vectors,which means no additional FFT(IFFT) operation is needed.

APPENDIX GCOMPUTATION PROCESS OF λv AND Sx(l)

We understand that G =∑2N

i=1 aififHi = 1

2N FH:,1:N Diag

(2Na)F:,1:N . Following [29] and [6, Lemma 3], λv is set tobe N(max1≤i≤N a2i + max1≤i≤N a2i−1), which is of complex-ity O (N).

Then, we compute Sx(l) , where S=∑2N

i=1 2|fHi x(l) |2

fifHi = 1

2N FH:,1:N Diag(2Nv)F:,1:N and v = 2|FFT([x(l)T ,

0TN ×1 ]

T )|2 = 2|t|2 (t [cf. (79)]). So 1 FFT for computing tis needed. Thus,

Sx(l) =[

12N

FH Diag (2Nv)F[

x(l)

0N ×1

]]

1:N

= [IFFT (2Nv � t)]1:N (85)

v=2|t|2= 4N

[IFFT

(|t|2 � t

)]

1:N.

Here comes 1 IFFT.


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Licheng Zhao received the B.S. degree in informa-tion engineering from Southeast University, Nanjing,China, in 2014. He is currently working toward thePh.D. degree in the Department of Electronic andComputer Engineering, Hong Kong University ofScience and Technology, Hong Kong. His researchinterests include optimization theory and fast algo-rithms, with applications in signal processing, ma-chine learning, and financial engineering.

Junxiao Song received the B.Sc. degree in controlscience and engineering from Zhejiang University,Hangzhou, China, in 2011, and the Ph.D. degree inelectronic and computer engineering from the HongKong University of Science and Technology, HongKong, in 2015.

His research interests include convex optimizationand efficient algorithms, with applications in big data,signal processing, and financial engineering.

Prabhu Babu received the Ph.D. degree in electrical engineering from theUppsala University, Sweden, in 2012. From 2013–2016, he was a Post-DoctorialFellow with the Hong Kong University of Science and Technology. He is cur-rently with the Center for Applied Research in Electronics (CARE), IndianInstitute of Technology Delhi, New Delhi, India.

Daniel P. Palomar (S’99-M’03-SM’08-F’12) re-ceived the Electrical Engineering and Ph.D. de-grees from the Technical University of Catalonia,Barcelona, Spain, in 1998 and 2003, respectively.

He is a Professor in the Department of Electronicand Computer Engineering, Hong Kong Universityof Science and Technology (HKUST), Hong Kong,which he joined in 2006. Since 2013, he has been aFellow of the Institute for Advance Study, HKUST.He had previously held several research appoint-ments, namely, at King’s College London, London,

U.K.; Stanford University, Stanford, CA, USA; Telecommunications Techno-logical Center of Catalonia, Barcelona, Spain; Royal Institute of Technology,Stockholm, Sweden; University of Rome “La Sapienza,” Rome, Italy; andPrinceton University, Princeton, NJ, USA. His current research interests in-clude applications of convex optimization theory, game theory, and variationalinequality theory to financial systems, big data systems, and communicationsystems.

Dr. Palomar received the 2004/06 Fulbright Research Fellowship, the 2004and 2015 (coauthor) Young Author Best Paper Awards by the IEEE Signal Pro-cessing Society, the 2015–16 HKUST Excellence Research Award, the 2002/03best Ph.D. prize in Information Technologies and Communications by the Tech-nical University of Catalonia (UPC), the 2002/03 Rosina Ribalta first prize forthe Best Doctoral Thesis in Information Technologies and Communications bythe Epson Foundation, and the 2004 prize for the best Doctoral Thesis in Ad-vanced Mobile Communications by the Vodafone Foundation and COIT. He isa Guest Editor of the IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESS-ING 2016 Special Issue on Financial Signal Processing and Machine Learningfor Electronic Trading and has been Associate Editor of IEEE TRANSACTIONS

ON INFORMATION THEORY and of IEEE TRANSACTIONS ON SIGNAL PROCESS-ING, a Guest Editor of the IEEE Signal Processing Magazine 2010 SpecialIssue on Convex Optimization for Signal Processing, the IEEE JOURNAL ON

SELECTED AREAS IN COMMUNICATIONS 2008 Special Issue on Game Theoryin Communication Systems, and theIEEE JOURNAL ON SELECTED AREAS IN

COMMUNICATIONS 2007 Special Issue on Optimization of MIMO Transceiversfor Realistic Communication Networks.

Documents

A Uniﬁed Framework for Low Autocorrelation Sequence Design ... · Abstract—In this paper, we consider the low autocorrelation sequence design problem. We optimize a uniﬁed metric