Multi User Detection I

Transcription

1 January 12, 2005

2 Outline Overview Multiple Access Communication Motivation: What is MU Detection? Overview of DS/CDMA systems Concept and Codes used in CDMA CDMA Channels Models Synchronous and Asynchronous Channel Model Single User Matched Filter 1

3 Overview Multiple Access Communication Random Multi Access (e.g. ALOHA) one user can work simultaneously, otherwise collision occur. FDMA many user can work simultaneously, but in non overlapping frequency bands. TDMA many user in one frequency band, but one user per time. CDMA many user simultaneously in the same frequency band. The user are separated by orthogonal or non orthogonal signature waveforms. 2

4 Access Techniques 3

5 Motivation: What is MU Detection? y k = A }{{ k b k} + A j b j ρ jk 1 term j k }{{} MAI first term: desired information MAI: Multi Access Interference n: noise + n k }{{} n Cross correlation: ρ ij = s i, s j = T 0 s i (t)s j (t) d t s i... signature waveform of the i th user 4

6 Where MU Detection can be used? Answer: For nonorthogonal multiple access due to design or due to unwanted crosstalk! nonorthogonal CDMA TDMA with Multipath bundle of twisted pair magnetic recording in adjacent tracks 5

7 6

8 Properties of Random Binary Sequences (Codes in CDMA) Autocorrelation only one single peak Cross-correlation should be low Frequency Spectrum the spread frequency should be flat 7

9 Orthogonal / Non Orthogonal Orthogonal code e.g. WALSH bad spreading properties more peaks in the autocorrelation Orthogonality hold only for perfect synchronized and no multi-path Non Orthogonal code e.g. Gold, Kasami good spreading properties single peak in the autocorrelation limited (low) cross-correlation 8

10 Synchronous/Asynchronous CDMA Model 9

11 Basic Synchronous CDMA Model (K-user) y(t) = K A k b k s k (t) + σn(t), t [0, T] k=1 s k (t) deterministic signature waveform assigned to the k th normalized energy user, with A k amplitude of the k th user b k { 1,+1} is the bit transmitted by the k th user n(t) is white Gaussian noise with unit power 10

12 Discrete Time Synchronous CDMA Model (K-user) Matched Filter Output y 1 =. y k = T 0 T 0 y(t)s 1 (t) dt y(t)s k (t) d t 11

13 Cross Correlation ρ ij = s i, s j = T 0 s i (t)s j (t) d t Cross correlation matrix for example in the 3-user case: R = 1 ρ 12 ρ 13 ρ 12 1 ρ 23 ρ 13 ρ

14 y k = A k b k + j k A j b j ρ jk + n k As a Matrix: y= RAb + n R = normalized cross correlation y = [y 1, y 2,...,y k ] T b = [b 1,b 2,...,b k ] T A = diag (A 1, A 2,...,A k ) n = n is zero mean Gaussian random vector 13

15 Asynchronous Model Cross correlation ρ kl (τ) = ρ lk (τ) = T τ τ 0 ρ kl := ρ kl (τ l τ k ) s k (t)s l (t τ) d t s k (t)s l (t + T τ) d t 14

16 Basic Asynchronous CDMA Model Assuming (without losing generality) that a user sends data packets with 2M + 1 bits. continuous time CDMA Model for K user: y(t) = K M k=1 i= M A k b k [i]s k (t it τ k ) + σn(t) t [ MT, MT + 2T] 15

17 special case: A 1 =... = A K = A y(t) = K M s 1 τ k = =... = s K = s (k 1)T K Ab k [i]s(t it (k 1)T/K) + σn(t) k=1 i= M single user channel with ISI: y(t) = j Ab[j]s(t jt/k) + σn(t) 16

18 ISI as an asynchronous CDMA channel with 4 users 17

19 Discrete Time Asynchronous CDMA Model (K-user) Output of the matched Filter: τ 1 τ 2... τ k y k = A k b k [i] + j<k A j b j [i + 1]ρ kj + j<k A j b j [i]ρ jk + j>k A j b j [i]ρ kj + j>k A j b j [i 1]ρ jk + n k [i] 18

20 Asynchronous Cross correlation Matrix Cross correlation matrix for example in the 3-user case: R [0] = 1 ρ 12 ρ 13 ρ 12 1 ρ 23 ρ 13 ρ 23 1 R [1] = 0 ρ 21 ρ ρ Time discret asynchronous model in matrix form: y [i] = R T [1] Ab [i + 1] + R [0] Ab [i] + R [1] Ab [i 1] + n [i] 19

21 Matched Filter for Single User y(t) = A b s(t) + σ n(t), t [0, T] We want to find a linear Filter, which minimize the probability of error. b = sgn( y,h ) = sgn ( T due to linearity of decision statistic Y = y,h = Ab s,h }{{} signal 0 y(t)h(t)d t + σ n, h }{{} noise ) 20

22 Some Properties: If h is finite-energy deterministic signal and n(t) is withe noise with unit spectral density, then: 1. E [ n,h ] = 0 2. E [ n, h 2] = h 2 3. If n(t) is a Gaussian process, then n, h is a Gaussian random variable. 21

23 Maximize SNR of Y Cauchy-Schwarz max h A 2 ( s, h ) 2 σ 2 h 2 ( s,h ) 2 h 2 s 2 A 2 ( s, h ) 2 σ 2 h 2 A2 σ 2 s 2 with equality if and only if h is a multiple of s! The matched filter can be implemented either as a correlator (mult. with s(t) and integration) or by a linear filter with an impulse response s(t-t) sampled at time T. 22

24 Proof: Hypothesis Testing Problem: H 1 : Y f Y 1 = N(A s, h,σ 2 h 2 ) H 1 : Y f Y 1 = N( A s,h,σ 2 h 2 ) 23

25 Probability of Error P = 1 2 = 1 2 = A s,h A s,h A s,h σ h f Y 1 (v) d v πσ h exp 1 2π exp 0 ( 1 2πσ h exp ( v2 2 ) f Y 1 (v) d v v2 2σ 2 h 2 ( v2 ) d v + ) 2σ 2 h 2 dv ( ) A s, h dv = Q σ h 24

26 Probability of Error(2) ( ) A s, h P = Q σ h The Q function is monotonic decreasing. We assume that argument of Q-function is non negative and so we can maximize the square of the argument to minimize the error probability. And that we already did before. ( ) A P C = Q σ 25

27 References Multiuser Detection, Serio Verdu Spread Spectrum Technique and its Application to DS/CDMA, Bernard H. Fleury and Alexander Kocian all 2.pdf THE END 26