multi user detection
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Multiuser Detection
Mohit Garg
Under the guidance ofProf. U. B. Desai
SPANN Lab
Department of Electrical EngineeringIIT-Bombay
Group Members: Prof. S. N. Merchant
Aditya Dua, Ritesh Sood, Prateek Dayal, Umesh Nimbhorkar
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Outline ...
Recapitulating CDMA
Standard single user detector Optimum multiuser detector Non-optimal multiuser detectors
An adaptive Minimum Probability of Error based multiuser detector pro-posed by our group
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Code Division Multiple Access (CDMA)
All users transmit at the same time and across the entire frequency band
Users separated on the basis of their signature waveforms
sk(t) The signature waveforms may be Orthogonal
Non-orthogonal
Orthogonal CDMA does not give any capacity improvement over TDMA
or FDMA in a cellular system
No. of orthogonal signature waveforms is limited!
Non-orthogonal CDMA is therefore used
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The Single User Detector . . .
The simplest way: Apply a bank of matched filters, one for each user, to the
received signal
Demodulate all users independent of each other Consider the first time interval (i = 0) and the jth user,
r(t) = Ajbjsj(t) + Kk=1k=j
Akbk(i)sk(t)
+ n(t), 0 t < Tb= Signal + M AI + Noise
(6)
Multiple Access Interference (MAI) In-band interference unlike noise which is wideband
Cannot be rejected through a band-pass filter
Occurs in different forms in other systems also e.g. Multi-Carrier in-
terference in OFDM
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Issues with the Single User Detector
+ Simple to implement
+ Does not require knowledge of the channel or the user amplitudes
Multiple Access Interference (MAI) Kk=1k=j
Akbk(i)kj
Gives non-zero probability of error even with zero noise due to MAI Near-Far Effect: Strong users overwhelm the weak ones. Thus stringent
power control is necessary
The single user detector forms the core of almost all CDMA handsets currently in use
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Optimality of the Single User Detector
The matched filter is optimum only for AWGN interference The MAI term is not Gaussian in general
But can be approximated by a Gaussian random variable for large no.
of users Central Limit Theorem
Even if the MAI was Gaussian, the single user detector is still not optimum yj is not a sufficient statistic for bj
But [y1, . . . , yK]T is a sufficient statistic for [b1, . . . , bK]T
The single user detector would have been optimum Ifij = 0
i, j
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Therefore,
Multiuser Detection!
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Multiuser Detection
It is clear that the single user detector is not optimum
The optimum detector should take into account the information available inall yks to estimate the bit of a particular user This is known as Multiuser Detection and was proposed by Sergio Verdu in
early 1980s.
Any multiuser detector will utilise the information available in the MAI
term to demodulate the user and will not treat it like a noise term
Processing the interference term to extract useful information
Ideologically similar to utilising multipaths for diversity!
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Individually Optimum Multiuser Detector
Consider the simple 2-user case
r(t) = A1b1s1(t) + A2b2s2(t) + n(t), 0 t < Tb (10) The optimum estimate ofb1 will minimise the probability of error It is obtained by choosing b1 {1, +1} such that the aposteriori proba-
bility P(b1
|r(t), 0
t < Tb) is maximised.
Similarly for user 2, i.e. we need to choose b2 such that P(b2|r(t), 0 t
||y||2y
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Reducing Complexity Further . . .To further reduce the computational cost we consider:
MPOE Implementation using ISI Cancellation: Efficient MJPOE Minimizing the probability of error for each user separatelyMCPOE
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Efficient MJPOEA proposed modification to the MJPOE algorithm proposed will now be dis-
cussed which
Reduces the computation allowing pre-computation of weights Weight computation is now done on channel variation timescales rather than
symbol timescales
Improves the BER performance Using the same signal model as before, the received signal at thepth antenna
can be written as
rp(i) = SpLAb(i 1) + SpRAb(i) + np
Demodulation
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Demodulation
The receiver structure consists of a linear filter.
Augmented received vector r = [rT1 , . . . , rTP]
T (N P 1) Weights for the kth user, wk = [w
Tk1, . . . ,w
TkP]
T (N P 1) . Filter Output at the ith bit interval is
yk(i) = wH
kr(i)
=
Pp=1
wHkprp
=P
p=1 wHkpSpLAb(i 1) +P
p=1 wHkpSpRAb(i) + wHk n The Decision Rule is
bk = sgn[
(yk)]
Modification
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Modification
E(
[yk]) =
[k]
= 0
Hence, a threshold of0 is not optimum Subtracting an estimate of this term from yk, we obtain the proposed deci-
sion statistic zk
zk = yk kwhere,
k =
P
p=1 wHkpSpLAb(i 1) Since, E([zk]) = 0, we get the proposed decision rule as
bk = sgn[(zk)]
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Modification . . .
Some observations on the proposed demodulation scheme Does not involve any ISI term. Thus, we can pre-compute the weights
for MJPOE
(Cannot pre-compute for MCPOE since it requires training)
E([zk] ) = 0, which is the decision threshold. Thus, reduction inBER is expected
Simulation results support the claim
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MCPOE: Conditional Probability of Error
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MCPOE: Conditional Probability of Error
If bits +1 and -1 are equiprobable, the probability of error for the
k
th user,
conditioned on the transmitted bit vector b given by :
Pk|b =1
2P((yk) < 0|bk = 1) + 1
2P((yk) 0|bk = 1)
Density function ofyk conditioned on b given by:f(yk)|b(y|b) =
1
k
2exp
(y k)
2
22k
where:
k = k + Pp=1
wHkpSpRAb
k = wHk wk
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Conditional Probability of Error . . .
Let k = k|1 when bk = 1 and k = k|1 when bk = 1
k|1 = k + Pp=1
wHkp Ki=1
i=k
Aibi
Mm=1
aim,pgimsiL(nim) + Ak
Mm=1
akm,pgkmskL(nkm)
k|1 =
k +
P
p=1wHkp
K
i=1i=k AibiM
m=1aim,pgimsiL(nim) Ak
M
m=1 akm,pgkmskL(nkm)
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The conditional probability of error can be expressed as :
Pk|b =1
2
0
1
k
2exp
(y k|1)
2
22k
dy
+1
2
0
1
k2exp
(y k|1)2
22k dy The above expression can be simplified to obtain :
Pk|b =1
2+
1
2Q
k|1k
12
Q
k|1
k Minimize Pk|b with respect to wk
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Figure 3: Schematic of the Space-Time MCPOE Adaptive Detector
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MCPOE Adaptive Algorithm MCPOE : Minimum Conditional Probability of Error Minimizes probability of error for each user (conditioned on
transmitted bit vector) individually
Training sequence required for adaptation Gradient descent based adaptation of filter weights Ifw(i)k denotes the filter for detecting the kth user during the ith bit interval,
then :
w(i+1)k = w
(i)k Pk|bwk wk=w(i)k
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MCPOE Adaptive Algorithm . . .
Derivative of
Pk|bw.r.t w
kcan be computed using:
Pk|bwk
=12
exp
2k|1
2k
wk
k|1k
1
2exp2k|1
2k
wkk|1
k
MCPOE Adaptive Algorithm
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MCPOE Adaptive Algorithm . . .
Define ukp = k|1wkp
and vkp = k|1wkp
uk = [uHk1, . . . ,uHkP]H and vk = [vHk1, . . . ,vHkP]H
ukp = SpLAb(i 1) +K
j=1j=k
Ajbj
M
m=1 ajm,pgjms(njm)jR + AkM
m=1akm,pgkms(nkm)kR vkp =
SpLAb(i 1) + Kj=1
j=k
Ajbj
Mm=1
ajm,pgjms(njm)jR Ak
Mm=1
akm,pgkms(nkm)kR
wk
k|1k
= wk2uk k|1wkwk3/2
wk
k|1
k
=
wk2vk k|1wkwk3/2
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MCPOE Adaptive Algorithm . . .Blind Version
MCPOE requires a training sequence in adaptation mode We tried to make it blind by conditioning the probability of error
expression on the output of a matched filter rather than on the
training bits
Convergence rate not affected by the modification
Slight degradation in BER performance Similar modification works poorly for other training based adaptivedetectors such as LMS and RLS
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Figure 4: Schematic of the Blind MCPOE Adaptive Detector : Output of Matched filter Bank used as training bits
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Simulation Results
Results for both synchronous and asynchronous multipath channels (with
multiple antennas)
BER performance compared with non-adaptive MMSE Convergence rates compared with training based LMS and RLS
Flat Rayleigh faded channel assumed for simulations
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Dec. 06, 2004 Mohit Garg ISTE-STTP, TSEC First Prev Next Last Go ToFull Screen QuitFigure 5: Convergence curves for training based LMS and RLS (Multipath)
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Dec. 06, 2004 Mohit Garg ISTE-STTP, TSEC First Prev Next Last Go ToFull Screen QuitFigure 6: Convergence Curves for MJPOE (Multipath)
100
Convergence Curves at 15dB Transmit SNR
Theoretical Probability of Error: MJPOE
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0 50 100 15010
3
102
101
No. of Iterations
Prob.ofEr
ror(on
logscale)
Theoretical Probability of Error: MJPOETheoretical Probability of Error: Efficient MJOPE
Figure 7: Convergence performance comparison of MJPOE and Efficient MJPOE, K = 4, M = 3, N = 15, P = 4
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Figure 8: Schematic of the Diversity Combining MCPOE Adaptive Detector
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Figure 9: Convergence Curves for Training Based MCPOE (Multipath)
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Figure 10: Comparison of MCPOE and Non-Adaptive MMSE (Multipath)
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Figure 11: BER performance comparison of MJPOE, MCPOE and MMSE
100 Prob. of Error Vs. SNR
Simulated Recursive Least Squares (RLS)
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0 2 4 6 8 10 1210
4
103
102
101
Transmit SignaltoNoise Ratio (SNR) (in dB)
Prob.ofEr
ror(on
logscale)
Simulated Recursive Least Squares (RLS)Theoretical Prob. of Error: MJPOETheoretical Prob. of Error: Efficient MJPOESimulated Prob. of Error: Efficient MJPOE
Figure 12: BER performance comparison of MJPOE and Efficient MJPOE, K = 4, M = 3, N = 15, P = 4
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Figure 13: Effect of number of antennas on performance of MJPOE, for a fixed number of multipaths
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Conclusion
+ MPOE based adaptive multiuser detection algorithms were proposed
+ Performance better than MMSE based approaches
+ Similar to the optimum multiuser detector without the overhead of expo-nential computation at each bit interval
High computational complexity
Work has been extended to OFDM-SDMA and MC-CDMA with encourag-ing results
References
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Sergio Verdu, Multiuser Detection, Cambridge University Press, 1998.
Aditya Dua, U.B. Desai and R.K. Mallik, Minimum probability of error-based methods for adaptive multiuser detection in multipath DS-CDMA
channels, IEEE Transactions on Wireless Communications, May 2004.
R. Sood and U. B. Desai, Minimum probability of error demodulation for
multipath OFDM-SDMA systems, IEEE International Conference on Com-
munications, Jun. 2004.
P. M. Dayal, U. B. Desai and A. Mahanta, Minimum conditional probabil-ity of error detection for MC-CDMA, IEEE International Symposium on
Spread Spectrum Techniques and Applications, Aug. 30-Sept. 2 2004.
Mohit Garg, Umesh Nimbhorkar, U. B. Desai and S. N. Merchant, Efficientminimum probability of error demodulation for DS-CDMA systems, To ap-
pear in IEEE Wireless Communications and Networking Conference, Mar.
2005.
Prof. U. B. Desai
SPANN Lab
Dept. of Electrical Engineering
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p g g
Indian Institute of Technology - Bombay
Mumbai 400076
Mohit Garg
Dept. of Electrical Engineering
Indian Institute of Technology - Bombay
Mumbai 400076
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