# Sequence pairs with asymptotically optimal aperiodic correlation

The Pursley-Sarwate criterion of a pair of finite complex-valued sequences measures the collective smallness of the aperiodic autocorrelations and the aperiodic crosscorrelations of the two sequences. It is known that this quantity is always at least 1 with equality if and only if the sequence pair is a Golay pair. We exhibit pairs of complex-valued sequences whose entries have unit magnitude for which the Pursley-Sarwate criterion tends to 1 as the sequence length tends to infinity. Our constructions use different carefully chosen Chu sequences.

## Authors

• 1 publication
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## 1. Introduction and Results

In this paper, a sequence of length is a finite sequence of complex numbers; such a sequence is called binary if each entry is or and it is called unimodular if each entry has unit magnitude.

Let and be two sequences of length , where und contain at least one nonzero entry. The aperiodic crosscorrelation of and at shift is defined to be

 CA,B(u)=∑j∈Zaj¯¯¯¯¯¯¯¯¯bj+u,

with the convention that if or . The aperiodic autocorrelation of at shift is then .

There is sustained interest in sequences with small correlation (see [15] for a survey of recent developments), mainly because small correlation helps to separate a useful signal from noise or unwanted signals. In particular, small crosscorrelation is usually required to ensure that sequences can be distinguished well from each other and small autocorrelation is usually required to keep the transmitter and the receiver synchronised.

The collective smallness of the aperiodic crosscorrelations of the sequences and is measured by the crosscorrelation demerit factor of and , which is defined to be

 CDF(A,B)=∑u∈Z|CA,B(u)|2CA,A(0)⋅CB,B(0).

Accordingly, the collective smallness of the aperiodic autocorrelations of is measured by the autocorrelation demerit factor of , which is defined to be

Notice that if and are unimodular. The reciprocals of and are known as the crosscorrelation merit factor of and and the autocorrelation merit factor of , respectively.

A fundamental relationship between the three quantities , , and is given by

as proved by Pursley and Sarwate [13] for binary sequences and generalised by Katz and Moore [8] for general sequences. Following Boothby and Katz [2], we define the Pursley-Sarwate criterion of  and  to be

From (1) we obtain . Hence in order to design pairs of sequences with simultaneously small aperiodic autocorrelations and crosscorrelations, we would like to have close to .

Katz [7] and Boothby and Katz [2] studied the Pursley-Sarwate criterion of sequence pairs derived from m-sequences and Legendre sequences and generalisations of them. This gives pairs of unimodular and binary sequences whose Pursley-Sarwate criterion is close to , but strictly bounded away from , as the sequence length tends to infinity. (Unlike claimed in [7], the crosscorrelation demerit factors of pairs obtained from Legendre sequences and m-sequences have been studied before, namely in [16, Theorem 3] and [6, Theorem 9].)

In fact, pairs of unimodular sequences with

were recently classified by Katz and Moore

[8] to be exactly the Golay pairs, namely pairs of unimodular sequences satisfying

 CA,A(u)+CB,B(u)=0for all u≠0.

Golay pairs of unimodular sequences are known to exist for infinitely many, though not for all, lengths (see [4] for the existence for small lengths). The classification in [8] does however not say anything about the individual quantities , , and for a Golay pair . These values are known for the Rudin-Shapiro pairs, which are Golay pairs of binary sequences of length and satisfy

(see [15], for example). Since , we have

 CDF(Am,Bm)=13(2+(−1/2)m).

Therefore, as , we have and and .

In this paper we exhibit unimodular sequences whose Pursley-Sarwate criterion is asymptotically and for which we can control the autocorrelation and crosscorrelation demerit factors. Our results involve Chu sequences [3], which are unimodular sequences of length  of the form

 Z(a)n=(z0,z1,…,zn−1),zj=eπiaj2/n,

where  is an integer (Chu [3] used a slightly different definition when

is odd).

Several authors [11][17][10] have shown independently that grows like  and the exact constant has been determined by the second author [14] by showing that

which confirms previous numerical evidence obtained by Littlewood [9] and Bömer and Antweiler [1]. In fact, since tends to zero, this immediately implies that

 limn→∞PSC(Z(1)n,Z(1)n)=1.

However, the pair would be a bad choice when good crosscorrelation is required since the crosscorrelation at the zero shift equals the sequence length . This problem is avoided by taking the pair . Indeed it can be shown using Lemma 6 that in this case the crosscorrelation at the zero shift is of the order . Our first result is the following theorem.

###### Theorem 1.

For each , let and be Chu sequences of length . Then, as ,

1. , ,

2. ,

3. .

In our second result we construct a pair of unimodular sequences from two Chu sequences of even length such that the Pursley-Sarwate criterion is asymptotically and the autocorrelation and crosscorrelation demerit factors are asymptotically balanced, which means that they all tend to the same constant .

###### Theorem 2.

For each , let and be Chu sequences of length . Then, as ,

1. , ,

2. ,

3. .

We shall prove the first parts of Theorems 1 and 2 in Section 2 and the second parts in Section 3. Of course the third parts are trivial consequences of the first two parts.

## 2. Autocorrelation of Chu sequences

In this section we prove our results on the autocorrelations of the Chu sequences in question. We begin with a lemma, which gives an expression for the autocorrelation demerit factor of Chu sequences.

###### Lemma 3.

Let and be integers and write . Then

where if and otherwise.

###### Proof.

Write , , and . Straightforward manipulations give

 |CZ,Z(u)|=|CZ,Z(−u)|=∣∣∣n−u−1∑j=0e2πibuj/m∣∣∣.

for . Hence, if is not a multiple of , then

 |CZ,Z(u)|=∣∣∣sin(πbu2/m)sin(πbu/m)∣∣∣.

Therefore

since for and for even . Finally, note that

 d−1∑k=1|CZ,Z(km)|2 =d−1∑k=1(n−km)2 =m2d−1∑k=1k2 =n26d(d−1)(2d−1),

which completes the proof. ∎

The second author obtained (2) from Lemma 3 with and the first identity in the following lemma.

###### Lemma 4 ([14]).

We have

 limn→∞1n3/2∑1≤u≤n/2(sin(πu2/n)sin(πu/n))2 =12π, limn→∞1n3/2∑1≤u≤n/2(sin(πu2/n)πu/n)2 =12π.

From Lemmas 3 and 4 we also obtain

which implies Theorem 1 (i). We now use Lemmas 3 and 4 to prove Theorem 2 (i).

###### Proof of Theorem 2 (i).

We distinguish the cases that runs through the set of even and odd positive integers. First we show that

It is readily verified that the aperiodic autocorrelations of and have equal magnitudes at all shifts, so that it is sufficient to establish that . Noting that is coprime to and using trigonometric addition formulas, Lemma 3 with and shows that equals

 2m∑u=1u even(sin(πu2/(4m))sin(πu/(4m)))2+2m∑u=1u odd(cos(πu2/(4m))cos(πu/(4m)))2.

By Lemma 4, the first sum is , so that it is sufficient to show that

 (4) limm→∞1m2m∑v=1(cos((π(2v−1)2/(4m))cos(π(2v−1)/(4m)))2=2.

Let be a real number satisfying . From the Taylor series of at we find that

 −(x−π/2)+16(x−π/2)3

from which it follows that

 0<1(cosx)2−1(x−π/2)2<1.

Therefore,

 ∣∣∣m∑v=1(cos(π(2v−1)2/(4m))cos(π(2v−1)/(4m)))2−m∑v=1(cos(π(2v−1)2/(4m))π(2v−1)/(4m)−π/2)2∣∣∣

Put to obtain

 m∑v=1(cos(π(2v−1)2/(4m))π(2v−1)/(4m)−π/2)2=m∑w=1(cos(π(2w−1)2/(4m))π(2w−1)/(4m))2.

Apply to the right hand side and use Lemma 4 to obtain

 limm→∞1m2m∑w=1(cos(π(2w−1)2/(4m))π(2w−1)/(4m))2 =16π2∞∑w=11(2w−1)2 =16π2(∞∑w=11w2−∞∑w=11(2w)2) =12π2∞∑w=11w2=2,

using Euler’s evaluation . This gives (4), and so completes the proof of (3).

Next we prove that

Note that , so that Lemma 3 with and gives

On the other hand, , so that Lemma 3 with and gives

Comparing the two preceding equations and using

we conclude that . To complete the proof, we show that

 (6) limm→∞1m2m∑u=1(sin(πmu2/(2m+1))sin(πmu/(2m+1)))2=0.

For even , we have

so that

 m∑u=1u even(sin(πmu2/(2m+1))sin(πmu/(2m+1)))2=O(m3/2)

by Lemma 4. For odd , we have

so that

 m∑u=1u odd(sin(πmu2/(2m+1))sin(πmu/(2m+1)))2=m∑u=1u odd(cos(πu2/(4m+2))cos(πu/(4m+2)))2.

The summands on the right hand side are at most , so that the entire sum is at most . This proves (6), and so completes the proof of (5). ∎

## 3. Crosscorrelation of Chu sequences

In this section we prove our results on the crosscorrelations of the Chu sequences in question. A straightforward computation shows that the modulus of the aperiodic crosscorrelations between two Chu sequences of equal length are given by generalised Gauss sums, which are for real and and integral defined to be

 SN(x,θ)=N∑j=1eπixj2+2πiθj.

An asymptotic expansion of these sums was obtained by Paris [12]

using an asymptotic expansion of the error function. We deduce an estimate for generalised Gauss sums from this expansion. To state the result, define for real

and ,

 E(x,θ)=e−πiθ2/xerfc(eπi/4θ√π/x),

where, for complex ,

 erfc(z)=1−2√π∫z0e−t2dt

is the complementary error function and the integral is over any path from  to .

###### Proposition 5 ([12, Theorem 1]).

Let be a positive integer, let , and let . Write , where is integral and . Then

 SN(x,θ) =e−πiθ2/x+πi/4√xSM(−1/x,θ/x)+μ−12 +eπi/42√x(E(x,θ)−μE(x,ϵ))+i2(g(θ)−μg(ϵ))+R,

where and and is given by

 g(t)={0for t=0cot(πt)−(πt)−1otherwise.

Fiedler, Jurkat, and Körner [5] obtained the following estimate from a slightly weaker version of Proposition 5.

###### Lemma 6 ([5, Lemma 4]).

Let , , and be integers such that and and let be real. Then

 |SN(k/m,θ)|=O(√m),

where the implicit constant is absolute.

From Proposition 5 we can also deduce the following result, which will be the key ingredient to our proofs of Theorem 1 and 2.

###### Lemma 7.

Let be a positive integer and let be an integer such that either or is in the set

 (7) {w∈Z:m2/3≤w≤m/2−m2/3}.

Then

 (8) ∣∣Sm−u(2/m,u/m)∣∣=√m2+O(m1/3),

where the implicit constant is absolute.

###### Proof.

Throughout the proof, all implicit constants are absolute. We first prove the desired bound when is in the set (7). This is an application of Proposition 5 with , , , so that and and . We have

 cot(πθ)−cot(πϵ)−1πθ+1πϵ =2cot(πu/m)−2mπu =O(m1/3),

using that is nonnegative on and, for ,

 cot(πu/m) ≤cot(πm−1/3) ≤1sin(πm−1/3) ≤12m1/3.

Now use the identity

 erfc(−z)=2−erfc(z),

to obtain

 E(x,θ)−E(x,−θ)=2E(x,θ)−2e−πiθ2/x.

Since

 S2(−1/x,θ/x)=(−i)m(−1)u+1,

we then find from Proposition 5 that

 |SN(x,θ)|=√m2∣∣(−i)m(−1)u+E(x,θ)∣∣+O(m1/3).

From the asymptotic expansion [12, (1.3) and (1.4)] of we find that

 ∣∣∣E(x,θ)−e−2πiθ2/x−πi/4√xπθ∣∣∣≤x3/22π2θ3,

so that is and (8) follows, as required.

Now assume that is in the set (7). Put and apply Proposition 5 with , , and , so that and and . Proceeding similarly as in the first case, we obtain

 |SN(x,θ)|=√m2∣∣e−πiθ2/x−E(x,θ)∣∣+O(m1/3),

which again implies (8). ∎

We conclude by proving the second parts of Theorems 1 and 2.

###### Proof of Theorem 1 (ii).

For each , we have

 |CXn,Yn(u)|=|CXn,Yn(−u)|=∣∣∣n−u−1∑j=0e2πi(j2+ju)/n∣∣∣.

Since the first and the last term in the sum are both equal to , we find that

 n2CDF(Xn,Yn)=|Sn(2/n,0)|2+2n−1∑u=1|Sn−u(2/n,u/n)|2.

Now use Lemma 7 to find that when or is in the set

 {w∈Z:n2/3≤w≤n/2−n2/3}

and use Lemma 6 to bound the remaining values of by . This gives

 CDF(Xn,Yn)=1+O(n−1/6),

as required. ∎

###### Proof of Theorem 2 (ii).

For each , we have

 |CXn,Yn(u)|=|CXn,Yn(−u)|=∣∣∣2n−u−1∑j=0(−1)jueπi(j2+ju)/n∣∣∣.

If  is odd, then the summands corresponding to and add to zero, leaving the summand corresponding to . Therefore for each odd satisfying . Hence

 (2n)2CDF(Xn,Yn)=|S2n(1/n,0)|2+2n−1∑v=1|S2n−2v(1/n,v/n)|2+O(n).

Now use Lemma 7 to find that when or is in the set

 {w∈Z:(2n)2/3≤w≤n−(2n)2/3}

and use Lemma 6 to bound the remaining values of by . This gives

 CDF(Xn,Yn)=12+O(n−1/6),

as required. ∎

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