New results on the statistical propertiesof pseudonoise sequences

A.A.A. Al-Mukhtar, B.Sc.(Eng), D.I.C., Ph.D., A.M.I.E.E.. Mem.I.E.E.E., andProf. L.F. Turner, Ph.D., D.Sc.(Eng.), C.Eng., F.I.E.E.

Indexing terms: Mathematical techniques, Modelling, Filters and filtering

Abstract: A mathematical model for a general periodic correlation function of pseudonoise (PN) sequences isproposed. The model is used to derive expressions for the probability density function of the correlation func-tion, when its argument is a random variable with a uniform distribution. The model is also used to deriveexpressions for the autocorrelation function and power spectral density of lowpass filtered PN sequences.

1 Introduction

Properties of pseudorandom or pseudonoise (PN)sequences have been investigated extensively in the liter-ature [1]. In this paper, new results relating to the proper-ties of PN sequences are presented. A mathematical modelfor a general periodic correlation function is presented inSection 2. The model is sufficiently general to represent theautocorrelation or crosscorrelation functions of anymaximal length, Gold, or any other PN sequence. InSection 3, the model is used to derive expressions for theprobability density function (PDF) of the general correla-tion function for the case when the argument of the latteris a uniformly distributed random variable. The results ofSection 3 have important applications in spread spectrumsystems employing maximal length, or Gold sequences,and operating over multipath channels, where the argu-ment of the correlation function could represent the timedelay of the received signal. Another application of theresults could be in multi-user spread spectrum systemswhere the different users might transmit information atrandom instants in time. In Sections 4 and 5, respectively,the correlation model is used to derive expressions for theautocorrelation function and the power spectral density ofthe signal at the output of a lowpass filter, when the inputto the filter is a general PN sequence. The results of Sec-tions 4 and 5 are new generalisations of results previouslyobtained [2] for the special case of maximal-lengthsequences.

2 General correlation function

Fig. 1 shows a general correlation function, 4>XY{X\ whichis periodic with a period Ld seconds. This general form of<PXY(X)

c a n be used to represent the autocorrelation orcrosscorrelation functions of any maximal length, Gold, orrelated sequence. If x(t) and y(t) represent two binarysequences which are periodic with period Ld seconds,0xr(T) can be defined as

1 Ld

= TT x(f)y(t + x) dt (1)

By inspecting Fig. 1, the following features are noted:(i) The function is periodic with period Ld(ii) It has a constant DC level of — 1/L(iii) There is one positive peak of height (Ao + 1/L)

above the DC level and centred at T = 0 (mod Ld). Thiscan be considered as the 'principal peak'

Paper 4475G (E10), first received 14th May 1985 and in revised form 8th January1986

Prof. Turner is, and Dr. Al-Mukhtar was formerly, with the Department of Electri-cal Engineering, Imperial College, Exhibition Road, London SW7 2BT, UnitedKingdom. Dr. Al-Mukhtar is now with Technocrat Ltd., 62 King Street, Maiden-head, Berks. SL6 1EQ, United Kingdom

(iv) In addition to the principal peak, there are P othergroups of positive peaks. The 7th group contains Nj peaks(Nj ^ 1) each with height {Aj + 1/L) above the DC level.These peaks are centred at x = dJti, dj 2 , . . . , dJtN where; = 1, 2, . . . , P. The order of each group is related to theheight of its peaks, i.e. A0>Al>A2>--->AP. Thetotal number of positive peaks is Yj=o Ny, where No ^ 1

(v) There are Q groups of negative peaks. The jth groupcontains Vj peaks (Vj > 1) each with height (Bj — 1/L)below the DC level. These peaks are centred at T = 61Jt lt

Slj2, ••-, &lj,Vj- Here 7 takes the values j= 1, 2, . . . , Q.Again it is assumed that B^ > B2 >number of negative peaks is £ ? = x Vj

(vi) Each peak positive or negative has a base which isof width 25 at the DC level

(vii) Usually peaks can overlap by exactly one 3(viii) An autocorrelation function, 0yy(T), is X symmetri-

cal about the axis x = 0 (mod Ld), whereas a cross-correlation function ^ ^ ( T ) ,

x ^ y> n e e ( i n o t be.

> BQ. The total

Thus

(2)

Note that properties (i) and (viii) are true for any binarysequence.

2.1 Mathematical representationDefine the function go(x) such that

go(x) = u_2(x + d)- 2 U _ 2 ( T ) + W _ 2 ( T - 5) (3)

where w_2() is a unit ramp function defined [2] asfollows:

W _ 2 ( TT

= s= 0

(4)otherwise

It follows that the entire correlation function 0Xy(T) c a n

represented mathematically as follows:

\1 (\= -T + Z (-

L fc=-oo V1

\A0)g0(x-kLS)

kLS) 1 (5)

IEE PROCEEDINGS, Vol. 133, Pt. G, No. 2, APRIL 1986 99

PDF of the general correlation function

In this Section, expressions for the PDF of

In evaluating the PDF of (j)XY(x), the range for 0^y(r) isdivided into different intervals, and each interval is

, . , T' a r e analysed separately. Consider the interval Ai+1 < $ < A-.derived as x varies randomly. Consideration is given to the s h o w n s h a d e d i n F i g j a n d m a g n i f i e d i n F i g

J 2. Using the

•XY(T)

(L6.AJ

Fig. 1 General correlation function

special case where x is uniformly distributed. Thus, if/£(x) is terminology adopted in Fig. 2, the cumulative distributionthe PDF of T, function (CDF) for this interval is given by

1

I ' (6)

ffl

(8)k = l

The problem is to evaluate the PDF of ^ ^ ( T ) , where0xy(r) has become a function of the random variable T.Define the random variable <£ such that

^ = (j)XY(x) (7) The PDF /̂ (</>) for this interval can now be obtained by

to (Lfi.A.)

A i . «

to

( d j . 2 5 . A j > Cdj 3 S . A j

d 6i. 1

Fig. 2 Detail of the interval AJ+l < (f> < /47- o

h;l(cP) = LS(A0-<t>)/(l +A0L)(1 + AjL)}/(\ + AjL)

L48 «}/(! 4 ^1X1 + ^ 1 ^^ W + dj 25{l + AjL)}/(\ + A}L)

1X1 + /1;L) - 0^5 - 5}/(l + AjL)» { / I JW + ^. 3<5(1 + AjL)}/(i + ^-L)

/»,"'(</>) = { ^ 3 + 'X I + Aj'L) - ct>L5 - 5}/(l + AjL)L d A L S d 5(1 AL)}/(1

{<5(rfj_,,, + 1X1 + - 4 ^ , L{4>L5 + LS + A0L8(L -

j

L5 - 6}l(\+ A0L)

d &1 .3

d 61 - 1 . 1

L6

differentiating eqn. 8 with respect to (f). The general formis given by

k=j 2N

Eqn. 9 is true for the interval above, provided that all thepeaks in that interval are simple (non-overlapping). Using

100 IEE PROCEEDINGS, Vol. 133, Pt. G, No. 2, APRIL 1986

the same procedure, the PDF /0(0) can be derived forother intervals with the result being seen to be:

2N, 1(10)

Similarly, for negative intervals

k j 2V

= Z

Q_ y

-Bj<<t><-BJ+i,

2Vk -BQ<(f)< - 7

(11)

(12)

If the DC level 0 = — 1/L is represented by a Dirac deltafunction, w_i(.), then

/,«>) = •!' - ( J

The results contained in eqns. 9-13 apply when all thepeaks are non-overlapping. The effect of overlap amongthe peaks will now be considered.

3.1 Overlapping peaks of equal height (Type I)Fig. 3a illustrates the effect of a cluster of n peaks of height{Aj + 1/L) above the DC level. Note the existence of a newDC level at 0 = A}.For - 1/L < 0 < A,-,the PDF [3] is

Fig. 3 Effect of a cluster ofn peaks of height Aj + 1/L

a Above DC level b Below DC level(i) T = \<}>L5 -AjL6 + Sdj ,(1 + AjL)}/{\ + AjL)

(ii) T = {-<t>L5-5 + d(dj „+ IK1 + AjL)}/(l + AjL)(iii) t = {<t>L6 + S + {1- BjL)(ej , 1)5}/(1 BL)( i v ) T = { -<t>Ld - S + <5(e, „ + l ) ( l

It follows from eqn. 14 that a cluster of peaks of the sameheight can be counted as a single peak. Also for 0 = Aj

(15)

where c, is the width of the domain T for which 0 has theDC value Aj. The results for a cluster of negative peaks ofequal height, (see Fig. 36), are similar. For 0 = — Bk

(16)

where c'k is the width of the domain T for which </> has theDC value - Bk.

3.2 One-sided overlapping peaks of opposite sign(Type II)This situation is shown in Fig. 4. Consider Fig. 4a first.For - 1/L < 0 < Aj, the PDF is

For -Bk < (f>< - 1 / L

(18)

Results for Fig. 4b are similar,

( i )

-1/L\

-1/L

(vi)

Fig. 4 One-sided overlapping peaks of opposite sign

(i) T = {<t>LS - AjLd + djid(l+ AjL))/(\ + AjL)(ii) r = {-<l>6 + Aj6 + dj tS(Aj + Bk)}/(Aj + Bk)

(iii) T = {-5 - <t>LS + 5(et , + 1X1 - BkL)}/{\ - BkL)(iv) { }(v)

(vi)

t{<t>L6 + 3 + 8(ek r - 1)(1 - Bk L)}/(\ - Bk L)

= {(f>6 - AjS + Sd] {A; + Bk))/(Aj + Bk)= {-(f)L8 -6 + 8(dj ,. + 1 HI + AjL)}/(l + Aj

IEE PROCEEDINGS, Vol. 133, Pt. G, No. 2, APRIL 1986 101

3.3 Double-sided overlap between a positive peak and For Ak < 0 < Aj} the PDF istwo negative peaks of equal height, or vice-versa(Type III) f{(j)) = ... f

This situation is shown in Fig. 5. Consider first Fig. 5a. - J , . 1_,For - 1/L < 0 < Aj, the PDF is ) l + [AJ ~ k~ L

2

(23)

j + Bk)}(19)

whereas for Fig. 5b, for the interval —Bk<(f)< — 1/L, thePDF is

(20)

-1/L

-1/L

(iv)

( in)

Fig. 5 Double-sided overlap between three peaks of equal height

a Positive and two negative peaksb Negative and two positive peaks

(i) r = { # - AjS + ddj i(Aj + Bk)}/(Aj + Bk)(ii) T = {-<t>S + Aj5 + 5dj fiAj + Bk)}/(Aj + Bk)

(iii) t = {-4>d + Ajd + 5(lk , - lUj + Bk)}/{Aj + Bk)(iv) T = {(j>6 - Ajd + d(lk,,. + 1X/1; + Bk)}HAj + Bk)

3.4 Double-sided overlap between a positive peak andtwo negative peaks of different heights, orvice-versa {Type IV)

This situation is shown in Fig. 6. Consider first Fig. 6a.For - 1/L < (f) < Aj, the PDF is

(2Aj + Bk + Br)

{2L(Aj + Bk)(Aj + Br)}(21)

whereas for Fig. 6b, for the interval —Bk<(j)< — 1/L, thePDF is

U(4>) = • • • 2(2Bk

{2L(Ar + Bk)(Aj + Bk)}(22)

3.5 Overlap between peaks of the same polarity(Type V)

-1/L

-1/L

Fig. 6 Double-sided overlap between three peaks of different heights

a Positive and two negative peaksb Negative and two positive peaks

(i)

-1/L

-1/L

This situation is shown in Fig. 7. Consider first Fig. la. (iv) T = {-

-B.

Fig. 7 Overlap between peaks of the same polarity

(i) T = {0.5 + AjHdj , - I) - Ak6dj ,}/(Aj - Ak)(ii) T = {-<t>5 + Aj8(dj ,- + 1) - Akdj i8}/(Aj - Ak)

(iii) t = {<f>LS + 6 + 5{dk , - l)(l + AkL)}/(\ + AkL)( }k

+dkr+l)}/(\ + AkL)

102 IEE PROCEEDINGS, Vol. 133, Pt. G, No. 2, APRIL 1986

The peak thus 'appears' to have a height of Aj — Ak abovethe DC level, whereas for - 1 / L < 0 < Ak the PDF is

• • • • ( T T * i $ - ( 2 4 )

which is the same result as would be obtained with onesimple peak of height (Ak + 1/L) above the DC level.Results for Fig. 1b are obtained in the same way. For con-venience, the results of above relating to the PDF of 0xr(T)can be summarised as follows:

The autocorrelation function 0Z Z(T), is given [2] by

(t>zz(s) = E{z{t)z{t + s)}

0

k%(l+AkL)

IN,

2K

j IV,

\ 0

Ao < 0 < oo

0 = Ap

1< d> < AP

L

Q P Q

- J3 Q <0< - -

I 4>=-Bn

= - B .

— 00 < 0 < — £ j

(28)

Note that 4>YY(X) is periodic with period LSj the function

= 0,1,...

(25)

In applying eqn. 25, care must be exercised to ensure thatthe different types of overlap that may occur among thepeaks are taken into account and that the derived correc-tions are applied.

4 Autocorrelation of smoothed PN sequences

In Reference 2, calculations were carried out to determinethe correlation properties at the output of a first-orderlowpass filter when the input was a maximal lengthsequence. In this Section, the results are generalised toapply to a sequence with general correlation properties ofFig. 1 (e.g. Gold sequences) with maximal length sequencesbeing a special case.

Let h(x) be the impulse response of a lowpass filterwhere

(26)h(x) = \-)e

and 1/TC is the 3 dB cutoff frequency of the filter. If y(t) isthe binary sequence at the input to the filter, the filteroutput z(t) is given by

z(t) = h(r)g(tJo

x)dx (27)

0zz(s) is also periodic with the same period. Furthermore,by using eqn. 2,

(29)

When S ^ s < (L — 1)<5 it can be shown [3] that

(f>YY(u - T j e - ' 1 ' * dx,

k = 0

[.I Jo (30)

In arriving at eqn. 30, use is made of eqn. 5 with onlynonpositive values of k considered [3] and the substitutionyk = u + kL is used.

IEE PROCEEDINGS, Vol. 133, Pt. G, No. 2, APRIL 1986 103

The integral in the second term on the right-hand side Note further that as 0yy(T) is an even function of x accord-ing to eqn. 2, it follows that the positive and negativepeaks (apart from the principal peak) generally occur inpairs. This means that Nj and Vj are generally even. Hence

of eqn. 30 was evaluated in Reference 2 by noting that

0 , — co < x^ <yk — s

> 7k - <5 < Tj < yfc

\0 ,yk + 3 <xx<oo (31)

Thus, by using eqn. 31, it follows that

/IZ T 9o(yk - *i)e-zlllc dx,

- - 1

and

ei i =

= ejt Vj+! _,(mod L), j = 1, 2 , . . . , Q

(37)

(38)

On substituting eqns. 37 and 38 in eqn. 36, the following isobtained:

- (|){cosh (i) -

,rc}(32)

By considering the integral in the third term on the right-hand side of eqn. 30 and using the substitution y'k = yk

— dj {d, it follows that

(33)

On considering the integral in the fourth term on the right-hand side of eqn. 30 and using the substitution y'k' = yk —Iji • 3, it follows that

TllZc dx,

H"f'« exp

±1-1 (34)

As noted in Reference 2 there is an error in the integrals ofequation (28) when 0 ^ s ^ <5. This error term was evalu-ated in Reference 2 and shown to be

£(s) =

0 otherwise (35)

Thus, by using eqns. 32-35 in eqn. 30 and by evaluatingthe integrals over u, it follows that

/=!

5d{

E(s) (36)

x < -

~Nil2 fbd. •

x I £ cosh ' ^

SL / l

x cosh E(s) (39)

Eqn. 39 applies to the general correlation functions of Fig.1 even with overlap, provided that the overlapping peaksare counted separately.

5 Power spectral density of smoothed PNsequences

In Reference 2 an expression was derived for the powerspectral density at the output of a lowpass filter, when theinput was a maximal length sequence. In this Section, theresult is generalised to the case of a sequence with generalautocorrelation function shown in Fig. 1.

Using the notation of Section 4, let Sy(f) be the powerspectral density of y(t). It follows that

(40)= FT{<f>¥Y(x)} = <t>YY{x)e-»^dx

where FT {.} denotes YY the Fourier transform. Let Sz(f)be the power spectral density of z(t). Then

(41)

where H(f) is the transfer function of the lowpass filter.For the filter

H(f) = FT{h(x)} =1

+J2nfxc)(42)

Thus, provided that Sy(f) is available, S3(f) can easily becalculated. Note that

FT{go(x)} = 3 sin = 3 sine2 (f3) (43)

104 IEE PROCEEDINGS, Vol. 133, Pt. G, No. 2, APRIL 1986

and that [4, p. 37, eqn. 2.15]

f 00 1 /i

FT] £ gQ{T-kLS)\ = [-k= - o o

k

By using eqns. 41-44, applying the shift theorem to eqn. 44and using the result in eqn. 5, and further, noting that theFourier transform of a constant is a Dirac delta function,the following expressions can be obtained for Sz(f),

£ exp{-j2nditmdf)

r v,

E\_m= 1

Qxp(-j2nei<mSf) (45)

Finally, by combining eqns. 37 and 38 and using the factthat the second Dirac delta function in eqn. 45 is zeroeverywhere except a t / = k/(LS),

fc= - 0 0

s in°2(z

X M _ , < / —

COS (46)

6 Illustrative example

The results set out above can be illustrated by consideringthe Gold sequence of length 31(=25) generated using alinear feedback shift register with feedback connections atpositions [10, 9, 3, 1] and with an initial loading(1000000000). The autocorrelation function of thissequence is shown in Fig. 8 and the PDF of the autocor-relation function is shown in Fig. 9. From eqn. 39, the

2/31

13/31

1/12

-9/31 -1/31 7/31 1 p

Fig. 9 PDF of the autocorrelation function of the Gold sequence [70, 9,

Fig. 8 Autocorrelation function of the Gold sequence [70, 9, 3, 7] with loading (1000000000)L = 3 l , N o = 1, Ao= I, N , = 6 , / J , = 7 / 3 1 , c , = 4 , d ,_ , = 1, dl 2 = 2, du3 = 3, rf,_4 = 28, dui = 29, d1>

< - i . 4 = 2 1 , < - , , , = 2 2 , < ? , , 6 = 2 4 .V, = 6 , B, =9/31 , c', = 2 , e, , = 7 , e, 2 =9, v e , 3 = 10,

/££ PROCEEDINGS, Vol. 133, Pt. G, No. 2, APRIL 1986 105

autocorrelation function of the smoothed sequence is

31

J-] - 1 | exp (-?

x < 1 — exp32

r (&x cosh —

L W4- cosh I — + cosh —

T, / V T

- (C( c o s h (? + c o s h(?where

0 otherwise

On application of eqn. 46, the power spectral density of thesmoothed sequence is found to be

961 k=-oosine' I —

C 0 S

4TE/C 6TC/C

\6_961 c o s c o s

7 Conclusions

A general mathematical model for the correlation func-tions of PN sequences has been proposed. The model isgeneral enough to represent the autocorrelation or cross-correlation functions of any maximal length, Gold, orrelated sequence. The model has been applied to deriveexpressions for the PDF of the correlation function, whenthe argument of the latter is a uniformly distributedrandom variable. The model has also been used to evalu-ate the autocorrelation function and the power spectraldensity of a general PN sequence, when the latter under-goes lowpass filtering.

The above results presented in the paper have impor-tant application in spread spectrum systems using maximallength, of Gold sequences, and operating over multipathchannels. Another important application is in the analysesof multi-user spread spectrum systems where differentusers might transmit information at random instants intime.

8 References

1 SARWATE, D.V., and PURSLEY, M.B.: 'Crosscorrelation propertiesof pseudorandom and related sequences', Proc. IEEE, 1980, 68, (5), pp.593-619

2 ROBERTS, P.D., and DAVIS, R.H.: 'Statistical properties of smoothedmaximal-length linear binary sequences', Proc. IEE, 1966, 113, (1), pp.190-196

3 AL-MUKHTAR, A.A.A.: 'Spread spectrum systems operating throughmultipath channels'. Ph.D. Thesis, University of London, 1984, pp.82-130

4 TORRIERI, D.J.: 'Principles of military communication systems'(Artech House Inc., Dedham, MA, 1981)

106 IEE PROCEEDINGS, Vol. 133, Pt. G, No. 2, APRIL 1986