low correlation sequences for cdma - suraj @...

Low Correlation Sequences for CDMA

Gagan Garg

Indian Institute of Science, Bangalore

International Networking and Communications ConferenceLahore University of Management Sciences

Gagan Garg Low Correlation Sequences for CDMA

Acknowledgement

I Prof. Zartash Afzal Uzmi,Lahore University of Management Sciences

I Indian Institute of Science, Bangalore

Outline

I Motivation

I Binary Sequences

I Quaternary Sequences

Motivation

There are a large number of problems in communications thatrequire sets of signals with one or both of the following twoproperties:

I each signal in the set is easy to distinguish from a time-shiftedversion of itself

I ranging systems, radar systems

I each signal in the set is easy to distinguish from (a possiblytime-shifted version of) every other signal in the set

I code-division multiple-access (CDMA) communication systems

Mean-Squared Difference

Assume the signals to be periodic with common time period T

We want to distinguish between a(t) and b(t)

We also want to distinguish between −a(t) and b(t)

Measure of distinguishability is

0[b(t)± a(t)]2 dt

{∫ T

0[b2(t) + a2(t)] dt ± 2

0a(t)b(t) dt

}↓ ↓

energy in b(t) + energy in a(t) r

Cross-correlation function 1/2

ra,b(τ) :=

0a(t)b(t + τ) dt

If a(t) = b(t), we want ra,b(τ) to be small for τ ∈ (0,T )If a(t) 6= b(t), we want ra,b(τ) to be small for τ ∈ [0,T ]

For complex signals, we replace b(t + τ) by its complex conjugate

Assume a(t) and b(t) to be periodic signals, which consist ofsequences of elemental time-limited pulses.

a(t) :=∞∑

n=−∞x(n)ϕ(t − nTc)

where ϕ(t) is the basic pulse waveform and Tc is the time durationof this pulse and Tc |T ⇒ sequence {x(n)} is periodic

Cross-correlation function 2/2

Let N = T/Tc . Then, period of {x(n)} divides N.

b(t) :=∞∑

n=−∞y(n)ϕ(t − nTc)

If τ = mTc

ra,b(τ) = λ

N−1∑n=0

x(n) y(n + m)

where the constant

∫ Tc

0ϕ2(t) dt

Periodic and Aperiodic Correlation

Let {u(t)} and {v(t)} be complex valued sequences of period N.

Periodic correlation between them:

θu, v (τ) =N−1∑t=0

u(t + τ)v∗(t) , 0 ≤ τ ≤ N − 1

Aperiodic correlation between them:

ρu, v (τ) =

min{N−1, N−1−τ}∑t=max{0,−τ}

u(t + τ)v∗(t) , −(N − 1) ≤ τ ≤ N − 1

An example

Let N = 3.

θu,v (0) = u(0)v∗(0) + u(1)v∗(1) + u(2)v∗(2)θu,v (1) = u(1)v∗(0) + u(2)v∗(1) + u(0)v∗(2)θu,v (2) = u(2)v∗(0) + u(0)v∗(1) + u(1)v∗(2)

ρu,v (−2) = u(0)v∗(2)ρu,v (−1) = u(0)v∗(1) + u(1)v∗(2)ρu,v (0) = u(0)v∗(0) + u(1)v∗(1) + u(2)v∗(2)ρu,v (1) = u(1)v∗(0) + u(2)v∗(1)ρu,v (2) = u(2)v∗(0)

Binary sequences

I Binary m-sequences

I Finite Fields

I Gold sequences

Binary m-sequence of Period 7

Let α be a root of x3 + x + 1 = 0

Let {s(t)} have characteristic polynomial x3 + x + 1, i.e.,

s(t + 3) + s(t + 1) + s(t) = 0

i.e.,s(t + 3) = s(t + 1) + s(t)

s(t) = 1 0 0 1 0 1 1 1 0 0 1 0 . . . is an m-sequence

Periodic autocorrelation

θs(τ) =6∑

(−1)s(t+τ)−s(t) =

{7 τ = 0 (mod 7)

−1 else

Binary m-sequences 1/2

An irreducible binary polynomial f (x) is said to be primitive if thesmallest positive exponent k for which f (x)|xk − 1 is k = 2m − 1.

Let f (x) be a primitive binary polynomial of degree m.

For example, when m = 3, the polynomial f (x) = x3 + x + 1 isprimitive since f (x) divides x7 − 1 but does not divide xk − 1 forany 1 ≤ k ≤ 6.

Let f (x) =∑m

i=0 fi x i and let {s(t)} be a binary sequencesatisfying the recursion

m∑i=0

fi s(t + i) = 0 ∀ t

⇒ {s(t)} can be generated using an m-bit linear feedback shiftregister (LFSR).

Binary m-sequences 2/2

f (x) is a primitive polynomial ⇒ {s(t)} has period 2m − 1.

Such a sequence is called a maximum-length LFSR sequence, i.e.,an m-sequence.

For example, when f (x) = x3 + x + 1, {s(t)} satisfies therecursion

s(t + 3) + s(t + 1) + s(t) = 0

s(t+3) s(t)s(t+2) s(t+1)

Properties of binary m-sequences

Periodic autocorrelation function θs of a binary {0, 1} sequence{s(t)} of period N is defined by

θs(τ) =N−1∑t=0

(−1)s(t+τ)−s(t) 0 ≤ τ ≤ N − 1

An m-sequence of period N = 2m − 1 has the following properties:

1. Balance property: In each period of the m-sequence, there are2m−1 ones and 2m−1 − 1 zeroes.

2. Run property: Every nonzero binary s-tuple, s ≤ m occurs2m−s times and the all-zero tuple occurs 2m−s − 1 times.

3. Two-level autocorrelation function:

θs(τ) =

{N τ = 0−1 τ 6= 0

Sketch of the proof

I Property 1 and 2 can be proved using the fact that everym-sequence occurs precisely once in each period of them-sequence.

I For Property 3, when τ 6= 0, consider the difference sequence{s(t + τ)− s(t)}.

1. This sequence is not the all zero sequence.

2. It satisfies the same recursion as the sequence{s(t)}⇒ {s(t + τ)− s(t)} ≡ {s(t + τ ′)} for some 0 ≤ τ ′ ≤ N − 1 ,i.e., it is a different cyclic shift of the m-sequence {s(t)}.

3. Balance property of {s(t + τ ′)} proves Property 3.

Field Axioms

Two operations – addition and multiplication – that satisfy

Field Axioms -- from Wolfram MathWorld http://mathworld.wolfram.com/FieldAxioms.html

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12,805 entriesLast updated: Thu Apr 3 2008

Algebra > Field Theory >Foundations of Mathematics > Axioms >

Field Axioms

The field axioms are generally written in additive and multiplicative pairs.

Name Addition Multiplication

Commutativity

Associativity

Distributivity

Identity

Inverses

SEE ALSO: Algebra, Field

REFERENCES:

Apostol, T. M. "The Field Axioms." §I 3.2 in Calculus, 2nd ed., Vol. 1: One-VariableCalculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 17-19, 1967.

CITE THIS AS:

Weisstein, Eric W. "Field Axioms." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FieldAxioms.html

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Weisstein, Eric W. ”Field Axioms.” From MathWorld–A Wolfram Web Resource.http://mathworld.wolfram.com/FieldAxioms.html

A Finite and an Infinite Field

I The set R of all real numbers

I The setF2 = {0, 1}

We shall call this the binary field.

1⊕ 1 = 0

1 · 1 = 1

Check that all the field axioms are satisfied.

The Field of Complex Numbers

Introduce an imaginary element ı such that ı2 = −1

C = {a + ıb | a, b ∈ R}

(a + ıb)(c + ıd) = (ac + ı(ad + bc) + ı2 bd)

= (ac − bd) + ı(ad + bc)

Check that all the field axioms are satisfied.

The Finite Field of 4 Elements – F4

Introduce an imaginary element α such that α2 = α⊕ 1

F4 = {a⊕ αb | a, b ∈ F2}= {0, 1, α, 1 + α}

(a⊕ αb)(c ⊕ αd) = ac ⊕ α(ad ⊕ bc) ⊕ α2(bd)

= (ac ⊕ bd) ⊕ α(ad ⊕ bc ⊕ bd)

(1⊕ α)(1⊕ α) = (1⊕ 1) ⊕ α(1⊕ 1⊕ 1)

(1⊕ α) ⊕ α = 1 ⊕ α(1⊕ 1)

Learn your Tables!

addition 0 1 α 1 + α

0 0 1 α 1 + α

1 1 0 1 + α α

α α 1 + α 0 1

1 + α 1 + α α 1 0

multiplication 0 1 α 1 + α

0 0 0 0 0

1 0 1 α 1 + α

α 0 α 1 + α 1

1 + α 0 1 + α 1 α

Representation as powers

Since α2 = 1 + α, we replace (1 + α) by α2

addition 0 1 α α2

0 0 1 α α2

1 1 0 α2 α

α α α2 0 1

α2 α2 α 1 0

multiplication 0 1 α α2

0 0 0 0 0

1 0 1 α α2

α 0 α α2 1

α2 0 α2 1 α

Best of both

Addition table in terms of 1 + αMultiplication table in terms of α2

addition 0 1 α 1 + α

0 0 1 α 1 + α

1 1 0 1 + α α

α α 1 + α 0 1

1 + α 1 + α α 1 0

multiplication 0 1 α α2

0 0 0 0 0

1 0 1 α α2

α 0 α α2 1

α2 0 α2 1 α

The Finite Field of 8 Elements – F8

F8 ={0, 1, α, α2, · · · , α6

}binary addition 1 + 1 = 0 αi + αi = 0

α satisfies α3 + α + 1 = 0

We also have α7 = 1

Multiplication α3 · α6 = α9 = α2

Addition α3 + α6 = (α + 1) + (α2 + 1)

= α2 + α = α(α + 1) = α1+3 = α4

The Finite Field of 2m Elements – F2m

Let f (x) be a binary primitive polynomial of degree m and let αsatisfy f (α) = 0.

{m−1∑i=0

ciαi | ci ∈ {0, 1}

}= {0} ∪

{αi | 0 ≤ i ≤ 2m − 2

}In F2m , f (x) has one zero, viz. α.

F2m has the remaining m − 1 zeroes of f (x) as well.{α2j | j = 1, 2, . . . ,m − 1

Binary m-sequence in terms of the trace function

Power series expansion gives us that the m-sequence {s(t)} has aunique expression of the form

s(t) =m−1∑i=0

ciα2i t =

m−1∑i=0

= tr(c0αt)

where tr(·) : F2m → F2 is the trace function given by

tr(x) =m−1∑i=0

It turns out that every m-sequence {b(t)} of period 2m − 1 can beexpressed in the form

b(t) = tr(aαdt)

where a ∈ F2m and 1 ≤ d ≤ 2m − 2 is an integer such that(d , 2m − 1) = 1.

Trace Function over F8

maps from F8 → {0, 1}

tr(x) = x + x2 + x4

tr(α) = α + α2 + α4 = (α + α2) + α(1 + α)= (α + α2) + (α + α2) = 0

{tr(αt)}6t=0 = 1 0 0 1 0 1 1 is m-sequence s(t){

tr(α3t)}6

t=0= 1 1 1 0 1 0 0 is another m-sequence u(t)

Periodic crosscorrelation

θs,u(τ) =6∑

(−1)s(t+τ)−u(t) ∈ {−5, −1, 3}

Family of Gold Sequences

Contains the 9 sequences

F = {s(t)} ∪ {u(t)} ∪ {u(t + τ) + s(t) | 0 ≤ τ ≤ 6}

θmax = max { |θa,b(τ)| | a, b ∈ F either a 6= b or τ 6= 0}= 5

General parameters

Family Period Size θmax

Gold 2n − 1, n odd 2n + 1 ≈√

Gold sequences have been employed in GPS satellites

General case

Let {u(t)} be an m-sequence of period 2m − 1, m odd.

Let {v(t)} = {u(dt)}, where d = 2k + 1 for some k such that(k,m) = 1.

Then G is a family of 2m + 1 cyclically distinct sequences{gi (t)}, 0 ≤ i ≤ 2m given by

gi (t) = tr(αt + αiαdt), 0 ≤ i ≤ 2m − 2,

g2m−1(t) = tr(αt),

g2m(t) = tr(αdt).

Correlation properties

The cross correlation of sequences in G assumes values from the set{−1,−1±

√2m+1

It can be shown that the Gold family G is optimal in the sense ofhaving the lowest value of θmax possible for a family of binarysequences for the given length N = 2m − 1 and family sizeM = 2m + 1.

Quaternary Sequences

Galois Rings

Family A

A Polynomial with Z4 coefficients

Z4 = Z/4Z = {0, 1, 2, 3} = integers (modulo 4)

f (x) = x3 + x + 1 ∈ F2[x ]

g(x2) = (−1)f (x)f (−x) (mod 4)

= (−1)(x3 + x + 1)(−x3 − x + 1)

= x6 + 2x4 + x2 + 3

⇒ g(x) = x3 + 2x2 + x + 3

g(x) = x3 + x + 1 = f (x) (mod 2)

So g(x) is irreducible (mod 2) and therefore irreducible (mod 4).

Sequence Family Generated by g(x)

Let A be the family of all nonzero sequences having characteristicpolynomial g(x) = x3 + 2x2 + x + 3, i.e.,

A = {s(t) | s(t + 3) + 2s(t + 2) + s(t + 1) + 3s(t) = 0 (mod 4)}

Turns out A contains 9 sequences of period 7:

s(t)s(t+1)s(t+2)s(t+3)

+X X2 3

… 3 2 2 1 2 1 1 …

… 1 2 2 3 2 3 3 …

… 1 0 2 1 0 1 3 …

Better than Gold !

Periodic (quaternary) correlations of family A defined by

θs,u(τ) =6∑

(i)s(t+τ)−u(t), i =√−1.

are better than those of Gold sequences!

Family Period Size θmax

Gold 2n − 1 2n + 1 ≈√

Z4 Family A 2n − 1 2n + 1 ≈√

(Of course, A uses a larger alphabet)

Galois Ring of 64 elements – GR(4, 3)

Let β be a root of g(x), i.e.,

β3 + 2β2 + β + 3 = 0 (mod 4)

β has multiplicative order 7 since g(x2) = (−1)f (x)f (−x).

GR(4, 3) =

ciβi | ci ∈ Z4

}An alternate description:

GR(4, 3) = {x + 2y | x , y ∈ T }

whereT = {0} ∪ {1, β, β2, . . . , β6}

is the set of Teichmuller representatives.

T ≡ F8 under (mod 2) arithmetic

Addition in the Galois Ring

GR(4, 3) ={x + 2y | x , y ∈ T = {0, 1, β, . . . , β6}

}Let z1 + 2z2 = (x1 + 2y1) + (x2 + 2y2)

⇒ z1 = x1 + x2 (mod 2) and z2 = y1 + y2 +√

x1x2 (mod 2)

Eg. z1 + 2z2 = (β3 + 2β2) + (β6 + 2β5)

z1 = β3 + β6 (mod 2) = α3 + α6 (mod 2) = α4 ⇒ z1 = β4

z2 = β2 + β5 +√

β9 (mod 2) = α2 + α5 + α (mod 2)= 1 (mod 2) ⇒ z2 = 1

Thus, (β3 + 2β2) + (β6 + 2β5) = β4 + 2 · 1

(Multiplication (x1 + 2y1) · (x2 + 2y2) is now straightforward)

Family A and the Trace Function

The trace function on GR(4, 3) maps GR(4, 3) → Z4

T (x + 2y) = (x + x2 + x4) + 2(y + y2 + y4)

x T (x)

β4 β2 β 2

β5 β6 β3 1

t → 0 1 2 3 4 5 6 0 1 2 . . .

T (βt) → 3 2 2 1 2 1 1 3 2 2 . . .

Trace description of Family A:

A = {T (βt)} ∪{T ([1 + 2δ]βt) | δ ∈ T

}Gagan Garg Low Correlation Sequences for CDMA

Galois Ring GR(4, m) 1/2

Let f (x) be a primitive polynomial over F2 of degree m.

g(x2) = (−1)m f (x) f (−x)

GR(4,m) = Z4[x ]/ (f (x))

⇒ GR(4,m) ∼= Z4[ζ]

where ζ belongs to some extension ring of Z4 and satisfies

f (ζ) = 0

Galois Ring GR(4, m) 2/2

SetT = {0} ∪ {1, ζ, ζ2, . . . , ζN−1}

where N = 2m − 1 is the order of ζ in GR(4,m).

The trace function T (·) : GR(4,m) → Z4 is defined by

T (x + 2y) =m−1∑i=0

+ 2y2i)

, x , y ∈ T

Family A 1/2

LetT = {γi | i = 1, 2, . . . , 2m}

be a labeling of the elements in T .

si (t) = T ([1 + 2γi ]ζt), 1 ≤ i ≤ 2m,

s2m+1(t) = 2T (ζt).

Then the sequences {si (t)} are a set of cyclically distinctrepresentatives for Family A.

Family A 2/2

If i 6= j or τ 6= 0, the correlation values of sequences in Family Abelong to the set

θij(τ) ∈{−1,−1± 2

m−12 ± ı2

m−12

}, m odd

orθij(τ) ∈

{−1,−1± 2

m2 ,−1± ı2

}, m even

⇒ θmax ≤ 1 +√

As compared to the binary family of Gold sequences, Family Aoffers a lower value of θmax for the same period and family size.

low correlation sequences for cdma - suraj @...

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