Low Correlation Sequences for CDMA

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1 Indian Institute of Science, Bangalore International Networking and Communications Conference Lahore University of Management Sciences

2 Acknowledgement Prof. Zartash Afzal Uzmi, Lahore University of Management Sciences Indian Institute of Science, Bangalore

3 Outline Motivation Binary Sequences Quaternary Sequences

4 Motivation There are a large number of problems in communications that require sets of signals with one or both of the following two properties: each signal in the set is easy to distinguish from a time-shifted version of itself ranging systems, radar systems each signal in the set is easy to distinguish from (a possibly time-shifted version of) every other signal in the set code-division multiple-access (CDMA) communication systems

5 Mean-Squared Difference 1 T 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 T 0 [b(t) ± a(t)] 2 dt = 1 T { T [b 2 (t) + a 2 (t)] dt ± 2 0 T 0 } a(t)b(t) dt energy in b(t) + energy in a(t) r

6 Cross-correlation function 1/2 r a,b (τ) := T 0 a(t)b(t + τ) dt If a(t) = b(t), we want r a,b (τ) to be small for τ (0, T ) If a(t) b(t), we want r a,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 of sequences of elemental time-limited pulses. a(t) := n= x(n)ϕ(t nt c ) where ϕ(t) is the basic pulse waveform and T c is the time duration of this pulse and T c T sequence {x(n)} is periodic

7 Cross-correlation function 2/2 Let N = T /T c. Then, period of {x(n)} divides N. b(t) := y(n)ϕ(t nt c ) n= If τ = mt c N 1 r a,b (τ) = λ x(n) y(n + m) n=0 where the constant λ = Tc 0 ϕ 2 (t) dt

8 Periodic and Aperiodic Correlation Let {u(t)} and {v(t)} be complex valued sequences of period N. Periodic correlation between them: θ u, v (τ) = N 1 t=0 Aperiodic correlation between them: ρ u, v (τ) = min{n 1, N 1 τ} t=max{0, τ} u(t + τ)v (t), 0 τ N 1 u(t + τ)v (t), (N 1) τ N 1

9 An example Let N = 3. θ u,v (0) = u(0)v (0) + u(1)v (1) + u(2)v (2) θ u,v (1) = u(1)v (0) + u(2)v (1) + u(0)v (2) θ u,v (2) = u(2)v (0) + u(0)v (1) + u(1)v (2) ρ u,v ( 2) = u(0)v (2) ρ u,v ( 1) = u(0)v (1) + u(1)v (2) ρ u,v (0) = u(0)v (0) + u(1)v (1) + u(2)v (2) ρ u,v (1) = u(1)v (0) + u(2)v (1) ρ u,v (2) = u(2)v (0)

10 Binary sequences Binary m-sequences Finite Fields Gold sequences

11 Binary m-sequence of Period 7 Let α be a root of x 3 + x + 1 = 0 Let {s(t)} have characteristic polynomial x 3 + 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) = is an m-sequence Periodic autocorrelation 6 θ s (τ) = ( 1) s(t+τ) s(t) = t=0 { 7 τ = 0 (mod 7) 1 else

12 Binary m-sequences 1/2 An irreducible binary polynomial f (x) is said to be primitive if the smallest positive exponent k for which f (x) x k 1 is k = 2 m 1. Let f (x) be a primitive binary polynomial of degree m. For example, when m = 3, the polynomial f (x) = x 3 + x + 1 is primitive since f (x) divides x 7 1 but does not divide x k 1 for any 1 k 6. Let f (x) = m i=0 f i x i and let {s(t)} be a binary sequence satisfying the recursion m f i s(t + i) = 0 i=0 {s(t)} can be generated using an m-bit linear feedback shift register (LFSR). t

13 Binary m-sequences 2/2 f (x) is a primitive polynomial {s(t)} has period 2 m 1. Such a sequence is called a maximum-length LFSR sequence, i.e., an m-sequence. For example, when f (x) = x 3 + x + 1, {s(t)} satisfies the recursion s(t + 3) + s(t + 1) + s(t) = 0 + s(t+3) s(t+2) s(t+1) s(t)

14 Properties of binary m-sequences Periodic autocorrelation function θ s of a binary {0, 1} sequence {s(t)} of period N is defined by N 1 θ s (τ) = ( 1) s(t+τ) s(t) 0 τ N 1 t=0 An m-sequence of period N = 2 m 1 has the following properties: 1. Balance property: In each period of the m-sequence, there are 2 m 1 ones and 2 m 1 1 zeroes. 2. Run property: Every nonzero binary s-tuple, s m occurs 2 m s times and the all-zero tuple occurs 2 m s 1 times. 3. Two-level autocorrelation function: { N τ = 0 θ s (τ) = 1 τ 0

15 Sketch of the proof Property 1 and 2 can be proved using the fact that every m-sequence occurs precisely once in each period of the m-sequence. For Property 3, when τ 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.

16 Field Axioms Field Axioms Two operations addition and multiplication that satisfy The field axioms are generally written in additive and multiplicative pairs. Name Addition Multiplication Commutativity Associativity Distributivity Identity Inverses SEE ALSO: Algebra, Field Weisstein, Eric W. Field Axioms. From MathWorld A Wolfram Web Resource. REFERENCES:

17 A Finite and an Infinite Field The set R of all real numbers The set F 2 = {0, 1} We shall call this the binary field. 1 1 = = 1 Check that all the field axioms are satisfied.

18 The Field of Complex Numbers Introduce an imaginary element ı such that ı 2 = 1 C = {a + ıb a, b R} (a + ıb)(c + ıd) = (ac + ı(ad + bc) + ı 2 bd) = (ac bd) + ı(ad + bc) Check that all the field axioms are satisfied.

19 The Finite Field of 4 Elements F 4 Introduce an imaginary element α such that α 2 = α 1 F 4 = {a αb a, b F 2 } = {0, 1, α, 1 + α} (a αb)(c αd) = ac α(ad bc) α 2 (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

20 Learn your Tables! addition 0 1 α 1 + α α 1 + α α α α α 1 + α α 1 + α α 1 0 multiplication 0 1 α 1 + α α 1 + α α 0 α 1 + α α α 1 α

21 Representation as powers Since α 2 = 1 + α, we replace (1 + α) by α 2 addition 0 1 α α α α α 2 α α α α α 2 α 2 α 1 0 multiplication 0 1 α α α α 2 α 0 α α 2 1 α 2 0 α 2 1 α

22 Best of both Addition table in terms of 1 + α Multiplication table in terms of α 2 addition 0 1 α 1 + α α 1 + α α α α α 1 + α α 1 + α α 1 0 multiplication 0 1 α α α α 2 α 0 α α 2 1 α 2 0 α 2 1 α

23 The Finite Field of 8 Elements F 8 F 8 = { 0, 1, α, α 2,, α 6} binary addition = 0 α i + α i = 0 α satisfies α 3 + α + 1 = 0 We also have α 7 = 1 Multiplication α 3 α 6 = α 9 = α 2 Addition α 3 + α 6 = (α + 1) + (α 2 + 1) = α 2 + α = α(α + 1) = α 1+3 = α 4

24 The Finite Field of 2 m Elements F 2 m Let f (x) be a binary primitive polynomial of degree m and let α satisfy f (α) = 0. F 2 m = { m 1 i=0 c i α i c i {0, 1} } = {0} { α i 0 i 2 m 2 } In F 2 m, f (x) has one zero, viz. α. F 2 m has the remaining m 1 zeroes of f (x) as well. { } α 2j j = 1, 2,..., m 1

25 Binary m-sequence in terms of the trace function Power series expansion gives us that the m-sequence {s(t)} has a unique expression of the form s(t) = m 1 i=0 c i α 2i t = m 1 i=0 ( c0 α t) 2 i = tr(c 0 α t ) where tr( ) : F 2 m F 2 is the trace function given by tr(x) = m 1 It turns out that every m-sequence {b(t)} of period 2 m 1 can be expressed in the form i=0 x 2i b(t) = tr(aα dt ) where a F 2 m and 1 d 2 m 2 is an integer such that (d, 2 m 1) = 1.

26 Trace Function over F 8 maps from F 8 {0, 1} tr(x) = x + x 2 + x 4 tr(α) = α + α 2 + α 4 = (α + α 2 ) + α(1 + α) = (α + α 2 ) + (α + α 2 ) = 0 {tr(α t )} 6 t=0 = is m-sequence s(t) { tr(α 3t ) } 6 t=0 = is another m-sequence u(t) Periodic crosscorrelation θ s,u (τ) = 6 ( 1) s(t+τ) u(t) { 5, 1, 3} t=0

27 Family of Gold Sequences Contains the 9 sequences F = {s(t)} {u(t)} {u(t + τ) + s(t) 0 τ 6} θmax = max { θ a,b (τ) a, b F either a b or τ 0} = 5 General parameters Family Period Size θmax Gold 2 n 1, n odd 2 n n+1 Gold sequences have been employed in GPS satellites

28 General case Let {u(t)} be an m-sequence of period 2 m 1, m odd. Let {v(t)} = {u(dt)}, where d = 2 k + 1 for some k such that (k, m) = 1. Then G is a family of 2 m + 1 cyclically distinct sequences {g i (t)}, 0 i 2 m given by g i (t) = tr(α t + α i α dt ), 0 i 2 m 2, g 2 m 1(t) = tr(α t ), g 2 m(t) = tr(α dt ).

29 Correlation properties The cross correlation of sequences in G assumes values from the set { 1, 1 ± 2 m+1 } It can be shown that the Gold family G is optimal in the sense of having the lowest value of θmax possible for a family of binary sequences for the given length N = 2 m 1 and family size M = 2 m + 1.

30 Quaternary Sequences Galois Rings Family A

31 A Polynomial with Z 4 coefficients Z 4 = Z/4Z = {0, 1, 2, 3} = integers (modulo 4) Let f (x) = x 3 + x + 1 F 2 [x] g(x 2 ) = ( 1)f (x)f ( x) (mod 4) = ( 1)(x 3 + x + 1)( x 3 x + 1) = x 6 + 2x 4 + x g(x) = x 3 + 2x 2 + x + 3 g(x) = x 3 + x + 1 = f (x) (mod 2) So g(x) is irreducible (mod 2) and therefore irreducible (mod 4).

32 Sequence Family Generated by g(x) Let A be the family of all nonzero sequences having characteristic polynomial g(x) = x 3 + 2x 2 + 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: + X X s(t+3) s(t+2) s(t+1) s(t)

33 Better than Gold! Periodic (quaternary) correlations of family A defined by 6 θ s,u (τ) = (i) s(t+τ) u(t), i = 1. t=0 are better than those of Gold sequences! Family Period Size θmax Gold 2 n 1 2 n n+1 n odd Z 4 Family A 2 n 1 2 n n any n (Of course, A uses a larger alphabet)

34 Galois Ring of 64 elements GR(4, 3) Let β be a root of g(x), i.e., β 3 + 2β 2 + β + 3 = 0 (mod 4) β has multiplicative order 7 since g(x 2 ) = ( 1)f (x)f ( x). { 2 } GR(4, 3) = c i β i c i Z 4 i=0 An alternate description: GR(4, 3) = {x + 2y x, y T } where T = {0} {1, β, β 2,..., β 6 } is the set of Teichmuller representatives. T F 8 under (mod 2) arithmetic

35 Addition in the Galois Ring GR(4, 3) = { x + 2y x, y T = {0, 1, β,..., β 6 } } Let z 1 + 2z 2 = (x 1 + 2y 1 ) + (x 2 + 2y 2 ) z 1 = x 1 + x 2 (mod 2) and z 2 = y 1 + y 2 + x 1 x 2 (mod 2) Eg. z 1 + 2z 2 = (β 3 + 2β 2 ) + (β 6 + 2β 5 ) z 1 = β 3 + β 6 (mod 2) = α 3 + α 6 (mod 2) = α 4 z 1 = β 4 z 2 = β 2 + β 5 + β 9 (mod 2) = α 2 + α 5 + α (mod 2) = 1 (mod 2) z 2 = 1 Thus, (β 3 + 2β 2 ) + (β 6 + 2β 5 ) = β (Multiplication (x 1 + 2y 1 ) (x 2 + 2y 2 ) is now straightforward)

36 Family A and the Trace Function The trace function on GR(4, 3) maps GR(4, 3) Z 4 T (x + 2y) = (x + x 2 + x 4 ) + 2(y + y 2 + y 4 ) x T (x) β 4 β 2 β 2 β 5 β 6 β 3 1 t T (β t ) Trace description of Family A: A = {T (β t )} { T ([1 + 2δ]β t ) δ T }

37 Galois Ring GR(4, m) 1/2 Let f (x) be a primitive polynomial over F 2 of degree m. Let g(x 2 ) = ( 1) m f (x) f ( x) GR(4, m) = Z 4 [x]/ (f (x)) GR(4, m) = Z 4 [ζ] where ζ belongs to some extension ring of Z 4 and satisfies f (ζ) = 0

38 Galois Ring GR(4, m) 2/2 Set T = {0} {1, ζ, ζ 2,..., ζ N 1 } where N = 2 m 1 is the order of ζ in GR(4, m). The trace function T ( ) : GR(4, m) Z 4 is defined by T (x + 2y) = m 1 i=0 ( x 2i + 2y 2i ), x, y T

39 Family A 1/2 Let T = {γ i i = 1, 2,..., 2 m } be a labeling of the elements in T. Let s i (t) = T ([1 + 2γ i ]ζ t ), 1 i 2 m, s 2 m +1(t) = 2T (ζ t ). Then the sequences {s i (t)} are a set of cyclically distinct representatives for Family A.

40 Family A 2/2 If i j or τ 0, the correlation values of sequences in Family A belong to the set } θ ij (τ) { 1, 1 ± 2 m 1 2 ± ı2 m 1 2, m odd or θ ij (τ) } { 1, 1 ± 2 m m 2, 1 ± ı2 2, m even θ max m As compared to the binary family of Gold sequences, Family A offers a lower value of θ max for the same period and family size.

41 Further Reading D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE, vol. 68, pp , May T. Helleseth and P.V. Kumar, Sequences with low correlation, in Handbook of Coding Theory, V.S. Pless and W.C. Huffman, Eds., Amsterdam: Elsevier, 1998.

Helleseth, T. & Kumar, P.V. Pseudonoise Sequences Mobile Communications Handbook Ed. Suthan S. Suthersan Boca Raton: CRC Press LLC, 1999

Helleseth, T. & Kumar, P.V. Pseudonoise Sequences Mobile Communications Handbook Ed. Suthan S. Suthersan Boca Raton: CRC Press LLC, 1999 c 1999byCRCPressLLC Pseudonoise Sequences Tor Helleseth University