A Nonlinear Sum of Squares Search for CAZAC Sequences Mark Magsino

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A Nonlinear Sum of Squares Search for

CAZAC Sequences

Mark Magsino

Department of Mathematics

U.S. Naval Academy

magsino@usna.edu

Yixin Xu

Department of Mathematics

Ohio State University

xu.3520@buckeyemail.osu.edu

Abstract—We report on a search for CAZAC se-

quences by using nonlinear sum of squares optimiza-

tion. Up to equivalence, we found all length 7 CAZAC

sequences. We obtained evidence suggesting there are

ﬁnitely many length 10 CAZAC sequences with a total

of 3040 sequences. Last, we compute longer sequences

and compare their aperiodic autocorrelation proper-

ties to known sequences. The code and results of this

search are publicly available through GitHub.

I. INTRODUCTION

Given x∈Cn, we deﬁne the periodic ambiguity

function by

Ap(x)[k, `] := 1

n−1

j=0

xj+kxje−2πij`/n,(1)

where 0≤k, ` ≤n−1, and indices are taken

modulo n. The periodic autocorrelation is the case

with no frequency shift, i.e. when `= 0. In this

case it is convenient to suppress the second input

and write

Ap(x)[k] := Ap(x)[k, 0] = 1

n−1

j=0

xj+kxj.(2)

ACAZAC sequence of length nas a vector x∈

Cnwith the properties

(i) |xj|= 1 for 1≤j≤n,

(ii) Ap(x)[k]=0for 1≤k≤n−1.

The ﬁrst property is known as the constant ampli-

tude property and the second is known as the zero

autocorrelation property, which gives rise to the

acronym CAZAC.

These sequences have several interpretations and

applications which motivate their study. In com-

munication theory they have been used to reduce

the cross-correlation of signals, for uplink synchro-

nization [9], and OFDM for 5G communication

[7][15]. It is also studied as an idealized waveform

with regards to the narrowband ambiguity function

in radar [14]. In general, the constant amplitude

allows one to encode information purely in terms of

phase and the zero autocorrelation ensures no inter-

ference with shifted copies of the signal. CAZAC

sequences are also known as perfect polyphase

sequences and are also well studied under that name

[11][12][13].

CAZAC sequences have perfect periodic autocor-

relation, but many applications require their ape-

riodic autocorrelation. The aperiodic ambiguity

function of a sequence x∈Cnis given by

Aa(x)[k, `] :=

n−1

j=1

x(a)

j+kx(a)

je−2πij`/n,(3)

where 0≤k, ` ≤n−1and x(a)

jis deﬁned by

x(a)

j=(xj,if 0≤j≤n−1,

0,otherwise.(4)

The key difference is instead of taking indices

modulo n, we set the value to zero when the index

falls outside of the usual range. The aperiodic

autocorrelation corresponds to `= 0 and it is

convenient to write

Aa(x)[k] := Aa[k, 0] =

n−1

j=1

x(a)

j+kx(a)

j.(5)

arXiv:2210.14827v2 [cs.IT] 2 Nov 2022

In particular, we use aperiodic autocorrelation to

study two properties of interest. We deﬁne the peak

sidelobe level (PSL) of x∈Cnby

PSL(x) = 1

|Aa(x)[k]|max

k6=0 |Aa[k]|,(6)

and the integrated sidelobe level (ISL) by

ISL(x) = 1

|Aa(x)[k]|2

n−1

k=1 |Aa[k]|2.(7)

Although CAZAC sequences are well studied,

many things about them are still unknown. Given

two CAZAC sequences, xand y, we say they are

equivalent if there exists a complex scalar cwith

|c|= 1 so that y=cx and make the representative

of the equivalence class the sequence whose ﬁrst

entry is 1. With this in mind, it is natural to

ask: For each n, how many CAZAC sequences

of length nare there? As a partial answer, it is

known that if nis prime, then there are at most

2n−2

n−1CAZAC sequences [8]. If nis composite

and divisible by a perfect square, then there are

inﬁnitely many sequences [6][10]. If nis composite

and not divisible by any perfect square, then it

is unknown how many there are. A brute force

calculation veriﬁes that there are ﬁnitely many for

n= 6 [5]. Beyond that, it is currently unknown.

There are additional transformations under which

CAZAC sequences are closed [3]. They are a ﬁnite

set of transformations so it does not fundamentally

change the question of whether the set of CAZAC

sequences of a given length is ﬁnite. We use these

to ﬁlter out known length 7 CAZAC sequences.

Proposition 1. Let x∈Cnbe a CAZAC sequence

and let ω=e2πi/n. Then, the following sequences

are also CAZAC sequences:

(i) (Tkx)j=xj+k,0≤k≤n−1,

(ii) (M`x)j=ω`j xj,0≤`≤n−1,

(iii) (Dmx)j=xmj ,gcd(m, n) = 1,

(iv) (x)j=xj.

II. CAZAC SEQUNECES OF LENGTH 7

The CAZAC sequences of length 7 can be split

into quadratic phase sequences and non-quadratic

phase sequences. Suppose x∈Cnis deﬁned by

xj=eπip(j)/n,

where p(j)is a quadratic polynomial. In this case,

we say that xis a quadratic phase sequence. The

polynomials associated with the known quadratic

phase CAZAC sequences are

Zadoff-Chu: p(j) = j(j−1),(nodd)

P4: p(j) = j(j−n).

Wiener: p(j)=2kj2,(gcd(k, n) = 1, n odd),

p(j) = kj2,(gcd(k, 2n) = 1, n even).

When nis prime, there are at least n(n−1) CAZAC

sequences comprised of roots of unity, including the

quadratic phase sequences [2]. When n= 7, this

gives at least 42 roots of unity sequences. Moreover,

the transformations described in Proposition 1 will

keep the sequence a root of unity sequence. On

the other hand, in [4] Bj¨

orck constructed CAZAC

sequences comprised of non roots of unity for each

prime p > 5. The construction is as follows.

Given an odd prime p, let j

pdenote the Leg-

endre symbol deﬁned by

j

p=









0,if j≡0 mod p,

1,if j≡x2mod p, for some x6= 0,

−1,if j6≡ x2mod p, for any x6= 0.

We deﬁne the Bj¨

orck sequence of length pby

xj=eiθ(j),0≤j≤p−1,(8)

where if p≡1 mod 4, then θ(j)is given by

j

parccos 1

1 + √p,(9)

and if p≡3 mod 4, then θ(j)is given by

θ(j) = (arccos 1−p

1+pif j

p=−1,

0,otherwise.(10)

Since 7≡3 mod 4, the Bj¨

orck sequence of length

7 is

x= (1,1,1, eiθ7,1, eiθ7, eiθ7),(11)

where θ7= arccos(3/4). Since CAZAC sequences

are closed under the operations outlined in Propo-

sition 1 the Bj¨

orck sequence generates up to 252

CAZAC sequences which begin with 1 when p =

7. Some combinations of transformations result in

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ANonlinearSumofSquaresSearchforCAZACSequencesMarkMagsinoDepartmentofMathematicsU.S.NavalAcademymagsino@usna.eduYixinXuDepartmentofMathematicsOhioStateUniversityxu.3520@buckeyemail.osu.eduAbstractWereportonasearchforCAZACse-quencesbyusingnonlinearsumofsquaresoptimiza-tion.Uptoequivalence,wefoundalll...

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A Nonlinear Sum of Squares Search for CAZAC Sequences Mark Magsino

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