Generalised Sarwate bounds on the aperiodic correlation of sequences over complex roots of unity

Generalised Sarwate bounds on the aperiodiccorrelation of sequences over complex roots of unity

D.Y. Peng and P.Z. Fan

Abstract: General aperiodic correlation bounds named generalised Sarwate bounds are derived,for sequence sets over complex roots of unity with zero or low correlation zone (ZCZ/LCZ), withrespect to family size, sequence length, maximum autocorrelation sidelobe, maximum cross-correlation value and the zero or low correlation zone. It is shown that the existing aperiodic binarysequence bounds, such as Sarwate bounds, Welch bounds, Levenshtein bounds, Tang–Fan boundsand Peng–Fan bounds, are only special cases of the presented generalised Sarwate bounds. Inaddition, the new bounds are also, in general, stronger than the existing aperiodic binary sequencebounds, as well as Boztas aperiodic correlation bounds for normal complex roots-of-unitysequences.

1 Introduction

In a typical DS-CDMA system, it is desirable to design setsof the spreading sequences having zero or very lowautocorrelation sidelobes and cross-correlation values[1–3]. However, it has been proved that it is impossible todesign a set of spreading sequences with ideal impulsiveautocorrelation function and ideal zero crosscorrelationfunction, thus resulting in co-channel interference in practicalCDMA systems. In order to overcome this difficulty, therehas recently been much investigation of quasi-synchronousCDMA (QS-CDMA) systems which make use of zerocorrelation zone (ZCZ) sequence, or low correlation zone(LCZ) sequences, or generalised orthogonal sequences [4–15]. In QS-CDMA systems, also called ‘approximatelysynchronous CDMA’ (AS-CDMA) systems, the correlationfunctions of the spreading sequences employed take zero orvery low values for a continuous correlation shift zonearound the in-phase shift. The significance of ZCZ/LCZsequences to QS-CDMA systems is that, even if there arerelative delays between the received spreading signals due tothe inaccurate access synchronisation and the multipathpropagation, the orthogonality between the signals is stillmaintained, as long as the relative delay does not exceedcertain limits. In order to evaluate the theoretical perfor-mance of the spreading sequences, it is important to find thetight theoretical limits that set bounds among the sequencelength, sequence family size, maximum aperiodic (periodic)autocorrelation sidelobe and maximum aperiodic (periodic)cross-correlation value. In fact, the tight constraint relationamong these parameters has been a key and active issue ininformation theory and communication engineering.

For the periodic correlation function of sequences overcomplex roots of unity, early in 1971 Sidelnikov obtained alower bound [16, 17]. In 1990, Kumar and Liu provided a

new bound for this class of sequences [18], which is a slightimprovement to the Sidelnikov bound, although morecomplicated. For the aperiodic correlation function ofsequences over complex roots of unity, Welch [19] derived abound by using the properties of inner products, in 1974,which can be considered as a special case of Sarwateaperiodic bounds [20]. Later, Levenshtein [21] derivedseveral bounds by introducing ‘weights’ for shifts ofsequences for binary sequence sets, which are tighter thanWelch bounds. Recently, Peng and Fan [6] obtained a fewaperiodic bounds based on Levenshtein’s technique, butwhich are stronger than the Welch bounds, the Sarwatebounds and the Levenshtein bounds. In 1998, Boztasgeneralised Levenshtein’s results to complex roots-of-unitysequences [22].

For the LCZ/ZCZ spreading sequences, first, Tang andFan established bounds on the periodic and aperiodiccorrelations based on Welch’s technique [8, 9], whichincluded Welch bounds as special cases. For periodiccorrelations of LCZ/ZCZ sequences, generalised Sarwatebounds were obtained [4, 7], which included all the previousperiodic sequence bounds as special cases, such as Welchbounds, Sarwate bounds and Tang–Fan LCZ bounds. In2001, D.Y. Peng et al. [5] obtained new lower bounds onaperiodic correlation of the LCZ sequences, which arestronger than Tang–Fan aperiodic bounds. In this paper, itwill be shown that even tighter aperiodic bounds for LCZsequences over complex roots of unity can be derived.

In this paper, only the aperiodic correlation bounds, andcomplex roots-of-unity sequences are considered.

2 Preliminaries

Throughout this paper, a* denotes the complex conjugate ofa, Jkn denotes the smallest integer which is greater than orequal to k and Ikm denotes the greatest integer which issmaller than or equal to k for any positive real number k.Let q be an arbitrary, positive, integer greater than 1,

Zq ¼ {0, 1,y,q�1}, i ¼ffiffiffiffiffiffiffi�1p

, o¼ exp(2pi/q), E¼{1, o1,y,oq�1}, then x¼ (x0, x1,y,xn�1)AE n is called acomplex roots-of-unity sequence of length n. When q¼ 2, itis clear that the complex roots-of-unity sequencebecomes the binary sequence. For any two such sequences

The authors are with the Institute of Mobile Communications, SouthwestJiaotong University, Chengdu 610031, People’s Republic of China

r IEE, 2004

IEE Proceedings online no. 20040372

doi:10.1049/ip-com:20040372

Paper first received 1st May and in revised form 18th November 2003.Originally published online: 29th June 2004

IEE Proc.-Commun., Vol. 151, No. 4, August 2004 375

x¼ (x0, x1, y, xn�1) and y¼ (y0, y1,y, yn�1) theiraperiodic correlation function A(x, y;l ) is defined as follows

Aðx; y; lÞ ¼Xn�1�l

i¼0xiy�iþl; l ¼ 0; 1; � � � ; n� 1 ð1Þ

where subscripts are performed modulo n and only positivetime shifts l are considered here. The inner product of x andy is defined by

hx; yi ¼Xn�1i¼0

xiy�i

Let C denote a set of M complex roots-of-unity sequencesof length n. Given that daZ0, dcZ0, and dm¼max{da, dc},the aperiodic low correlation zone LA of C is defined asfollows:

LA ¼ minfLAA; LACg ð2Þ

where

LAA ¼maxfT : jAðx; x; lÞ � da;

x 2 C; 0ol � TgLAC ¼maxfT : jAðx; y; lÞj � dc; x; y 2 C;

x 6¼ y; 0 � l � Tg

A sequence set C with a LA40 is called the aperiodic lowcorrelation zone (LCZ) sequence set. When dm¼ 0, then theLCZ sequence set becomes the aperiodic zero correlationzone (ZCZ) sequence set and the aperiodic low correlationzone LA becomes aperiodic zero correlation zone ZA. Thedefinition of LCZ sequence set or ZCZ sequence set withrespect to periodic correlation is similar [7].

In particular, if LA¼ n�1, then the LCZ sequencesbecome normal sequences. Therefore, the LCZ sequencescan be considered as very general sequence, which includethe ZCZ sequences and the normal sequences as specialcases. Having obtained the general bounds for LCZsequences, it is relatively easy to derive the bounds forZCZ sequences and normal sequences.Lemma 1: For any sequence xAE n and any integer l¼ 0,1,y, n�1, we haveX

y2En

jAðx; y; lÞj2

¼Xy2En

jAðy; x; lÞj2 ¼ ðn� lÞqn ð3Þ

Proof: Let x ¼ fou0 ; ou1 ; � � � ;oun�1g, y ¼ fov0 ; ov1 ;� � � ; ovn�1g, where ui, viAZq(i¼ 0, 1,y,q�1) we have:X

y2En

jAðx; y; lÞj2 ¼Xy2En

Aðx; y; lÞA�ðx; y; lÞ

¼Xy2E�

Xn�1�l

i¼0xiy�iþl

! Xn�1�l

j¼0x�j y�jþl

!

¼Xy2En

Xn�1�l

i¼0oui�viþl

! Xn�1�l

j¼0o�ujþvjþl

!

¼ qlXq�1

vl;vlþ1;��;vn�1¼0

Xn�1�l

i¼0oui�vjþl

! Xn�1�l

j¼0o�ujþvjþl

!

¼ qlXn�1�l

i¼0

Xn�1�l

j¼0oui�uj

Xq�1vl;vlþ1;��;vn�1¼0

ovjþl�viþl

!

If i¼ j, then


ovjþ1�viþl ¼ qn�l

If iaj, then


ovjþl�viþl ¼ qn�l�2Xq�l

vjþl;viþl¼0ovjþl�viþl ¼ 0

Therefore

Xy2En

jAðx; y; lÞj2 ¼ qnXn�l�1

i¼0oui�ui ¼ ðn� lÞqn

Q.E.D.From now on, let w ¼ ðw0; w1; � � � ;wLAÞ be a weight

vector, where

wi � 0; ði ¼ 0; 1; � � � ; LAÞPLA

i¼0wi ¼ 1

8<:

For any sequence x¼ (x0, x1,y,xn�1)AE n, let T denote theoperator which shifts sequence cyclically to the left by oneplace, i.e. Tx¼ (x1, x2,y,xn�1, x0), and let T 0x¼ x,T i+1x¼T(T ix) for position integer iZ1. Given anypositive integer k, a sequence x0 k¼ (x0, x1,y,xn�1, 0,0,y,0) is obtained by appending k zeros to the right-handof x.

For any sequence xAE n, any sequence sets A, BDE n,7A77B740, let

W ðxÞ ¼fT iðx0LAÞji ¼ 0; 1; . . . ; LAgW ðAÞ ¼

[x2A

W ðxÞ

F ðA; BÞ ¼ 1

jAjjBjXx2A

Xy2B

XLA

s¼0

XLA

t¼0jhT sðx0LAÞ;

T tðy0LAÞij2wswt

Lemma 2: For any sequence xAE n, sequence set ADE n,

F ðfxg; EnÞ ¼F ðA; EnÞ ¼ F ðEn; EnÞ

¼XLA

s;t¼0ðn� js� tjÞwswt

Proof: For any x, yAE n, 0rs, trLA, It is noted that

hT sðx0LAÞ; T tðy0LAÞi

¼Aðx; y; s� tÞ; s � t;

A�ðy; x; t � sÞ; sot;

�

376 IEE Proc.-Commun., Vol. 151, No. 4, August 2004

where A*(y, x, t�s) denotes the complex conjugate ofA(y, x, t�s). We have by, lemma 1,

F ðfxg; EnÞ ¼ 1

qn

Xy2En

XLA

s¼0

XLA

t¼0jhT sðx0LAÞ;

T tðy0LAÞij2wswt

¼ 1

qn

X0�t�s�LA

Xy2En

jAðx; y; s� tÞj2wswt

(

þX

0�sot�LA

Xy2En

jAðy; x; t � sÞj2wswt

)

¼ 1

qn

X0�t�s�LA

½n� ðs� tÞ�qnwswt

(

þX

0�sot�LA

½n� ðt � sÞ�qnwswt

)

¼X

0�s;t�LA

ðn� js� tjÞwswt

This gives the formula for F({x}, E n), which does notdepend on x, and thus completes the proof. Q.E.D.Lemma 3: For any sequence sets A, BDE n, 7A77B740, wehave

jF ðA;BÞj2 � F ðA;AÞ � F ðB;BÞ

Proof: For any positive integers i, j¼ 0, 1, y, n+LA�1,s¼ 0, 1,y,LA, define function f (i, j, s; X) on W(E n) asfollows. If X ¼ T sðx0LAÞ ¼ ðxs

0; xs1; � � � ; xs

nþLA�1Þ 2 W ðEnÞ,where xAE n, then

f ði; j; s; xÞ ¼ xsi ðxsjÞ�ws

We can verify that

F ðA; BÞ ¼ 1

jAjjBj

�X

X2W ðAÞ

XY2W ðBÞ

XnþLA�1

i;j¼0f ði; j; s; X Þf �ði; j; t; Y Þ ð4Þ

In fact, for any xAA, yAB, let

X ¼T sðx0LAÞ ¼ ðxs0; xs

1; � � � ; xsnþLA�1Þ;

Y ¼T tðy0LAÞ ¼ ðyt0; yt

1; � � � ; ytnþLA�1Þ

then

XLA

s;t¼0jhT sðx0LAÞ; T tðy0LAÞij2wswt

¼XLA

s;t¼0jXnþLA�1

k¼0xskðyt

kÞ�j2wswt

Therefore,

XX2W ðxÞ

XY2W ðyÞ

XnþLA�1

i;j¼0f ði; j; s; X Þf �ði; j; t; Y Þ

¼XLA

s;t¼0

XnþLA�1

i;j¼0f ði; j; s; T sðx0LAÞÞf �ði; j; t; T tðy0LAÞÞ

¼XLA

s;t¼0

XnþLA�1

i;j¼0½xsi ðxsjÞ

�ws�½yti ðyt

jÞ�wt��

¼XLA

s;t¼0

XnþLA�1

i;j¼0½xsi ðyt

i Þ��½ðxsjÞ

�ytj�wswt

¼XLA

s;t¼0

XnþLA�1

i¼0xsi ðyt

i Þ�

" # XnþLA�1

j¼0xsjðyt

jÞ�

" #�wswt

¼XLA

s;t¼0jXnþLA�1

i¼0xsi ðyt

i Þ�j2wswt

¼XLA


and

Xx2A

Xy2B

XLA


¼Xx2A

Xy2B

XX2W ðxÞ

XY2W ðyÞ

XnþLA�1


¼X

X2W ðAÞ

XY2W ðBÞ

XnþLA�1


From (4), we have

F ðA; BÞ ¼ 1

jAjjBjX

X2W ðAÞ

XY2W ðBÞ

XnþLA�1

i;j¼0f ði; j; s; X Þ

� f �ði; j; t; Y Þ

¼ 1

jAjjBjXnþLA�1

i;j¼0

XX2W ðAÞ

f ði; j; s; X Þ

24

35

�X

Y2W ðBÞf �ði; j; t; Y Þ

24

35

and

F ðA;AÞ ¼ 1

jAj2XnþLA�1

i;j¼0jX

X2W ðAÞf ði; j; s; X Þj2

Using the Cauchy inequality we have

jF ðA; BÞ2 � 1

jAj2XnþLA�1

i;j¼0jX

X2W ðAÞf ði; j; s; X Þj2

8<:

9=;

� 1

jBj2XnþLA�1

i;j¼0jX

Y2W ðBÞf �ði; j; t; Y Þj2

8<:

9=;

¼F ðA; AÞ � F ðB; BÞ

Q.E.D.


Lemma 4: For any sequence set CDE n, we have

F ðC;CÞ �XLA


Proof: Let A¼E n, B¼C, we have by lemmas 3 and 2

jF ðEn;CÞj2 � F ðEn;EnÞ � F ðC;CÞ

F ðC;CÞ � F ðEn;EnÞ � n�XLA

s;t¼0js� tjwswt

Q.E.D.Lemma 5: For any sequence set CDE n, M¼ 7C740, wehave

F ðC; CÞ � n2

M�XLA

s¼0w2

s þ1

M1�

XLA

s¼0w2

s

!

� d2aðCÞ þ 1� 1

M

� �� d2cðCÞ

Proof: We have

M2F ðC; CÞ ¼X

x;y2C

XLA


¼Xx2C

XLA

s¼0jhT sðx0LAÞ; T sðx0LAÞij2wsws

þXx2C

XLA

s 6¼t;s;t¼0jhT sðx0LAÞ; T tðx0LAÞij2wswt

�Mn2XLA

s¼0w2

s þMd2aðCÞXLA

s6¼t;s;t¼0wswt

þMðM � 1Þd2cðCÞSince

1 ¼XLA

s;t¼0wswt ¼

XLA

s¼0w2

sþXLA

s6¼t;s;t¼0wswt

the lemma follows. Q.E.D.

3 Lower bounds on the aperiodic correlation ofLCZ complex roots-of-unity sequences

Let C be a set of M complex roots-of-unity sequences oflength n, maximum aperiodic autocorrelation sidelobe da,maximum aperiodic cross-correlation value dc and aperiodiclow correlation zone LA. Compared with the Sarwatebound [20], it is desirable to find functions P(n, M, LA) andD(n, M, LA) which depend on variables n, M and LA suchthat

P ðn;M ; LAÞd2a þ Dðn;M ; LAÞd2c � 1 ð5Þand it is expected that both P(n, M, LA) and D(n, M, LA)should be as small as possible due to the requirements ofsequence design for CDMA systems [12, 14]. The bound (5)is named here as the generalised Sarwate bound. Inparticular, let dm¼max{da, dc}, the inequality (5) can bewritten as

d2m � Hðn;M ; LAÞ ð6ÞIt is generally hoped that H(n, M, LA) is large. The bound(6) is named here as the generalised Welch bound.

The bounds in the form of (5) and (6) on periodiccorrelation were obtained by Peng and Fan [4, 7]. In thisSection, we will derive the required functions P(n, M, LA)

and D(n, M, LA) for (5) and (6) and then, based on the twofunctions, establish the tight lower bounds as desired.

For any real number a, let

Qða; LA; wÞ ¼ aXLA

s¼0w2

s þXLA

s;t¼0js� tjwswt

be a quadratic form on the weight vector w ¼ðw0;w1; � � � ;wLAÞ. We now state the main theorem, whichdepends on the quadratic form Q.Theorem 1: For any sequence set CDE n

, M¼ 7C740, wehave

1

M1�

XLA

s¼0w2

s

!� d2a þ 1� 1

M

� �

� d2c � n� Qðn2=M ; LA; wÞ ð7Þ

1� 1

M

XLA

s¼0w2

s

!� d2m � n� Qðn2=M ; LA; wÞ ð8Þ

Qðn2=M ; ZA; wÞ � n ð9Þ

Proof: By the lemmas 4 and 5, we have

n2

M�XLA

s¼0w2

s þ1

M1�

XLA

s¼0w2

s

!

� d2aðCÞ þ 1� 1

M

� �� d2cðCÞ

� F ðC; CÞ �XLA


namely,

1

M1�

XLA

s¼0w2

s

!� d2aðCÞ þ 1� 1

M

� �� d2cðCÞ

�XLA

s;t¼0ðn� js� tjÞwswt �

n2

M�XLA

s¼0w2

s

�n� n2

M�XLA

s¼0w2

s þXLA

s;t¼0js� tjwswt

!

¼Qðn2=M ; LA; wÞ

Noting dm¼max{da, dc}, the inequality (8) followsimmediately from (7). Putting dm¼ 0 in the inequality (8),we see that LA becomes ZA, and that (9) holds. Q.E.D.

Based on theorem 1, we can derive some useful results bycalculating the value of the quadratic form Q for somespecial cases.Corollary 1: For any 0rLrLA

3Ld2a þ 3ðLþ 1ÞðM � 1Þ � d2c � 3Mn

� 3n2 þ 3MnL� 2ML�ML2 ð10Þ

d2m �3Mn� 3n2 þ 3MnL� 2ML�ML2

3ðMLþM � 1Þ ð11Þ

Proof: Put the weight vector w ¼ ðw0; w1; � � � ;wLAÞ, where

ws ¼1

Lþ1 ; 0 � s � L0; Los � LA

�


then

XLA

s¼0w2

s ¼1

Lþ 1; Qða; LA; wÞ ¼ a

Lþ 1

þXL

s;t¼0js� tj 1

ðLþ 1Þ2¼ a

Lþ 1þ LðLþ 2Þ

3ðLþ 1Þ

We have by theorem 1,

1� 1

Lþ 1

� �� d2a þ ðM � 1Þ � d2c

� nM � n2

Lþ 1� LMðLþ 2Þ

3ðLþ 1Þthus, the corollary holds. Q.E.D.

Corollary 2: If MZ3, nZ2, and LA4ffiffiffiffiffiffiffiffiffiffi3=M

pn� 1, thenffiffiffi

3p

n�ffiffiffiffiffiMp

ðffiffiffi3p

M � 2ffiffiffiffiffiMpÞn2� d2a

þffiffiffi3pðM � 1Þ

ðffiffiffi3p

M � 2ffiffiffiffiffiMpÞn� d2c � 1 ð12Þ

d2m �ffiffiffiffiffiffiffi3Mp

� 2ffiffiffiffiffiffiffi3Mp

n� 1� n2 ð13Þ

Proof: Let L ¼ffiffiffiffiffiffiffiffiffiffi3=M

pn

j k� 1, then L ¼

ffiffiffiffiffiffiffiffiffiffi3=M

pn� 1þ

e � LA, where �1oer0. Put the weight vectorw ¼ ðw0; w1; . . . ;wLAÞ, where

ws ¼1

Lþ1 ; 0 � s � L0; Los � LA

�

We have

XLA

s¼0w2

s ¼1

Lþ 1�

ffiffiffiffiffiMpffiffiffi3p

n

Qn2

M; LA; w

� �¼ 1

3ðLþ 1Þ3n2

Mþ LðLþ 2Þ

� �

¼ 1

3ðffiffiffi3p

nþ effiffiffiffiffiMpÞffiffiffiffiffiMp

� ð6n2 þ 2ffiffiffiffiffiffiffi3Mp

neþ e2M �MÞ

� 1

3ðffiffiffi3p

nþ effiffiffiffiffiMpÞffiffiffiffiffiMp

� ð6n2 þ 2ffiffiffiffiffiffiffi3Mp

neÞ ¼ 2nffiffiffiffiffiffiffi3Mp

We have by theorem 1

1�ffiffiffiffiffiMpffiffiffi3p

n

� �� d2a þ ðM � 1Þ � d2c � Mn� 2nffiffiffiffiffiffiffi

3Mp M

thus, the results hold. Q.E.D.Corollary 3: For any 0rLrLA.

2ð4L � 1Þ � d2a þ 3ðM � 1Þ4L � d2c� ð3Mn� n2 � 4MÞ4L

þ 6ðL� 2Þ2LM þ 6MLþ 16M � 2n2 ð14Þ

Proof: Let the weight vector w ¼ ðw0; w1; . . . ;wLAÞ, where

ws ¼2�L; s ¼ 0;2�s; 1 � s � L;0; Los � LA:

8<:

Thus we have,

XLA

s¼0w2

s ¼1

3þ 2

3

1

4

� �L

; Qða; LA; wÞ

¼a1

3þ 2

3

1

4

� �L( )

þ 4

3� 2ðL� 2Þ 1

2

� �L

� 2

3ð3Lþ 8Þ 1

2

� �2L

and by theorem 1,

3

21� 1

4

� �L( )

� d2a þ ðM � 1Þ � d2c

� Mn� n2 1

3þ 2

3

1

4

� �L( )

� 4

3M

þ 2ðL� 2ÞM 1

2

� �L

þ 2

3ð3Lþ 8ÞM 1

2

� �2L

2f4L�1 � 1g � d2a þ 3ðM � 1Þ4L�1 � d2c� ð�4M þ 3LM � L2Þ4L�1 þ 3ðL� 3ÞM2L

þ 2ð5M þ 3LM � L2Þ

This completes the proof. Q.E.D.Based on theorem 1, one can also derive some useful

results by calculating the minimum of the quadratic form Qunder some special conditions. We need the followinglemma.Lemma 6: Let aZ1, 0oj¼ arccos(1–1/a)rp/2, rr(p+j)/(2j), and w ¼ ðw0; w1; � � � ;wLAÞ be a weight vector withwr�s�1¼wr+s (0rsrr�1) and ws¼ 0 (2rrsrLA), thenthe minimum of quadratic form Q(a, LA; w) is given by

Q0ðaÞ ¼ r þ a2� sin ðrjÞ � sin fðr � 1Þjg

sin ðrjÞ � 1

2ð15Þ

and we have

XLA

i¼0w2

i ¼cos2 j

4 cos2 ðj=2Þ sin2 ðrjÞ

� r þXr�1i¼0

cos ð2iþ 1Þj !

ð16Þ

Proof: The proof of (15) can be found in [21], and we haveassumed that the notations denoted in [21] are already wellknown, thus

XLA

i¼0w2

i ¼1

2�Xr�1i¼0

y2i ¼

1

2� y2

0 �Xr�1i¼0

u2i ¼

2

U 2r�1ðZÞ

� sin2 ðj=2Þsin2 j

�Xr�1i¼0

cos2 ðijþ j=2Þ

¼ cos2 j

4 cos2 ðj=2Þ sin2 ðrjÞr þ

Xr�1i¼0

cos ð2iþ 1Þj !

Q.E.D.


Theorem 2: Let Mrn2, 0oj¼ arccos (1�M/n2)rp/2, thenfor any integer r such that 0r2r�1rmin{p/j, LA}

1

M�ðn2 �MÞ2 r þ

Pr�1i¼0

cos ð2iþ 1Þj� �

2Mn2ð2n2 �MÞ sin2 ðrjÞ

8>>><>>>:

9>>>=>>>;

� d2a þ 1� 1

M

� �� d2c � n� r þ 1

2� n2

2M

� sin ðrjÞ � sin ðr � 1Þjsin ðrjÞ

ð17Þ

1�ðn2 �MÞ2 � r þ

Pr�1i¼0

cosð2iþ 1Þj� �

2Mn2ð2n2 �MÞ sin2ðrjÞ

8>>><>>>:

9>>>=>>>;� d2m

� n� r þ 1

2� n2

2M� sinðrjÞ � sinðr � 1Þj

sinðrjÞ ð18Þ

Proof: By theorem 1 and lemma 6, we have

1

M� cos2 j

4M cos2 ðj=2Þ sin2 ðrjÞ

( )

� r þXr�1i¼0

cos ð2iþ 1Þj !)

� d2a þ 1� 1

M

� �

� d2c � n� r þ 1

2� n2

2M� sin ðrjÞ � sin ðr � 1Þj

sin ðrjÞ

Note that cos ðj=2Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�M=ð2n2Þ

p, because cosj¼

1�M/n2, thus (17) holds. Q.E.D.We can derive the following results by simplifying the

expressions on the left-hand sides of (17) and (18). All theproofs are omitted due to lack of space.Corollary 4: Let Mrn2, 0oj¼ arccos(1�M/n2)rp/2,LAZp/j, r¼Jp/(2j)�1/2n,

hðn; MÞ ¼12p ; 2jrop;14; 2jr � p;

(

gðr; jÞ ¼ 1

sin2 ðrjÞ� fr þ ðp� 2rjÞhðn; MÞjr2

þ 1� ð�1Þr

2½cos ðrjÞ

þ ð1� 2rÞðp� 2rjÞhðn; MÞj�g

P0ðn; M ; LAÞ ¼1

M� ðn2 �MÞ2

2Mn2ð2n2 �MÞ

� gðr; jÞ; D0ðn; M ; LAÞ ¼ 1� 1

M

R0ðn; M ; LAÞ ¼ n� r þ 1

2� n2

2M

� sin ðrjÞ � sin ðr � 1Þjsin ðrjÞ

then,

P0ðn; M ; LAÞ � d2a þ D0ðn; M ; LAÞ� d2c � R0ðn; M ; LAÞ ð19Þ

1

M� ð32� 3p2Þðn2 �MÞ2r

64Mn2ð2n2 �MÞ

( )� d2a

þ 1� 1

M

� �� d2c � R0ðn; M ; LAÞ ð20Þ

1

M� ð32� 3p2Þðn2 �MÞ2ð2n2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Mn2 �M2p

Þ128n2Mð2n2 �MÞ


( )

� d2a þ 1� 1

M

� �� d2c � n� pnffiffiffiffiffiffiffi

8Mp�

ð21Þ

1� ð32� 3p2Þðn2 �MÞ2ð2n2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Mn2 �M2p



( )

� d2m � n� pnffiffiffiffiffiffiffi8Mp�

ð22Þ

4 Lower bounds on the aperiodic correlation ofnormal complex roots-of-unity sequences

Because the normal correlation operation can be consideredas a special case of the operation for LCZ sequences forLA¼ n�1, based on the general results presented in Section3, new lower bounds for normal complex roots-of-unitysequences can be established.Corollary 5: Let

P1ðn; MÞ ¼ 3ðn� 1Þ2Mn2 � 3n2 þM

; D1ðn; MÞ

¼ 3nðM � 1Þ2Mn2 � 3n2 þM

We have

P1ðn; MÞ � d2a þ D1ðn; MÞ � d2c � 1 ð23Þ

d2m �2Mn2 � 3n2 þM

3ðMn� 1Þ ð24Þ

Equation (24) was first obtained by Boztas [22].Corollary 6: If MZ3, nZ2, thenffiffiffi

3p


ðffiffiffi3p

M � 2ffiffiffiffiffiMpÞn2� d2a þ

ffiffiffi3pðM � 1Þ

ðffiffiffi3p

M � 2ffiffiffiffiffiMpÞn

� d2c � 1

ð25Þ


� 2ffiffiffiffiffiffiffi3Mp

n� 1� n2 ð26Þ

Boztas obtained the following bound [22]:

d2m � n� 2nffiffiffiffiffiffiffi3Mp ð27Þ

It is clear that the bound (26) is stronger than the Boztasbound (27).


Corollary 7: For any integers M, n, n42, M4n/3+1, let

Rðn; MÞ ¼ð3Mn� n2 � 4MÞ4n�1

þ 3ðn� 3ÞM2n þ 2ð3Mn� n2 þ 5MÞ

P2ðn; MÞ ¼ 2ð4n�1 � 1ÞRðn;MÞ ; D2ðn; MÞ

¼ 3ðM � 1Þ4n�1

Rðn;MÞwe have

P2ðn; MÞ � d2a þ D2ðn; MÞ � d2c � 1 ð28ÞCorollary 8: Let Mrn2, 0oj¼ arccos(1�M/n2)rp/2, thenfor any integer r such that rjop/2+j/2, 1r2rrn,

1� ðn2 �MÞ2

2Mn2ð2n2�MÞ sin2 ðrjÞ� rþ

Xr�1i¼0

cos ð2iþ 1Þj" # !

� d2m � n� r þ 1

2� n2


sin ðrjÞð29Þ

Boztas [22] obtained the following bound

d2m � n� r þ 1

2� n2


sin ðrjÞ ð30Þ

It is clear that the bound (29) is stronger than the Boztasbound (30), since

r þXr�1i¼0

cosð2iþ 1Þj4 0

Corollary 9: For all integers M, n, and 6rMrn2, let

P3ðn; MÞ

¼

1

M� ð32� 3p2Þðn2 �MÞ2ð2n2 �




n� pnffiffiffiffiffiffiffi8Mp�

:D3ðn; MÞ ¼1� 1

M

n� pnffiffiffiffiffiffiffi8Mp�

we have

P3ðn; MÞ � d2a þ D3ðn; MÞ � d2c � 1 ð31Þ

1� ð32� 3p2Þðn2 �MÞ2ð2n2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Mn2 �M2p



( )

� d2m � n� pnffiffiffiffiffiffiffi8Mp�

ð32Þ

Boztas [22] obtained the following bound:

d2m � n� pnffiffiffiffiffiffiffi8Mp�

ð33Þ

Thus the bound (32) is better than the Boztas bound (33)for all integers M, n, 6oMon2.

5 Conclusions

We have established several new lower bounds on theaperiodic autocorrelation and cross-correlation for LCZand normal sequence sets over the complex roots of unity. Ithas been shown that these bounds are tighter than the

previous Boztas bounds for the aperiodic correlations ofnormal complex roots-of-unity sequences. As a specificcomparison, if n¼ 20 and M¼ 8, then the minimum value

of d2m given by the Boztas bound (33) is 12.0000, while the

minimum value of d2m given by the new bound (32)presented in this paper is 12.1243.

Because the complex roots-of-unity sequences include thebinary sequence as a special case, all the previous aperiodicbinary sequence bounds, such as Sarwate bounds, Welchbounds, Levenshtein bounds, Tang–Fan bounds and Peng–Fan bounds, can be considered as special cases of thispaper. Moreover, for the binary sequence set, the newbounds are stronger than the well known Welch bounds,Sarwate bounds and Levenshtein bounds. For the aperiodiccorrelation function of binary sequences, Levenshteinderived the following bound [21]:


� 2ffiffiffiffiffiffiffi3Mp � n; ðn � 2; M � 3Þ ð34Þ

It is clear that the new bound (26) presented in this paper isstronger than the Levenshtein bound (34). For example, let

n¼ 10 and M¼ 15, then the minimum value of d2m given bythe Levenshtein bound (34) is 7.0186, while the minimum

value of d2m given by the new bound (26) is 7.1248.Let

Sðn; MÞ ¼ 2ðn� 1ÞðM � 1Þn2

; T ðn; MÞ ¼ 2n� 1

n2;

Sarwate [20] established the following aperiodic bound:

Sðn; MÞ � d2a þ T ðn; MÞ � d2c � 1 ð35ÞLet

P0ðn; MÞ ¼ffiffiffi3p


ðffiffiffi3p

M � 2ffiffiffiffiffiMpÞn2

;

D0ðn; MÞ ¼ffiffiffi3pðn� 1Þ

ðffiffiffi3p

M � 2ffiffiffiffiffiMpÞn

Equation (25) becomes

P0ðn; MÞ � d2a þ D2ðn; MÞ � d2c � 1 ð36ÞWe see that the bound (36) is stronger than the Sarwatebound (35), since S(n,M)4P0(n,M) and T(n,M)4D0(n,M)for all integers M47 and n46. If M¼ o(n2) tends toinfinity as n-N, then

Sðn;MÞP0ðn;MÞ

¼ 2ðn� 1Þffiffiffi3p

n�ffiffiffiffiffiMp �

ffiffiffi3p

M � 2ffiffiffiffiffiMp

M � 1

and

T ðn;MÞD0ðn;MÞ

¼ 2ðn� 1Þffiffiffi3p

n�

ffiffiffi3p

M � 2ffiffiffiffiffiMp

M � 1

are approximately equal to two. Therefore, the Sarwatebound (35) is twice as large as the bound (36). As a specificcomparison, let n¼ 9 and M¼ 10, then it can be seen that

the coefficients of d2aand d2c in the Sarwate bound (35) are0.0219 and 0.2099 respectively, while the coefficients of

d2aand d2c in the new bound (36) presented in this paper are0.0140 and 0.1575 respectively. Similarly, it can be shownthat our new bounds are tighter than Welch bound [19].

6 Acknowledgments

This work was supported by the National ScienceFoundation of China (NSFC) and the Research Grants


Council of Hong Kong (RGC) joint research scheme (NO.60218001/N_HKUST617-02), the National Key Labora-tory of Communications (UESTC), the Royal SocietyFellowship, UK, and the Millennium Project of MEXT,Japan.

7 References

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Generalised Sarwate bounds on the aperiodic correlation of sequences over complex roots of unity

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