IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 1, JANUARY 2013 51
Low-Complexity Soft-Output Sphere Decoding with
Modified Repeated Tree Search Strategy
Shin-Lin Shieh, Member, IEEE, Rong-Dong Chiu, Shih-Lun Feng, and Po-Ning Chen, Senior Member, IEEE
Abstract—Many solutions for detecting signals transmitted
over flat-faded multiple input multiple output (MIMO) channels have been proposed, e.g., the zero-forcing (ZF), minimum mean squared error (MMSE), sphere decoding (SD) algorithms, to name a few. These approaches however suffer from either unsatisfactory performance or high complexity. In this paper, we focus on the soft-output SD algorithm and propose a modification on the repeated tree search (RTS) strategy. It is shown that our modification can maintain a fixed upper limit in decoding com-plexity and results in a good performance-comcom-plexity tradeoff.
Index Terms—MIMO, sphere decoding, repeated tree search.
I. INTRODUCTION
I
N recent years, wireless transmission with multiplean-tennas at the transmitter and receiver, also being referred to as the multiple input multiple output (MIMO) system, has attracted enormous interest. It is considered to be the technology that can provide significant capacity improvement over existing communication systems.
In an MIMO fading channel suffering additive white Gaus-sian noises (AWGN), different data streams are transmitted from different antenna elements via the same channel; so the receiver has to separate these data streams in order to recover them. Some detection algorithms for MIMO systems have thus been proposed and are reviewed in the following.
Linear detection methods, such as zero-forcing (ZF) or min-imum mean-squared error (MMSE), estimate the information of channel matrix and then use the estimate to compensate the channel effect. Although having usually low computational complexity, the linear detection methods cannot totally remove the inter-stream interference and may induce noise enhance-ment; they accordingly could result in significant perfor-mance degradation. As a contrary, the brutal-force maximum likelihood (ML) detector is optimal in performance but its computational complexity is high. Being regarded as a balance of the previous two, the sphere decoding (SD) algorithm [1], [2] smartly reduces the number of candidate symbol vectors during the codeword search and can still statistically guarantee the finding of the ML solution with greatly reduced complexity.
Unlike uncoded systems where single hard-decision ML solution is directly outputted, the soft information for each information bit is required for iterative decoding. The soft-output SD algorithm [3], [4], [5] thus draws research attention
Manuscript received August 2, 2012. The associate editor coordinating the review of this letter and approving it for publication was M. Lentmaier.
S.-L. Shieh is with the Graduate Institute of Comm. Eng., Nat’l Taipei Univ., Taipei, Taiwan 23741, R.O.C. (e-mail: [email protected]).
R.-D. Chiu, S.-L. Feng, and P.-N. Chen are with Nat’l Chiao-Tung Univ., Taiwan 30010, R.O.C.
Digital Object Identifier 10.1109/LCOMM.2012.112012121728
recently. In comparison with the hard-output SD algorithm, the soft-output SD algorithm generally requires more compu-tational complexity; therefore a particular method to reduce its complexity is necessary [4]. Some known complexity-reduction methods in literature include log-likelihood ratio (LLR) clipping, channel matrix regularization, and run-time constraint (i.e., imposing a constraint on the maximal compu-tational complexity of the decoder). As expected, these meth-ods reduce the decoding complexity at a price of performance degradation. The above complexity-reduction methods still yield a varying complexity; yet a hardware implementation may prefer one with a fixed complexity, which motivated the work of a fixed complexity soft-output SD in [5].
In this paper, we propose a modification on the repeated tree search (RTS) in [3], resulting in less performance degradation and also less complexity than those of the single tree search (STS) in [4], and the smart ordering and candidate adding (SOCA) algorithm in [5]. Moreover, our modification can maintain a fixed upper limit in decoding complexity and thus meet the requirement of hardware implementation.
The remaining of the paper are organized as follows. Section II introduces the system model and the existing MIMO detection approaches. Section III gives the detail of the soft-output SD algorithm. Section IV presents the idea of our proposed algorithm. Section V summarizes the simulation results, and Section VI concludes the paper.
Throughout the paper, superscripts “T” and “H” are reserved to denote the transpose and Hermitian transpose of a matrix, respectively.
II. SYSTEMMODEL ANDKNOWNMIMO DETECTORS
Consider an MIMO system withNT transmit antennas and
NRreceive antennas, whereNT≤ NR. At the transmitter end,
Q coded information bits are mapped to a complex
constel-lation O (e.g., QPSK, 16-QAM, etc); hence, the number of
constellation points is |O| = 2Q. The set of system vectors
being transmitted is then given by ONT. Assume that the
channel suffers flat fading. The received symbol vector thus can be written as:
y = Hx + n, (1)
where x ∈ ONT is the transmitted symbol vector with
covariance matrix E{xxH} = INT; y ∈ CN
R denotes the
received symbol vector; n ∈ CNR is an independent
zero-mean Gaussian-distributed complex noise vector with common
varianceN0per entry; andH ∈ CNR×NTdenotes theN
R×NT
channel matrix with each element in H being a complex
Gaussian variable with zero mean and variance 1/NT. In the
above expression,C denotes the domain of complex numbers,
and each channel matrix realization H is assumed perfectly
52 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 1, JANUARY 2013
estimated by the receiver. Note that in our setting, the
signal-to-noise ratio (SNR) per receive antenna is equal to 1/N0.
The SD algorithm has recently been used to solve the signal detection problem for MIMO systems because it can signif-icantly reduce the computational complexity in comparison with the brutal force ML detector while maintaining the same ML performance. The idea behind the SD algorithm can be described as follows. It first sets a sphere centered at the received symbol vector with a properly chosen radius. Then instead of searching the entire system constellation, it only searches the constellation points inside the sphere.
The SD algorithm can be separated into two steps: 1) preprocessing step and 2) tree search step. The preprocessing step is mainly on the construction of the tree structure.
Specifically, the channel matrix H is QR-decomposed with
sorting and regularization [4] as:
H
αINT
P = QR (2)
whereQ is an (NR+NT)×NTunitary matrix,R is an NT×NT
upper triangular matrix with diagonals being real-valued, and
theNT×NTpermutation matrixP and the real parameter α are
carefully chosen to fit the need of sorting and regularization
[4], [5]. Partition Q into Q1 andQ2 according to
Q =QT 1 QT2
T
whereQ1 andQ2are respectivelyNR× NT matrix andNT×
NT matrix. Then, multiplying (1) byQH1 leads to a modified
input-output relation as
˜y = QH
1y = QH1Hx + QH1n = R˜x + ˜n
where ˜x = PTx and ˜n = QH
1n. In matrix form, the above
equation can be written as ⎡ ⎢ ⎣ ˜y1 .. . ˜yNT ⎤ ⎥ ⎦= ⎡ ⎢ ⎢ ⎢ ⎣ r1,1 r1,2 · · · r1,NT 0 r2,2 · · · r2,NT .. . . .. ... ... 0 · · · 0 rNT,NT ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎣ ˜x1 .. . ˜xNT ⎤ ⎥ ⎦+ ⎡ ⎢ ⎣ ˜n1 .. . ˜nNT ⎤ ⎥ ⎦.
By treating ˜n to be independent Gaussian distributed, it can
be obtained that ˆxML = arg min ˜ x∈ONT ˜y − R˜x 2 = arg min ˜ x∈ONT NT i=1 ˜yi− NT j=i ri,j˜xj 2 . (3)
The second step then performs tree traversal based on (3). There are three major tree traversal algorithms that have been proposed in the literature, which are depth-first search [4], breadth-first search [6], and best-first search [7] algorithms.
III. SOFT-OUTPUTSPHEREDECODING ANDMETHODS
FORCOMPLEXITYREDUCTION
Denote by ˜xj,b the bth bit in the constellation point
corre-sponding to the jth entry of vector ˜x. In order to reduce the
computational burden, we approximate the true LLR for bit
˜xj,b by its max-log approximation [4]:
L(˜xj,b)= min ˜ x∈Xj,b(0) ˜y − R˜x 2− min ˜ x∈Xj,b(1) ˜y − R˜x 2, (4)
whereXj,b(0)andXj,b(1)are sets of vectors that have thebth bit in
the jth entry equal to 0 and 1, respectively. The computation
of Eq. (4) can be done via a tree search process, after which
the resulted LLRs should be permuted back to thex-domain
using the relation of ˜x = PTx.
Several tree traversal strategies have been proposed for the generation of the LLR values. They are described below.
1) Repeated Tree Search (RTS): The idea behind the RTS strategy [3] is to repeat the tree search to compute the value of (4) for every bit in the symbol vector based on the ML solution located by the hard-output SD algorithm. Its main drawback is that some branch computations may be performed more than once, resulting in significant complexity waste.
2) Single Tree Search (STS): The STS is a more efficient tree search strategy when it is compared with the RTS. It ensures that every node in the tree is visited at most once; hence it reduces considerably the computational complexity. In particular, the STS [4] searches the ML solution as well as its corresponding counter-hypothesis paths concurrently in a depth-first fashion. Whenever a leaf is reached, two situations will be considered: If the leaf updates the temporary ML solution, the metric of the former temporary ML solution can serve as the metric of a counter-hypothesis of the new temporary ML solution, and if however no new temporary ML solution is found, the algorithm checks whether better counter-hypotheses of the current temporary ML solution have been traversed and the LLRs are updated accordingly.
3) Smart Ordering and Candidate Adding (SOCA) Algo-rithm: In [5], a soft-output SD algorithm named SOCA was proposed. By performing QR decomposition with a smart ordering criterion and also by adding pre-defined numbers of layer-by-layer candidates for its breadth-first search, it achieves a good performance-complexity tradeoff. A striking characteristic of the SOCA is that it has a fixed complexity. This makes it well suited for hardware implementation.
There are many subsequent researches focusing on further complexity reduction of the tree search algorithms mentioned previously. Additional complexity reduction enhancements such as LLR clipping, sorting and regularization are subse-quently proposed [3], [4], [5].
IV. MODIFIEDRTS TRAVERSALSTRATEGY
In this work, we propose to modify the RTS soft-output SD algorithm such that the computational complexity can be limited by a pre-defined number. It is observed that better performance-complexity tradeoff can be resulted in compari-son with the STS and the SOCA.
Similar to the original RTS, the proposed soft-output SD algorithm is separated into two stages. In the first stage, a hard-output solution is found by an SD algorithm. We set
a maximum allowable complexity T1 for the first stage so
that only near-ML hard-output is guaranteed. After finding the near-ML hard-output path, we repeat the tree traversal in the second stage to generate soft output with a fixed complexity
upper limit T2. By pre-defining T1 and T2, our modified
RTS algorithm can guarantee to generate soft output with
complexity no more than (T1+ T2). Details are given below.
One suitable candidate for the first stage is the Schnorr-Euchner sphere decoder (SESD) with radius reduction [10].
SHIEH et al.: LOW-COMPLEXITY SOFT-OUTPUT SPHERE DECODING WITH MODIFIED REPEATED TREE SEARCH STRATEGY 53 root node R A B C counter hypotheses single best path only counter hypotheses counter
hypotheses single best path only single best path only Level 1
Level 2
Fig. 1. Illustration of the second stage withNT= 3 and b = [b1b2b3] =
[432] for the proposed modification. The thick solid line corresponds to the near-ML path obtained from the first stage.
It is known that the SESD has variable and potentially high decoding complexity; hence it is not suitable for hardware
implementation. We therefore propose to set an upper limitT1
such that the first stage is terminated when either the SESD
finds the ML solution within complexityT1 or the maximum
allowable complexity T1 is reached. With a complexity
con-straint, the hard-decision output no longer guarantees to be ML. Nevertheless, we will later show by simulations that a
small to medium value of T1 is adequate to secure a good
performance-complexity tradeoff.
We next describe the second stage. First, a vector b =
[b1, · · · , bNT] that specifies the number of counter hypothesis
paths to be extended at each level is given, where 1≤ bi≤ Q.
Note that at each level, there are Q counter hypothesis paths
corresponding to the near-ML path obtained from the first
stage; but only the best bi of them are extended at level i.1
Further extension along thesebicounter-hypothesis paths only
include the best path. For a better understanding, a simple illustration of the second stage is given in Fig. 1.
For all soft-output SD algorithms, clipping the LLR to make
it within ±Lmax plays an important role for performance,
complexity or both [4], [5]. In this work, the counter hypoth-esis paths traversed by our modified RTS strategy may not be
the ones required by (4) since usually bi < Q. In such case,
the max-log approximated LLR will be infinity for these
non-traversed paths. The selection of clipping limit Lmax thus is
essential in our modification. Notably, except for the RTS, the situation that some of the counter hypothesis paths required are not visited could also happen to the STS and the SOCA; hence, these two schemes also need a clipping limit to prevent from overestimation of the LLR values. In all cases, LLR clipping can at the same time help reducing the complexity of the second stage.
The upper complexity limit of the second stage can be
1Take 16-QAM (hence, Q = 4) as an example. If ˆx
i = 0000 is the
i-th symbol of the near-ML path, then the four counter hypothesis paths
are specified by {1000, 0100, 0010, 0001}. Among them, the bi counter hypothesis paths selected are the ones that are closest in distance to the received complex value ˜yi.
It should be mentioned that the SOCA also pre-defines numbers of layer-by-layer candidates for its breadth-first search; however, by considering the performance-complexity tradeoff, the selective choice for the SOCA isb2= b3 = · · · = bNT = 1. That is why only b1 is identified for the SOCA in
Figs. 2 and 3. computed as follows: T2= NT i=1 bi(NT+ 1 − i) . (5)
By only expending those nodes with branch metric within
Lmax, the complexity of the second stage can be further
reduced and is usually smaller thanT2.
We close this section by stress the differences between our modified RTS and the SOCA. First, our modified RTS finds the near-ML hard-output path and compute the LLR values in different stages, whereas the SOCA obtain both concurrently in one tree search. A second difference is that the SOCA adds Q counter hypothesis paths at all levels in a simple bit-flip fashion as only a portion of the MAP path is known before the selection of these counter hypotheses, whereas our modified
RTS extends onlybi counter hypothesis paths at leveli with
respect to a completely known near-ML path. V. SIMULATIONRESULTS
In our simulations, we assume that the MIMO channels are Rayleigh faded without spatial or temporal correlation, and all channel matrix realizations can be perfectly estimated
by the receiver. Also, NT = 4 transmit antennas, NR = 4
receive antennas, and 16-QAM constellation are considered. In addition, the channel realizations do not change during the transmission of an entire codeword in the simulated slow fading scenario, as identically assumed in [5].
The outer codes adopted [8] are 3GPP-specified (2, 1, 8)
convolutional code and punctured turbo code of code rate R = 1/2. For the convolutional coded simulations, 180
16-QAM symbols are fed into a 15× 48 block interleaver before
they are sent, while for the turbo coded simulations, 500 16-QAM symbols are transmitted after being passed through
a (40 × 50)-block interleaver. As a result, the codeword
lengths for the convolutional and turbo coded systems are
respectively 180× 4 = 720 bits and 500 × 16 = 2000
bits. At the receiver, the Viterbi decoder and the 8-iteration Max-Log-MAP decoder are used respectively for the decoding of convolutional code and turbo code. Simulation results for convolutional and turbo coded systems are then summarized in Figs. 2 and 3, respectively.
In our simulations, variousLmax values are tested for the
STS, while Lmax = 0.3 and Lmax = 0.25 are chosen as
suggested by our trial simulations not shown in this paper for the SOCA and our modified RTS, respectively. The channel regularization algorithm used in our modified RTS is the same as that in [4]. As for the sorting approach in QR decomposition, the SQRD [4] is used for both the STS and our modified RTS, while the SOQR [5] is implemented for the SOCA. The maximum allowable complexity for the first
stage isT1= 30, and various vectors b required for the second
stage are examined, which are b = [4444], [4442], [4422],
[4222], [2222]; they respectively result in T2 = 40, 38, 34,
28, 20. Finally, the metric for the system performance after decoding is the SNR required to achieve a block error rate
(BLER) of 10−2. The index for computational complexity
is the number of visited nodes during the tree search; this complexity index is widely adopted for one-node-per-cycle
54 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 1, JANUARY 2013 16.5 17 17.5 18 100 101 102 103
Minimum required SNR [dB] for BLER = 0.01
Average and 99.9%−percentile complexity
0.200 0.175 0.150 0.125 0.100 0.075 0.050 0.025 0.200 0.175 0.150 0.125 0.100 0.075 0.050 0.025 16 14 12 10 8 6 4 40 38 34 28 20 40 38 34 28 20 STS average SOCA modified RTS average STS 99.9% modified RTS 99.9%
Fig. 2. Performance versus complexity for the STS, the SOCA, and the modified RTS in slow Rayleigh fading channels. The numbers beside the STS marks are theLmaxused. The numbers next to the SOCA curve correspond tob1. The number next to each modified RTS mark isT2. The channel code adopted is the convolutional code.
16.8 17 17.2 17.4 17.6 17.8 18 100 101 102 103 0.250 0.225 0.200 0.175 0.150 0.125 0.100 0.075 0.050 0.025 16 14 12 10 8 6 4 40 38 34 28 20 0.250 0.225 0.200 0.175 0.150 0.125 0.100 0.075 0.050 0.025 40 38 34 28 20
Minimum required SNR [dB] for BLER = 0.01
Average and 99.9%−percentile complexity
STS average SOCA
modified RTS average STS 99.9% modified RTS 99.9%
Fig. 3. The same simulations as Fig. 2 except that the channel code adopted is the turbo code.
hardware implementation architecture [9]. Based on the above setting, we are ready to present our simulation results.
As observed from Fig. 2 and Fig. 3, the simulation results for convolutional and turbo coded systems are quite similar. A result not shown in these two figures is that the performance
degradation fromT1= ∞ down to T1= 30 is less than 0.05
dB; this is the basis of our claim that a small to mediumT1is
adequate to secure a good performance for our modified RTS. For a complete comparison, we also show the 99.9th percentile complexities in the two figures. Notably, a highly variant decoding complexity will add difficulties to the hardware implementation of a decoding algorithm. From this regard,
the 99.9th percentile complexity can serve as an assessment
index for hardware design.
These two figures then show that the STS achieves the best performance-complexity tradeoff in slow fading scenario when only the average complexity is considered; however, its 99.9th percentile complexity is the worst among all simulated schemes. In particular, the 99.9th percentile complexity of the STS is six times more than its average complexity in both figures. Thus, the STS has a large complexity variation.
On the contrary, our modified RTS, whose 99.9th percentile
complexity almost approaches its strict upper complexity
bound (T1+T2), turns out to have a much smaller complexity
variation than the STS. Note that the SOCA has a fixed decoding complexity so that its 99.9th percentile complexity is identical to its average complexity. Since the curve of the 99th percentile complexity (equivalently, the upper complexity bound) of our modified RTS is only slightly above the curve of the SOCA, and since a hardware designer may use this upper complexity bound as its design criterion, we can conclude that our modified RTS requires a similar hardware complexity to the SOCA. As a result, the SOCA and our modified RTS remain to be more attractive solutions for hardware implementation because of their low complexity variation.
VI. CONCLUSION
In this paper, we present a modified RTS algorithm for soft detection in an MIMO system as a support to an outer code. A visible performance-complexity tradeoff improvement has been obtained by our proposed modification. The guaranteed decoding complexity upper limit makes this modified RTS algorithm a suitable candidate for hardware implementation.
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![Fig. 1. Illustration of the second stage with N T = 3 and b = [b 1 b2 b3 ] =](https://thumb-ap.123doks.com/thumbv2/9libinfo/7667201.140970/3.918.114.404.85.233/fig-illustration-second-stage-n-t-b-b.webp)
