ISSN 1064-2269, Journal of Communications Technology and Electronics, 2017, Vol. 62, No. 11, pp. 1248–1254. © Pleiades Publishing, Inc., 2017.
THEORY AND METHODS
OF SIGNAL PROCESSING
Low-Complexity SIC-MMSE
for Joint Multiple-input Multiple-output Detection1
F. C. Kamaha Ngayahalaa, S. Ahmedb, D. M. Saqib Bhattib, *, N. Saeedc,
N. A. Kaimkhanid, and M. Rasheed b
aElectronics
and Information Engineering Chonbuk University, Republic of Korea, 54896 Jeonju
Computer and Telecommunication Engineering Dawood University of Engineering and Technology,
Islamic Republic of Pakistan, 74800 Karachi
cDepartment of Computer Science Iqra National University, Islamic Republic of Pakistan, Peshavar
d
Electrical Engineering Bahria University of Engineering and Technology, Islamic Republic of Pakistan, Karachi
*e-mail: saqibbhatti@ieee.org
b
Received May 31, 2017
Abstract⎯Iterative detection and decoding based on a soft interference cancellation–minimum mean
squared error (SIC-MMSE) scheme provides efficient performance for coded MIMO systems. The critical
computational burden for a SIC-MMSE detector in a MIMO system lies in the multiple inverse operations
of the complex matrix. In this paper, we present a new method to reduce the complexity of the SIC-MMSE
scheme based on a MIMO detection scheme that uses a single universal matrix with a non-layer-dependent
inversion process. We apply the Taylor series expansion approach and derive a simple non-layer-dependent
inverse matrix. The simulation results reveal that the utilization of the universal matrices presented in this
paper produces almost the same performance as the conventional SIC-MMSE scheme but with low computational complexity.
Keywords: multiple-input multiple-output (MIMO), minimum mean squared-error (MMSE), iterative
detection and decoding, soft detection
DOI: 10.1134/S1064226917110146
INTRODUCTION
The demand for continuous increases in data rates
and quality-of-service (QoS) in modern wireless communication systems can be met only through novel
technologies that provide higher spectral efficiency
and improved link reliability. Multiple-input multiple-output (MIMO) systems can improve the reliability and spectral efficiency of wireless links through
additional transmit and receive antennas [1–3]. Spatial multiplexing in MIMO schemes can improve their
spectral efficiency, but QoS is only guaranteed with a
good detection scheme at the receiver. Iterative detection and decoding (IDD) is one of the promising concepts that can enhance the performance of future
wireless systems. In this paper, we focus on soft interference cancellation–minimum mean squared error
(SIC-MMSE) scheme, combining a linear MMSE
receiver with successive decoding and soft cancellation for an IDD system [4–6].
The SIC-MMSE scheme has received attention
because of its good performance-complexity tradeoff
for coded MIMO systems [3]. The basic concept of
1 The article is published in the original.
IDD is feeding back the a priori log-likelihood ratio
(LLR) estimated by the channel decoder to the
MIMO detector for the next iteration. At the MIMO
detector, the a priori LLR is used to cancel interference and find the estimate of the transmitted symbol
at each transmit antenna. Then, the estimated symbol
is sent to a symbol-level maximum likelihood (ML)
detector to extract soft bit information.
There have been attempts to enhance the performance of SIC-MMSE method in the literature
[7‒13]. The critical computational burden of a SICMMSE detector in a coded MIMO system lies in the
multiple inverse operations of the complex matrix.
Therefore, several researchers have attempted to
reduce the complexity by using various matrix decomposition methods, such as singular value decomposition [10], Eigenvalue decomposition [11], and the
Cholesky method [12]. On the other hand, a method
that requires only a single MMSE filtering process was
proposed to reduce complexity by deriving an approximated non-layer-dependent matrix for inversion [13].
Despite the advances that have been made, the complexity of SIC-MMSE method remains a challenging
issue.
1248
LOW-COMPLEXITY SIC-MMSE
Data
source
a
Channel
encoder
c
Data
sink
ã
Channel
decoder
Lac
IDD scheme
π–1
Led
–
d
π
+
π
Lec
+
1249
Modulator
–
Soft bits
estimation
SIC-MMSE
Lad
Fig. 1. Block diagram of a MIMO system with a BICM transmitter and iterative receiver.
In this paper, we present a new approach that
reduces the complexity of the SIC-MMSE scheme
without any performance degradation. We derive a
new mathematical expression to reduce the complexity of SIC-MMSE scheme. Using the Taylor series
expansion approach for a complex matrix, we derive a
single universal matrix for the inversion process.
The remainder of the paper is organized as follows.
In Section 1, we introduce the basic concept of SICMMSE scheme applied to IDD for coded MIMO systems and the conventional method for using a nonlayer-dependent single matrix inversion proposed in
[13]. Section 2 presents the proposed SIC-MMSE
scheme with a universal matrix inversion. Simulation
results and comparisons are provided in Section 3, and
our conclusions follow in the last section.
1. CONVENTIONAL
SIC-MMSE DETECTION
A. System Model
We consider a MIMO system based on the bitinterleaved coded modulation (BICM) transmission
strategy with Nt transmitters and Nr receivers, as
shown in Fig. 1. During the transmission process, an
information bit vector, a, is first channel-encoded to a
sequence vector, c, with an error correction code. In
the second step, after bit-interleaving of c to d, the
coded sequence is divided into Nt independent
streams. Each stream consists of M bits, where M
denotes the number of bits per symbol. Therefore,
Nt × M information bits are transmitted in a MIMO
frame [3].
The information bits in each frame are then
mapped onto symbols for transmission, denoted by
s = [s1, s 2,..., s N t ]T , where si (i = 1, 2, 3, …, Nt) is chosen from a complex constellation, Φ , with a cardinality of Φ = 2 M , [.]T is the transpose symbol. We
assume that energy is equally distributed among all
transmit antennas, and that channel coefficients are
known at the receiver. The received signal, denoted by
y = [ y1, y2, , y N r ]T , can be represented as an Nr × Nt
complex channel matrix, H , as follows [3]:
(1)
y = Hs + n,
where n is an Nr × 1 complex noise vector.
At the receiver, the SIC-MMSE detector and soft bit
estimator calculate LLRs for the Nt × M bits using y.
Then the extrinsic information, Lec, is de-interleaved
and sent to the channel decoder as a priori information. After that, either the LLR values are used to make
decisions for information bits, or the bit-interleaved
version of the extrinsic information, Led, is fed-back to
the MIMO detector as the a priori information. The
same process is repeated for a fixed number of iterations.
B. Conventional SIC-MMSE Algorithm
MMSE detection minimizes the average meansquared error between the transmitted vector, s , and
2
its estimate, ŝ , i.e., min E ⎡⎣ s − ˆs ⎤⎦ , where ŝ is
obtained using
ˆs = Py ,
(2)
where P is an MMSE filter that can be found using
P = H H (HH H + σ 2I N r ) −1,
(3)
where (.)H is the Hermitian operator, σ2 is the variance
of complex Gaussian noise, and I N t is an Nt × Nt identity matrix.
The basic concept of SIC-MMSE detection is to
compute estimates of the transmitted symbols based
on the a priori information provided by the channel
decoder at each iteration and cancel the interference.
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During the detection process, the SIC-MMSE algorithm requires the following steps in order to estimate
the soft bit information. Initially, the soft symbols si
for i = 1, …, Nt are estimated for the transmitted symbol s i as follows [6]:
M
si = E[s i ] =
∑ z∏ 12 (1 + x
z∈Φ
i,t
tanh(La( x i,t ))),
(4)
t
where x i,t is set to be ±1 according to the complex
symbol value z in Φ , which represents a set of 2M complex symbols. The error between the transmitted symbol s i and the estimated soft symbol si is given as
ei = s i − si . The variance, Var[i], that characterizes the
reliability of the soft symbol si can be obtained as [6],
2
Var[ i] = E ⎡⎣ ei ⎤⎦
M
=
∑z ∏
2
z∈Φ
t
1 (1 + x tanh(L ( x ))) − s 2 .
i,t
a i,t
i
2
(5)
After computing the estimated soft symbols and
their variance, the second step is to cancel the interference from all the other layers. The interference cancelled ith vector is given by:
Nt
yˆ i = y − Hs i = y −
∑ h s
j j
= h i s i + n i ,
(6)
j, j ≠ i
where s i = [s1,… , si −1,0, si +1,… , sN t ]T , and
Nt
n i =
∑h e
j j
+ n.
(7)
In the next step, a linear MMSE filter is applied to
yˆ i to obtain the MMSE estimate of the ith processing
layer, w i , as follows:
(8)
where Σ i is a diagonal matrix that can be found using
Σ i = diag {Var[1],… , Var[ i − 1],
E s , Var[ i + 1],… , Var[N t ]} ,
(9)
where Es is the transmitted symbol energy. Applying
the linear MMSE filter in (8) to yˆ i provides the estimate of the i-th symbol, sˆi , as follows:
sˆi = w i yˆ i .
the variance of σ i2 given by
2
σ i = μ i − μ i .
2
(13)
Finally, the a posteriori LLR, L( x i,m ), is approximated for each symbol, sˆi . This results in
⎛ sˆi − μ i s 2 M xLa ( x i,m ) ⎞
L ( x i,m y ) ≈ min
−
⎜
⎟
2
(0)
⎟
2
s∈Φ i , m ⎜
⎝ σ i
⎠
m =1
⎛ sˆi − μ i s 2 M xLa ( x i,m ) ⎞
− min
−
⎜
⎟,
2
(1)
⎟
2
s∈Φ i , m ⎜
⎝ σ i
⎠
m =1
∑
(10)
To estimate the noise variance, we use the fact that (10)
can be equivalently expressed as
sˆi = μ i s i + ηi ,
(11)
μi = w ih i ,
(12)
where
(14)
∑
(1)
where Φ (0)
i , m and Φ i , m denote the candidate symbol vectors
corresponding to x i,m = 0 and x i,m = 1, respectively.
C. Conventional Non-Layer-Dependent
SIC-MMSE Scheme
In the conventional SIC-MMSE detection
scheme, estimating w i in (8) requires inverse operation
of the complex matrix, (HΣ i H H + σ 2I N r ) −1. Because
the value of Σ i is subject to the ith layer, the inverse
operations in (8) should be performed Nt times, which
causes a high complexity burden. Reference [13] presents an approach to reduce the computational complexity by approximating the computation of (8) in a
non-layer-dependent matrix operation. Thereby, only
one matrix inversion per MIMO iteration is required.
The method in [13] approximates (8) into the following form:
w i = h iH (HΣ i H H + σ 2I N r ) −1 ≅ a i H H ,
j, j ≠ i
w i = h iH (HΣ i H H + σ 2I N r ) −1,
and ηi is a zero-mean Gaussian random variable with
(
(15)
where a i is the ith row vector of H H HΣ + σ 2I N t
and
0
0 ⎤
⎡ Var[1] 0
⎢ 0 Var[2] 0
0 ⎥
⎥.
Σ=⎢
⎥
⎢ ⎢ 0
0
0 Var[N t ]⎥⎦
⎣
(
)
)
−1
,
(16)
−1
Because Σ and H H HΣ + σ 2I N t
are non-layerdependent matrices, the algorithm needs to estimate
only single matrix inversion as in (15). Therefore, we
refer to this method as the conventional single matrix
inversion (CSMI) method.
2. PROPOSED SIC-MMSE METHOD
WITH UNIVERSAL MATRIX INVERSION
Our investigation of the CSMI method showed that
it degraded performance compared to the conventional SIC-MMSE scheme. To reduce the perfor-
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LOW-COMPLEXITY SIC-MMSE
mance degradation, we propose a new method for
non-layer-dependent matrix inversion. The Taylor
series expansion for a complex matrix, Z , with a size of
n × n can be expressed as [14]:
(I + ε Z) −1 = I − (ε Z)1 + (ε Z)2
∞
∞
− (ε Z) + + (ε Z) =
3
(17)
∑ (−εZ) ,
k
k =0
provided that the eigenvalues λ p for Z should satisfy
the following condition
λ p < 1, for p = 1, , n .
wi =
h iH (HΣ i H H
⎛
= h iH ⎜
⎜
⎝
(18)
+ σ I Nt )
2
−1
=
h iH
j
H
j
+ (E s − Var[i])h i h iH
w i = h i (HΣ i H + σ I N t )
=
=
(C +
+
(22)
(27)
0 ≤ ε ≤ Es.
We divide ε by a given positive value, ϕ, which ensures
that
−1
)
−1
ε h i h iH
−1
H −1
ε h i h i C )C) ,
E
0 ≤ ε ≤ s < Es.
ϕ ϕ
where
ε = E s − Var[ i].
(23)
Z i = h i h iH C −1,
(24)
Let
then
wi =
H −1
hi C
(I N
r
+ εZ i )
−1
=
H −1
hi C
∞
∑ ( −εZ )
i
k
. (25)
k =0
To minimize the complexity without any serious performance degradation, we limit k to 1; then we can
rewrite (25) as
wi ≅
h iH C −1(I
− εZ i ).
(20)
⎞
+ σ 2I N r ⎟ .
⎟
⎠
(21)
Representing w i in terms of C , we have
2
−1
required. However, as MIMO iterations increase, the
reliability of the soft symbols increases, and the elements of Σ can approach zero. In that situation, the
elements of C −1 become very large values. Because the
convergence condition of Taylor series expansion
defined in (19) might not be satisfied [14], we use a
scaling approach to prevent the problem.
Because the variance, Var[ i], in (5) is always
between 0 and Es, we can derive the range of ε in (23) as
)
H
h i ((I N r
∑
∑
j =1
∑
h iH
∑
−1
⎛ Nt
⎞
C = ⎜ Var[ j]h j h Hj + σ 2I N r ⎟
⎜
⎟
⎝ j =1
⎠
= H Σ H H + σ 2I N r .
H
⎧
⎛ n
⎞
R ( z pq ) + I ( z pq ) ⎟ < 1,
⎪max ⎜
⎟
⎪1≤ p ≤ n ⎜⎝ q =1
⎠
(19)
⎨
n
⎛
⎞
⎪
⎜
R ( z pq ) + I ( z pq ) ⎟ < 1,
⎪max
⎟
1≤ q ≤ n ⎜
⎩
⎝ p =1
⎠
where z pq represents the element of the pth row and
qth column of Z , and R ( z pq ) and I ( z pq ) represent
the real and imaginary parts of z pq , respectively.
To convert w i in (8) to a non-layer-dependent
matrix inversion, we represent (8) as follows:
⎛ Nt
⎞
⎜
Var[ j]h j h Hj + E s h i h iH + σ 2I N r ⎟
⎜
⎟
⎝ j =1, j ≠i
⎠
Let us define a non-layer-dependent matrix C as
follows:
H
This condition requires that
Nt
∑ Var[ j]h h
(
1251
(26)
Using (26), estimation of w i involves C −1, which is
universal to all i; thus, only single matrix inversion is
(28)
Then, we approximate w i as
(29)
w i ≅ h iH C −1(I − ( ε ϕ) Z i ).
Compared to the CSMI method, computation
of (29) in the proposed method also requires only a
single matrix inversion process to estimate w i , but
requires a slightly more layer by layer computations. In
the next section, we demonstrate that this increased
complexity contributes to enhance the performance.
3. SIMULATION RESULTS
In this section, we compare the performance of
conventional SIC-MMSE (C-SIC-MMSE) schemes
with that of the proposed method in terms of bit error
rate (BER) for a 4 × 4 MIMO system over a Rayleigh
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(Mit, Dit) C-SIC-MMSE Proposed scheme II
ϕ = 4 ϕ = 10 ϕ = 20
(1, 8)
(2, 8)
(4, 8)
10–1
BER
10–2
10–3
10–4
10–5
5.0
5.5
6.0
6.5
7.0
7.5 8.0
Eb/N0, dB
8.5
9.0
9.5
10.0
Fig. 2. BER performance comparison with a turbo code with a code rate of 1/3, according to scaling value ϕ .
(Mit, Dit) C-SIC-MMSE Proposed
(1, 8)
(2, 8)
(4, 8)
10–1
CSMI
BER
10–2
10–3
10–4
10–5
5.0
5.5
6.0
6.5
7.0 7.5 8.0
Eb/N0, dB
8.5
9.0
9.5
10.0
Fig. 3. BER performance comparison of the proposed and conventional schemes when using a turbo code with a code rate of 1/3.
fading channel. We first compare the BER performance of the proposed method with conventional
methods by using a turbo code with 16-quadrature
amplitude modulation (QAM). For the turbo code, we
used the 3GPP-defined turbo code with an information block size of 378 bits and a code rate of 1/3; the
constraint length of each recursive systematic convolutional component code is 3.
Figure 2 compares the BER performance of the
proposed single matrix inversion method with different values of ϕ to that of the conventional SIC-MMSE
method in [6] in order to determine the optimum
value of ϕ. In the figure, Mit denotes the number of
inner iterations performed between the channel
decoder and SIC-MMSE detector and Dit denotes the
number of outer iterations performed at the turbo
decoder. The proposed method produces nearly the
same performance as the conventional SIC-MMSE
method at a ϕ value of 10. Therefore, we use that value
in the following simulations.
Figure 3 shows a BER performance comparison of
the proposed single matrix inversion method with the
conventional SIC-MMSE method and the CSMI
method in [13]. The proposed method produces
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LOW-COMPLEXITY SIC-MMSE
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(Mit, Dit) C-SIC-MMSE Proposed
scheme II
(1, 20)
(2, 20)
(4, 20)
CSMI
10–1
BER
10–2
10–3
10–4
10–5
4.0
4.5
5.0
5.5
6.0
6.5 7.0 7.5
Eb/N0, dB
8.0
8.5
9.0
9.5
Fig. 4. BER performance comparison of the proposed and conventional methods when using the (64800, 16200) LDPC code.
number of antenna increases, we can have an exponential increase in computational complexity differences between the single matrix inversion based methods and conventional SIC-MMSE scheme. Even
though the complexity of the proposed method is
slightly higher than that of the CSMI method, the
complexity orders of them are the same. In addition,
the proposed method produces better BER performance than the CSMI method, as shown in Fig. 3.
Z
almost the same performance as the conventional
SIC-MMSE method and a better performance than
the CSMI method. We note that the proposed method
produces a performance gain of about 0.5 dB around
the BER range of 10–3 to 10–5 compared to the CSMI
scheme. This performance improvement was resulted
from the adoption of slight increase in computations
in (29), compared to that in (15).
Figure 4 shows the BER performance of the 4 × 4
MIMO system with 16-QAM when using another
channel coding scheme. We used the (64800, 16200)
low-density parity check (LDPC) code with a code
rate of 1/4 and the sum-product decoding algorithm
as an iterative decoding method for the LDPC code.
All the methods produce nearly the same performance
because the LDPC code has much higher error correction capability than the turbo code used in Figs. 2
and 3.
Figure 5 compares the complexity of the SICMMSE algorithms in terms of floating point operations (FLOPS) per MIMO frame. The main difference among the three algorithms investigated in this
paper lies in the estimation of filtering matrix w i ; all
the other parts of the SIC-MMSE process have the
same complexity, though the complexity is highly
reduced as the antenna size increases.
The conventional SIC-MMSE detector requires Nt
matrix inversion processes, whereas the proposed and
the CSMI methods require only one matrix inversion
process for each MIMO detection process. As the
219
218
217
216
215
214
213
212
211
210
29
28
C-SIC-MMSE
Proposed
CSMI
2
4
6
8
N r = Nt
10
12
Fig. 5. Complexity comparison in terms of the number of
FLOPS Z.
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CONCLUSIONS
In this work, we presented a new complexity
reduced SIC-MMSE scheme for iterative MIMO
detection. The proposed method uses Taylor series
expansion and needs only a single matrix inversion per
iteration. Thus it reduces complexity by Nt times compared with the conventional SIC-MMSE scheme.
The proposed method produces almost the same BER
performance as the conventional SIC-MMSE scheme
but has lower computational complexity.
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