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A lecture note from the university of new mexico, ece595: multiuser communications course, fall 2007. It covers adaptive linear detectors and subspace methods for multiuser communications. The derivation of the mmse solution for linear detectors and the use of subspace methods for multiuser detection.
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ECE595: Multiuser Communications
Dr. Sudharman K. Jayaweera
Assistant Professor
Department of Electrical and Computer Engineering
University of New Mexico
Lecture 13 - November
th
, Thursday
Fall 2007
ECE595: Multiuser Communications
Revisit Part II
Adaptive Multiuser Detection - Outline
Discrete-time Signal Model
Re-derivation of Linear MUD’s
Adaptive Multiuser Detection
Subspace Methods
ECE595: Multiuser Communications
Discrete-time DS-CDMA Model
so-called chip-matched filtered output modelAdaptive multiuser detection can best be formulated using the
DS-CDMA channel in whichConsider again our baseband, synchronous, non-dispersive
s k ( t ) = A k
N − 1
= 0 c k ( (^) j ) ϕ
( t −
jT
c )
for
k
In each observation interval
i , we may generate
chip-matched
filtered
observables as
z
j^ ( i ) = Z ϕ ( t −
(^) iT
jT
c ) r ( t )
dt
for
j
=
ECE595: Multiuser Communications
Discrete-time DS-CDMA Model (ctd...)
If we were to form a vector
z ( i )
of those chip-matched filter outputs,
then it is easy to see that
z ( i ) = K
= 1 A k b k ( i
s k
(^) η ηη
( i )
where
z ( i )
is an
-vector,
ηηη
2
I N
)
and the
j -th element
of the
-vector
s k
is
s k , (^) j
=
1
√
N (^) c k ( (^) j )
for
c k ( (^) j ) =
Note that the filtered noise
ηηη
( i )
vector has iid elements (unlike in the
matched filtered output model)
One can show that
z ( i )
is also a sufficient statistic for detecting
b
( i )
ECE595: Multiuser Communications
Another View of Linear Detectors - In Two Steps
θθθˆ ( i ) = W T
z^ ( i )
where
w
1
w
2
w
K
bˆ k ( i )
sgn
θˆ k ( i ) ) =
sgn
w
k T z ( i ) )
observation model (2)We can re-derive the linear detectors considered earlier based on the
ECE595: Multiuser Communications
Matched Filter
Note that, this agrees with our earlier matched filter detector:
θθθˆ ( i ) = W T
z^ ( i ) =
T z^ ( i ) =
T
( S θθθ ( i ) +
(^) ηηη
( i ))
(from (2))
T S^ θθθ ( i ) +
T η^ ηη ( i ) =
θθθ ( i ) +
(^) n
( i )
RAb
i ) +
(^) n
( i ) =
y ( i )
where last step follows by noting that
T S^
and
θθθ
=
Ab
E Finally, it easy to see that
{ nn
T }^
=
T ηη^ η ( S T η^ ηη
) T }^
=
T E^
{ ηηηη
ηη } S
=
T (^ N 0
2
I N (^) ) S
=
N 0
2
R
, as
required
ECE595: Multiuser Communications
Decorrelator (Least squares) (ctd...)
Let us choose
θθθ¯
so that,
T z^
T S^ ) θθθ¯
θθθ¯
T S^ ) − 1 S^ T z^
Then,
θˆ ( i )
arg min
θθθ ∈ R K ( z T
z^ (^) + (
θθθ (^) −
θθθ¯ ) T S^ T S^ ( θθθ (^) −
θθθ¯ ) (^) −
θθθ¯ T S^ T S^ θθθ¯ )
It is easily seen that the minimum is achieved when
θθθ
θθθ¯
. Hence,
θθθˆ ( i ) = ( S T
− 1 S^ T z^
Comparing (3) and (4) we see that the decorrelator for (2) is given by,
T
S^
) − 1
ECE595: Multiuser Communications
MMSE or MMOE
MMSE estimator
θθθˆ ( i )
of
θθθ ( i )
in (2) is given by
argmin
W
θθθ ( i ) (^) −
T z^ ( i ) ∥∥ 2 }
argmin
W
tr
(^) ( θθθ ( i ) (^) −
T z^ ( i ) ) (
θθθ ( i ) (^) −
T z^ ( i ) ) T (^) }
argmin
W
tr ( E
{ θθθ ( i ) θθθ ( i ) T
−
(^) W
T z^ ( i ) θθθ T (^ i ) (^) −
(^) θ θθ ( i ) z T (^ i ) W
T z^ ( i ) z T (^ i ) W
argmin
W
tr
(
A
2 −
(^) W
T SAA^
T
−
(^) AA
T S T W^
T
(
SAA
T S^ T
0
Let us define a matrix
such that,
2
− 1
N × N
2
11
ECE595: Multiuser Communications
MMSE or MMOE
W applying the matrix inversion lemma:We can re-write the above MMSE solution in a different form by
0
2
0
T
(
0
− 1
0
0
− 2
(^) S
T S^ ) − 1 S T ( N 0
0
− 1 SA
2 −
0
− 2 +^
(^) S
T S^ )
− 1 S T SA^
2 ]
13
ECE595: Multiuser Communications
MMSE or MMOE
W Hence,
0
− 2 +^
T S^ )
− 1 (^) [(
0
− 2
(^) S
T S^ )
2 −
(^) S
T SA^
2 ]
0
− 2
(^) S
T S^ ) − 1 [ N 0
− 2 +^
(^) S
T S^ )
−
Thus, the MMSE solution is
0
Or from (5):
− 1 SA
2
14
ECE595: Multiuser Communications
LMS-adaptive MMSE Multiuser Detector for DS-CDMA
The basic LMS algorithm is (for
x k , (^) j ( i (^) +
x k , (^) j ( i ) (^) −
2 μ
x k , (^) j
where the cost function
is defined as
e k 2 (^) ( i ) =
θ k ( i ) (^) −
(^) w
k T z^ ( i ) ) 2
The required gradient of the cost function is:
x k , (^) j
= − 2 e k ( i ) ∂ [ ( s k +
(^) x
k ) T z ( i ) ]
x k , (^) j
= − 2 e ( i ) z
j^ ( i )
Hence, the LMS adaptive algorithm is
x k , (^) j ( i (^) +
x k , (^) j ( i ) +
(^) μe
k ( i ) z j^ ( i )
ECE595: Multiuser Communications
LMS-adaptive MMSE Multiuser Detector for DS-CDMA
In vector form we can write:
x k ( i (^) +
) = x k ( i
(^) μe
k ( i ) z ( i )
detector solution (also see the text for details of convergence).Clearly, the LMS algorithm in (9) converges to the MMSE multiuser
requires training symbolsAbove implementation of the adaptive MMSE multiuser detector
ECE595: Multiuser Communications
LMS-based Blind Adaptive MMSE Multiuser Detection
Often, we are interested in
blind adaptation
, in which we know only
one received waveform
f k ( t ) , or one transmitted waveform
s k ( t )
In order to do that, again write
w k = s k +
(^) x
k
where
x k T s k
0 and
adjust
x k
output signal power:optimization problem of minimizing the MOE subject to a fixedRecall that MMSE estimator is the solution to the constrained
w min ∈ R
w
T z^ ( i )
| 2 }
subject to
w
T s^ k
The above formulation renders itself for blind adaptation
The cost function to be minimized is:
( x k ) = E {
z T (^ i ) (
s k
(^) x
k ( i (^) −
2 }
ECE595: Multiuser Communications
LMS-based Blind Adaptive MMSE Multiuser Detection (ctd...)
The gradient of the cost function is
( x k ) = 2 E
z T (^ i ) (
s k
(^) x
k ( i (^) −
(^) z ( i ) }
The (noisy) stochastic gradient is
( x k ) = 2
z T (^ i ) (
s k
(^) x
k ( i (^) −
z^ ( i )
Observe that the above gradient is in the direction of
z ( i )
orthogonal toWe want to adapt the gradient along the component of the gradient
s k
But the component of
z ( i )
in the direction orthogonal to
s k
is,
z ( i ) (^) −
z T (^ i ) s k ) s^ k
