2016 13th IEEE Annual Consumer Communications & Networking Conference (CCNC) 1
Reduced Complexity 𝐾 -Best Sphere Decoding Algorithms for Ill-Conditioned MIMO Channels Ibrahim Al-Nahhal1 , Masoud Alghoniemy2 , Osamu Muta3 , Adel B. Abd El-Rahman1 Univ. of Science and Technology, Egypt. ({ibrahim.al-nahhal},{adel.bedair}@ejust.edu.eg). 2 University of Alexandria, Egypt. (
[email protected]). 3 Center for Japan-Egypt Cooperation in Science and Tech., Kyushu Univ., Fukuoka, Japan. (
[email protected]).
1 Egypt-Japan
Abstract—The traditional 𝐾-best sphere decoder retains the best 𝐾-nodes at each level of the search tree; these 𝐾-nodes, include irrelevant nodes which increase the complexity without improving the performance. A variant of the 𝐾-best sphere decoding algorithm for ill-conditioned MIMO channels is proposed, namely, the ill-conditioned reduced complexity 𝐾-best algorithm (ill-RCKB). The ill-RCKB provides lower complexity than the traditional 𝐾-best algorithm without sacrificing its performance; this is achieved by discarding irrelevant nodes that have distance metrics greater than a pruned radius value, which depends on the channel condition number. A hybrid-RCKB decoder is also proposed in order to balance the performance and complexity in both well and ill-conditioned channels. Complexity analysis for the proposed algorithms is provided as well. Simulation results show that the ill-RCKB provides significant complexity reduction without compromising the performance. Index Terms—Sphere decoder, 𝐾-best decoder, ill-conditioned MIMO channels.
I. I NTRODUCTION In spatial multiplexing multi-input multi-output (MIMO) communication systems, the 𝐾-best sphere decoder (KB) memorizes the best 𝐾-nodes at each level of the search tree in order to produce fixed throughput which is suitable for parallel implementations [4]. On other hand, the traditional sphere decoder (SD) algorithm adopts the depth-first tree search resulted from variable throughput, which limits the decoding efficiency. Instead of the depth-first tree search, the 𝐾-best sphere decoder adopts a breadth-first search and keeps only the 𝐾-best nodes, at each tree level (layer). It should be noted that the KB-SD does not guarantee the Maximum Likelihood (ML) performance [4]. In order to achieve high performance close to ML, the KB-SD algorithm needs to visit more nodes at each tree level. Hence, a trade-off between decoding complexity and decoding performance is unavoidable. This has led to several variants which have been proposed in order to further mitigate the KB-SD complexity or/and improve its performance [1]– [3], [5]–[7], [10]–[12]. It should be noted that, up to our knowledge, the proposed algorithms in the literature have not considered the condition of the channel. In particular, for ill-conditioned channels, the channel is likely to be very sensitive to noise perturbations. In this paper, we propose a reduced complexity 𝐾-best sphere decoder for ill-conditioned channels (ill-RCKB) that only keeps nodes that have distance metrics greater than pruned
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radius value, predetermined at each tree level. The pruned radius value is provided by a heuristic formula that depends on the operating Signal to Noise Ratio (SNR) and the condition number of the channel matrix. We also propose a hybrid algorithm that alternates between the ill-RCKB and the improved KSD (IKSD) (proposed in [5]) to balance the performance and the complexity in both well and ill-condition channels. The contribution of this paper lies in reducing the complexity of the KB decoder without sacrificing its performance in ill-conditioned channels. It should be noted that, up to our knowledge, the previous work in the literature have achieved improvements only in the case of well-conditioned channels. The rest of the paper is organized as follows: Section II reviews the background of the traditional algorithms. Sections III & IV discuss the proposed KB algorithms and its complexity analysis. Section V presents simulation results. Finally, the paper is concluded in Sec. VI.
II. BACKGROUND Consider the complex-valued baseband MIMO model in Rayleigh fading channels with 𝑀 transmit and 𝑁 receive antennas. Let the 𝑁 × 1 received signal vector y, ˙ ˙ x˙ + w, y˙ = H ˙
(1)
𝑀 with the transmitted signal vector x˙ ∈ 𝑍𝐶 whose elements are drawn from q-QAM constellation and 𝑍𝐶 is the set of complex ˙ whose elements ℎ𝑖𝑗 integers, the 𝑁 × 𝑀 channel matrix H represent the Rayleigh complex fading gain from transmitter 𝑗 to receiver 𝑖 with ℎ𝑖𝑗 ∼ 𝐶𝑁 (0, 1) . In this paper, it is assumed that channel realization is known to the receiver through preamble and/ or pilot signals, and 𝑁 ≥ 𝑀 . The 𝑁 ×1 complex noise vector w ˙ has independent complex Gaussian elements with variance 𝜎 2 per dimension. Throughout the paper, we will consider the real model of (1)
y = Hx + w,
(2) 𝑇
𝑛
where, 𝑚 = 2𝑀 , 𝑛 = 2𝑁 , y = [𝑅(y) ˙ 𝐼(y)] ˙ ∈ 𝑅 , x = 𝑇 𝑚 𝑇 𝑛 ∈ 𝑍 , 𝑤 = [𝑅( w) ˙ 𝐼( w)] ˙ ∈ 𝑅 , and H = [𝑅( x) ˙ 𝐼( x)] ˙ ( ) ˙ ˙ 𝑅(H) −𝐼(H) ∈ 𝑅𝑛×𝑚 , where 𝑅(.) and 𝐼(.) are the ˙ ˙ 𝐼(H) 𝑅(H) real and imaginary parts, respectively.
2016 13th IEEE Annual Consumer Communications & Networking Conference (CCNC) 2
𝑥⊂Λ
where Λ is the lattice whose points represent all possible codewords at the transmitter. The sphere decoder reduces the computational complexity by limiting the search space inside a sphere of radius 𝜌 centered at the received signal vector y. The estimated signal ˆ , should satisfy the radius constraint ∥y − Hˆ vector, x x∥ < 𝜌 [11]. The SD transforms the closest-point search problem into a tree-search problem by factorizing the channel matrix, H = QR, where Q is a 𝑛×𝑛 unitary matrix and R is an upper triangular matrix of size 𝑁 × 𝑀 . Thus, (3) can be rewritten as ˆ 𝑆𝐷 = 𝑎𝑟𝑔 𝑚𝑖𝑛 ∥˜ y − Rx∥ , (4) x
0
Node metric
1.2
0.15 -3
(3)
Root
Branch
.15
Branch Label (symbol) 1
Level (i)
14
-1
+1
1.2
1
+3
4
14
Group 1 (if K = 6)
ˆ 𝑀 𝐿 = 𝑎𝑟𝑔 𝑚𝑖𝑛 ∥y − Hx∥2 , x
Branch distance metric
Pruned node 8.9 ʌ3 = 14.8 dmin = 2.97 9
ʌ4 = 9.89 dmin = 79.1
7.8
0.05
12.8
11
6
8.8
9
8
10
13
8 8
4
5 9.1
12
10
9
3
11
2
3
18
Leaf
2 12
7 9
8.9 9
4 14
9
8.9 10
13
8
3
9.9
.2
2 10
3 11
9
8 12
Group 2 (if K = 6)
The ML decoder is the optimum decoder, in Gaussian noise environment, where the ML solution finds the symbol estimate ˆ 𝑀 𝐿 that minimizes the 2-norm of the error x
17
18
1
KB solution = ill-RCKB solution
Figure 1. An example of tree representation of the KB & ill-RCKB for 16-QAM 2 × 2 MIMO, 𝐾 = 2, 𝑐𝑜𝑛𝑑(𝐻) = 8 and SNR = 5dB.
˜ 𝑥⊂Λ
˜ is the subset of the lattice that lies where y ˜ = QH y and Λ inside the sphere of radius 𝜌 centered at the received signal vector y. The revised problem of (4) can be well represented by a tree structure with 𝑚+1 levels, and each node in the tree expanded √ into 𝑞 child nodes where 𝑞 represents the constellation size. It is noted here that level 𝑚 represents the highest level of tree, level 𝑚 − 1 represents the second highest level, etc [11]. The partial Euclidean distance or node distance metric vector, di , at tree level, 𝑖, can be calculated in a recursive manner as follows 2
di = di+1 + ∣ei ∣ ,
𝑓 𝑜𝑟 𝑖 = 𝑚, 𝑚 − 1, . . . 1
with the distance increments 2 𝑚 ∑ 2 ∣ei ∣ = 𝑦˜𝑖 − 𝑟𝑖𝑗 𝑥𝑗 , 𝑗=𝑖
(5)
(6)
˜ , 𝑟𝑖𝑗 where 𝑦˜𝑖 represents the 𝑖𝑡ℎ element of the vector y denotes the (𝑖, 𝑗)𝑡ℎ element of the matrix R, and 𝑥𝑗 is the 𝑗 𝑡ℎ element of the vector x. III. P ROPOSED A LGORITHMS A. Ill-conditioned reduced complexity 𝐾-best sphere decoder (ill-RCKB) Fixing the number of nodes that survive at each tree level may results in visiting unnecessary nodes, which increases the decoding complexity. For example, consider the following eight distance metrics in the third tree level in Fig. 1, 𝐷 = [ 0.2 8 9 9 9 9 10 10 ], the KB with 𝐾 = 2 will choose nodes with the smallest two values which are [0.2 8]. Since the distance metrics are accumulated from top to bottom of the tree [11], the probability of finding the smallest error at the end of branch starting with value of eight is very low, compared to the probability of finding the smallest error from the branch starting with 0.2. By discarding this unlikely node, especially near the end of the tree, the complexity will be reduced significantly without sacrificing the performance.
Note that the element values of 𝐷 are assumed to be amplified for illustration purpose. In order to vary the number of surviving nodes at each tree level adaptively, we provide a heuristic formula for determining the pruned radius, 𝜌𝑖 , at a specific tree level, 𝑖, based on the condition number of the channel matrix. 1) Pruned radius rule: The choice of the pruned radius, 𝜌𝑖 , is the key to the proposed ill-RCKB algorithm. In particular, if 𝜌𝑖 is too large, more nodes are visited and the complexity increases. Depending on the parametrization of 𝜌𝑖 , a flexible performance-complexity trade-off could be achieved. To keep the same performance as the traditional KB, the number of discarded nodes at each tree level should be directly proportional to the order of the tree level (in breadth-first search strategy), the value of 𝐾, and the channel condition number. In essence, for ill-conditioned channel, data recovery at the receiver becomes progressively more sensitive to errors. On the other hand, the number of discarded nodes at each tree level should be inversely proportional to the operating SNR. Then, the decoder needs to increase the number of visited nodes (VNs) in order to achieve the desired performance. In particular, the pruned radius can be computed using the following heuristic formula 𝜌𝑖 =
𝑐𝑜𝑛𝑑(𝐻) 𝑖 𝐾 𝑑𝑚𝑖𝑛 √𝑖 , 𝑆𝑁 𝑅 + 1
𝑖 = 𝑚 − 1, 𝑚 − 2, ⋅ ⋅ ⋅ , 2
(7)
where 𝑑𝑚𝑖𝑛 is the minimum distance metric survived at tree 𝑖 level 𝑖 and 𝑐𝑜𝑛𝑑(H) = ∥H+ ∥ ∥H∥ is condition number of the channel matrix, where H+ = (H∗ H)−1 H∗ is pseudo-inverse of the channel. The square-root of SNR in (7) and the initial value of the tree level, 𝑖, are determined experimentally. By applying (7) on the tree illustrated in Fig. 1, the illRCKB achieves the same solution as the KB while saving one node (at level 𝑖 = 3). In order to illustrate the performance of this formula, a comparison between the proposed formula and the IKSD formula in [5] is presented in Sec. V. 2) Complexity reduction based on the pruned radius: In order to discard visiting irrelevant nodes, only 𝑁𝑖𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 nodes survive at each tree level 𝑖, where
2016 13th IEEE Annual Consumer Communications & Networking Conference (CCNC) 3
Algorithm 1 ill-RCKB Pseudo-Code
𝑁𝑖𝑖𝑙𝑙−𝑅𝐶𝐾𝐵
is given by
𝑁𝑖𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 = min(card(Δ𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 ), 𝐾), 𝑖 = 𝑚−1, ⋅ ⋅ ⋅ , 2. 𝑖 (8) where 𝑁𝑖𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 is the number of best nodes at level 𝑖 that will survive to the next tree level 𝑖 − 1 and ) is the cardinality of the set card(Δ𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 𝑖 𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 √ = {𝑗∣𝑑𝑗𝑖 < 𝜌𝑖 , 𝑗 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑁𝑖+1 𝑞}}, Δ𝑖 (9) 𝑖 = 𝑚 − 1, 𝑚 − 2, ⋅ ⋅ ⋅ , 2. In essence, card(Δ𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 ) is the number of nodes at 𝑖 level 𝑖 that have distance metrics smaller than the pruned radius 𝜌𝑖 given by (7). The pseudo-code in algorithm 1 summarizes the ill-RCKB. B. Hybrid reduced complexity 𝐾-best sphere decoder (hybridRCKB) As will be shown in Sec. V, the IKSD is better than illRCKB in well-conditioned channels, while the IKSD fails to decode symbols in ill-conditioned channels. For lower complexity in both well- and ill-conditioned channels, without performance degradation, we propose a hybrid-RCKB decoder that switches between the ill-RCKB and IKSD according to a threshold on the channel condition number which is specified empirically based on the simulation result shown in Fig. 2. In particular, Fig. 2 shows that the complexities of the KB, IKSD and ill-RCKB, as a function of the channel condition number, where the complexities of these algorithms will be derived in the next section. It is obvious from this figure that the IKSD has the lowest complexity when the channel condition number below 20; while the proposed ill-RCKB algorithm has the lowest complexity when the channel condition number is greater than 20. Thus, it is clear that the threshold on the condition number, in this case, is 20. Thus, the hybrid-RCKB decoder alternates between the IKSD decoder and the ill-RCKB according to the state of the channel and can be described by the following pseudo-code.
700
The average number of visited nodes
Set H and y in the real form (model) as in eq. (2). Compute 𝑄𝑅 decomposition for H. Compute y ˜ = QT y. Set 𝑑𝑚+1 = 0 𝑓 𝑜𝑟 𝑖 = 𝑚 : 1 𝑠𝑡𝑒𝑝 = −1 √ Expand each survived node into 𝑞 new possible children. Compute the distace metric vector, 𝑑𝑖 from eq. (5) 𝑖𝑓 𝑖 ≤ 𝑚 − 1 Compute pruned radius, 𝜌𝑖 from eq. (7) Compute the number of survived nodes in the next tree level, 𝑁𝑖𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 from eq. (8) 𝑒𝑛𝑑 𝑖𝑓 Sort the distance metric vector, 𝑑𝑖 ascending Select the smallest 𝑁𝑖𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 nodes from the sorted distance metric vector, 𝑑𝑖 . 𝑒𝑛𝑑 ∙ Select the smallest 𝑑1 at the last tree level. ∙ Trace the rout between the smallest 𝑑1 and 𝑑𝑚+1 . ∙ Combine the estimated vector x ˆ. ∙ ∙ ∙ ∙
600
K = 16 IKSD ill−RCKB16
500 400 300 200 cond(H) = 20
100 0 0 10
1
2
10 10 Channel Condition Number
3
10
Figure 2. Complexity vs. channel condition number for different algorithms for 16-QAM 3 × 3 MIMO channels.
1) 2) 3) 4)
Compute the condition number of the channel 𝑐𝑜𝑛𝑑(𝐻). Set a threshold 𝛾 = 20. ˆ is estimated using IKSD. If 𝑐𝑜𝑛𝑑(𝐻) < 𝛾, then x ˆ is estimated using ill-RCKB. If 𝑐𝑜𝑛𝑑(𝐻) > 𝛾, then x
IV. C OMPLEXITY A NALYSIS The complexity in this paper is defined as the average number of VNs necessary to find the solution. A. Complexity of the 𝐾-best algorithm In order to determine the complexity of the KB algorithm, tree levels are divided into two groups. The first group contains tree levels, where the number of available nodes per tree level is such that, 𝑁𝑖𝐾𝐵− ≤ 𝐾, whereas the second group contains tree levels, where the number of available nodes per tree level is such that 𝑁𝑖𝐾𝐵+ ≥ 𝐾 (see Fig. 1 for 𝐾 = 6). √ Note that, each survived node is expanded into 𝑞 child nodes in the next tree level. Then, the number of VNs in the first group,∑which equals to the total available nodes is √ 𝑝𝐾 𝑁𝑗𝐾𝐵− which equals to 𝑁 𝐾𝐵− = 𝑞 𝑗=0 𝑁 𝐾𝐵− =
√
𝑃𝐾 −1
𝑞
∑ √ 𝑗 ( 𝑞) ,
√ ( 𝑞)𝑃𝐾 ≤ 𝐾,
(10)
𝑗=0
where 𝑃𝐾 − 1 is the number of tree levels in the first group √ √ for a specific 𝐾. Given that ( 𝑞)𝑃𝐾 ≤ 𝐾 < ( 𝑞)𝑃𝐾 +1 and knowing 𝑞 and 𝐾, one can determine 𝑃𝐾 [3]. ⌊ ⌋ 𝑙𝑛(𝐾) 𝑃𝐾 = , (11) √ 𝑙𝑛( 𝑞) where, ⌊.⌋ is the floor operation. For the second group, each tree level has a fixed number of √ VNs, namely, 𝑁𝑖𝐾𝐵+ = 𝐾 𝑞 nodes. Then, the total number of VNs in the second group √ (12) 𝑁 𝐾𝐵+ = (2𝑀 − 𝑃𝐾 − 1)𝐾 𝑞.
2016 13th IEEE Annual Consumer Communications & Networking Conference (CCNC) 4
𝐾𝐵 Using 𝑃𝐾 from (11), the complexity of the KB, 𝐶𝐾 , is the total number of VNs [ ] √ √ 1 − ( 𝑞)𝑃𝐾 +1 + (2𝑀 − 𝑃𝐾 − 1)𝐾 . (13) 𝐶 𝐾𝐵 = 𝑞 √ 1− 𝑞
4
10
BER
B. Complexity of the ill-RCKB
K=4 ill−RCKB4
3
−2
10
10 K=2 ill−RCKB2
10
Complexity Curves
The average number of visited nodes
BER−Performance Curves
−1
According to (8), the maximum number of survived nodes is 𝐾, then it is clear that the complexity of the ill-RCKB is upper 10 bounded by that of the KB. In order to find the lower bound, 10 the lower bound is achieved when the minimum number of 𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 survived nodes equals to one, 𝑁𝑖 = 1, at all tree levels of the second group. Hence, 0 5 10 15 20 25 30 [ ] √ SNR (dB) √ 1 − ( 𝑞)𝑃𝐾 +1 𝑖𝑙𝑙−𝑅𝐶𝐾𝐵 𝐾𝐵 + (2𝑀 − 𝑃𝐾 − 1) ≤ 𝐶 𝑞 ≤𝐶 . √ 1− 𝑞 Figure 3. Performance and complexity comparison of ill-RCKB and KB for (14) 64-QAM 6 × 6 MIMO. −3
2
V. S IMULATION R ESULTS AND D ISCUSSION A. Simulation setup In this section, the performance and the complexity of the ill-RCKB are assessed for uncoded MIMO systems. We assume that the transmitted power is independent of the number of transmit antennas, 𝑀 , and equals to the average symbols energy in a frequency-flat Rayleigh fading channel. Performance and complexity are evaluated by the bit-error-rate (BER) and the average number of visited nodes, respectively. Channel is considered ill-conditioned if the channel condition number exceeds 20 (e.g., cond(H) > 20). It should be noted that the threshold value of the channel condition number is empirically determined as an optimal one by comparing required complexities of IKSD and ill-RCKB for various condition numbers as in Fig. 2. B. Results and discussion Figure 3 illustrates the performance and complexity of the ill-RCKB for 64-QAM and 6 × 6 uncoded MIMO compared to the KB for different 𝐾 values. As can be seen, the complexity of the KB in (13) is the same regardless of the SNR. As expected from (14), the ill-RCKB decoder has identical performance to the KB but with significant complexity reduction which , in some cases, reaches 56% (e.g., 𝐾 = 4 ). Figure 4 compares between the ill-RCKB with the IKSD and the KB in the case of well-conditioned channels for 16QAM, uncoded 4 × 4 MIMO system. From Fig. 4, the IKSD achieves lower complexity than that of the ill-RCKB, while both IKSD and ill-RCKB shows similar BER performance to the traditional KB with 𝐾 = 16 in well-conditioned channels. To show the advantage of the ill-RCKB compared to the IKSD, consider 50% ill-condition channels with condition number 103 16-QAM uncoded 3 × 3 MIMO as in Fig. 5. The illRCKB provides about 32% lower decoding complexity than the traditional KB and IKSD without performance degradation at higher SNRs. It should be noted that, in Fig. 5, the BERperformances of traditional KB, ill-RCKB16 (ill-RCKB with 𝐾 = 16) and hybrid-RCKB algorithms are identically matching. Note that, unlike the ill-RCKB algorithm, the performance
of the IKSD in ill-conditioned channels case, is improved but with high complexity cost; the hardware capability may fail to bare such complexity cost, and then it will not be able to decode the symbols correctly [8], [9]. In Fig. 5, the hybrid-RCKB, with condition number threshold equals to 20, provides the lowest decoding complexity without sacrificing the performance compared to the traditional KB algorithm.
C. Advantage of the proposed algorithms It is obvious from Fig. 5 that the IKSD algorithm will fail to decode the desired symbols in ill-conditioned channels. This is because, in case of ill-conditioned channels, the hardware capability cannot operate at a complexity level higher than the allowable complexity range by approximately 3.5 times (e.g., 3.5 time of the traditional KB with 𝐾 = 16, in this example). According to the authors of the IKSD [5], the receiver end cannot do anything better if the channel is ill-conditioned. Thus, in case of ill-conditioned channels, the IKSD will fail to decode the symbols. On the contrary, the proposed ill-RCKB takes into account complexity reduction within the allowable complexity range, especially in ill-conditioned channels. It is obvious that the hybrid-RCKB is the best remedy for both well and ill-conditioned MIMO channels.
VI. C ONCLUSIONS We have proposed the ill-RCKB decoding algorithm, which achieves the same performance as the traditional 𝐾-best decoder with significant complexity reduction in case of illconditioned channels. In particular, the ill-RCKB algorithm have provided around 32% complexity reduction at higher SNR values for ill-conditioned channels. Also, the proposed hybrid-RCKB alternates between the ill-RCKB and the IKSD algorithms according to channel condition number in order to gain the benefits of complexity reduction of both ill-RCKB and IKSD algorithms.
2016 13th IEEE Annual Consumer Communications & Networking Conference (CCNC)
−1
K=1 4 10 K = 16 IKSD ill−RCKB16
BER−Performance Curves
10
Complexity Curves 3
−2
10
BER
10
−3
10
2
10
0
5
10
15 SNR (dB)
20
25
The average number of visited nodes
5
30
K = 16 IKSD ill−RCKB16
−1
10
BER−Performance Curves
hybrid−RCKB
−2
10
3
BER
10
−3
Complexity Curves
10
−4
10
The average number of visited nodes
Figure 4. Performance and complexity comparison of different SD algorithms for 16-QAM 4 × 4 well-conditioned MIMO channels.
2
0
10
20
SNR (dB)
30
40
10 50
Figure 5. Performance and complexity comparison of different SD algorithms for 16-QAM 3 × 3 50% ill-conditioned MIMO channels.
VII. ACKNOWLEDGMENT This work was partially supported by the JSPS KAKENHI (25420376).
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