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
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
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
(1) 
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 PursleySarwate 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 PursleySarwate criterion of sequence pairs derived from msequences and Legendre sequences and generalisations of them. This gives pairs of unimodular and binary sequences whose PursleySarwate 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 msequences 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 satisfyingGolay 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 RudinShapiro pairs, which are Golay pairs of binary sequences of length and satisfy
(see [15], for example). Since , we have
Therefore, as , we have and and .
In this paper we exhibit unimodular sequences whose PursleySarwate 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
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
(2) 
which confirms previous numerical evidence obtained by Littlewood [9] and Bömer and Antweiler [1]. In fact, since tends to zero, this immediately implies that
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 ,

, ,

,

.
In our second result we construct a pair of unimodular sequences from two Chu sequences of even length such that the PursleySarwate 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 ,

, ,

,

.
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
for . Hence, if is not a multiple of , then
Therefore
since for and for even . Finally, note that
which completes the proof. ∎
Lemma 4 ([14]).
We have
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
(3) 
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
By Lemma 4, the first sum is , so that it is sufficient to show that
(4) 
Let be a real number satisfying . From the Taylor series of at we find that
from which it follows that
Therefore,
Put to obtain
Apply to the right hand side and use Lemma 4 to obtain
using Euler’s evaluation . This gives (4), and so completes the proof of (3).
Next we prove that
(5) 
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) 
For even , we have
so that
by Lemma 4. For odd , we have
so that
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
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 ,where, for complex ,
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
where and and is given by
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
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) 
Then
(8) 
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
using that is nonnegative on and, for ,
Now use the identity
to obtain
Since
we then find from Proposition 5 that
From the asymptotic expansion [12, (1.3) and (1.4)] of we find that
so that is and (8) follows, as required.
Proof of Theorem 1 (ii).
Proof of Theorem 2 (ii).
References
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