A simplified MMSE-based iterative receiver for MIMO systems

Abstract

A simplified minimum mean square error (MMSE) detector is proposed for joint detection and decoding of multiple-input multiple-output
(MIMO) systems. The matrix inversion lemma and the singular value decomposition (SVD) of the channel matrix are used to simplify
the computation of the coefficient of the MMSE filter. Compared to the original MMSE detector, the proposed detector has a
much lower computational complexity with only a marginal performance loss. The proposed detector can also be applied to MIMO
systems with high order modulations.

Full text

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Yang et al. / J Zhejiang Univ Sci A 2009 10(10):1389-1394
1389




A simplified MMSE-based iterative receiver for MIMO systems

Yuan YANG†1,2, Hai-lin ZHANG1
(1State Key Lab of Integrated Services Networks, Xidian University, Xi’an 710071, China)
(2Datang Linktech Infosystem Co., Ltd., Beijing 100191, China)
†E-mail: isn_yuan@163.com
Received Mar. 27, 2009; Revision accepted July 19, 2009; Crosschecked Aug. 14, 2009; Online published Sept. 3, 2009

Abstract: A simplified minimum mean square error (MMSE) detector is proposed for joint detection and decoding of multi-
ple-input multiple-output (MIMO) systems. The matrix inversion lemma and the singular value decomposition (SVD) of the
channel matrix are used to simplify the computation of the coefficient of the MMSE filter. Compared to the original MMSE
detector, the proposed detector has a much lower computational complexity with only a marginal performance loss. The proposed
detector can also be applied to MIMO systems with high order modulations.

Key words: Multiple-input multiple-output (MIMO), Minimum mean square error (MMSE), Matrix inversion lemma, Singular
value decomposition (SVD)
doi:10.1631/jzus.A0920172 Document code: A CLC number: TN 914


INTRODUCTION

The design of an efficient and effective multiple-
input multiple-output (MIMO) detector is a chal-
lenging task in coded MIMO systems (Foschini and
Gans, 1998; Hassibi, 2000; Zhu et al., 2004; Ronto-
giannis et al., 2006). The maximum a posterior
probability (MAP) algorithm achieves optimal per-
formance. However, its high computational com-
plexity is a major obstacle to its application in real
MIMO systems. Other complexity-reduced subopti-
mal detectors have been proposed, such as the sphere
decoder (SD) (Hochwald and Brink, 2003), Markov
chain Monte Carlo method (Farhang-Boroujeny et al.,
2006) and tree search algorithm (TS) (de Jong and
Willink, 2005). These can achieve near-optimal per-
formance but their complexities are still too high for
real applications.
Among the soft-input soft-output detectors, the
simplest are those that use the minimum mean square
error (MMSE) criteria (El Gamal and Geraniotis,
2000; Sellathurai and Haykin, 2002; Choi, 2006). The
original soft-input soft-output MMSE requires the
computation of Nt (number of transmit antennas)
matrix inversions for each iteration, which is com-
putationally costly. In this letter, a simplified
soft-input soft-output MMSE algorithm is proposed.
The matrix inversion lemma and the singular value
decomposition (SVD) of the channel matrix are used
to avoid the matrix inverse calculation per iteration
and time period. The simplified method retains almost
the same performance as the original method while
greatly reducing the computational complexity.


SYSTEM MODEL

Consider a joint detection-decoding MIMO
(Tonello, 2001; Sellathurai and Haykin, 2002) wire-
less system with Nt transmit antennas and Nr receive
antennas. The information sequence x2 is encoded to a
sequence x2′ with an error correction code. Then, x2′
is interleaved to produce x1, which is then divided into
several frames, each consisting of NtMc bits. Each
frame is then mapped onto a MIMO symbol, denoted
as
t
12
[] ,
s
N
s s ... s
where si (i=1, 2, ···, Nt) is ob-
tained by mapping Mc coded bits out of the frame onto
a modulation constellation of size
number of bits per constellation symbol. Assuming
T
=
c
2 .
M
Mc is the
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Yang et al. / J Zhejiang Univ Sci A 2009 10(10):1389-1394
1390
the total transmit power is Es, the power constraint on
each antenna is E{||si||2}=Es/Nt. Let
t
T
12
[]
=
y
N
y y ... y

be the received signal vector. The MIMO channel is
assumed to be flat fading, modeled as

=
y Hs
+
n , (1)

where H is the channel matrix, and each element hi,j
of the matrix denotes the channel fading coefficient
between the jth transmit antenna and the ith receive
antenna. The elements of the channel matrix are in-
dependent, complex-valued Gaussian random vari-
ables with zero mean and unit variance. n is an Nr×1
vector whose elements are independent zero-mean
complex Gaussian random variables with variance
σ per dimension.


SOFT-INPUT SOFT-OUTPUT MMSE DETECTOR

Soft-input soft-output MMSE can provide good
performance in the high signal-to-noise ratios (SNR)
region and its computational complexity is much
lower than that of MAP detectors (Benesty et al.,
2003; Lee et al., 2006; Tomasoni et al., 2006). An
MMSE detector does not perform joint detection
among all transmit antennas; it detects signals on each
transmit antenna individually. The principle of such a
detector is to view the Nr×Nt channel as Nt interfering
Nr×1 sub-channels. It uses the information provided
by the decoder in each iteration to cancel the inter-
ference of the sub-channels.
First, we compute the a priori based symbol
mean
An MMSE detector for the symbol si on the ith
transmit antenna is a linear filter wi that provides the
symbol estimation (Sellathurai and Haykin, 2002;
Choi, 2006)

⎛
=
⎜
⎝

by minimizing the mean square error
We have

(
{
i
E
=
wyy
2
n
is and variance var{si} of si.
t
1,
H
i
=≠
⎞
⎟
⎠
−
∑
h
wy
N
jj
j j i
i
s
s ?
(2)
2
{|| || }
ii
Ess
− ?
.
)
1
H
}{}
i
E s
−
y , (3)
where
r
HH2
n ss,
{},
σ
=+
yyHR HI
iN
E
(4)
{
}
t
ss,01s
11
diag { }, , {}, ,
E
{ }, , {} .
−
+−
=
R
ii
iN
var svar s
var svar s
?
?
(5)

We can evaluate Eq.(2) as

t
r
1,
H
i
H2
n
1
s ss,
().
N
jj
jj i
≠
iiN
s
s
?
E
σ
=
−⎛⎞
⎟
⎠
=+−
⎜
⎝
∑
h
hHR HIy
(6)

Thus, the matrix inversion is required for the
estimation of each symbol during each iteration and
each time period. Finally, the soft information on each
coded bit is calculated. We assume that the estimated
signal is transmitted on an equivalent AWGN channel
with the equivalent Gaussian noise
2
~(0,),
i
i
N
η
ησ

where
2
s
(1)
i
iiE
η
σμμ=−
, (7)
H
iii
μ = w h . (8)

According to the AWGN channel assumption
and the Bayes rule, the a posterior probability of the
symbol si can be obtained:

( | ) ( )
( )
P s
?
( | )
i
p s
( | )
i
p s s
iii
i
i
p s s P s
?
≈=
y
?
, (9)
where
2
22
1
σ
( |
p s s
?
)exp
π
ii
iim
iim
s
?
a
a
ηη
μ
σ
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
−
==−
. (10)

Then the a posterior information about the bit xk
can be written as

(
(| )ln
(
k
p x
p s s
∈ =+
=
∑
D
T
i
A,
,1
T
i
A,
,1
1| )
1| )
s
( | )exp(
ii
?
/ 2)
ln.
( | )exp(
ii
p s s
?
/ 2)
ik
ik
ki
ki
i
i
sA x
i
sA x
p xs
?
?
Lxs
?
∈ =−
= +
= −
=
∑
x L
x L
(11)

Here the bit xk belongs to the bit vector xi from which
the symbol si is mapped, and (i−1)Mc≤k≤iMc−1. LA,i
denotes the vector of a priori information corre-
sponding to the bit vector xi.

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Yang et al. / J Zhejiang Univ Sci A 2009 10(10):1389-1394
1391
SIMPLIFIED MMSE DETECTOR

The soft-input soft-output MMSE requires the
computation of Nt matrix inversions for each iteration
and time period. Some simplified algorithms have
been proposed (Choi, 2006; Lee et al., 2006; Wang et
al., 2007). These algorithms use a special form of the
correlation matrix and the matrix inversion formula to
avoid the computation of a matrix inversion per
symbol and per iteration. We propose a new algorithm
that can maintain the performance gain per iteration
while greatly reducing the computational complexity
associated with matrix inversion. Compared with
other simplified algorithms, our proposed algorithm
can be applied to high order constellations.
According to Gresset (2004) and Wang et al.
(2007), the matrix inversion in Eq.(6) can be rewritten
as

H21
ss,n
()
σ+
HR HI
iN
r
r
t
r
1
H
j
H
i
2
ns
1
H
j
H
i
2
ns
1
{ }
{ }({ })
i
.
σ
σ
−
−
≠
−
=
⎛
⎜
⎝
⎛
⎜
⎝
⎞
⎟
⎠
=++
⎞
⎟
⎠
(12)
=+−+
∑
j i
∑
j
h h
j
hh
i
I
h h
j
hh
i
I
jN
N
jN
var sE
var sE var s


Denote
t
r
H
j
2
n
1
{ }
N
jvar s
=
jjN
σ
=+
∑
B h hI
and ai=B−1hi.
By applying the matrix inversion formula (Zhang,
2004), we obtain

r
H
H2
n
11
s
E
ss,
H
i
s
(
+
{ })
{ })
var s
().
1 (
σ
−−
−
+=−
−
aa
i i
HR HIB
h a
i
iN
ii
Evar s
(13)

From Eq.(13), we can see that we need to compute
only one matrix inversion in each iteration. For each
antenna, we have to compute one ‘matrix by vector’
multiplication and two ‘vector by vector’ multiplica-
tions. Let

{
ss 1
diag{ }, ,
=
R
var s
?
}
t
1
1
{}, { },
i
{}, , { } .
−
+
i
iN
var svar s
var svar s
?
(14)

Then B−1 can be rewritten as

()
r
1
1H2
n ss,
iN
σ
−
−=+
BHR HI
. (15)
Without loss of generality, we assume Nt=Nr=N. The
SVD of the channel matrix H (Liang et al., 2006) is
H=USVH, where


1
diag{ ,
δ=
S
[
12
=
Uuu
[
12
=
Vvv

Substituting the decomposed matrix H into HRssHH,
we obtain

HHH
ssss
δ
δ
⎢
⎢⎥
=
⎢
⎢ ⎥⎢
⎢⎥⎢
⎣⎦⎣
⎡
⎢
⎢
⋅⎢
⎢
⎣
⎡
⎢
⎢
⋅
⎢
⎢
⎣
⎡ ⎤⎡
⎢⎥
⎢⎥
=
⎢⎥
⎢⎥
⎢⎥
⎣⎦
v
NN
[
1 12 2

δδδ
⋅
vvv
N
?
2
, , },
δδ
]
,
]
.
N
?
uN
vN

?
?
[]
H
1
H
2
?
1
2
H
N
1
2
1
2
H
12
H
1
1 1
v
v
?
H
2
2 2
H

{ }
{ }

{}

{ }
{ }

{}
δ
δ
δ
δ
δ
δ
δ
=
⎡
⎢
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎤
⎥
⎤
⎥
⎥
⎥
⎥
⎦
⎤
⎥
⎥
⎥
⎥
⎦
HR HUSV R VSU
v
v
U
v
vvvU
U
N
N
N
N
N
var s
var s
var s
var s
var s
var s
?
?
?
?
?
]
H
. (16)
⎤
⎥
⎥
⎥
⎥
⎦
⎢
⎢
⎢
⎢
⎣
U
N

When MMSE is implemented, it can be observed
that when the detector works at the first iteration, the
variance of all symbols is one. When the SNR is high
and the iteration number increases, the variance on all
antennas approaches zero. If we assume that the
variance of each signal of the same iteration is equal,
that is var{s1}=var{s2}=···=var{sN}=c, then Eq.(16)
can be simplified as

c
c
⎢
=
⎢
⎢
⎢
⎣
2
1
2
2
HH
ss
2
N
c
δ
δ
δ
⎡
⎢
⎤
⎥
⎥
⎥
⎥
⎥
⎦
HR HUU
?
. (17)

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Yang et al. / J Zhejiang Univ Sci A 2009 10(10):1389-1394
1392
We can evaluate the calculation of B−1 as

∑
1H
j
2
j
2
n
1
1
+
N
j
j
cδσ
−
=
=
Bu u . (18)

In fact, the variance of each signal is different in most
cases. So we let

1
c
N
=

Then we can use Eq.(18) to calculate B−1. The com-
putational complexity associated with the matrix
inversion is thus greatly reduced. If the channel is
block fading (i.e., the channel is constant during L
frames of time), the algorithm can be further simpli-
fied. We use the mean over L frames:

1
c
LN
=

Then for the L frames, the calculation of matrix in-
version B−1 is performed only once. Thus, the calcu-
lation of Nt matrix inversions per iteration and time
period is avoided.
When constant module constellation is em-
ployed, we have

1
{ }
j
ij
LN
==

Compared with Choi’s algorithm (Choi, 2006),
our proposed algorithm can be applied to various
constellations and its usage is not limited to constant
modulus constellations. Choi’s algorithm assumes a
constant module transmit signal and this is the basis
for its derivation. However, there is no such assump-
tion in this paper.
In the quasi-static channel, channel varies per L
frames of time. Then, in the original soft-input
soft-output MMSE, the complexity of performing
matrix inversions per L frames of time is
Using our simplified algorithm, only one SVD of
channel matrix is performed and its complexity is
()
N
O
. Using Choi’s algorithm, an eigenvalue de-
composition (EVD) is performed per L frames and its
t
1
t
{ }
N
j
j
var s
=
∑
. (19)
t
11
t
{ }
N
L
j
ij
var s
=
=
∑∑
. (20)
tt
2
j
1111
tt
1
(1)
NN
LL
ij
c var ss
LN
==
==−
∑∑∑∑
. (21)
3
rt
().
N N L
O

3
r
complexity is also
rithm has almost the same complexity as Choi’s al-
gorithm but it can be applied to various constellations.
Table 1 shows a comparison of the complexity of the
different algorithms in a time period of L frames. Here
we consider only the computational complexity as-
sociated with the calculation of the MMSE filter wi.
Channel is assumed to be constant during L frames
and the receiver performs Niter outer iterations.

















RESULTS

In this section, we compare the BER perform-
ance of the original algorithm, the proposed algorithm,
and Choi’s algorithm. We considered a MIMO system
with four transmit antennas and four receive antennas.
Channel was assumed to be block fading and constant
during 21 frames (L=21). Each element of the channel
matrix H was an independent circularly symmetric
complex Gaussian random variable with mean zero
and variance unity. We denoted the average transmit
power on each antenna as Es/Nt and the average re-
ceived power on receive antenna as Es. The SNR per
bit on each receive antenna was defined as
Eb/N0=EsNr/(NtMcRN0), where R is the coding rate. A
half-rate (R=1/2) turbo code was used with generating
polynomial (5, 7) in octal. For turbo coding, a random
bit interleaver with 2048 length was employed. For
the receiver, 5 iterations were assumed between the
detector and the decoder and 8 iterations for the turbo
decoder. For simulations, natural mapping QPSK,
16QAM, and 64QAM were employed.
3
r
( ).
N
O
So our proposed algo-
Table 1 Complexity comparison between different MMSE
algorithms
Algorithm Multiplication
Matrix
inversion
SVDEVD
Original
MMSE
titer
+
2
r
2
rt
rt
(4
+
(
(4
5
+
4
))
⋅
LN N
N N
N N
N
O
3
rtiter
()
LN NN
O

– –
Proposed
MMSE
3
riter
2
r
2
rt
tr
(4
(
4
8
8
))
NN
N
N N
N N
+
+
O
–
3
r
()
N
O
–
Choi’s
2
ritert
3
rrt
2
r
(
412
2 ))
NN N
N N N
N
+
+
+
O
– –
3
r
()
N

O
SVD: singular value decomposition; EDV: eigenvalue decomposition

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Yang et al. / J Zhejiang Univ Sci A 2009 10(10):1389-1394
1393
Fig.1 shows the BER performance of the dif-
ferent algorithms when QPSK modulation was used.
Fig.1 shows that the three algorithms had the same
performance in the first iteration. This is because
when there is no a priori information, the three algo-
rithms use the same formulas. In the fifth iteration, the
proposed algorithm showed a slight loss of perform-
ance compared with the original algorithm.













Fig.2 shows the BER performance of the dif-
ferent algorithms using 16QAM. In the first iteration,
the three algorithms had identical performance. In the
fifth iteration, our proposed algorithm showed a slight
loss of performance compared with the original
MMSE. Choi’s algorithm showed the worst per-
formance: the BER of Choi’s algorithm at the fifth
iteration was higher than that of our proposed algo-
rithm even at the first iteration at 5~10 dB. The BER
performance of the three algorithms using 64QAM
showed a similar result (Fig.3). Choi’s algorithm is
not applicable to non-constant module modulation
and it cannot make the iterative system convergent.
Our proposed algorithm can be employed with vari-
ous modulations and has better applicability.












channel














CONCLUSION

A simplified MMSE detector is proposed for it-
erative detection/decoding MIMO systems. Com-
pared with the original algorithm, the proposed algo-
rithm can greatly reduce computational complexity
while retaining a good BER performance. Compared
with Choi’s algorithm, it has better applicability and
can be used for various constellations.

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0
24
Eb/N0 (dB)
6
8
10−4
10−3
10−2
10−1
100
BER

Proposed, 1st iter.
Proposed, 5th iter.
Choi’s, 1st iter.
Choi’s, 5th iter.
Original, 1st iter.
Original, 5th iter.
Fig.1 BER performance of QPSK over 4×4 MIMO channel
Fig.3 BER performance of 64QAM over 4×4 MIMO
channel
10 1520 25
Eb/N0 (dB)
BER
Proposed, 1st iter.
Proposed, 5th iter.
Choi’s, 1st iter.
Choi’s, 5th iter.
Original, 1st iter.
Original, 5th iter.
10−4
10−3
10−2
10−1
100
Fig.2 BER performance of 16QAM over 4×4 MIMO
510 15
Eb/N0 (dB)
BER

Proposed, 1st iter.
Proposed, 5th iter.
Choi’s, 1st iter.
Choi’s, 5th iter.
Original, 1st iter.
Original, 5th iter.
10−4
10−3
10−2
10−1
100