Topic 3. Design of Sequences with Low Correlation

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1 Topic 3. Design of Sequences with Low Correlation M-sequences and Quadratic Residue Sequences 2 Multiple Trace Term Sequences and WG Sequences 3 Gold-pair, Kasami Sequences, and Interleaved Sequences 4 Case Study: Examples of Orthogonal Codes 5 Applications in CDMA G. Gong

2 M-sequences and Quadratic Residue Sequences A. M-sequence Generator: Input: - A finite field F = GF(q); n n" - f ( x) = x " cn" x " L" cx " c, ci F, a primitive polynomial over F. Output: a = a, a, a2,l an m-sequence over F of period q n -. Procedure (F, f(x), a): Compute : a quit i n- + n = c ja j+ i, i j= =,, G. Gong 2

3 B. Profile of Randomsess of Binary M-sequences of Degree n Period 2 n Balance Run property 2 n- s and 2 n- - s () For k n-2 runs of s ( s) of length k occurs 2 n-2-k times (2) Zero run of length n- occurs once; no runs of s of length n- (3) Runs of s of length n occurs once As good as it could be Span n property or ideal n-tuple distribution Autocorrelation Linear span and ρ Each nonzero n-tuple occurs once 2-level n, the normalized linear span n " = 2 n Only need to know 2n bits to reconstruct the entire G. Gong 3

4 Property (Binary m-sequences). Let an LFSR have a primitive characteristic polynomial f(x). Then there exists an initial state for the LFSR such that the corresponding output sequence a = {a i }, i.e., an m-sequence, satisfies a i = a 2i, i =,,. In other words, there exists an initial state of the LFSR such that the output sequence is identical to its 2-decimation G. Gong 4

5 C. Quadratic Residue (or Legendre) Sequences Note. Starting from now, we only consider binary sequences. Some concepts in number theory: Quadratic Residue: Let p be an odd prime number. A number i: < i < p is called a quadratic residue modulo p if there is an integer x for which x 2 i mod p. Fact. A set defined by QR = {i 2 mod p < i (p-)/2} contains all quadratic residue modulo p. The Legendre symbol: The Legendre symbol (i/p) is defined as follows: * ( ) i p ' % & # if i is a quadratic residue modulo p = " $ G. Gong 5

6 Quadratic Sequence Generator: Input: A prime number p with p 3 mod 4 Output: a = a, a,... a binary sequence of period p Procedure(p, a): + i ( Compute :) & and set a * p ' a i = and set $ + i (, ) & = * p = ' # + i (, ) & = % " * p ' Return: a G. Gong 6

7 @ G. Gong 7 Example 8. Let p =. Then 2,, = " # $ % & ' = " # $ % & ' = " # $ % & ' p p p 5, 4, 3 = " # $ % & = " # $ % & = " # $ % & p p p 8, 7, 6 = " # $ % & ' = " # $ % & ' = " # $ % & ' p p p, 9 = " # $ % & ' = " # $ % & ' p p So, a has period and a = (). For p = 3, then a = () which has period 3. Verify it for the autocorrelation

8 Profile of Quadratic Residue (or Legendre) Sequences:. Period: p 2. Balance: (p+)/2 's and (p-)/2 's One should know entire 3. 2-level autocorrelation bits to reconstruct the 4. Linear span: (p-)/2 sequence 5. N(QR,p), the number of shift distinct quadratic sequences modulo p, N(QR, p) = 2. Sound randomness However, it cannot be implemented efficiently Compare it with G. Gong 8

9 2 Multiple Trace Term Sequences and WG Sequences A. Three-Term Sequences Construction A: For odd n 5, and n = 2m+ with period N = 2 n, let b = {b i } be an m-sequence of period N with the constant-on-coset property. Assume q = 2 m + and q 2 =2 m + 2 m- +. Define a i = b i + b q i +b q 2 i, i =,, L Then a = {a i } is a binary sequence with 2-level G. Gong 9

10 LFSR LFSR 2 + Output LFSR 3 Architecture for Implementation of the 3-Term G. Gong

11 Profile of 3-term sequences. Period: 2 n 2. Balance: 2 n- 's and 2 n- 's 3. 2-level autocorrelation 4. Linear span: 3n 5. N(T3, n), the number of shift distinct 3-term sequences, n N( T3, n) = "(2 ) / G. Gong

12 LFSR + LFSR 2 + Output + LFSR3 + Implementation of 3-term sequences of n = 5: (, 5, G. Gong 2

13 LFSR + LFSR 2 + Output + LFSR3 + Implementation of 3-term sequences of n = 7: (, 9, G. Gong 3

14 5-Term Sequences and WG Transformation Sequences Some property of finite fields Some property of finite G. Gong 4

15 B. 5-Term Sequences and WG Transformation Sequences Some property of finite fields Trace Functions Definition. A trace function Tr (x) of GF(pn ) is a function from GF(p n ) to GF(p) defined as follows Tr( x) = x + x p + L+ x p n n Tr( x) : GF( p ) GF( p) Property. (a) The trace function is a linear function. (b) p p p n ( x + y) = x + y," x, y GF( p ). Primitive Elements of Finite Fields Let the finite field GF(p n ) be defined by a primitive polynomial over GF(p) with degree n. Then the defining element is called a primitive element in GF(p n G. Gong 5

16 5-Term Sequences and WG Transformation G. Gong 6

17 @ G. Gong 7

18 f 2 (x) = f 3 (x) = f 5 (x) = Remark. The initial states of the LFSRs should satisfy the constan-on-coset G. Gong 8

19 There are two methods to implement a WG sequence. For small n, a WG sequence can be implemented by Method, as shown G. Gong 9

20 Remark: WG sequences have important applications in generating key streams in stream ciphers because their good randomness G. Gong 2

21 3. Gold-pair, Kasami Sequences, and Interleaved Sequences Signal Set Let s j = ( s j,, s j,, L, s j, v" ), j < r be r shift-distinct p-ary sequences of period v, S = r { s, s, L, s } Let # = max C, (" ) for any " < v, i, j r, s < i s j " v where if i= j. If " c where c > is a constant, then we say that the set S has low cross-correlation. In this case, we call S a ( v, r, ) signal set. When we consider the cross correlation of two sequences s i, s j in S, we simply write C i, j (τ) for ( ). C si, s G. Gong 2

22 A. Gold-pair Sequences Construction: Let a = {a i } be an m-sequence of degree n odd. Select d as one of the following values: - d =2 k + (Gold exponents), (k, n) =, k < (n + )/2. - d = 2 2k - 2 k + (Kasami Large Set Exponents), (k, n) =, k < (n + )/2. n - d = (the Welch Conjecture Exponent) n For " j < 2, let s j = {s j, i } be a binary sequence whose elements are given by Then s j is called a Gold-pair sequence. s j,i = a i + a j +di, i =,,L,2 n " 2 Let s 2 n " = a and s 2 n = a (d ). Set S = { s j j 2 n }. Then S is called a Gold-pair (signal) set.

23 Note that we can also write s j as a sum of a shift of a and b = a (d), i.e., s j = L j n a + b, " j < 2. Remark. A useful formula for the relationship between the cross correlation and the Hamming weight. Let a and b be two binary sequences with period 2 n -, then the cross correlation between these two sequences can be determined by the Hamming weight of a + L τ b as follows: C a,b n ( ) = 2 " " 2w( a + L ( b)) where w(.) denotes the Hamming weight of the G. Gong 23

24 Profile of the Gold-pair Signal Set: n n ( n+)/ 2. S is a (2, 2 +, + 2 ) -signal set. 2. The cross correlation C i, j (τ) of any two sequences s i and s j in S is three-valued and it takes values: ( n+ )/ 2, ± 2 3. In S, there are 2 n- +2 sequences are balanced, i.e., each has 2 n- 's. Moreover, each sequence in S has its Hamming weight taking one of the three values: n n ( )/ 2 2, 2 ± 2 n 4. b is also an m-sequence, i.e., (d, 2 n - )=. 5. Linear span is either 2n or G. Gong 24

25 LFSR Implementation:. Select n odd, f(x), a primitive polynomial of degree n over GF(2), as a characteristic polynomial of LFSR. 2. Select d and compute the characteristic polynomial of the LFSR2, which is the minimal polynomial of d-decimation of a sequence generated by LFSR. 3. We obtain all sequences in S by varying different initial states of the LFSR. LFSR: varying initial state LFSR2: a fixed initial state LFSR implementation of Gold pair G. Gong 25

26 Example. Design of a (3, 33, 9) Gold-pair signal set. Method : LFSR Implementation: 5 3. Select f ( x) = x + x +, a primitive polynomial over GF(2) of degree 5 as the characteristic polynomial of LFSR. 2. Select d = = 5 and compute the minimal polynomial of in GF( 2 5 ), defined by f(α) =. This gives g( x) = x + x + x + x + Use g(x) as the characteristic polynomial of LFSR2. 3. Fixing an initial state of one these two LFSRs and varying another, we get all 33 shift distinct sequences in G. Gong 26

27 Method 2: Software Implementation 5 3. Select f ( x) = x + x +, a primitive polynomial over GF(2) of degree 5 to generate a sequence a: 2. Select d = = 5 and perform 5-desimation on a, we obtain (5), i 5i b = a b = a, i =,,L 3. Generate L i ( a) + b, i < 3 Together with a and b, this gives all 33 sequences in G. Gong 27

28 : : 2: 3: 4: 5: 6: 7: 8: 9: : : 2: 3: 4: G. Gong 28

29 6: 7: 8: 9: 2: 2: 22: 23: 24: 25: 26: 27: 28: 29: G. Gong 29

30 Profile of S Period 3. There are 33 shift distinct sequences in S. The cross correlation of any two sequences in S only takes three values: -, -9, 7. S is a (3, 33, 9)-signal set. There 8 sequences in S are balanced, i.e. each has weight 6. The rest 5 has their weights either 2 or 2. Linear span or G. Gong 3

31 B. Kasami Sequences Construction: Let n = 2m and and a = {a i } be an m-sequence of degree n. Let s j = {s j, i } be a binary sequence whose elements are given by s j,i = a i + a j +di, i =,, L where d = 2 m +. Let s 2 m " = a and S = {s j " j " 2 m #}. Then S is called a Kasami small signal set. In other words, if we write then b = a (d ) s j = a + L j b, " j < 2 m G. Gong 3

32 Profile of the Kasami small signal set. The Kasami small signal set S is a signal set. (2 n, 2,+ 2 n / 2 n / 2 ) 2. The cross correlation C i, j (τ) of any two sequences s i and s j in S is three-valued and it takes values: n / 2, ± 2 3. b is an m-sequence of period 2 n/ Linear span: 3n/2 or G. Gong 32

33 Example 2. Design a (63, 8, 9) Kasami signal set. LFSR Implementation: - Select n = 6 and f(x) = x 6 + x + ; set α be a root of f(x) in GF(2 6 ). - Compute the minimal polynomial of α 9 which gives g(x) = x 3 + x 2 +. c b a a, b, and c are taken as or The correlation values are: -, 7, - G. Gong 33

34 b = a = a + b = Complement the columns corresponding to s in b: Write it row by row from the most left upper corner: s G. Gong 34

35 C. Interleaved Sequences Definition of Interleaved Sequences Constructions of Signal Sets from Interleaved Sequences Profiles of the Interleaved Signal Sets G. Gong 35

36 @ G. Gong 36 Definition of Interleaved Sequences Let be a binary sequence of period st. We can write the elements of the sequence A into a s by t array as follows: ),,, ( = st u u u L u " # $ $ $ $ $ $ % & ' + ' + ' ' ' + + ' + + ' ) ( ) ( ) ( t t s t s t s t t t t t t t t t u u u u u u u u u u u u L M L L L If all these column sequences are phase shifts of a binary sequence, say a, of period s, then we say that u is an (s, t)-interleaved sequence. Let u j denote the jth column of the above array. Then we have where L is the left shift operator and the e j are nonnegative integers with e j <s. t j L i e j < = (a), u

37 Example 3. Let e = (3, 6, 5, 5, 2, 3, 5) and a = (,,,,,, ), a binary m-sequence of period 7. Then a (7, 7) array is as follows: A (7, 7) interleaved sequence u G. Gong 37

38 Procedure : Constructions of (v 2, v+, 2v+3) signal sets. Choose a = (a, a,..., a v- ) and b = (b, b,..., b v- ), two binary sequence of period v with 2-level auto-correlation. 2. Choose e =(e, e,..., e v- ), an integer sequence whose elements taken from Z v, a set consisting of integers modulo v. 3. Construct u = ( u, u, L, u ) v 2, a (v, v) interleaved sequence whose e jth column sequence is given Lby j (a). 4. Set = ( s, s, L, s ), j v whose elements are defined by " s j j, j, 2 < j, v j s j, i = ui + bj+ i, or s = u + L ( b), j < v 5. A signal set S is defined by S = {u, s j : j =,,..., G. Gong 38

39 Example 4. Let v = 7.. Choose a = ( ) and b = ( ), m-sequences of period Choose e = (3, 6, 5, 5, 2, 3, 5). 3. Construct the interleaved sequence u: 4. Construct s j : j =,,..., 6. The column sequences of s can be obtained from the sequence u by complement of columns of u whose indexes correspond to 's in the sequence b; the column sequences of s obtained from u by complement of those in the sequence b with a phase shift, and so on. u = b = s = u + b: s = u + Lb G. Gong 39

40 s = u + b = s = u + b = s 2 = u + L 2 b = s 3 = u +L 3 b = s 4 = u +L 4 b = s 5 = u +L 5 b = s 6 = u +L 6 b = 5. Set S = {u, s, s,..., s 6 G. Gong 4

41 Constructions for (v 2, v+, 2v+3) signal sets (Cont.) Theorem. If the shift sequence e satisfies the following condition, (*) { e j " e j + s j < v " s} = v " s, for all s < v then S is a (v 2, v+, 2v+3) signal set. Moreover, the cross-correlation and out of phase auto-correlation values of any two sequences in S belong to the set {, -v, v+2, 2v+3, -2v-}. * Gong (995) introduced the idea of Procedure for constructing ((2 n -) 2, 2 n, +2 n ) signal sets where v = 2 n - and both a and b are m-sequences of period 2 n -, and proved Theorem. We can verify that the exponent sequence in Example 4 satisfies (*). From Theorem, it is a (49, 8, 7) signal G. Gong 4

42 Shortened M-sequence Construction Procedures for constructing the exponent sequence e which satisfies the condition (*). Construction ( Shortened M-sequence Construction) ((2 n -) 2, 2 n, +2 n+ ) signal set:. Choose a primitive polynomial f(x) over GF(2) of degree 2n as the characteristic polynomial of an LFSR and generate GF(2 2n ) by f(α)=. 2. Generate an m-sequence {w i } of period 2 2n - by the LFSR which satisfies the constant-on-coset property. 3. Arrange {w i } into an (r, v) array where r = 2 n + and v = 2 n -: # w w 2 L w s"2 w s" & % ( w s+ w s+2 L w s+s"2 w 2s" W = % ( % M ( % ( $ w (v")s+ w (v")s+2 L w (v")s+s"2 w vs" G. Gong 42

43 Let U be a matrix obtained from W by deleting the first and the last columns, then U gives an interleaved sequence u of period (2 n -) 2 which has the base sequence a which is r-decimation of b and the exponent sequence satisfied (*). 4. Select b a binary 2-level sequence of period 2 n -. Continue the Procedure, we get the signal set G. Gong 43

44 Example 5. Interleaved Generator (49, 8, 7):. Choose a primitive polynomial f(x) = x 6 +x + for generating m- sequence {w i } with the initial state (,,,,, ) and arrange it into a 9 by 7 array W: Cut Cut 2. By deleting the first and the last columns of W, we have which is the sequence u given in Example G. Gong 44

45 Profile of Randomness of (49, 8, 7). Period: shift distinct sequences 3. The maximal magnitude of the cross correlation: 7 4. The cross correlation takes five values: {, -7, 9, 7, -5} 5. Balance: 25 's and 24 's 6. Linear span: 24 except for u where u has linear span 2. Compare it with the Kasami set with parameters (63, 8, 9) (An example of scarified the correlation for linear G. Gong 45

46 4. Case Study: Examples of Orthogonal Code Design Review for Inner Product Hadamard Matrices Orthogonal Code Design from 2-Level Autocorrelation Sequences G. Gong 46

47 Review for Inner Product: let C be the complex field; F = C (r), contains vectors of dimension r whose elements are taken from C. The inner product of any two vectors in F, say and b c = r is defined by b ( b, b, L, b ) = r ( c, c, L, c ) " c = b c + bc + L+ b r cr If b.c =, then b and c are said to be orthogonal. Example. Let b = (, -, -, ) and c = (-, -,, ). Since b c = (" ) + (" )( " ) + (" ) + = then b and c are orthogonal. Definition. A finite subset M of F is said to be an orthogonal code if any two vectors in M, called codewords, are orthogonal. How to construct a binary orthogonal G. Gong 47

48 Hadamard Matrices: Let H be a m by m matrix whose elements are taken from {, -}. If T HH = mim where I m is the m by m identity matrix, then H is called a Hadamard matrix. The following matrix is a Hadamard matrix: H & $ $ = $ $ % ' ' ' ' # ' ' " Existence of Hadamar matrices: m is a multiple of 4 A set consisting of row vectors of a Hadamard matrix constitutes an orthogonal code, i.e., orthogonal code Hadamard matrix (it is also called a Walsh code in G. Gong 48

49 Orthogonal Code Design from 2-Level Autocorrelation Sequences Construction: Let a = {a i } be a binary 2-level autocorrelation sequence of period m = 2 n and Then is a Hadamard matrix.,,,, ) ( = = m i b i a i L " # $ $ $ $ $ $ % & = ' ' ' 2 2 m m m b b b b b b b b b H L M L L L Question: Which construction of 2-level autocorrelation sequence has a simple implementation? Example. Let a = (), an m-sequence of period 7. Then the following matrix is a Hadamard matrix, so we obtain an orthogonal code from the m- sequence of degree 3.

50 5 Applications to DS-CDMA System System Basics Physical Layer of Forward and Reverse Links Use of Hadamard matrices for orthogonality Forward: Orthogonal CDMA Channels Reverse: Orthogonal Coding Use of cyclic Hadamard matrices for 2-level auto-correlation function property PN Long Code and Short Code Unique Identification Data Scrambling and Authentication Spectrum Spreading

51 The CDMA Cocktail Party This is great stuff Where is she Who called How long will this take Who knows Where is the meeting You know that How can I get there Can I go home? Where is the office

52 Multiple Access Methods

53 CDMA System Design Voice Coding Forward Link Generation CODEC 96 bps 48 bps 24 bps 2 bps R = /2 Convolutional Encoder and Repetition User Address Mask (ESN) Block Interleaver Long Code PN Generator 9.2 ksps.2288 Mcps Decimator 9.2 ksps Power Control Bit Decimator MUX 4 8 Hz Wt.2288 Mcps I PN Q PN Cell VOCODER Power Control Voice Coding Reverse Link Generation I PN CODEC 96 bps 48 bps 24 bps 2 bps R = /3 Convolutional Encoder and Repetition 28.8 ksps Block Interleaver User Address Mask 28.8 ksps Walsh Cover Long Code PN Generator 37.2 khz.2288 Mcps Data Burst Randomizer.2288 Mcps QPN /2 PN Chip Delay D Mobile

54 The CDMA Rate Families IS-95 defines the 96 bps family of rates (Rate Set ) 96, 48, 24, and 2 bps Can select one of the four rates every 2 ms frame 44 bps family of rates (Rate Set 2) 44, 72, 36, and 8 bps Can select one of the four rates every 2 ms frame Extended rates (extended Rate Set ) Adds 92, 384, and 768 bps At most four rates can be active Can select one of the four active rates every 2 ms frame

55 Link Waveform CDMA Forward Link Waveform Pilot Channel Sync Channel Paging Channel Traffic Channel QTSO CDMA REVERSE Link Waveform Access Channel Traffic Channel QTSO

56 Forward CDMA Channel A 64 x 64 Hadamard matrix is used to produce 64 orthogonal channels. Each row of a Hadamard matrix is used to uniquely identify channels from to 63. This is possible since each row (as a binary vector of length 64) of the 64 x 64 Hadamard matrix is orthogonal to any other row. If we call 64 row vectors as W,W,...,W 63, then the waveform for Channel #k is obtained by multiplying W k to the input signal.

57 Forward CDMA Channel FORWARD CDMA CHANNEL (.23 MHz channel transmitted by base station) Pilot Chan W Sync Paging Paging Traffic Ch Chan Up Ch 7 Ch W32 W to W7 W8 Traffic Ch N Traffic Traffic Traffic Ch 25 Up Ch 24 Up Ch 55 to W3 W33 to W63 W = Code Channel Traffic Data Personal Station Power Control Sub-Channel

58 Reverse CDMA Channel A Reverse Channel is uniquely identified by an initial phase of a binary PN sequence (so called, the long code) of length Different initial phases correspond to different shift of the PN sequence, hence they are orthogonal. Only /256 positions of the length are used to address all the users in the system.

59 Reverse Traffic Channel Structure for Rate Set Reverse Traffic Channel Information Bits (72, 8, 4, or 6 bits/frame) 8.6 kbps 4. kbps 2. kbps.8 kbps Add Frame Quality Indicators (2, 8,, or bits/frame) Add 8 bit Encoder Tail 9.6 kbps 4.8 kbps 2.4 kbps.2 kbps Convolutional Encoder r=/3, K=9 Code Symbol 28.8 ksps 4.4 ksps 7.2 ksps 3.6 ksps Symbol Repetition Code Symbol 28.8 ksps Block Interleaver Code Symbol 28.8 ksps I-channel Sequence.2288 Mcps 64-ary Orthogonal Modulator Modulation Symbol (Walsh chip) 4.8 ksps (37.2 kcps) Frame Data Rate Data Burst Randomizer Long Code Generator PN chip.2288 Mcps Q-channel Sequence.2288 Mcps /2 PN chip Delay = 46.9 ns D I Q Baseband Filter Baseband Filter I(t) cos(2"f c t) Q(t) sin(2"f c t) s(t) Long Code Mask

60 Orthogonal Modulation (R. Link) chip pattern of Walsh Code # 35 Symbols Walsh Lookup Table... For every 6 symbols in, 64 Walsh chips are out. Here, 64 Walsh chips are 64 components of a row of the 64 x 64 Hadamard matrix. Note the different role of the same 64 x 64 Hadamard matrix in Reverse Link.

61 PN Long Code Spreading 64 chip pattern of Walsh Code # k... (37.2 khz).2288 Mcps Channel Mask Long Code (2 42 -) PN Generator.2288 Mcps

62 PN Long Code Long Code Maximal Linear Feedback Shift Register Sequence Length (Period) is Unique identifier for: Access and Traffic Channel (R. Link) Paging and Traffic Channel (F. Link) Functions Access, Paging, Traffic Channel spreading DBR on the Reverse Traffic Channel Power Control Bit randomization Data Scrambling (encryption) and Authentication

63 PN Short Codes of length 2 5 I PN.2288 Mcps To Tx Q PN

64 Short Code Short Code (2 5 ) Unique identifier for a Cell or a Sector Repeats every ms Consists of I and Q codes (different polynomials) It is called a Modified Maximal Length Sequence or de bruijn sequence or span-n sequence. It is obtained by adjoining one extra bit after the unique run of of length 4 in the PN sequence of length

65 Summary of Application to DS-CDMA systems Orthogonality -- Channelizing and Coding Scrambling and Authentication Spectrum G. Gong 65

Design of Signal Sets with Low Intraference for CDMA Applications in Networking Environment

Design of Signal Sets with Low Intraference for CDMA Applications in Networking Environment Guang Gong Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario N2L 3G1,