IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 9, SEPTEMBER 2013 6083

Merit Factor of Chu Sequences and Best MeritFactor of Polyphase Sequences

Idris Mercer

Abstract—Chu sequences are a family of polyphase sequencesthat have perfect periodic autocorrelations and good aperiodic au-tocorrelations. It has previously been proved that the maximumoffpeak (aperiodic) autocorrelation (in absolute value) of the Chusequence of length is asymptotically equal to . It hasalso been empirically observed that the merit factor of Chu se-quences appears to grow like a constant times . In this note,we provide an analytic proof that the merit factor of the Chu se-quence of length is bounded below by a constant multiple offor all . To the author’s knowledge, this is the first time a family ofpolyphase sequences of all lengths has been proved to have meritfactor growing at least like order .

Index Terms—Autocorrelation, Chu sequence, merit factor.

I. INTRODUCTION

B Y a complex sequence of length , we mean a finite se-quence of the form

where for each , we have and . The sequenceis called a polyphase sequence if there exists a positive integersuch that each is an th root of 1.For , we define the aperiodic autocorre-

lations of :

and the periodic autocorrelations of :

where the bar denotes complex conjugation. In the definition of, the addition in the subscript is modulo .We have , which we call the “trivial” autocorre-

lations. It is not hard to show that . Also notethat .We can either seek complex sequences with periodic auto-

correlations near zero, or with aperiodic autocorrelations nearzero. If for all , we say is a perfect sequence.

Manuscript received February 17, 2012; revised October 09, 2012; acceptedMarch 08, 2013. Date of publication April 23, 2013; date of current versionAugust 14, 2013.The author is with the Department of Mathematical Sciences, University of

Delaware, Newark, DE 19716 USA (e-mail: idmercer@math.udel.edu).Communicated by A. Ashikhmin, Associate Editor for Coding Techniques.Digital Object Identifier 10.1109/TIT.2013.2259537

If for all , we say is a generalized Barker se-quence. If we further have for all , we say is apolyphase Barker sequence.Given a complex sequence , two natural measures of

the closeness to zero of the aperiodic autocorrelations are asfollows:

which we respectively call the peak sidelobe level (PSL) andtotal sidelobe energy (TSE). (Some authors define the TSE to betwice our value, since they also define .) One can alsodefine the merit factor of the sequence , which in our notationis

We define three functions of :

where the extremum is taken over all complex sequences oflength . These are mathematically well defined (the “space”of all complex sequences of length is the product of copiesof the unit circle, and is hence compact) but explicit computa-tion of the extrema appears very difficult.It would be of interest to have good bounds for the growth

rates of the functions and .If is a generalized Barker sequence of length , then

and . Therefore, if there is an infinitefamily of generalized Barker sequences, their merit factorwould be at least .Polyphase Barker sequences have been found for all lengths

up to , where the value of has been gradually increasing.For example, Friese [4] found polyphase Barker sequences ofall lengths up to 36, but conjectured that they do not exist forsignificantly higher lengths. However, Borwein and Ferguson[2] found polyphase Barker sequences of all lengths up to 63,and Nunn and Coxson [7] found polyphase Barker sequences oflengths 64 to 70, as well as 72, 76, and 77. It is unknownwhetherthere exist generalized Barker sequences for every length .Thus, we have for all , and possibly

for larger . However, the best-known asymptotic upper bound

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6084 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 9, SEPTEMBER 2013

in the literature appears to be of the form .The existence of an infinite family of polyphase sequences withPSL bounded above by was shown by Turyn [9], andthe fact that the Chu sequence of length has PSL asymptoticto was shown by Mow and Li [5].It has been empirically observed [1], [8] that Chu sequences,

and some related sequences, appear to have merit factor thatgrows like a constant times , or equivalently, their TSE ap-pears to grow like a constant times .The main purpose of this note is to provide an analytic proof

that, if is the Chu sequence of length , we have a bound ofthe form

that holds for all (where is constant). This implies.

II. PROOF OF MAIN RESULT

Main Result: Let be the Chu sequence of length , as de-fined in [3]. Then the TSE of is bounded by

and the merit factor of is bounded below by

where is asymptotic to when.Proof: As in [3], for each positive integer , the Chu se-

quence of length is the sequence ,where

if is even,

if is odd,

where is shorthand for .Assume . As proved in [3], the Chu sequences satisfy

for all . This implies that . Now note thatin general, is a sum of terms of absolute value at most1. Therefore, .Using some relatively straightforward manipulations (see,

e.g., the beginning of [6, Sec. IV], or [10, Proof of Th. 2]) onecan show that the Chu sequences satisfy

For convenience, we define the half-energy

Using the relation , one can see that

if is odd,

if is even.

Now note that

The key is now to break the sum into two pieces

where is a fixed positive real number yet to be determined.Let and let . Since ,

we have

We now consider

This latter sum is equal to

which can be regarded as a Riemann sum. If the interval of realnumbers is divided into subintervals of width , thenas ranges through the integer values , the num-bers are the right endpoints of the subintervals.Note that , and that . Therefore,

is a subset of the interval , on which the functionis strictly decreasing. It follows that we have

and the integral evaluates to

(1)

On the interval , the function is positive and satisfies. The quantity (1) can therefore be bounded above

by

Combining our bounds, we conclude

MERCER: MERIT FACTOR OF CHU SEQUENCES AND BEST MERIT FACTOR OF POLYPHASE SEQUENCES 6085

Since , we conclude

(2)which is a polynomial in of degree 3. The coefficient of thedominant power is

Using elementary calculus, we find that the positive that min-imizes this quantity is

and the minimum value itself is

Choosing , we then have an upper bound of the form

which implies

This implies that the merit factor asymptotically grows atleast like

If we seek explicit bounds that are valid for all , we can startby evaluating the bound (2) when . This gives

and then since , we have

A computation then reveals that we have, for example:

which implies

Antweiler and Bömer [1] observed that the merit factor of Chusequences appears empirically to grow like a constant times ,but to the best of the current author’s knowledge, the existingliterature does not contain proofs that the Chu sequences satisfybounds of the form and that holdfor all .

III. CONCLUSION AND FURTHER QUESTIONS

We have proved that the Chu sequences satisfy in an asymp-totic sense

Numerical evidence suggests that the constants in front are notthe best constants. If we recall that

then we can calculate numerically for certain values ofand compare with . Doing this, for example, with from1000 to 1024 suggests

The empirical result was already noted in [1].Furthermore, it was shown in [8] that certain other sequences,related but not identical to Chu sequences, appear to satisfy

for a larger constant .We have shown that the function is bounded above

by a constant times , and therefore that the functionis bounded below by a constant times . It would

be interesting to know whether this can be improved. If thereare polyphase Barker sequences of every length, thengrows at most linearly in .

REFERENCES

[1] M. Antweiler and L. Bömer, “Merit factor of Chu and Frank se-quences,” Electron. Lett., vol. 26, pp. 2068–2070, 1990.

[2] P. Borwein and R. Ferguson, “Polyphase sequences with low autocor-relation,” IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1564–1567, Apr.2005.

[3] D. C. Chu, “Polyphase codes with good periodic correlation proper-ties,” IEEE Trans. Inf. Theory, vol. IT-18, no. 4, pp. 531–532, Jul. 1972.

[4] M. Friese, “Polyphase Barker sequences up to length 36,” IEEE Trans.Inf. Theory, vol. 42, no. 4, pp. 1248–1250, Jul. 1996.

6086 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 9, SEPTEMBER 2013

[5] W. H. Mow and S.-Y. R. Li, “Aperiodic autocorrelation properties ofperfect polyphase sequences,” in Proc. Int. Symp. Inf. Theory Appl.,Singapore, Nov. 16–20, 1992, pp. 1232–1234.

[6] W. H. Mow and S.-Y. R. Li, “Aperiodic autocorrelation and crosscor-relation of polyphase sequences,” IEEE Trans. Inf. Theory, vol. 43, no.3, pp. 1000–1007, May 1997.

[7] C. J. Nunn andG. E. Coxson, “Polyphase pulse compression codeswithoptimal peak and integrated sidelobes,” IEEE Trans. Aerosp. Electron.Syst., vol. 45, no. 2, pp. 775–781, Apr. 2009.

[8] P. Rapajic and R. Kennedy, “Merit factor based comparison of newpolyphase sequences,” IEEE Commun. Lett., vol. 2, no. 10, pp.269–270, Oct. 1998.

[9] R. Turyn, “The correlation function of a sequence of roots of 1,” IEEETrans. Inf. Theory, vol. IT-13, no. 3, pp. 524–525, Jul. 1967.

[10] N. Zhang and S. Golomb, “Polyphase sequence with low autocorrela-tions,” IEEE Trans. Inf. Theory, vol. 39, no. 3, pp. 1085–1089, May1993.

Idris Mercer received his Ph.D. under Peter Borwein and is currently a VisitingAssistant Professor at the University of Delaware.