Low Correlation Sequences for CDMA - Suraj @...
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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
Gagan Garg Low Correlation Sequences for CDMA
Outline
I Motivation
I Binary Sequences
I Quaternary Sequences
Gagan Garg Low Correlation Sequences for CDMA
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
Gagan Garg Low Correlation Sequences for CDMA
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
1
T
∫ T
0[b(t)± a(t)]2 dt
=1
T
{∫ T
0[b2(t) + a2(t)] dt ± 2
∫ T
0a(t)b(t) dt
}↓ ↓
energy in b(t) + energy in a(t) r
Gagan Garg Low Correlation Sequences for CDMA
Cross-correlation function 1/2
ra,b(τ) :=
∫ T
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
Gagan Garg Low Correlation Sequences for CDMA
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
λ =
∫ Tc
0ϕ2(t) dt
Gagan Garg Low Correlation Sequences for CDMA
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
Gagan Garg Low Correlation Sequences for CDMA
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)
Gagan Garg Low Correlation Sequences for CDMA
Binary sequences
I Binary m-sequences
I Finite Fields
I Gold sequences
Gagan Garg Low Correlation Sequences for CDMA
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)
Then
s(t) = 1 0 0 1 0 1 1 1 0 0 1 0 . . . is an m-sequence
Periodic autocorrelation
θs(τ) =6∑
t=0
(−1)s(t+τ)−s(t) =
{7 τ = 0 (mod 7)
−1 else
Gagan Garg Low Correlation Sequences for CDMA
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).
Gagan Garg Low Correlation Sequences for CDMA
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)
Gagan Garg Low Correlation Sequences for CDMA
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
Gagan Garg Low Correlation Sequences for CDMA
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.
Gagan Garg Low Correlation Sequences for CDMA
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
Gagan Garg Low Correlation Sequences for CDMA
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.
Gagan Garg Low Correlation Sequences for CDMA
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.
Gagan Garg Low Correlation Sequences for CDMA
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)
= α
(1⊕ α) ⊕ α = 1 ⊕ α(1⊕ 1)
= 1
Gagan Garg Low Correlation Sequences for CDMA
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
Gagan Garg Low Correlation Sequences for CDMA
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
Gagan Garg Low Correlation Sequences for CDMA
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
Gagan Garg Low Correlation Sequences for CDMA
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
Gagan Garg Low Correlation Sequences for CDMA
The Finite Field of 2m Elements – F2m
Let f (x) be a binary primitive polynomial of degree m and let αsatisfy f (α) = 0.
F2m =
{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
}
Gagan Garg Low Correlation Sequences for CDMA
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
(c0α
t)2i
= tr(c0αt)
where tr(·) : F2m → F2 is the trace function given by
tr(x) =m−1∑i=0
x2i
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.
Gagan Garg Low Correlation Sequences for CDMA
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∑
t=0
(−1)s(t+τ)−u(t) ∈ {−5, −1, 3}
Gagan Garg Low Correlation Sequences for CDMA
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 ≈√
2n+1
Gold sequences have been employed in GPS satellites
Gagan Garg Low Correlation Sequences for CDMA
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).
Gagan Garg Low Correlation Sequences for CDMA
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.
Gagan Garg Low Correlation Sequences for CDMA
Quaternary Sequences
Galois Rings
Family A
Gagan Garg Low Correlation Sequences for CDMA
A Polynomial with Z4 coefficients
Z4 = Z/4Z = {0, 1, 2, 3} = integers (modulo 4)
Let
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).
Gagan Garg Low Correlation Sequences for CDMA
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 …
Gagan Garg Low Correlation Sequences for CDMA
Better than Gold !
Periodic (quaternary) correlations of family A defined by
θs,u(τ) =6∑
t=0
(i)s(t+τ)−u(t), i =√−1.
are better than those of Gold sequences!
Family Period Size θmax
Gold 2n − 1 2n + 1 ≈√
2n+1
n odd
Z4 Family A 2n − 1 2n + 1 ≈√
2n
any n
(Of course, A uses a larger alphabet)
Gagan Garg Low Correlation Sequences for CDMA
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) =
{2∑
i=0
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
Gagan Garg Low Correlation Sequences for CDMA
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)
Gagan Garg Low Correlation Sequences for CDMA
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)
0 0
1 3
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.
Let
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
Gagan Garg Low Correlation Sequences for CDMA
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
(x2i
+ 2y2i)
, x , y ∈ T
Gagan Garg Low Correlation Sequences for CDMA
Family A 1/2
LetT = {γi | i = 1, 2, . . . , 2m}
be a labeling of the elements in T .
Let
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.
Gagan Garg Low Correlation Sequences for CDMA
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
m2
}, m even
⇒ θmax ≤ 1 +√
2m
As compared to the binary family of Gold sequences, Family Aoffers a lower value of θmax for the same period and family size.
Gagan Garg Low Correlation Sequences for CDMA
Further Reading
I D. V. Sarwate and M. B. Pursley, “Crosscorrelation propertiesof pseudorandom and related sequences,” Proc. IEEE, vol.68, pp. 593–619, May 1980.
I T. Helleseth and P.V. Kumar, “Sequences with lowcorrelation,” in Handbook of Coding Theory, V.S. Pless andW.C. Huffman, Eds., Amsterdam: Elsevier, 1998.
Thank You!
Gagan Garg Low Correlation Sequences for CDMA