1 multi-user detection gwo-ruey lee. wireless access tech. lab. ccu wireless access tech. lab. 2...

1

Multi-user Detection

Gwo-Ruey Lee

2

Wireless Access Tech. Lab.

CCU Wireless Access Tech. Lab.

Outlines

Multiple Access Communication Synchronous CDMA Model/ Asynchronous CDMA Model Single-user Matched Filter Optimum Multi-user Detection Decorrelating Detector Non-Decorrelating Linear Multi-user Detection Decision-Driven Multi-user Detection

3



Multiple Access Communication

Several transmitters share a common channel, e.g., mobile telephones transmitting to a base station ground stations communicating with a satellite, ...

4




The receiver obtains the superposition of the signals sent by the active transmitters

Receiver

User 1

User 2

User 3

User 4

User K Noise

5




Frequency Division Multiple Access (FDMA) FDMA assigns a different carrier frequency to each user so that

the resulting spectra so not overlap

6




Time Division Multiple Access (TDMA) In TDMA, time is partitioned into slots assigned to each

incoming digital stream in round-robin fashion. Synchronization is required.

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Code Division Multiple Access (CDMA) Users are assigned different signature waveforms.

Each transmitter send its data stream by modulating its own signature waveform as in a single-user digital communication system.

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Code Division Multiple Access (CDMA) Direct Sequence Spread Spectrum (DS-SS)

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Code Division Multiple Access (CDMA) Frequency Hopping Spread Spectrum (FH-SS)

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Near-far problem: Any interferer that is sufficiently powerful receiver

causes arbitrarily high performance degradation.

The objective of multi-user detection is: the design and analysis of digital demodulation in the

presence of multi-access interference (MAI).

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Synchronous CDMA Model

Basic Synchronous CDMA Model

where is the inverse of the data rate. is the deterministic signature waveform assigned to the

k-th user. It is normalized such that

is the received amplitude of the k-th user's signal. is the bit transimitted by the k-th user. is the white Gaussian noise, which is uncorrelated with

the transmitted signals, and has unit power spectral density.

1

, 0,K

k k kk

y t A b s t n t t T

T ks t

2 2

01

T

k ks t s t dt

kA

n t 1, 1kb

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b1

A1s1(t)

b2

A2s2(t)

bK

AKsK(t)

...

y(t)

n(t)

1

( ) ( ) ( ), [0, ].K

k k kk

y t A b s t n t t T

13




The crosscorrelation of two signature waveforms, and , is

By Cauchy-Schwarz inequality, the crosscorrelation satisfies

The cross correlation matrix, defined by

has diagonal elements equal to 1 [see (29) and (30)], and is symmetric nonnegative definite, i.e.,

is t js t

0

,T

ij i j i js t s t s t s t dt

, 1ij i j i js t s t s t s t

ij

, , 1, 2,..., ijR i j K

2

1

0K

Tk k

k

a Ra a s t

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Asynchronous CDMA Model

Basic Asynchronous CDMA Model

where are the time offsets that correspond to users

One special case happens when then asynchronous model reduces to synchronous model

Another special case happens when and (a single user undergoes multipaths), it becomes

1 1

, - 2K M

k k k kk i

y t A b i s t iT n t MT t MT T

1 20 K T 1,2, , K

1 20 ,K

1 2 ,KA A A A 1 2 ,Ks k s k s k s t

1k

K T

K

1 1

1

1, - 2

,

K M

kk i

MK K

j MK

K Ty t Ab i s t iT n t MT t MT T

K

TAb j s t j n t

K

1 kb iK k b i

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CDMA Model

Direct-sequence spread spectrum Direct-sequence waveforms

where is the chip waveform that satisfies

and

N is the number of chips per bit N,

1 1

1 11 1i

c i c

N Nc

T c k T c ci i

s t P t i T d P t iT TN N

cTP t

1, 0

0, 1,2,...p c

nR nT

n

c cP T TR P t P t dt

1 2, , , , 0,1 , , is the binary sequence (code)N ic c c c i 1, 1 , 1,2, , and 1,2, ,

ikd k K i N

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Single-user Matched Filter

Consider the synchronous CDMA model, where only a single user exist:

The signal listed above is passed through a linear filter, the output of which is then sampled at T

, 0y t Abs t n t t T

, , , 0y t Ab s t h t n t h t t T

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One problem is: Find the linear filter h(t) that maximize the signal-to-noise ratio at the filter output Y , i.e.,

By Cauchy-Schwarz inequality, we have

22

2

222

, ,max

,h t

E Ab s t h t s t h tAJ

h tE n t h t

2 2 2 2,s t h t s t h t h t

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The objective function satisfies , where the equality holds when

Notice that in this derivation, we did not invoke the fact that noise is Gaussian.

Note that is a Gaussian r.v. with zero-mean and unit variance.

2

2

AJ

, 0h t s t t T

,n t s t

, ,

,

Y Ab s t s t n t s t

Ab n t s t

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The probability of error, in determining from , is

b Y

2 2

2 2

0

| 1 | 10

2 2

1 1 1 1

1 1

2 2

1 1 1 1

2 22 2

Y b Y b

v vA

A

AQ

P error P b P error b P b P error b

f v dv f v dv

Ae dv e dv Q

A

A

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Single-user Matched Filter in Rayleigh Fading single user model

Assuming that A and s(t) are given, we want to find the estimate of b, , that minimizes

The first and second terms on the RHS of above equation are irrelevant to b, and we can write the minimization problem as a maximization problem:


ˆ 1, 1b

2 2 2 *

0 0 0 0

T T T Ty t Abs t dt y t dt As t dt y t Abs t dt

*

* *

01, 1 1, 1max max

T

b b

y

Ab y t s t dt b Ay

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The solution to

*ˆ sgnb Ay

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Discrete-time Synchronous Models Multi-user detection commonly have a front-end,

whose objective is to obtain a discrete-time process from the received continuous-time waveform y(t).

Matched filter outputs

. . .Sync K

. . .Sync 3

. . .Sync 2

. . .Sync 1

Matched FilterUser 1


Matched Filter User 2




Matched Filter User K


y(t)

y1

y2

y3

yK

......

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In the synchronous case, the outputs of the bank of matched filters are

0

01

0 01

, 1, 2,...,

, 1, 2, ,

jk k

T

k k

KT

j j j kj

K T T

j j j k kj

n

k k j j jk kj k

y y t s t dt k K

A b s t n t s t dt

A b s t s t dt n t s t dt

A b A b n k K

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The vector form of above equation is

where

and n is a zero-mean Gaussian random vector with covariance matrix equal to , i.e.,

y RAb n

1 2

1 2

1 2

, , ,

, , ,

, ,

T

K

jk

K

T

K

y y y y

R

A diag A A A

b b b b

2TE nn R

0 0

2 2

0 0

j k

T T

j k j k

n n

T T

j k jk

E n n E n s d n s d

E n n s s d d

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Maximum A Posteriori (MAP) and Maximum Likelihood (ML) Detectors

The MAP-detector chooses the hypothesis that maximizes the a posteriori probability, and achieves the minimum probability of error.

The ML-detector chooses the hypothesis that maximizes the likelihood function, it achieves the minimum probability of error, when the hypotheses are equally probable (P0 = P1).

1

1 0

0

| 1 | 0| |

H

H r H r

H

P H r P H r

1

1 0

0

| 1 | 0| |

H

r H r H

H

P r H P r H

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Maximum Likelihood (ML) Detectors

27




28




29




30




Are they the same?

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Individual Optimum ML-Detector

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33




34



Joint and Individual Optimum ML-Detector for the K-User Scenario

Recall the discrete-time synchronous CDMA model that

The joint optimum ML-detector is the solution to


11/ 2 2/ 2 2

1

1 1max | exp

22

max

max 2

T

Kb

T

b

T T

b

P y b y RAb R y RAbR

y RAb R y RAb

b Ay b ARAb

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Joint and Individual Optimum ML-Detector for the K-User Scenario

The maximization problem is a combinatorial optimization one, which means that the set of possible arguments comprises a finite set.

Combinatorial optimization problems can always be solved by exhaustive search, i.e., we evaluate the objective function at all possible arguments, and select our detected value to be the argument that produces the maximum.

Joint optimum decisions would be preferable to minimum bit-error-rate decisions due to their complexity.

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Decorrelation Detector

Recall that the output vector of the bank of K matched filters is

Assume that R is invertible.

Premultiplying by give

In the absence of noise n, the k-th component of

is The decorrelating detector detects through

y RAb n

1R

1 1R y Ab R n

1R yk kA b

kb

1sgnk

R y

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. . .Sync 3

. . .Sync 2

. . .Sync 1









y(t) R-1R-1

......

][1 ib

][2 ib

][3 ib

][ˆ ibK . . .Sync K

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Note that the decorrelating detector is influenced by additive noise, and not by other interferers ( ).

Two features of the decorrelating detectors are 1. It does not need to know the received amplitudes (

). 2. Detection of each user can be implemented

independently.

Note that

, jb j k

, iA i

1 1 1

1 1

1

1

,

, ,

K K

j jk kj kjj j

K

j kkjj

R y R y R y t s t

y t R s t y t s t

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From the fact that

We know that is orthogonal to any linear combination of .

If is linearly independent, we can find from for all k, and can have the modified decorrelating detector.

1

1

1 1

1 1

1

1

, ,

,

, ,

K

k l j ljkj

K K

j l ljjk jkj j

K

lklkj

s t s t R s t s t

R s t s t R R

RR l k

ks t j j k

s t ks t

js t

, ky t s t

40




Modified decorrelating detector

. . .Sync 3

. . .Sync 2

. . .Sync 1

Matched FilterMatched Filter




y(t)

......

][1 ib

][2 ib

][3 ib


1s

2s

3s

Ks

41




In the two user scenario,

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Decorrelating Detector and ML-Criterion

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Non-Decorrelating Detector - LMMSE

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Non-Decorrelating Detector - LMMSE

. . .Sync 3

. . .Sync 2

. . .Sync 1









y(t) [R+2A-2]-1[R+2A-2]-1

......

][1 ib

][2 ib

][3 ib


45



Properties of the LMMSE Detector

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LMMSE Detector for the Bank of Orthonormalized Filters

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LMMSE Detector Maximizes Signal to Interference Ratio

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Minimum Output Energy Detector

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Successive Cancellation

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1 1

2 2 2 1

1 2 2

1 2 2

1 1 2 2 2 1

ˆ ˆsgn( , )

ˆ sgn ( ) ( ),

ˆ sgn( )

sgn( sgn( ))

ˆ sgn( ( )) , )

b y s

y t A b s t s

y A b

y A y

Ab A b b n s

2 2

ˆ sgnb y

2 2 2

1 1 1 2 2 2 2

ˆˆ( ) ( ) ( )

ˆ

y t y t A b s t

Ab s t A b b s t n t

1 1 2ˆ sgn( )b y y

2

1( ) ( ) ( )k k kk

y t A b s t n t

k k k ky A b n

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Equivalent implementation of successive cancellation for two synchronization users

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Successive Interference Cancellation (SIC)

It requires knowledge of the received amplitude. User weaker than the user (or users) of interest are

neglected. In contrast to the (nonadaptive) multi-user linear

detectors, successive cancellation require no arithmetic computations with the crosscorrelation beyond their product with the received amplitudes.

The time complexity per bit is linear in the number of user

It applies not only to the basic CDMA model (where signals

are linearly modulated) but to any multiple-access channel where the receiver observes the additive superposition of the transmitted signal.

The demodulation delay in successive cancellation grows linearly with the number of user.

53



Partial Parallel Interference Cancellation (PPIC)

Multistage PPIC detection scheme with the discrete-time equivalent complex

baseband representation for synchronous CDMA systems.

54



Partial Parallel Interference Cancellation (PPIC)

Discrete-time signal r(m) at the chip rate

Decision statistic of the ith bit of the conventional receiver for the kth user

),()(

),()(

)()()(

1

1

1

memabA

mmbA

mmAm

kjK

kkkk

K

kkkk

K

kkk

n

na

nsr

iN

Nimkik mm

NZ

1)1(

*, )()(

1ar

55



Adaptive NLMS-PPIC

.

.

.

)1(,1 ib )(ˆ )1(

1 ms

y(m-N)Delay

N

a1(m)exp(-j1)

aK(m)exp(-jK)

1A a1(m)exp(j1)

KA aK(m)exp(jK)

)1(,

ˆiKb

.

.

.

.

.

.

.

.

.

wK(1)

w1(1)

)(ˆ )1( mKs

Stage 1 Stage 2

)2(,1 ib

)2(,

ˆiKb

y(m)

Delay

N

.

.

.

…y1

(1)(m-2N)

yK(1)(m-2N)

aK(m)exp(-jK)

a1(m)exp(-j1)

Normalizeed LMS

algorithm

K

k 2

)(

Delay

N

Delay

N

…

iN

NiT 1)1(

)(1

1

1

)(K

k

iN

NiT 1)1(

)(1

iN

NiT 1)1(

)(1

iN

NiT 1)1(

)(1

.

.

.

56



Multi-user Detection

Readings SERGIO VERDU, Multi-user Detection, CAMBRIDGE,

1998. Chapter 2 – 2..1, 2.2, 2.9 Chapter 4 – 4.1 Chapter 5 – 5.1 Chapter 6 – 6.1, 6.2 Chapter 7 – 7.1

1 multi-user detection gwo-ruey lee. wireless access tech. lab. ccu wireless access tech. lab. 2...

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