Optimum Spreading Sequences for Asynchronous CDMA

Optimum Spreading Sequences for Asynchronous CDMA System Based on Nonlinear Dynamical and Ergodic Theory Kung Yao C.C. Chen UCLA Winter School in Chaotic Communications, UCSD, Jan. 13, 2003

Outline I. Information theory-Communication theorycommunication system and chaos II. Introduction to CDMA comm. system III. Introduction to chaos and ergodic theory IV. Optimum asynchronous CDMA sequences V. Comparion of CDMA system performance under ideal and practical constraints VI. Conclusions 2

I. Information Theory - Communication Theory - Communication System • The purpose of communication is to provide efficient transmission of information • Information theory provides the mathematical theory for information processing/transmission • Communication theory provides the concepts/ models for analysis/design of communication • Communication system deals with implementation at harware/sub- system levels of communication • There is a close inter-play among these three well-developed but different disciplines 3

Impact of Hardwares in Comm. Systems • Early practical communication systems included: digital transmission in telegraph (over wires) and radio (over free-space propagation); analog transmission telephony (over wires) and a.m./f.m. radio broadcasts (over free-space prop.) • These system designs were ad hoc and governed mainly by available hardware technology • As new hardware devices came along(e.g.,vacuum tube amplifier; transistor; microelectronics chip/microprocessor; laser; etc.) new communication systems become possible 4

Impact of Theory to Comm. System • As comm. systems became more complex there were greater needs for systematic treatments • Since the 1950’s, information theory and statistical decision/estimation theory provided the analytical tools for the successful analysis and design of advanced communication systems • Today, communication theory and systems (e.g., satellite comm;cell phones;etc) are mature • Any new comm. concepts/technologies have to compete with the existing comm. theory/systems 5

Chaos in Communications • Chaos theory is a branch of advanced nonlinear mathematics with a history of over 100 years • Chaos and fractals are intellectually challenging and have been popularized by Lorenz, Mandelbrot, others in the last 30 years • Work of Pecora-Carroll (early 1990s) on selfsynchronization of chaotic systems motivated the consideration of chaos in communications 6

Chaos in Communications (continued) • Most workers in “chaotic communication” in the last ten years (particularly in the early years) were fundamentally more interested in the nonlinear circuit/system/mathematical aspects and not really in the communication system aspects (e.g., most publications were in physics/mathematics/ circuit/sys. journals and not in information theory/communication theory-communication system journals) • Can “chaotic communication” concepts be translated to practical communication systems7 ?

Chaotic vs. Conventional Communications • Two classes of chaotic communications: 1. Those that exploit some chaotic properties not necessarily implementable nor competitive with conventional communication systems (e.g., sys. that are essentially baseband sys. (that have no equiv. carrier freq. translatable version); sys. that utilized some artificially imposed nonlinearity with “interesting chaotic” properties; sys. that are not competitive to conv. sys. in terms of SNR;data rate;complexity;interf. rejection, etc.) 8

Chaotic vs. Conventional Comm. (cont.) 2. Those chaotic comm. sys. that only use certain chaotic properties to replace some functional parts of “conventional comm.” systems; these sys. may still be implemented, analyzed, and shown to have some advantages with respect to their conventional equivalent counterparts (e.g, a chaotic pulse-position-modulation sys. (PPM) is a PPM sys.; a FM-DCSK sys. is a FM digital comm sys.; laser comm. sys. that exploit some intrinsic chaotic prop.;a “chaotic generated seq.” CDMA sys. is a CDMA sys. ; etc.) 9

II. Introduction to CDMA Systems • Freq. Division Multiplexing (FDM/FDMA) - subbands assigned to individual user; simple; used in telephony/digital microwave sys; inflexible • Time Div. Mult. (TDM/TDMA) - each user is assigned a time slot with full bandwidth; possible dynamic user assignment; used in sat. comm; cell phones (GSM-2G); more efficient than FDMA • Code Div. Mult. (CDM/CDMA) Spread Spectrum (SS) sys.; inform. data of bandwidth R is spread to a larger bandwidth of B for multiple access/interf. rej.;IS95;WCDMA-3G; more eff. than TDMA 10

Two Spectrum Communication Sys. • Direct Sequence System (DS; PN; CDMA)Original data of bandwidth R = 1/T Hz is modulated (spread) by a PN sequence code with a smaller chip duration of TC = T/PG to a larger bandwidth of B = R š PG, where T = data symbol duration, and PG is the Processing Gain • Frequency-Hopping (FH) System - Each chip duration corresponds to some freq. waveform; it is allowed to hop to various freq. within B Hz. 11

DS-Binary Phase Shift Keyed (BPSK) Spread Spectrum Transmitter (1)

Phase Modulator

(3)

(5)

x

(2)

(4)

• Binary data: m(t ) = bk = ±1, kT ≤ t < ( k + 1)T , (1) • Carrier waveform: q (t ) = 2P cos(ω 0t ),

(2)

• Data modulated waveform: s (t ) = q (t) m(t ) = 2Pm(t ) cos(ω 0t ),

(3)

• SS sequence (code): a(t ) = ak = ± 1, kTC ≤ t < (k + 1)TC ,

(4)

• Coded (Spread) Waveform: r (t ) = s(t )a(t ) = 2 Pm(t )a(t ) cos(ω 0t ).

(5)

12

BPSK SS Transmitted Waveforms T

Binary data (1)

Carrier waveform (2)

Data modulated waveform (3) TC

SS code (4)

Coded (Spread) Waveform (5) 13

BPSK SS Receiver

(5’)

x (6)

(7)

Bandpass filter

Data Phase Demodulator

• Received SS Waveform:

(9)

(8)

r '(t ) = 2 Pm(t )a (t ) cos(ω 0t ) + interf.+noise, (5') • Phase - locked (Locally generated) SS code: a '( t ) ≈ ak = ±1, kTC ≤ t < ( k + 1)TC ,

(6)

• SS decoded (Despread) waveform: u(t)=r'(t)a'(t) ≈ 2 Pm (t ) cos(ω 0t ),

(7)

• Phase locked (Locally generated) carrier waveform : q '(t ) ≈ co cos(ω 0t ),

(8)

• Demodulated data (using matched filter): T

m'(t)=∫ u(t)q'(t)dt ≈ m(t)=bk = ± 1. 0

(9)

14

BPSK SS Received Waveforms Phase - locked (Locally generated) SS code (6) SS decoded (Despread) waveform (7) Phase locked (Locally generated) carrier waveform (8)

Demodulated data (using matched filter)(9) 15

(3)

(5) 16

17

High Interference Power Results in Low SIR in a Conventional BPSK System

18

Advantage of BPSK-CDMA System over Conventional BPSK System with Interference

19

Binary Linear Feedback Shift Register Generator (LFSRG) for SS Coding

20

Analysis of Spreading Codes B

, with B = ( B0 ,..., BN-1 ), Bn ∈ {0,1}

a (t ) a (t ) =

∞

∑

n =−∞

an p (t − nTC ), an = (−1) Bn ∈{−1,1} B

Bn+N = Bn B

B'

1

a a' ∑ N

θ BB' (k ) =

θ B (k ) =

N −1 n =0

1

n

n+k

N −1

aa ∑ N n =0

n n+k

21

M-Sequence

θ B (k ) k = mN 1, θ B (k ) =   −1/ N , k ≠ mN

22

Gold Codes (Optimum minimax periodic crosscorrelation m-sequences) BB'

B' = B[q]

B'

B

{B, B', B ⊕ B',B ⊕ D B',B ⊕ D 2B',...,B ⊕ D N −1B'} D jB' represents a phase shift of the m- sequence B' by j units

23

An N = 31 Length Gold Code Generator

24

III. Introd. to Chaos and Ergodic Theory Basic Properties of Chaos • Chaos is deterministic and can be generated from appropriate NonLinear Dynamical Systems • Chaos is characterized by extreme sensitivity to variation in the initial condition of the NLDS • Chaos is unpredictable and thus “noise-like” • Chaotic NLDS can be used to generate pseudorandom (PN) sequences for simulations and CDMA spread spectrum code generation 25

Chaos and Bifurcation for the Logistic Map Function • Logistic map function has D = R = I= [0,1] xi = Axi-1(1-xi-1), 0 = 1,…, 0