Aperiodic correlation and the merit factor

Aina Johansen

02.11.2009

Correlation

The periodic correlation between two binary sequences {xt} and{yt} of length n at shift τ is defined as

θx ,y (τ) =

n−1∑

t=0

(−1)xt+τ−yt

xτ xτ+1 · · · xn−1 x0 x1 · · · xτ−1

y0 y1 · · · yn−1−τ yn−τ yn−τ+1 · · · yn−1

The aperiodic correlation between two binary sequences {xt} and{yt} of length n at shift τ is defined as

Cx ,y =n−τ−1∑

t=0

(−1)xt+τ−yt

x0 x1 · · · xτ−1 xτ · · · xn−1

y0 · · · yn−1−τ yn−τ · · · yn−1

Autocorrelation

When {xt} = {yt} we call it the autocorrelation, and we denote itCx(τ).

xτ xτ+1 · · · xn−1 x0 x1 · · · xτ−1

x0 x1 · · · xn−1−τ xn−τ xn−τ+1 · · · xn−1

θx(τ) = Cx(τ) + Cx(n − τ) when τ 6= 0

Example: Let st = 01011.

τ Cyclic shift θs(τ) Acyclic shift Cs(τ)

0 01011 5 01011 51 10101 −3 0101 −22 11010 1 010 13 01101 1 01 04 10110 −3 0 −1

CDMA

We have a set of sequences where all pairs of sequences from theset are orthogonal, i.e. the inner product is 0.

Each user i is given a sequence si . If the user wants to send a 1 hetransmits his sequence. If the user wants to send a 0 he transmitsthe complement of his sequence.

If two users i and j transmits a sequence simultaneously, thereceived signal will be the sum of the sequences, si + sj . Todetermine what was sent by user i , we can compute the innerproduct (si + sj) • si . As si • sj = 0, we are left with the data sentby user i .

When user i starts to transmit a sequence, and user j starts totransmit a sequence after i har started, but before i has finised,the received signal will be a partial overlap between the twosequences. In order to ease synchronization we need the sequencesto have low correlation.

Other applications

Binary sequences with low correlation have a number of uses:

◮ spread-spectrum communication systems

◮ stream ciphers

◮ pseudorandom number generation

◮ pulse compression

◮ synchronization

◮ theoretical physics and chemistry

◮ etc

The merit factorWe want binary sequences whose aperiodic autocorrelations arecollectively small. The merit factor can be used as a measure ofthis, and is defined as follows:

DefinitionLet {st} be a binary sequence of length n. The merit factor of{st} is given as

F (s) =n2

2∑n−1

u=1 [Cs(u)]2

If the aperiodic autocorrelation of {st} is small for all shifts, themerit factor will be high. So the higher the merit factor, the betterthe sequence.

Example: Let s = 01011. Then we get the merit factor

F (s) =52

2 ∗ ((−2)2 + 12 + 02 + (−1)2)=

25

2 ∗ 6= 2.08

The merit factor problem

Let Sn be the set of all binary sequences of length n. We define Fn

to be the optimal merit factor value for sequences of length n, thatis

Fn = maxs∈Sn

(F (s))

The meric factor problem

Determine the value of lim supn→∞ Fn.

Proposition

The mean value of 1/F , taken over all sequences of length n, isn−1n

.

The above proposition means that the expected asymptotic valueof F for a randomly-chosen binary sequence is 1.

Some results so far

The best asymptotic results so far are given by explicitlyconstructed families of sequences with a provable asymptotic meritfactor of 6.

In recent years new constructions have given seqeunces of very longlength (millions of elements) with asymptotic merit factor > 6.34.

Exhaustive computer search has shown that for1 ≤ n ≤ 40, n 6= 11, 13, 3.3 ≤ Fn ≤ 9.58. For lengths up to 117the highest known merit factor is between 8 and 9.56.

Fn has been computed for all n ≤ 60, and for larger values of n inthe range 61 ≤ n ≤ 217. Some large values of Fn has occured.

The only known sequences with merit factor ≥ 10 are the Barkersequences of length 11 and 13. These sequences have merit factorF11 = 12.1 and F13 = 14.08.

Golay’s postulate

The aperiodic autocorrelations C (1), C (2), . . . ,C (n − 1) of asequence that is chosen at random from the 2n binary sequences oflength n are clearly dependent random variables.

In 1977 Golay made a postulate that said that the correct value oflim supn→∞ Fn can be found by treating C (1), C (2), . . . ,C (n − 1)as independent random variables for large n. From this, he madethe following conjecture:

Conjecture

lim supn→∞

Fn = 12.32 . . .

Barker sequencesA Barker sequence is a binary sequence {st} of length n where

|Cs(τ)| ≤ 1 for all 0 < τ < n

Only the following Barker sequences are known so far:

n = 2 11 n = 7 1110010n = 3 110 n = 11 11100010010n = 4 1110 n = 13 1111100110101n = 5 11101

τ Acyclic shift C (τ)

0 1110010 71 111001 02 11100 −13 1110 04 111 −15 11 06 1 −1

The Barker property is preserved under the transformations

st → st

st →

{

st if t is even

st if t is odd

st → sn−1−t

Example

Let {st} = 1110010 be the Barker sequence of length 7. st → stgives us 0001101.

τ Acyclic shift C (τ)

0 0001101 71 000110 02 00011 −13 0001 04 000 −15 00 06 0 −1

Example

In the second transformation we complement the elements in allodd positions of the sequence. Hence 1110010 → 1011000.

τ Acyclic shift C (τ)

0 1011000 71 101100 02 10110 −13 1011 04 101 −15 10 06 1 −1

Example

In the last transformation, we simply reverse the order of thesequence. Hence 1110010 → 0100111.

τ Acyclic shift C (τ)

0 0100111 71 010011 02 01001 −13 0100 04 010 −15 01 06 0 −1

There are no Barker sequences of odd length greater than 13. If n

is even, it is necessary that n ≡ 0 (mod 4).

It has been shown that there are no Barker sequences for even n,4 < n ≤ 1 898 884.

It is believed that there exist no further Barker sequences otherthan those already listed, but so far noone has been able to provethis.

Rudin-Shapiro sequences

The Rudin-Shapiro sequence pair X (m), Y (m) is defined recursivelyby

X (m) = X (m−1) : Y (m−1)

Y (m) = X (m−1) : Y (m−1)

where X (0) = Y (0) = 1, a : b means appending b to a, and x is thecomplement of x .

Example

m X (m) Y (m)

0 1 11 11 102 1110 11013 11101101 11100010

The merit factor of Rudin-Shapiro sequences

TheoremThe merit factor of both sequences X (m) and Y (m) of a

Rudin-Shapiro pair of length 2m is

F =3

1 − (−1/2)m

Hence the asymptotic merit factor of both sequences is 3.

Example

Look at X (3) = 11101101. The aperiodic correlation values are(−1, 0, 3, 0, 1, 0, 1).

F =n2

2∑n−1

u=1[C (u)]2=

82

2 ∗ ((−1)2 + 32 + 12 + 12)=

8

3

F =3

1 − (−1/2)m=

3

1 − (−1/2)3=

8

3

Difference setsAmong the families of sequences whose merit factor has beendetermined we find several sequences corresponding to differencesets.

DefinitionLet G be a group of order v . Let D be a subset of G of size k . Wesay that D is a (v , k , λ) difference set if every nonzero element ofG has exactly λ representations as a difference d1 − d2, whered1, d2 ∈ D. The order of the difference set is defined as n = k − λ.

Example

Let G be the group Z11. D = {1, 3, 4, 5, 9} is a subset of G of size5. D is a (11, 5, 2) difference set of order n = 11 − 5 = 4.

1 4 − 3 5 − 42 3 − 1 5 − 33 1 − 9 4 − 14 5 − 1 9 − 55 3 − 9 9 − 4

6 4 − 9 9 − 37 1 − 5 5 − 98 1 − 4 9 − 19 1 − 3 3 − 510 3 − 4 4 − 5

Characteristic functionThe characteristic function of a difference set D is the binarysequence {st} of length v where

st =

{

1 t ∈ D

0 t /∈ D

Example

G = Z11, D = {1, 3, 4, 5, 9}, {st} = 01011100010

Hadamard difference setsDifference sets with parameters of the form

(v = 4t − 1, k = 2t − 1, λ = t − 1)

are called Hadamard difference sets. Only difference sets withHadamard parameters give rise to sequences with nonzeroastymptotic merit factor.

m-sequences

Let f (x) =∑m

i=0 fixi where fi ∈ 0, 1 be a primitive polynomial.

From this polynomial we get the recurrence relationst+m =

∑m−1i=0 fi st+i . This recurrence relation generates a binary

sequence of period 2m − 1, called an m-sequence.

Example

Let f (x) = x4 + x + 1. This gives us the recurrence relationst+4 = st+1 + st . For initial values s0 = s1 = s2 = 0, s3 = 1 we getthe sequence {st} = 000100110101111

TheoremThe mean value of 1/F , taken over all n rotations of an

m-sequence of length n = 2m − 1, is(n−1)(n+4)

3n2 .

TheoremThe asymptotic merit factor of an m-sequence is 3.

Legendre sequences

Let p be an odd prime. We define the “modified” Legendre symbolas[

a

p

]

=

{

1 if a is a quadratic residue mod p

0 if a ≡ 0 (mod p) or a is a quadratic nonresidue mod p

The Legendre sequence of length p is defined as

{st} =

[

t

p

]

When p ≡ 3 (mod 4) the Legendre sequence is the characteristicfunction of a Hadamard difference set, called the Legendredifference set.

Example

Let v = 11. We see that

1 ∗ 1 = 12 ∗ 2 = 43 ∗ 3 = 94 ∗ 4 = 55 ∗ 5 = 3

6 ∗ 6 = 37 ∗ 7 = 58 ∗ 8 = 99 ∗ 9 = 410 ∗ 10 = 1

Then {1, 3, 4, 5, 9} are the only quadratic residues, and we get theLegendre sequence {st} = 01011100010.

The merit factor of Legendre sequences

An “offset” sequence is one in which a fraction f of the elementsof a sequence of length n are chopped off at one end and appendedto the other end. This corresponds to a cyclic shift of nf places.

TheoremThe Legendre sequences, when offest by a fraction f of their

length, have asymptotic merit factor F satisfying

1

F=

2

3− 4|f | + 8f 2, |f | ≤

1

2

so that F reaches a maximum value of 6 when |f | = 14 .

Jacobi sequences

Let n = p1p2 · · · pk , where the pi ’s are primes. The “modified”Jacobi symbol is defined as

[a

n

]

=k

⊕

i=1

[

a

pi

]

The Jacobi sequence of length n is defined as

{st} =[ t

n

]

In the case when n = p(p + 2), there exists a modified version ofthe Jacobi sequences, called the Twin prime sequences.

Twin prime sequences

Let v = p(p + 2), where p and p + 2 are prime. The Twin prime

sequence is defined as

{st} =

0 if t ≡ 0 (mod p + 2)1 if t 6= 0, t ≡ 0 (mod p)[

tp

]

⊕[

tp+2

]

otherwise

Example of Twin prime setsLet v = 3 ∗ 5. QR3 = {1}, QR5 = {1, 4}.

{st} =

0 if t ≡ 0 (mod 5)1 if t 6= 0, t ≡ 0 (mod 3)[

t3

]

⊕[

t5

]

otherwise

t st0 01

[

13

]

⊕[

15

]

= 1 + 1 = 02

[

23

]

⊕[

25

]

= 0 + 0 = 03 14

[

13

]

⊕[

45

]

= 1 + 1 = 05 06 17

[

13

]

⊕[

25

]

= 1 + 0 = 1

t st8

[

23

]

⊕[

35

]

= 0 + 0 = 09 1

10 011

[

23

]

⊕[

15

]

= 0 + 1 = 112 113

[

13

]

⊕[

35

]

= 1 + 0 = 114

[

23

]

⊕[

45

]

= 0 + 1 = 1

{st} = 000100110101111

The merit factor of Jacobi sequences

TheoremLet {st} be a modified Jacobi sequence of length pq, where p and

q are distinct primes. If(p+q)5 log4(n)

n3 → 1 as n → ∞

1

F=

2

3− 4|f | + 8f 2, |f | ≤

1

2

Skew-symmetric sequencesA common strategy for extending the reach of merit factorcomputations is to impose restrictions on the structure of thesequence.

A skew-symmetric binary sequence is a binary sequence(a0, a1, . . . , a2m) of odd length 2m + 1 where

am+i =

{

am−i , i even, 1 ≤ i ≤ m

am−i , i odd, 1 ≤ i ≤ m

Example

Let {st} be the Barker sequence of length 7, {st} = 1110010. As7 = 2 ∗ 3 + 1, we have that m = 3. If the sequence isskew-symmetric, we must have that

sm−3 sm−2 sm−1 sm sm+1 sm+2 sm+3

s0 s1 s2 s3 s2 s1 s01 1 1 0 0 1 0

All odd-length Barker sequences are skew-symmetric.

Skew-symmetric sequences are known to attain the optimal meritfactor value Fn for most odd values of n < 60.

Proposition

A skew-symmetric binary sequence of odd length has C (u) = 0 for

all odd u.

Proposition

The mean value of 1/F , taken over all skew-symmetric sequences

of odd length n, is(n−1)(n−2)

n2 .

Periodic appending

Given a sequence A = (a0, a1, . . . , an−1) of length n and a realnumber k satisfying 0 ≤ k ≤ 1, we write Ak for the sequence(b0, b1, . . . , b⌊kn⌋−1) obtained by truncating A to a fraction k of itslength, that is, bi = ai for 0 ≤ i < ⌊kn⌋.

Let Xr be a Legendre sequence of prime length n, offset by afraction r of n. We have previously seen thatlim supn→∞ F (X 1

4) = 6.

Extensive numerical evidence suggests that

◮ for large n, the merit factor of the appended sequenceX 1

4: (X 1

4)k is greater than 6.2 when k ≃ 0.03

◮ for large n, the merit factor of the appended sequenceXr : (Xr )

k is greater than 6.34 for r ≃ 0.22 and r ≃ 0.72when k ≃ 0.06.

Stochastic search algorithms

The basic method of many of the stochastic search algorithms is tomove through the search space of sequences by changing only one,or sometimes two, sequence elements at a time. The merit factorof any close neighbour sequence can be computed in O(n)operations from knowledge of the aperiodic autocorrelations of thecurrent sequence. The search algorithm specifies when it isacceptable to move to a neighbour sequence, for example when ithas merit factor no smaller than the current sequence, or when ithas the largest merit factor amongst all close neighbours notpreviously visited. The search algorithm must also specify how tochoose a new sequence when no acceptable neighbour sequencecan be found.

So far no stochastic search algorithm has been found that reliablyproduces binary sequences with merit factor greater than 6 inreasonable time for large n.

The end