Low Correlation Zone Signal Sets Guang Gong Department of Electrical & Computer Engineering University of Waterloo CANADA Joint work with Solomon W. Golomb and Hong-Yeop Song
Outline of Presentation Outline of Presentation Requirements of Spreading Sequences in Quasi-synchronous (QS) CDMA Communications Definitions of Three Types Correlations, LCZ and Almost Signal Sets Correlation of Subfield Reducible Sequences Relationship between Subfield Constructions of LCZ Sequences and Complete Non-Cycle Hadamard Matrixes A New Construction of Subfield LCZ Sequences Open Questions @G. Gong 2
Code Division Multiplexing Access (CDMA) Code Division Multiplexing Access (CDMA) Multiple users share a common channel simultaneously by using different codes Narrowband user information is spread into a much wider spectrum by the spreading code The signal from other users will be seen as a background noise: Multiple access interference (MAI) The limit of the maximum number of users in the system is determined by interference due to multiple access and multipath fading: Adding one user to CDMA system will only cause graceful degradation of quality Theoretically, no fixed maximum number of users!
Code Division Multiplexing Access (CDMA) (Cont.) Code Division Multiplexing Access (CDMA) (Cont.) Received signal PSD Despread signal PSD for user 1 user 1 user M Despreading user M signal power Interference power Bandwidth Bandwidth user 1 user 2 user 3 user 4 user 2 user 3 user 4 CDMA is an interference-limited multiple access scheme The signal from other users will be seen as a background noise: Multiple access interference (MAI)
Spreading Sequences in CDMA Systems Spreading Sequences in CDMA Systems Walsh Codes: Basic spreading codes in CDMA systems H n x H nt = ni n n different Walsh codes: each row of an nxn Hadamard matrix Mutually orthogonal: inner product of different Walsh codes are zero Synchronization of all users are required to maintain the orthogonality: Otherwise, produce multiple access interference (MAI) Further, delayed copies received from a multipath fading are not orthogonal any more: Multipath fading interference MAI and multipath interference are major factors to limit the capacity of CDMA systems!
Quasi-synchronous CDMA Systems Synchronous CDMA Strict synchronization is required Spreading codes having perfect orthogonality at zero time delay are ideal Concern: What if multipath fading introduces nonzero time delay that can destroy the orthogonality of orthogonal codes? Asynchronous CDMA No synchronization between transmitted spreading sequences is required Relative delays between spreading sequences are arbitrary Nonzero interference due to multiple users and multipath fading is not avoidable: Interference cancellation or Multiuser detection process is required Quasi-synchronous CDMA Approximately synchronous CDMA (AS-CDMA) Originally proposed for satellite communications (1992) It may be feasible to maintain a chip or a few chips synchronization especially in microcell or indoor environment Relative time delay of a few chips is allowed Synchronized system with some synchronization errors Sequence design criterion: within a few chips, correlation between different spreading codes should be as low as possible
Three Types of Correlations Three Types of Correlations t n Notations: p a prime, q = p, N = q, 1 a = { a }, { } two i b = bi sequence over F q
Three Types of Correlations (Cont.) Three Types of Correlations (Cont.) (Golomb and Gong, 2005) @G. Gong 8
Three Types of Correlations (Cont.) Three Types of Correlations (Cont.)
Three Types of Correlations (Cont.) Three Types of Correlations (Cont.) An unaware fact about correlation of these three different alphabet sets: C = = C C a, b u,v s, t ( τ ) ( τ ) ( τ ) @G. Gong 10
LCZ and Almost LCZ Sequences LCZ and Almost LCZ Sequences ( N, r, δ, d) ( N, r, δ, d) where τ 0 for a = b. Then S is referred to as a ( N, r, δ, d) low correlation zone (LCZ) signal set. If the crosscorrelation/out-of-phase autocorrelation of any two sequences a and b in S satisfies C ( b τ ) < δ, 0 < < d a, τ Then S is called a ( N, r, δ, d) almost LCZ signal set.
LCZ and Almost LCZ Sequences (Cont.) LCZ and Almost LCZ Sequences (Cont.) Conventional low correlation δ δ 0 τ (Almost) Low correlation zone δ δ -d 0 d τ Low Correlation Zone (LCZ) Almost LCZ
Crossorrelation of Subfield Reducible Sequences Crossorrelation of Subfield Reducible Sequences From now on, we will also use trace representations of sequences over F q with period N.
where d n q = q m 1 1
Remarks on Theorem 1 1) Known binary cases ( Klapper, Chan and Goresky (1993), and Klapper (1995)) : - h(x) is a trace monomial function or a cascaded GMW function - h(x) is a k-form function 2) If both of f and g are balanced, then crosscorrelation/outof-phase autocorrelation between f and g takes value of -1 for n m m q q shifts, and the only at most q 1 correlation values may be irregular.
C f oh, g oh ( τ ) q n -1 (q m -1)q n-m -1 2q n-m -1 -q n-m -1 0 d (q-2)d -1 2d 3d τ -q n-m -1-2q n-m -1 -(q m -1)q n-m -1 -q n -1
-1 τ 15 0 7 47 55 ) (, τ h g h f C o o 23 31 39 63-57 -65-25 -17-49 -41-33 -9 9 18 27 36 45 54 62 ) ( ) ( and ), ( ) ( ), ( ) ( 3 1 3 1 6 3 x Tr x g x Tr x f x Tr x h = = =
Relationship between Subfield Constructions of of LCZ Sequences and Complete Non-Cycle Hadamard Matrixes
Subfield Constructions of of LCZ Seq. and Complete Non-Cycle Hadamard Matrixes (Cont.) For the known constructions: K < p m 1 for q = p (Tang-Fan, 2001); K = q m/2 for q = 2 2 (Kim etc. 2005); K = 2 m 1 for q = 2 or p (Jang etc. 2005; Tang and Udaya, 2005, Jang etc. 2006).
Subfield Constructions of of LCZ Seq. and Complete Non- Cycle Hadamard Matrixes (Cont.)
Subfield Constructions of of LCZ Sequences and Complete Non-Cycle Hadamard Matrixes (Cont.) An LCZ signal set with parameters ( N, q m 1,1, d) A complete non-cyclic Hadamard matrix, Classification of subfield reducible LCZ signal sets with the above parameters Classification of m m q q complete non-cyclic Hadamard matrices
A New Construction of Subfield LCZ Sequences A New Construction of Subfield LCZ Sequences Then K produces a subfield reducible LCZ signal sets for any q with parameters ( N, q m 1,1, d)
Open Problems for Characteristic 2 Case Open Problems for Characteristic 2 Case Open Question 1. Is that true that h(x), F n F m q q with 2- tuple-balance property is equivalent to that h(x) is k-form with difference balance property. Recall that h(x) is k-form if and only if h( yx) = y k h( x), y F q m, x F q m
For more about correlation of subfield reducible sequences, see Golomb and Gong s new book:
Reference Guang Gong, Solomon W. Golomb, and Hong-Yeop Song, A Note on Low Correlation Zone Signal Sets, - Proceedings of 40 th Annual Conference of Information Sciences and Systems (CISS 2006), March 22-24, 2006, Princeton University. - Technical Report, University of Waterloo, CACR 2006-06, January 2006. @G. Gong 28