Reduced-Complexity MIMO Detection Using
Adaptive Set Partitioning
Kuei-Chiang Lai
Department of Electrical Engineering National Cheng Kung University
Taiwan, R.O.C. Email: kclai@mail.ncku.edu.tw Abstract—In this paper, a reduced-complexity multiple-input
multiple-output (MIMO) detector for the Bell Labs Layered Space-Time Architecture (BLAST) is described. It incorporates the idea of set partitioning in the tree search to reduce the number of multiplications and comparisons. An algorithm that adapts the number of partitions based on the channel conditions is also proposed to achieve a better tradeoff between complexity and performance. Simulation results are presented to demonstrate the efficacy of the proposed algorithm under the assumptions of perfect channel and noise power estimates.
I. INTRODUCTION
Multiple-input multiple-output (MIMO) techniques are con-sidered by many as the most effective and promising tech-niques toward achieving high data rates and reliable link qual-ity. The Bell Labs Layered Space-Time Architecture (BLAST) [1] is an important MIMO transmission scheme that can boost the system capacity. In practice, however, this capacity increase is realizable only when the receiver employs a more sophisticated detection algorithm at the cost of a much greater computational complexity, which is not desirable for mobile devices due to concerns of power consumption and the form factor.
Although the complexity of a linear detector (e.g., a zero-forcing (ZF) or minimum mean-squared error (MMSE) detec-tor) and its advanced version that combines with successive interference cancelers (SIC) [2] is relatively low among the existent MIMO detectors, the performance lags significantly behind that of the (optimum) maximum-likelihood (ML) de-tector. Hence, there has been a lot of research effort going into investigating efficient, realizable algorithms that retain much of the performance of the ML detector. The sphere decoding algorithms [3] reduce the complexity of the ML detector by limiting the search of the most likely transmitted symbols to within a sphere of a specified radius that centers at the received signal vector. However, determination of a suitable radius that achieves a good tradeoff between complexity/latency and error-rate performance in fading channels is still an open problem.
A limited tree search algorithm referred to as QRM-MLD1
is proposed in [6]. The QR decomposition is used to impose a tree structure in the processed received signal vector, while the
1This was originally coined as QRD-M. In this paper, we will refer to
QRD-M and its variants (such as [4] and [5]) as QRM-MLD.
M-algorithm [7] is used to perform the limited, breadth-first tree search. An algorithm that adapts the value of M for each transmitted symbol based on the respective channel condition is proposed in [4] for the purpose of further complexity reduction. It is demonstrated that near-ML performance is achievable with the required number of multiplication opera-tions roughly in the same order of that of the MMSE receiver. In this paper, we propose an algorithm to further lower the complexity of [4] by using the idea of set partitioning [8]. Similar work toward the same goal is proposed in [5] that uses a technique termed quadrant detection. However, it works only for QAM constellations, but not for constellations such as M-ary PSK. In contrast, the method of set partitioning works as long as the constellation exhibits sufficient symmetry. In addition, this paper addresses an important aspect concerning complexity: the number of comparison operations required by the M-algorithm.
II. SIGNALMODEL
Consider a point-to-point communications link with ntand
nr antennas at the transmit (Tx) and receive (Rx) sites,
respectively. This is referred to as an nt-by-nrMIMO system.
The received signal samples at the Rx antennas at any given time instant can be stacked into a vector y whose i-th element yi is associated with the i-th Rx antenna. Assuming a flat
fading channel between any pair of Tx and Rx antennas, we can model the received signal vector y as
y= Hx + n (1)
where H is the nr-by-ntchannel matrix, x is the transmitted
symbol vector at that time instant, and n is the noise vector. Note that, with a flat fading channel model, the (i, j)-th element of channel matrix H is the gain of the scalar channel between the j-th Tx antenna and the i-th Rx antenna. Each entry in H is assumed to be an independent and identically distributed (i.i.d.) complex Gaussian random variable (r.v.) with zero mean and unit variance. The assumption of a frequency-flat channel model is a reasonable one for each sub-carrier in an orthogonal frequency division multiplexing (OFDM) system. The i-th entry of x, denoted as xi, is a
1 1= x 1 1=− x 1 1= x 1 1=− x 1 1= x 1 1=− x 1 1= x 1 1=− x 1 2= x 1 2=− x 1 2= x 1 2=− x 1 3= x 1 3=− x 3 z z2 z1
Fig. 1. Tree diagram of a 3-by-3 MIMO system with BPSK modulation
(i.e., c0= 1and c1=−1).
average signal power transmitted from each Tx antenna is normalized to 1/ntsuch that the total transmitted power across
Tx antennas is unity. It is assumed that the noise at each Rx antenna is an i.i.d. complex Gaussian r.v. with zero mean and variance σ2
n.
III. REVIEW OFQRM-MLD ALGORITHM
A. ML Detection via QR Decomposition and Tree Search Consider the QR decomposition of the channel matrix H, H = QR where Q is an nr-by-nt matrix with orthonormal
columns, and R is an upper-triangular matrix of size nt-by-nt.
and with real diagonal elements. Left multiplying the received signal vector y by QH, we obtain
z∆= QHy= Rx + w (2)
where w= Q∆ Hncan be shown to have the same distribution as n, and the superscript H denotes the complex conjugation
operator. It can be shown that the ML solution is ˆ
xML= arg min
x∈Cntkz − Rxk 2
(3) where k·k is the Euclidean norm. Denoting the (i, j)-th ele-ment of R as Ri,j, the squared Euclidean distance between z
and Rx, denoted by ξ(x), can be expressed as ξ(x) = nt X i=1 znt−i+1− nt X m=nt−i+1 Rnt−i+1,mxm 2 . (4)
With the following definitions and mappings, ˆxML can
be obtained via searching for the path with the minimum
Euclidean distance in a tree diagram. Figure 1 shows the corresponding tree diagram for a 3-by-3 MIMO system with BPSK modulation. Observe that the depth of the tree is nt, i.e.,
the tree consists of (nt+ 1) levels with level 0 corresponding
to the root node. Extending out of each node are |C| branches; the end of each branch is then attached with a child node. The entire tree is formed by iterating from level 0 (i.e., the root node) to level nt− 1; in each iteration, branch extension is
done for all the nodes at the tip of the current tree.
For each node at level i (i ∈ {0, · · · , nt− 1}), the j-th
outgoing branch (with the index j ∈ {0, · · · , |C| − 1} labeled from top to bottom) represents the hypothesized transmitted symbol xnt−i = cj where cj is the j-th point of C (refer
to Figure 1). Traversing from the root node to a given node at level i (i ∈ {1, · · · , nt}), the visited branches constitute
a path which determines a particular symbol vector xi by
concatenating along the way the symbol associated with each constituent branch. Specifically,
xi= [x∆ nt, · · · , xnt−i+1] (5)
where the subscript of xidenotes that the ending node of this
path is at level i, and the underline notation signifies a row vector. For notational convenience, we say that the state of the ending node of this path is xi. At level nt of the tree,
hence, there are a total of |C|nt n
t-branch paths, representing
all the possible transmitted symbol vectors x. For example, the path highlighted in Figure 1 corresponds to the sequence xnt = [x3= −1, x2= 1, x1= 1].
The squared Euclidean distance ξ(x) defined in (4) is associated with each nt-branch path in the tree diagram based
on the following rules. The j-th branch of a path xi (i.e., the
one connecting between levels j − 1 and j) is associated with a branch metric (BM) λj(x(1:j)i ) ∆ = znt−j+1− nt X m=nt−j+1 Rnt−j+1,mxm 2 (6) where x(1:j) i ∆
= [xnt, · · · , xnt−j+1] is the subpath of xi,
con-sisting of the first j branches of xi. Note that x (1:i) i = xi.
Here the inclusion of x(1:j)
i in the notation of λj stresses
the dependence on the hypothesized modulation symbols xnt, · · · , xnt−j+1only, due to the upper triangular structure of
R. Furthermore, each path xiis associated with a path metric (PM) ξi(xi), defined as the sum of the BMs of its constituent
branches: ξi(xi) ∆ = i X k=1 λk(x(1:k)i ) = ξi−1(x(1:i−1)i ) + λi(xi). (7)
Comparing (4) and (7), it follows that ξ(x) = ξnt(xnt). Thus,
it is clear that, at level nt, the path with the minimum path
0 A 0 B B1 0 C C2 C1 C3 0 E E8 E4 E12 E2 E10 E6 E14 E1 E9 E5 E13 E3 E11 E7 E15 0 D D4 D2 D6 D1 D5 D3 D7
Fig. 2. Ungerboeck’s set partitioning of the 16-QAM constellation into 1 (i.e.,
{A0}), 2 (i.e., {B0,B1}), 4 (i.e., {C0,· · · , C3}), 8 (i.e., {D0,· · · , D7}),
and 16 (i.e., {E0,· · · , E15}) subsets.
B. QRM-MLD
The QRM-MLD algorithm [6] uses the M-algorithm to reduce the complexity of the ML tree search. At the i-th level of the tree diagram, only M < |C|i paths – corresponding to
the M smallest PMs (i.e., ξi(xi)) – are retained for branch
extension to the next level. The retained paths are referred to as the survivor paths; the rest are pruned.
To mitigate the error propagation effects due to path prun-ing, the streams are detected in the decreasing order of the corresponding channel gain khkk2 where hk denotes the
k-th column of H. Thus, k-the following notation is adopted throughout the rest of the paper for algorithms related to QRM-MLD. Columns of H are re-ordered and re-labeled (and thus so do the elements of x, y, and z) such that kh1k ≤ · · · ≤ khntk. Also, xi, yi and zi now denote the
i-th entry of the vector x, y, and z, respectively, that has been re-ordered and re-labeled accordingly.
IV. PROPOSEDALGORITHM
A. Motivation
The complexity of the QRM-MLD algorithm lies primarily in (i) the multiplication operations, and (ii) the comparison operations needed to determine the survivor paths. To further reduce the complexity of QRM-MLD in these two areas, we propose to retain S out of the |C| branches extending out of each node (with S < |C|). Those paths (i.e., hypothesized transmitted symbols) consisting of the branches not retained are removed from further consideration, which essentially limits the tree search. To achieve the optimal tradeoff between complexity reduction and the risk of dropping the correct hypothesis, the S outgoing branches to be retained for each node ought to be the S most likely branches. Below, we determine the corresponding likelihood functions.
Consider the |C| branches leaving an (i − 1)-th level node whose state is denoted as xi−1. Here, i ∈ {1, · · · , nt} for an
nt-deep tree. Recall that the j-th branch represents xnt−i+1=
cj. Appending each of the |C| branches separately to xi−1
results in |C| extended paths into level i, each represented
by xi = [xi−1, xnt−i+1 = cj]. Denoting the corresponding
received signal samples as zi ∆
= [znt, · · · , znt−i+1], it can be
shown that the log likelihood function of the extended path xi
can be expressed as
log P (zi|xi, R) = log P (z (1:i−1)
i |xi, R) +
log P (znt−i+1|xi, R)
= log P (z(1:i−1)i |xi−1, R) +
log P (znt−i+1|xi, R)
= αξi−1(xi−1) + βλi(xi) (8)
where α and β are negative constants, and the assumptions on the noise distribution have been used to arrive at the first and third equalities. Due to the lower-triangular structure of R, we have used log P (z(1:i−1)
i |xi, R) = log P (z (1:i−1)
i |xi−1, R) to
arrive at the second equality.
Note that ξi−1(xi−1) is common for these |C| branches
since they stem out of the same node. It follows from (8) that picking out the S most likely extended paths that branch out from a given node is equivalent to selecting the S most likely outgoing branches of that node, which in turn is equivalent to choosing those corresponding to the S smallest BMs. For each node, therefore, computing all of the |C| BMs followed by sorting is required to determine which outgoing branches are to be retained. This, however, does not achieve any savings in the complexity required for computing BMs. By limiting S to an integer power of 2 and applying the idea of set partitioning, we achieve the goal of complexity reduction because branch selection can be done without computing the BMs and sorting, as described below.
B. Set Partitioning
Set partitioning divides the constellation points into S disjoint sets of equal sizes and with the same intra-subset distance, while at the same time maximizing the intra-subset distance. Figure 2 shows partitions of a 16-QAM constellation into 1, 2, 4, 8, and 16 subsets.
We now illustrate how set partitioning eliminates the need of computing and ranking all of the |C| BMs for the purpose of branch selection. Referring to (6), we can rewrite the BM term in (8) as λi(xi) = ˜znt−i+1(x (1:i−1) i ) − Rnt−i+1,nt−i+1cj 2 (9) where ˜ znt−i+1(x (1:i−1) i ) ∆ = znt−i+1− nt X k=nt−i+2 Rnt−i+1,kxk. (10)
Note that ˜znt−i+1(x
(1:i−1)
i )and Rnt−i+1,nt−i+1 are common
for all the branches leaving the same node. Defining ˆ znt−i+1(x (1:i−1) i ) ∆ = ˜znt−i+1(x (1:i−1) i )/Rnt−i+1,nt−i+1 (11) and following from (9), it can be seen that the S closest constellation points to ˆznt−i+1(x
(1:i−1)
BMs among the |C| outgoing branches, and are thus what should be retained.
As long as the constellation exhibits sufficient symmetry (as is the case for the modulation schemes used in practice), set partitioning enables an efficient two-step procedure of finding these S closest points without explicitly computing BMs: (a) For each signal subset defined by the S-subset partitioning, perform a slicing operation on ˆznt−i+1(x
(1:i−1)
i ). This yields
the winning constellation point that gives the minimum BM within each subset. (b) Collect the winning constellation point from each of the S subsets.
It should be noted that, for the proposed algorithm, only after the decisions of branch selection are made is computing BMs required – for the purpose of obtaining the PMs of the newly extended paths (refer to (7)). However, only S BMs (instead of |C| BMs) per node are required for computing PMs, which leads to complexity reduction. Although the value of S is restricted to be an integer power of 2, the simulation results shown in Section V-A suggest that little complexity increase is incurred by such a quantization operation. C. Adaptive Set Partitioning (ASP)
In time-varying channels typically encountered in wireless communication applications, a larger value of S should be used in anticipation of deep fades in order to minimize the chances of the correct hypothesis being eliminated prema-turely. Thus, it is advantageous to adapt S based on the channel condition experienced by each transmitted stream in order to achieve a better tradeoff between error-rate performance and complexity. Below, we formulate the criterion that is used to adapt S when the approach outlined in Section IV-B is used for branch selection.
1) Motivation: Denote Si as the number of retained
branches that leave a given node at level i − 1. We would like to choose the value of Si based on the following
crite-rion. For each node, the probability that the outgoing branch corresponding to the correct symbol hypothesis is excluded in the branch selection process can not exceed a pre-specified target level Pt. From the discussion in section IV-B, this is
equivalent to requiring the probability that the correct hypoth-esis falls within the most likely Si branches (or equivalently,
the probability that the BM of the correct hypothesis is ranked among the Si smallest ones) to be above 1 − Pt. It follows
from (9) that Si should be set to be the minimum integer that
satisfies Si X m=1 P (x∗ nt−i+1= x <m> nt−i+1 ˜znt−i+1(x (1:i−1) i ), γi) > (1 − Pt) (12) where γi ∆
= Rnt−i+1,nt−i+1, the superscript ∗ denotes the true
hypothesis, and x<m>
nt−i+1 corresponds to the outgoing branch
with the m-th smallest BM. However, the exact solution to (12) requires computation of BMs (i.e., (9)) for all |C| branches, which again defeats the purpose of complexity savings. We use the following approach to get around such a dilemma.
i dmin,′ ) ( ~ (1 : 1) 1 − + −i i i nt z x
Fig. 3. Geometric interpretation of the proposed algorithm for adaptive set partitioning.
2) Determination of the Number of Subsets: The probabil-ity that the BM of the correct symbol hypothesis x∗
nt−i+1 is
not ranked among the smallest Si BMs can be approximated
by the probability of event A that the correct hypothesis falls outside of the square box that centers at ˜znt−i+1(x
(1:i−1) i )and
encloses Si constellation points (as illustrated in Figure 3).
Assuming the constellation is sufficiently large such that the continuous approximation [9] holds and the effects of points at the edge of constellation are negligible, the side of such a box is 2pSi/4d0min where dmin and d0min,i
∆
= γidmin
are the minimum distances of the transmitted and received constellations, respectively. Setting P r(A) ≤ Pt to achieve
the specified performance level and using the assumption that each entry of w is an i.i.d. Gaussian r.v. with zero mean and variance σ2 n, we have P r(A) = 1 − " 1 − 2Q p Si/4d0min σn/ √ 2 !#2 ≤ Pt (13) where Q(x)∆ = 1/√2πR∞ x e −u2/2
du. With moderate to high SNRs, the second-order term of the Q-function in (13) can be ignored. Furthermore, let Smax denote the largest allowable
value of Si that is set by the implementation constraints and
the size of constellation, i.e., Smax ≤ |C|. It follows that
the minimum integer value for Si that satisfies (13) and the
constraints in complexity and constellation size is found to be Siraw= min 4 " σn √ 2d0 min Q−1 Pt 4 #2 , Smax (14) where due is the smallest integer that is greater than or equal to u. Finally, to enable set partitioning, we set
Si= dSirawe2 (15)
where d·e2is the operation that rounds up to the nearest integer
while the exact solution obtained from (12) could vary from one node to another.
D. Summary of Algorithm
It should be noted that the proposed algorithm can be used in conjunction with the method proposed in [4], which adapts the value of M at each level based on the channel conditions. To illustrate this point, we denote Mias the value of M used for
level i in the following. Below is a summary of the algorithm steps.
(A) Initialization: Set i = 1, M0= 1, x0= Ø(i.e., an empty
set), and PM(x0)=0.
(B) Determine Si using (14) and (15).
(C) Loop through each node at level i − 1.
(a) Determine the Si outgoing branches to retain.
(i) Compute ˆznt−i+1(xi−1)using (10) and (11).
(ii) Slice ˆznt−i+1(xi−1)to obtain the winning
con-stellation point cj within each of the Sisubsets.
(iii) Compute the BM of each of the winning con-stellation point cj using (9).
(b) Form Si candidate paths and compute the
corre-sponding PMs.
(i) Loop through each subset.
• For each subset, form a candidate path by
appending xi−1 (i.e., the path to the current
node) with its winning constellation point cj. • Compute the PM of this candidate path using
(7).
(D) If i = nt, then set Mi = 1; otherwise, determine Mi
based on the method proposed in [4].
(E) Rank the PMs of the Mi−1Si candidate paths in
as-cending order. Retain the first Mi paths, and store
the corresponding states (as xi) and PMs. Delete the
unselected paths. (F) Increment i. (G) If i = nt, Set Si= 1; Go to Step (C); elseif i = nt+ 1,
Set symbol estimate = minimum-PM path, and exit. else
Go to Step (B); end
Note that, in Step (G), i = ntcorresponds to the final level
of branch extension. After the extension is completed, the path with the smallest PM gives the symbol estimates. Therefore, there is no need to have more than one path other than the minimum-BM branch extending out of each node at level nt−
1. It follows that we set Snt = 1. For the same reason, we set
Mnt = 1 in Step (D).
V. SIMULATIONRESULTS
To evaluate the performance and complexity of the proposed algorithm, simulation results for a 4-by-4 BLAST system using 16-QAM are shown below, assuming perfect channel and
Exact Approx. Approx.
Eq. (12) Eq. (14) Eq. (15)
Level SNR mean std mean std mean std
1 23 dB 2.4 3.0 3.6 4.0 4.1 4.7 1 26 dB 1.8 2.3 2.4 3.2 2.7 3.7 2 23 dB 1.3 0.8 1.7 1.3 1.8 1.8 2 26 dB 1.1 0.4 1.2 0.7 1.2 0.9 3 23 dB 1.2 0.5 1.3 0.6 1.3 0.7 3 26 dB 1.0 0.2 1.0 0.2 1.0 0.3 TABLE I
STATISTICS OF THE NUMBER OF PARTITIONS ESTIMATED BASED ON THE PROPOSED APPROXIMATION VS.THOSE OF THE EXACT METHOD.
10 12 14 16 18 20 22 24 26 28 10−5 10−4 10−3 10−2 10−1 100 SNR (dB) SER MLMMSE ZF−SIC M=16, S=16 M=8, S=16 ASP, Pt=1e−2 ASP, Pt=1e−3 ASP, Pt=1e−4 ASP, Pt=1e−5
Fig. 4. Symbol error rates in a 4-by-4 MIMO system with 16-QAM
constellation in a flat fading channel.
noise power estimates. The flat fading channels were generated using the model in [10]. The SNR per Rx antenna is defined as SNR = 10log10(1/σn2). (Recall that the average signal power
from each Tx antenna is normalized to 1/nt.)
A. Accuracy of the Continuous Approximation
Table I compares the mean and standard deviation (std) of Siestimated based on the exact method defined by (12) and the
proposed method based on the continuous approximation (i.e., (15)). The SNR of 23 dB and 26 dB corresponds to an symbol error rate (SER) of interest that is approximately 1×10−3and
1 × 10−4, respectively, as shown by the simulation results in
Section V-B. It can be observed that the estimation based on the continuous approximation is quite accurate even for a 16-QAM constellation. It also suggests that the quantization in (15) does not introduce a significant increase in complexity because the resulting increase in mean and standard deviation is very small.
B. SER Performance
Figure 4 shows the symbol error rates (SERs) of several MIMO detectors. For the curves labeled as ASP, we set Smax=
16 and Mmax = 16 (which denotes the maximum allowable
10 12 14 16 18 20 22 24 26 28 102 103 104 SNR (dB) Multiplications (a) MMSE QRM−MLD in [6] ASP, Pt=1e−4 10 12 14 16 18 20 22 24 26 28 100 101 102 103 104 SNR (dB) Comparisons (b) QRM−MLD in [6] ASP, Pt=1e−4
Fig. 5. Comparisons of complexity counts. Same scenario as in Figure 4. (a) Number of real multiplications per symbol period. For a comparison purpose, the ML detector requires 525312 real multiplications per symbol period. (b) Number of comparisons per symbol period.
setting Pt= 1 × 10−4 gives a performance that is essentially
the same as that of the ML detector. C. Complexity
We focus on the two measures: number of real multiplica-tions and comparisons that are required to perform detection of ntsymbols (i.e., ntparallel streams) in one symbol period.
The results shown in Figure 5 (a) and (b), respectively, are obtained with the same simulation scenario in Figure 4. The approach of [4] is used as a reference because it is a reduced-complexity version of [6]. At the SNRs of interest (i.e., at an SER of 1 × 10−3 and 1 × 10−4), it can be seen that the
number of comparison operations required by the proposed algorithm is about an order of magnitude smaller than that of [4], and that there is about 25% saving in multiplication operations. Figure 6 shows the distributions of the values of Si estimated by the proposed algorithm across different tree
levels in the same scenario as Figure 5 2. Together with the
closeness of the SER to that of the ML detector, we see that in most cases, a relatively small Si is sufficient to cover the
correct hypothesis in branch extension, thus leading to the complexity reduction. This also explains why the quantization of the number of subsets into an integer power of 2 to exploit the nice structure of Ungerboeck’s set partitioning incurs little complexity increase, as indicated by the results in Table I. Finally, observed that the value of Si tends to concentrate
even more on the lower side at the deeper tree level, due to the distribution of γi.
VI. CONCLUSION
In this paper, we proposed an algorithm that reduces the number of multiplications and comparisons of the QRM-MLD
2The final level is not shown because S4= 1for the reason explained in
Section IV-D. 1 2 4 8 16 0 0.5 1 (a) Si Relative Frequency Level 1 Level 2 Level3 1 2 4 8 16 0 0.5 1 (b) Si Relative Frequency Level 1 Level 2 Level3
Fig. 6. Distributions of the values of Siacross different tree levels. Pt =
1× 10−4.Same scenario as in Figure 4. (a) SNR = 23 dB. The average
number of subsets is 2.1. (b) SNR = 26 dB. The average number of subsets is 1.5.
receiver for the BLAST system by employing the idea of set partitioning. An algorithm to perform dynamic partitioning based on the channel conditions is also described. Simulation results demonstrated a good tradeoff between performance and complexity. The effects of imperfect channel and noise power estimates are currently being investigated.
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