Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
A lecture note from the university of new mexico, ece595: multiuser communications course, focusing on optimal multiuser detection (optimal mud) and equalization. The multiple-access channel signal model, conventional detector, optimality criteria, dynamic programming solution, performance and complexity of the optimum detection, approximations to the ml mud, and adaptive implementations of ml mud using the em algorithm.
Typology: Study notes
1 / 64
This page cannot be seen from the preview
Don't miss anything!
ECE595: Multiuser Communications
Dr. Sudharman K. Jayaweera
Assistant Professor
Department of Electrical and Computer Engineering
University of New Mexico
Lecture 07 - September
th
, Thursday
Fall 2007
ECE595: Multiuser Communications
Optimal Multiuser Detection (Optimal MUD) and Equalization
Outline
Optimality Criteria
Dynamic Programming Solution
Performance and Complexity of the Optimum Detection
Approximations to the ML MUD
Adaptive Implementations of ML MUD (EM Algorithm)
ECE595: Multiuser Communications
The Conventional Detector
b ˆ k ( i )
sgn
y k ( i ))
for
i
=
and
k
where
y k ( i ) = Z ∞
− ∞ r ( t )
(^) f k ( t −
(^) iT
dt
As we discussed last time, this detector is:
requires complex signalling and, moreover, is not optimalmultiuser-inference-limited, suffers from the near-far problem,
-
What is optimal? Is it practical? Is it worthwhile?
ECE595: Multiuser Communications
The Conventional Detector
ECE595: Multiuser Communications
Optimality Criteria for General Multiple-access Channel
We can have different optimality criteria based on log
r ( t ))
Minimum Prob. of Error
) Detector: Ideally,
we would like to choose ˆ
b k ( i ) , 0
i ≤
(^) 1, for
each
user that would
minimize the prob. of error
k ( σ
)
of that user
Joint ML Detector
): Choose the
combined signal ˆ
b ( i ) = [
bˆ 1 ( i ) ,... ,
bˆ K (^) ( i )]
T , 0^
i ≤
(^) 1, that would
joint probability of error if priors are equal)maximize the joint likelihood function (This will also minimize the
-
equalThis will also minimize the joint probability of error if priors are
Note that, those two detectors are not the same!
ECE595: Multiuser Communications
Why Joint ML?
individually optimum detectorJoint ML detector is easier to characterize and analyze than the
probability of error detectorperformance of the joint ML detector is very close the minimumAlso it has been shown that, unless the SNR is very low, the
ECE595: Multiuser Communications
Sufficient Statistic
Main Conclusion:
From (2) and (3), we see that the vector
y
of
matched filter outputs is a
sufficient statistic
(see ECE642) for
detecting the vector
b
of user symbols, where,
y
y ( 0 )
y ( 1 )
y ( B
(^) −
y 1 ( 0 )
y 2 ( 0 )
y K (^) ( 0 )
y 1 ( 1 )
y K (^) ( B (^) −
KB
Thus, optimal detectors are algorithms that map
y
to
b
ECE595: Multiuser Communications
Optimal (Joint ML) Multiuser Detector
ECE595: Multiuser Communications
Joint Maximum-Likelihood Detection for Synchronous DS-CDMA
Recall, the baseband received signal:
r ( t ) = K
= 1 B − 1
0 A k b k ( i ) s k ( t −
(^) iT
(^) σ
n ( t )
where
n ( t ) =
and
Z
∞
− ∞
[ s k ( t )]
2 dt^
and
s k ( t ) 6
0 only if
t ∈
Hence,
Z
T
0 [ s k ( t
2 dt^
where
s k ( t ) =
N − 1
=
0
c k ( (^) j ) ϕ
( t −
jT
c )
ECE595: Multiuser Communications
Joint Maximum-Likelihood Detection for Synchronous DS-CDMA
Joint ML detector is:
b ˆ
arg
max
b ∈{
1 , − 1 } KB
Z
∞
− ∞ m b ( t ) r ( t )
dt
Z
∞
− ∞ [ m b ( t
2 dt^
Observations enter the decisions only through this term:
Z
∞
− ∞ m b ( t ) r ( t )
dt
K
= 1 B − 1
0 A k b k ( i
Z
∞
− ∞ s k ( t −
(^) iT
r ( t ) dt
B − 1
0
K
= 1 A k b k ( i
y k ( i )
B − 1
0 ( Ab
i ))
T
y ( i )
where
is a diagonal matrix with
k , (^) k ) =
k
ECE595: Multiuser Communications
Joint Maximum-Likelihood Detection for Synchronous DS-CDMA
y i.e. (we derived this before for conventional detector) k ( i ) = Z ∞
− ∞ s k ( t −
(^) iT
r ( t ) dt
Z
∞
− ∞
(
K
= 1 B − 1
= 0 A
j^ b j^ ( n ) s j^ ( t −
(^) nT
(^) σ
n ( t ) ) s k ( t −
(^) iT
dt
K
= 1 B − 1
= 0 A
j^ b
j^ ( n ) Z
T
0
s j^ ( t −
(^) nT
s k ( t −
(^) iT
dt
Z
T
0 σ n ( t ) s k ( t −
(^) iT
dt
K
= 1 A
j^ b j^ ( i ) ρ
j^ , k
(^) n
k ( i )
where, as before,
ρ
j^ , k
Z
∞
− ∞
s j^ ( t ) s k ( t )
dt
16
ECE595: Multiuser Communications
Sufficient Statistics for Synchronous DS-CDMA
matched filters at timeThen from (8) and (9), we can write the output of the bank of
i as:
y ( i )
RAb
i ) +
(^) n
( i )
where
is the normalized cross-correlation matrix with
(^) j , (^) k ) -th
element of
being equal to
ρ
j^ , k
and
(^) σ
2 R
)
Show that
(^) σ
2 R
)
(Note that from (9)
n k ( i ) =
R
T 0 σ n ( t ) s k ( t −
(^) iT
dt
where
n ( t )
is a zero-mean white
Gaussian noise process with
n ( t ) n ( t ′
) } = δ ( t −
(^) t
′ ) )
ECE595: Multiuser Communications
Joint Maximum-Likelihood Detection for Synchronous DS-CDMA
b ˆ Using (11) in (12), the joint ML detector becomes:
arg max
b ∈{
1 , − 1 } KB
B − 1
0 b ( i ) T
Ay^
i ) (^) −
B − 1
0
K
= 1
K
= 1 A
k A
j^ b k ( i ) b
j^ ( i ) ρ k ,
(^) j ]
Hence, joint ML detector is:
b ˆ
arg
max
b ∈{
1 , − 1 } KB
B − 1
0
b ( i ) T Ay^
i ) (^) −
B − 1
0 b ( i ) T
ARAb^
i ) ]
arg
max
b ∈{
1 , − 1 } KB
B − 1
0 [ b ( i ) T
Ay^
i ) (^) −
(^) b
( i ) T Hb^
i ) ]
where un-normalized cross-correlation matrix
defined as
19
ECE595: Multiuser Communications
Joint Maximum-Likelihood Detection for Synchronous DS-CDMA
be done independently for each symbol timeFrom (13), it is clear that the joint maximum likelihood decisions can
i (only in the
synchronous case):
Hence,
b ˆ ( i )
arg
max
b ∈{
1 , − 1 } K [ b T
Ay^
i ) (^) −
b T Hb^
arg
max
b ∈{
1 , − 1 } K (^) ΩΩΩ
b )
where we have defined
b ) = 2 b T
Ay^
i ) (^) −
(^) b
T
Hb^
