Optimal Binary Sequences for Spread Spectrum Multiplexing.pdf

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1/3

CORRESPONDENCE

619

of one communication link lock onto the cross-correlation peaks

obtained by correlating with the encoding sequence of a different

communication link. Thus the successful use of spread spectrum

communicati on syst ems in multiplexing applicati ons depends

upon the construction of large families of encoding sequences with

uniformly low cross-correlation values. In this paper we present

an analytical technique for the constructi on of such families of li near

binary encoding sequences.

II. NOTATION

Following the notation of Zierlex+l we denote by V(f) the vector

space of linear sequences generated by the recursion relation cor-

responding to the polynomial f of degree n and further identify

the members of V(f) with binary 2* - 1 tuples. If f is a primitive

polynomial over the field K = (0, 1) then h e V(f) implies h is a

maximal linear sequence. We denote by ]lhll the number of ones in

the sequence h and by & the sequence such that R(i) = h(z) + 1.

The correlation function 8 of two binary sequences a,

b

has been

defined as

Equation (17) is equivalent to the recursion

2 2

a, - an+

(SNR), 2

= ___ a,-,

N

whose solution is

a; = a;

[

1 + qq-.

Just as in Schalkwijklzl we generate the estimates recursively, i.e.,

1

a2, NW,

8'(n)

1+

@p

e"@1)

1 I (Sti-R)i % (20)

N

J. P. M. SCHALKWIJK

L. I. BLUESTEIN

Communication Systems Laboratories

Sylvania Electronic Systems

Div. of Sylvania Elec. Prods., Inc.

Waltham, Mass. 02154

REFERENCES

11 T. J. Gobliok, Theoretical limitat ions on the transmission of data from

an&g murces, IEEE Trans. Information Theory, vol. IT-l& pp. 558-567,

October 1965.

El J. P. M. Schalkwijk,

A coding scheme for additive noise chitnnels with

feedback, Part II: Band-limited signals,

IEEE Trans. Information Theory.

vol. IT-U, pp. 183-189, April 1966.

Ial T. Kailath, An application of Shannons r&e-distortion theory to analog

oommunicrttion over feedback channels,

Proc. Princeton Symp. on Systm Science,

Mrtrch 1967.

141T. J. Cruise, Achievement of rate-distortion bound over additive white

noise channel utili alng & noiseless feedback channel, Proc. IEEE (Letters), vol. 55,

pp. 583-584, April 1967.

Optimal Binary Sequences for Spread Spectrum Multi-

plexing

I. INTRODUCTION

Linear shit register sequences (see Zierlerlll and Gold121) have

found extensive applications in spread spectrum communication

systems. The binary sequences generated by shit register devices

serve as the encoding mechanism of such systems which, when

added to the baseband information, results in a wideband low-

power-densi ty signal which has statistical properti es similar t o

noise. The casual listener is thus denied access to the baseband

information which can be recovered from the wideband signal

only thr ough correlation with a stored reference sequence in the

receiver which is an exact replica of the original encoding sequence.

The usefulness of the maximal linear sequences n spread spectrum

communications depends in large part on their ideal autocorrelation

properties. The autocorrelation function of a binary sequence h is

defined as O,(T) = (number of agreements - number of disagree-

ments) when the sequence

h

is compared with a cyclic shift of itself.

It is well known that for maximal linear sequences s,(O) = period

of the sequence and &,(7) =

- 1 for 7 Z 0. The detection by the

receiver of the high i n-phase correlation value oh(O) determines

the synchronization between transmitter and receiver necessary

for t he removal of the encoding sequence and the recovery of t he

baseband information. In multiplexing applicati ons many systems

will be operating in the same neighborhood and each communication

link will employ a different maximal encoding sequence. In general,

the cross-correlation function between different maximal sequences

may be relatively large. Thus dierent systems operating in the

same environment can interfere with the successful attainment

and maintenance of proper synchronization by having the receiver

Manuscript received July 22, 1966; revised February 4, 1967.

where x is the unique isomorphiim of the additive group (0, 1)

onto the multiplicative group { 1, -1). We note that (?(a,

b) (T) is

simply described ss the number of agreements - number of dis-

agreements of t he sequences a and b for each 7 and that

O(a, b) = 2 - 1 - 2 I/a + bjj.

In what foll ows a will always denote a primitive 2 - 1 root of

unity in a splitting field of x2--1 + 1 and the minimal polynomial

of & will be denoted by fi. Finally, we note the following result of

Bose and Chaudhuri.131

Theorem 1

Let LY e any primiti ve element of the splitti ng field of zm-1 + 1.

Let fi be the minimal polynomial of at. Let

2n- + 1

= lcm 715, f2, . - - LJ

Then a, b E V(g) implies Ila + bj j > 2k.

III. STATEMENT AND PROOF OF RESULT

Our techniques for the construction of large families of encoding

sequences with uniformly low cross-correlation values is based on the

following result.

Theorem d

Let (Y be any primitive element of GF(2). Let fi be the minimal

polynomial of DI. Let ft be the minimal polynomial of CY~ here

t = p

(+1)2) + 1 (n odd)

b

(n+2)2) + 1 (n even).

Then a e V(f,) and b E V(ft) implies [@(a,b) 1 5 t.

The signif icance of this theorem is that it tells how to select

shit register tap connecti ons which will generat e maximal linear

sequences with a known bound on the cross-correlation function.

Since r~ is primitive, the sequence generated by t he shift register

corresponding to fi is maximal. Since

2(n+1,/2 + 1 a nd s-/n+z)/z 1

are both relatively prime to 2n - 1 for n $ 0 mod the 4 sequence

corresponding t o the polynomial ft is also maximal in these cases.

Theorem 2 thus permits the selection of pairs of maximal sequences

with known bound on the cross-correlation function. This result

is of practical importance since, for example, for n = 13 there are

630 maximal sequences and there exist pairs of t hese sequences


2/3

620

whose correlation values are as high as B = 703 while Theorem 2

guarantees the selection of pairs of sequences such that 101 5 129.

This result is a special case of the more general theorem stated

in the following which has been obtained independently by Goldfsl

and Kasami,f 41 and is related t o the weight distribution of error-

correcting codes.

Theorem

Let a and b be maximal linear sequences given by

a(i) = T(or-i) and b(i) = 5 ((~zL+1)-i)

where 01 s a primitive 2 - 1 root of unity (n odd), I is any integer

such that (I, k) = 1, and

T

is the trace of GF(2). Then e(a, b) (n) =

- 1 when u(7) = 0 and

IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER

1967

where

i (

2

(L+1)2+ 1)

qu, b)(T) =

I

or

when a(~) = 1.

(2

(n+1)/2

- 1)

The proof of this theorem is contained in Gold151.

In the remainder of this section, we proceed to the proof of

Theorem 2 by means of a series of lemmas.

Lemma 1

Let 01be any primitive 2 - 1 root, of unity in a splitting field of

z2-1 + 1.

Let

**-I+ 1

gk= lcm7fl, z,

**. fk)

where fi i s the minimal polynomial of (Y k

implies f i s a factor of gk.

Proof: f ir reducible of degree n implies f ( zin-l + 1 implies f 1 gk

lcm Vi, f2, . *. fkl,

and rnf > k implies ft j Icm (fi . . . fk) implies

f 1 gk.

Lemma .%?

Let 01be any primitive 2% - 1 root of unity in a splitting field of

X 2-1 + 1.

Let u = 2*-l - 1. Let

2-l - 1 - 2- for n odd

v=

1

y-1 - 1 - y/2

for n even.

Let fU be the minimal polynomial of 0%. Let f% be the minimal

polynomial of au. Then mf, = u and rnf, = a.

Proof: 01r and a* belong to t he same conjugate class of GF(2) if,

and only if, there exists an integer k such that (Y = (01~)~ f, and

only if, T = s.2k modulo 2% - 1 if, and only if, t here exists a cyclic

permutation p such that [r(O), r(l), . . . r(n - l)] = [s(p(O)),

4PU)) . . .

s (p(n - l))] where

n-l n-1

r = c ~(42 and s = c ~(232~.

i=o

i=

Now u = 2%

- 1 = c::h u(i)

2;

where [u(O), u(l), . . . u(n - l)] =

[l 1 **.

1 01. Clearly any permutation of [l 1 . 1 0] corresponds

to a larger integer and hence mf, = U. Now

n-1

2) = 2n-l

- 1 -

y-12

= c

v(4y

i=o

[

v(O), (l), * *

v(g ,v(+ )

n+1

fj __-

( >

. . . v(n - 2)v(?z - 1)

1

= [l, l,>;. * l,, 0, t ; 1; 01

n-1,2 ones n-1,2 ones

Again it is clear that any cyclic permutation of [v(O), . . . v(n - l)]

will result i n a larger integer and hence mf, = e. A similar argument

holds when a = 2n-1 - 1 - 2(n/2).

Lemma S

fU and f. are factors of g+l where u and a are as in Lemma 2

and

x2- + 1

-l = Icm (fl, fz, . . . f.-I)

then fv go-l,

Proof:

mfs

= u (by Lemma 2) = 2 - 1

>2n-1 > v - 1 implies fU / g,-l (by Lemma 1)

mf.

= v (by Lemma 2) > B - 1 impliesf, / g,,-l.

Lemma 4

Let ,fc denote the minimal polynomial of ai. Let

Then a, b e V(gh) implies ]@(a, b) < 2 - 1 - 21~.

Proof: a, b e V(gk) implies a + b e V(gk) implies Ijo + b] 1 > k by

Theorem 1. Since 1 is not a primitive root of unity, 1 + z is clearly a

factor of g, and hence

V(

1 + z) C

V(gk).

Thus the constant sequence

.of ones is a member of

V(gk),

and hence

V(gk)

is closed with respec

to the operation of complementation, i.e., h f

V(gk)

implies h c

V(hd

where A(i) = 1 + h(i).

Thus a,

b

e

V(gk)

implies a +

b

z

V(gk)

implies a +

b c V( gk

implies ]G]] = 2n - 1 - ]]a + bl > k implies k v - 1. Thus by Lemma 1 fi andft are factors of go-l. Hence,

V(fd C V(g& and v(fd C I%,-& Thus o e V(fd and b p V(ft)

implies a and

b

E (S.-I). Thus by Lemma 4

Ie(a, b)/ < 2 -

1 - 2(v - 1) =

7

- 1 -

q2:-

_

2

_

y-19

=

2(*+1)/z

+ 3

(2* - 1 - qy-

-

2

-

29

=

2(%+2)/z

+ 3.


3/3

CORRESPONDENCE

Since the value of the cross-correlation function is always odd we

have

621

Z* + 211 + ~12 + ~14and the corresponding 14-stage shift register

will generate 129 different linear sequences of period 127. The

cross-correlation function B of any pair of such sequences will

satisfy the inequality /e(7)] 5 17.

ROBERT GOLD

Magnavox Research Laboratories

Torrance, Calif. 90503

le(a, b)I 5 2n+12

+ 1 for n odd

le(a, b) 1 2 2r+22

+ 1 for n

even.

For 13-stage shift registers there are pairs of maximal sequences

with cross-correlation peaks as high as O(T) = 703 while pr oper

selection of shift registers in accordance with the above theorem

guarantees sequences whose cross correlation satisfies the inequality

ItI 1 < 23+)2 + 1 = 129.

We note further that for purely random s equences of length 23 -

1 = 8191 we would expect the cross-correlation function to exceed

2a=2-2

A 180 for 5 percent of the correlation values and hence

linear sequences chosen in accordance with our technique perform

bett,er with respect to their cross-correlation properties than purely

random sequences.

IV.

CONSTRUCTION OF ENCODING FAMILIES

In this section we show how to provide large families of encoding

sequences each of period 2n -

1 and such that the cross-correlation

function of any pair of sequences of the family has a cross-correlation

function 0 which satisfies the inequality

le(T)l 5 2(n+z)z + 1.

In a spread spectrum multiplexing application such families f orm

ideal codes which minimizes interlink interference. Inst ead of

having each communication link employ a different maximal

sequence we assign to eac h link a member of the encoding family

to be constructed below. These are nonmaximal linear sequences,

and hence their autocorrelation function will not be two-valued;

however, the out-of-phase value of the autocorrelation function

will satisfy the above inequality. Thus by slightly relaxing the

conditi ons on the autocorrelation function we obtain a family of

encoding sequences with the high cross-correlation peaks eliminated.

The procedure for generating t hese encoding families is embodied

in the following theorem.

Theorem

Let fi and ft be a preferred pair of primitive polynomials of

degree 12 whose corr esponding shift registers generate maximal

linear sequences of period 2 - 1 and whose cross-correlation fmlction

e satisfies the inequality.

jet< t =

2(n+1z + 1 for n odd

2(n+2)2 + 1 for n even n # mod 4.

Then the shift register corresponding to the product polynomial

fi.ft will generate 2* + 1 different sequences each period 2n - 1

and such that the cross-correlation f unction 0 of any pair of such

sequences satisfies the above inequality.

Proof:

u E V(fl.ft) = V(fd + V(fJ

implies a = b + c where

b

E V(fi) and c E V(fJ. Period

(b + c) =

lcm (period b, period cl =

2 - 1. Since degree i.ft = 2n, there are (azm l)/(an - 1) = 2 i-

1 essentially different sequences in V(fi .f t). Finally a, b P V(fl .f t)

implies

a = a, + a,

i

b = b, + b,

where al,

bl E V(fi),

and at, bt z

V(ft). \@(a, b)\ = IB(al -I - bl, at +

bt)

1 5

t

by Theorem 2 since al +

bl E V(fi) and UI $ b, E V(f t).

Thus, by way of illustration, if we consider the pair of polynomials,

fi(x) = 1 + z + x2 + x3 + x7 and f dz) = 1 + x -I- x2 i- x3 -I- x4 -I-

x5 + x7 then the product polynomial isfi(x) fz(x) = 1 -I- 2 -I- x6 f

REFERENCES

111N. Zierler, Linear recurring sequences,

@I R. Gold,

J. SIAM, vol. 7, March 1959.

Characteristi c linear sequences and their ooset functions,

(accepted for publication J. Sot. Ind. ii&. Math., May 1965).

[$I W. W. Peterson, Error C orrecting Codes. New York: Wiley, 1961.

141T. Kasami, Weight distributi on formula for some class of cyclic codes,

University of Illi nois? Urbana, Rept. R-265, April 1966.

[sl R. Gold, Maxunal recursive sequences wit h 3-valued recursive cross-correla-

tion functions (submitted for publication, January 1967).

Average Digit-Error Probability After Decoding Random

Codes

INTRODUCTION

When a transmitted code word is decoded incorrectly due to

channel errors, it does not follow that all of the digits in the decoded

word are incorrect. One way of getting an estimate of the actual

number of digit errors to be expected is to determine the average

over a whole cl ass of codes of the digit-error probability and to

compare this with the word-error probability averaged over the

same class. In this correspondence the ratio of these two averages

is determined for r andom bl ock codes and it is shown that, at fixed

code rate and channel-error probability, t,he ratio approaches a

nonzero limit with increasing code length; the value of this limit

is given as a function of code rate and channel-error probability.

It is pointed out that the result can be extended to random linear

codes and to the information digits of random parity check codes

is also given. A more detailed derivation for these two li near cases

is given elsewhere.

It should be stressed at the outset that both the average word-

error probability and the average digit-error probability can be

strongly i nfluenced (particularly at small channel probabiliti es)

by a very few codes with unusually large word- an d digit-error

probabilities. Hence, it is unlikely that there is any easy extension of

these results to statements about the distribution of the ratio of the

two probabilities over the cl asses of codes considered.

NOTATION AND ASSUMPTIONS

Let n = code length

R

= code rate

K =

2R

p = channel-error probability

H(t)

= -

t

logz

t

- (1 - t) log, (1 -

t)

pR = smaller solution of R + H&) = 1

PC = PR /(l - 2pR + 2PR)

pw = average word-error probability

pD = average digit-error probability

E = expected value of.

The symbol N used in equations of the form

f(t) N g(t) as t + 03

means that the ratio of the two functions approaches unity:

f(t)ldt) ---f 1.

The expression o(n) used in equations of the form f(n) = o(n)

as724 ~0 meansf(n)/n+Oasn+ m.

Manuscript received December 12, 1966; revised May 22, 1967.

1 J. N. Pierce, Average digit error probability after decoding random linear

codes, Air Force Cambridge Research Labs.,

October 1966.

Bedford, Mass.. Kept. 66-695,

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Optimal Binary Sequences for Spread Spectrum Multiplexing.pdf