Merit Factor Chu

7/29/2019 Merit Factor Chu

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Bounds on Asymptotic Merit Factor of Chu Sequences

Idris Mercer, University of Delaware, [email protected]

1 Abstract

Chu sequences are a family of polyphase sequences that have perfect periodicautocorrelations and good aperiodic autocorrelations. It has previously beenproved that the maximum offpeak (aperiodic) autocorrelation (in absolute

value) of the Chu sequence of length n is asymptotically equal to 0.480261n.It has also been empirically observed that the merit factor of Chu sequencesappears to grow like a constant times

n. In this note, we provide an analytic

proof that the merit factor of the Chu sequence of length n is bounded belowby a constant multiple of

n for all n.

2 Preliminaries

A polyphase sequence of length n is a finite sequence

S = (s0, s1, . . . , sn1),

where for each j, we have sj C and |sj| = 1.

For k = 0, 1, . . . , n 1, we define the aperiodic autocorrelations of S,

ck =nk1j=0

sjsj+k,

and the periodic autocorrelations of S,

k =n1j=0

sjsj+k,

where the bar denotes complex conjugation. In the definition of k, theaddition in the subscript is modulo n.

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We have c0 = 0 = n, which we call the trivial autocorrelations. It is not

hard to show that k = ck + cnk. Also note that |cn1| = |s0sn1| = 1.We can either seek polyphase sequences with periodic autocorrelations nearzero, or with aperiodic autocorrelations near zero. If k = 0 for all k = 0, wesay S is a perfect sequence. If |ck| 1 for all k = 0, we say S is a polyphaseBarker sequence.

Given a polyphase sequence S, two natural measures of the closeness to zeroof the aperiodic autocorrelations are as follows:

PSL(S) = maxk=0 |

ck|

,

TSE(S) =k=0

|ck|2 ,

which we respectively call the peak sidelobe level and total sidelobeenergy. (Some authors define the TSE to be twice our value, since they alsodefine ck = ck.) One can also define the merit factor of the sequence S,which in our notation is

MF(S) =n2

2

TSE(S).

We define three functions of n:

PSLmin(n) = minS

PSL(S),

TSEmin(n) = minS

TSE(S),

MFmax(n) = maxS

MF(S) =n2

2 TSEmin(S) ,

where the extremum is taken over all polyphase sequences of length n. Theseare mathematically well-defined (the space of all polyphase sequences oflength n is the product of n copies of the unit circle, and is hence compact)but explicit computation of the extrema appears very difficult.

It would be of interest to have good bounds for the growth rates of thefunctions PSLmin(n) and TSEmin(n).

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If S is a polyphase Barker sequence of length n, then PSL(S) = 1 and

TSE(S) n 1. Therefore, if there exists an infinite family of polyphaseBarker sequences, their merit factor would be at least n2/(2(n 1)) n/2.

Polyphase Barker sequences have been found for all lengths up to N, wherethe value of N has been gradually increasing. For example, Friese [4] foundpolyphase Barker sequences of all lengths up to 36, but conjectured that theydo not exist for significantly higher lengths. However, Borwein and Ferguson[2] found polyphase Barker sequences of all lengths up to 63. It is unknownwhether there exist polyphase Barker sequences for every length n.

Thus, we have PSLmin(n) = 1 for all n 63, and possibly for larger n.However, the best known asymptotic upper bound in the literature appearsto be of the form PSLmin(n) = O(

n). See, for example, Mow and Li [5].

It has been empirically observed [1, 6] that Chu sequences, and some relatedsequences, appear to have merit factor that grows like a constant times

n,

or equivalently, their TSE appears to grow like a constant times n3/2.

The main purpose of this note is to provide an analytic proof that, if S isthe Chu sequence of length n, we have a bound of the form

TSE(S) =k=0

|ck|2 Bn3/2

that holds for all n (where B is constant). This implies TSEmin(n) Bn3/2.

3 Proof of Main Result

We consider the sequences defined in [3]. For each positive integer n, theChu sequence of length n is the sequence S = (s0, s1, . . . , sn1), where

sj =

ej2

2n

if n is even,

ej(j +1)

2n

if n is odd,

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where e(t) is shorthand for ei2t.

Assume k = 0. As proved in [3], the Chu sequences satisfy k = 0 for all k.This implies that |cnk| = |ck|. Now note that in general, cnk is a sum of kterms of absolute value at most 1. Therefore |ck| = |cnk| k.

Using some relatively straightforward manipulations (see, e.g., the beginningof Section IV in [5], or the proof of Theorem 2 in [7]) one can show that theChu sequences satisfy

|ck

|=

sink2

n

sink

n .

For convenience, we define the half-energy

HE(S) =

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We now considerm


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which is a polynomial in

n of degree 3. The coefficient of the dominant

power n3/2 isA3

3+

1

2A.

Using elementary calculus, we find that the positive A that minimizes thisquantity is

A =1

and the minimum value itself is

A3

3+

1

2A=

4

33/2

0.2394494961.

Choosing A = 1/

, we then have an upper bound of the form

HE(S) 0.8933237710

n for all n 200,

MF(S) > 0.9969934260

n for all n 2000,MF(S) > 1.029283811

n for all n 20000.

Antweiler and Bomer [1] observed that the merit factor of Chu sequencesappears empirically to grow like a constant times

n, but to the best of the

current authors knowledge, the existing literature does not contain proofsthat the Chu sequences satisfy bounds of the form TSE(S) < Bn3/2 and

MF(S) > Cn that hold for all n.

4 Conclusions and Further Questions

We have proved that the Chu sequences satisfy, in an asymptotic sense,

TSE(S) 0.4788989922 n3/2,MF(S) 1.044061500

n.

Numerical evidence suggests the constants in front are not the best constants.If we recall that

TSE(S) =n1k=1

|ck|2 =n1k=1

sin2k2

n

csc2

kn

,

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then we can calculate TSE(n) numerically for certain values ofn and compare

with n3/2. Doing this, for example, with n from 1000 to 1024 suggests

TSE(S) 0.318 n3/2,MF(S) 1.57n.

The empirical result MF(S) 1.57n was already noted in [1]. Furthermore,it was shown in [6] that certain other sequences, related but not identical toChu sequences, appear to satisfy MF(S) Cn for a larger constant C.

We have shown that the function TSEmin(n) is bounded above by a constanttimes n3/2. It would be interesting to know whether this can be improved. If

there are polyphase Barker sequences of every length, then TSEmin(n) growsat most linearly in n.

References

[1] M. Antweiler & L. Bomer, Merit factor of Chu and Frank sequences,Electron. Lett. 26 (1990), 20682070.

[2] P. Borwein & R. Ferguson, Polyphase sequences with low autocorrelation,IEEE Trans. Inform. Theory 51 (2005), 15641567.

[3] D. C. Chu, Polyphase codes with good periodic correlation properties,IEEE Trans. Inform. Theory 14 (1972), 531532.

[4] M. Friese, Polyphase Barker sequences up to length 36, IEEE Trans. In-form. Theory 42 (1996), 12481250.

[5] W. H. Mow & S.-Y. R. Li, Aperiodic autocorrelation and crosscorrelationof polyphase sequences, IEEE Trans. Inform. Theory 43 (1997), 10001007.

[6] P. Rapajic & R. Kennedy, Merit factor based comparison of new polyphasesequences, IEEE Commun. Lett. 2 (1998), 269270.

[7] N. Zhang & S. Golomb, Polyphase sequence with low autocorrelations,IEEE Trans. Inform. Theory 39 (1993), 10851089.

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