1 multi-user detection gwo-ruey lee. wireless access tech. lab. ccu wireless access tech. lab. 2...
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Multi-user Detection
Gwo-Ruey Lee
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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
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Multiple Access Communication
Several transmitters share a common channel, e.g., mobile telephones transmitting to a base station ground stations communicating with a satellite, ...
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Multiple Access Communication
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
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Multiple Access Communication
Frequency Division Multiple Access (FDMA) FDMA assigns a different carrier frequency to each user so that
the resulting spectra so not overlap
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Multiple Access Communication
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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Multiple Access Communication
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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Multiple Access Communication
Code Division Multiple Access (CDMA) Direct Sequence Spread Spectrum (DS-SS)
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Multiple Access Communication
Code Division Multiple Access (CDMA) Frequency Hopping Spread Spectrum (FH-SS)
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Multiple Access Communication
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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Synchronous CDMA Model
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
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Synchronous CDMA Model
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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Single-user Matched Filter
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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Single-user Matched Filter
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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Single-user Matched Filter
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
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:
, 0y t Abs t n t t T
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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Single-user Matched Filter
The solution to
*ˆ sgnb Ay
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Single-user Matched Filter
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 FilterUser 1
Matched Filter User 2
Matched Filter User 2
Matched Filter User 3
Matched Filter User 3
Matched Filter User K
Matched Filter User K
y(t)
y1
y2
y3
yK
......
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Single-user Matched Filter
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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Single-user Matched Filter
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
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Maximum Likelihood (ML) Detectors
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Maximum Likelihood (ML) Detectors
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CCU Wireless Access Tech. Lab.
Maximum Likelihood (ML) Detectors
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Maximum Likelihood (ML) Detectors
Are they the same?
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Individual Optimum ML-Detector
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Individual Optimum ML-Detector
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Individual Optimum ML-Detector
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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
, 0y t Abs t n t t T
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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Decorrelation Detector
. . .Sync 3
. . .Sync 2
. . .Sync 1
Matched FilterUser 1
Matched FilterUser 1
Matched Filter User 2
Matched Filter User 2
Matched Filter User 3
Matched Filter User 3
Matched Filter User K
Matched Filter User K
y(t) R-1R-1
......
][1 ib
][2 ib
][3 ib
][ˆ ibK . . .Sync K
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Decorrelation Detector
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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Decorrelation Detector
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
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Decorrelation Detector
Modified decorrelating detector
. . .Sync 3
. . .Sync 2
. . .Sync 1
Matched FilterMatched Filter
Matched FilterMatched Filter
Matched FilterMatched Filter
Matched FilterMatched Filter
y(t)
......
][1 ib
][2 ib
][3 ib
][ˆ ibK . . .Sync K
1s
2s
3s
Ks
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Decorrelation Detector
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
Matched FilterUser 1
Matched FilterUser 1
Matched Filter User 2
Matched Filter User 2
Matched Filter User 3
Matched Filter User 3
Matched Filter User K
Matched Filter User K
y(t) [R+2A-2]-1[R+2A-2]-1
......
][1 ib
][2 ib
][3 ib
][ˆ ibK . . .Sync K
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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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Successive Cancellation
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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Successive Cancellation
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.
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Partial Parallel Interference Cancellation (PPIC)
Multistage PPIC detection scheme with the discrete-time equivalent complex
baseband representation for synchronous CDMA systems.
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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
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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
.
.
.
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Wireless Access Tech. Lab.
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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