In this section, we examine the performance of the codes proposed in Section 3.2. We also illustrate the decoding complexity of the maximum-likelihood priority-first search decoding algorithm presented in the previous section. For ease of comparison, the channel parame-ters used in our simulations follow those in [30], where h is zero-mean complex-Gaussian distributed with E[hhH] = (1/P )IP and P = 2. The average system SNR is thus given by
N Lσn2tr
E[hhH]1 NBTB
= N
Lσ2ntr 1 N PBTB
= N
Lσ2n, (3.17) since tr BTB
= N P for all simulated codewords.4
Figure 3.2 illustrates the simulation results of three codes: the computer-searched half-rate code obtained by the simulated annealing algorithm in [30] (SA-22), the constructed double-tree code with half of the codewords satisfying BTB = G−1 and the remaining half satisfying BTB= G1 (Double-22), and the constructed single-tree code whose codewords are all selected from the candidate sequences satisfying BTB = G−1 (Single-22). We observe from Figure 3.2 that the Double-22 code performs almost the same as the SA-22 code.
Actually, the simulations illustrated in Figure 3.3 provide evidence that the performance of the constructed double-tree half-rate codes is as good as the computer-searched half-rate codes for all N > 12. However, when N ≤ 12, the Double-N code performs slightly worse
4The authors in [30] directly define the channel SNR as 1/σ2n. It is apparent that their definition is exactly the limit of (3.17) as N approaches infinity.
Since it is assumed that an adequate guard period between two encoding blocks exists (so that there is no interference between two consecutive decoding blocks), the computation of the system SNR for finite N should be adjusted to account for this muting (but still part-of-the-decoding-block) guard period. For example, in comparison of the (6,3) and (20,10) codes over channels with memory order 1 (i.e., P = 2), one can easily observe that the former can only transmit 18 code bits in the time interval of 21 code bits, while the latter pushes out up to 20 code bits in the period of the same duration. Thus, under fixed code bit transmission power and fixed component noise power σ2n, it is reasonable for the (20,10) code to result in a higher SNR than the (6,3) code.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 3.2: The maximum-likelihood word error rates (WERs) of the computer-searched half-rate code by simulated annealing in [30] (SA-22), the constructed half-rate code with double code trees (Double-22), and the constructed half-rate code with single code tree (Single-22). The codeword length is N = 22.
0 5 10 15
Figure 3.3: The maximum-likelihood word error rates (WERs) of the computer-searched code by simulated annealing (SA-N ) and the constructed half-rate code with double code trees (Double-N ).
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 100
101 102 103 104
SNR (dB)
Average number of node expansion per information bit
EXH−SA−22 SEQ−Double−22−f1 SEQ−Single−22−f1 SEQ−Double−22−f2 SEQ−Single−22−f2
Figure 3.4: The average numbers of node expansions per information bit for the simulated-annealing-based computer-searched code in [30] by exhaustive decoding (EXH-SA-22), and the constructed single-tree (SEQ-Single-22) and double-tree (SEQ-Double-22) codes using the priority-first search decoding guided by either metric function f1 or metric function f2. than the SA-N code. This is because for N ≤ 12 the approximation in (3.9) can no longer be well maintained due to the restriction that |A(b(`)|G)| must be an integer.
In addition to the Double-22 code, Figure 3.2 also depicts simulation results of the Single-22 code. Since the pairwise codeword distance in the sense of (3.8) for the Single-Single-22 code is in general smaller than that of the Double-22 code, its performance has a 0.2 dB degradation compared with that of the Double-22 code. However, we will see in Figure 3.4 that the Single-22 code actually has the smallest decoding complexity among the three codes. This suggests that to select codewords uniformly from a single code tree should not be ruled out as a candidate design, especially when the decoding complexity becomes the main system concern.
In Figure 3.4, the average numbers of node expansions per information bit are illustrated for the codes examined in Figure 3.2. Since the number of node expansions is exactly equal to the number of tree branch metrics (i.e., one recursion of f -function values) computed,
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 10−4
10−3 10−2 10−1 100
Eb/N0 (dB)
BER
Single−22 Double−22 SA−22
Figure 3.5: Bit error rates (BERs) for the simulation of codes illustrated in Figure 3.2.
the equivalent complexity of exhaustive decoding is correspondingly plotted. It can then be observed that in comparison with the exhaustive decoder, a significant reduction in compu-tational burden is achieved at moderate-to-high SNRs by adopting the Double-22 code and the priority-first search decoder with on-the-fly computable metric f1 (see (3.11)). Further reduction can be achieved if the Double-22 code is replaced with the Single-22 code. This is because performing the sequential search over multiple code trees introduces extra node expansions for those code trees that the transmitted codeword does not belong to. An ad-ditional order-of-magnitude reduction in node expansions can be achieved when the metric f2 = g + ϕ2 (see (3.12)) is used instead.
The authors in [7] and [30] only focus on the word error rate (WER). No bit error rate (BER) performances that involve the mapping design between the information bit patterns and the codewords are presented. Yet, in certain applications, such as voice transmission and digital radio broadcasting, the BER is generally considered a more critical performance index. In addition, the adoption of the BER performance index, as well as the signal-to-noise ratio per information bit, facilitates the comparison of codes of different code rates.
Figure 3.5 depicts the BER performance of the same codes whose WER performances were depicted in Figure 3.2. The corresponding Eb/N0 is computed according to Eb/N0 = SNR/R, where R = K/N is the code rate. The mapping between the bit patterns and the codewords of the given computer-searched code is obtained through simulated annealing by minimizing the upper bound of
BER≤ 1 2K
2K
X
i=1 2K
X
j=1,j6=i
d(mi, mj)
K Pr
bˆ= bj
bi transmitted ,
where, other than the notations defined in (3.7), mi is the information sequence correspond-ing to the i-th codeword, and d(·, ·) is the Hamming distance. For the constructed codes of Section 3.2.3, the binary representation of the index of the requested codeword in Step 1 is directly taken as the information bit pattern corresponding to the requested codeword.
The result illustrated in Figure 3.5 then indicates that the BER performance of the three curves are almost the same. Hence we conclude that taking the binary representation of the requested codeword index as the information bit pattern for the constructed code not only makes its implementation easy, but also yields a BER performance similar to that of the best simulated-annealing-based computer-searched codes.
Lastly, we demonstrate the WER and BER performances, respectively, of Single-26, Double-26, Single-30, and Double-30 codes, together with those of Single-22 and Double-22 codes, over the quasi-static fading channels in Figures 3.6 and 3.7. Both figures show that the Double-30 code has the best maximum-likelihood performance not only in WER but also in BER. This result concurs with the intuition that a longer code will perform better provided that the channel coefficients remain unchanged in a coding block. The decoding complexities of the codes are listed in Table 3.1, from which we observe that the saving of decoding complexity of metric f2 with respect to metric f1 increases as the codeword length increases. It is worth mentioning that at very high SNR, the priority-first search decoding over the AWGN channels will directly go all the way down to the terminal nodes, and result in a decoding complexity of approximately two node expansions per information
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10−4
10−3 10−2 10−1 100
SNR (dB)
WER
Double−30 Single−30 Double−26 Single−26 Double−22 Single−22
Figure 3.6: Word error rates (WERs) for the codes of Single-22, Double-22, Single-26, Double-26, Single-30 and Double-30.
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
10−4 10−3 10−2 10−1 100
Eb/N0 (dB)
BER
Double−30 Single−30 Double−26 Single−26 Double−22 Single−22
Figure 3.7: Bit error rates (BERs) for the codes of Single-22, 22, Single-26, Double-26, Single-30 and Double-30.
bit. However, for fading channels, the decoding complexity cannot reach the ideal two node expansions per information bit even with zero additive noise, as shown in the last column of Table 3.1. In this regard, metric f2 still reaches a better ultimate decoding complexity than metric f1.
Table 3.1: Average number of node expansions per information bit for the priority-first search decoding of the constructed half-rate codes of length 22, 26, and 30.
SNR 5dB 6dB 7dB 8dB 9dB 10dB 11dB 12dB 13dB 14dB 15dB ∞ dB Double-22-f1 671 590 506 436 375 320 274 236 204 178 156 54
Double-22-f2 68 55 42 32 26 20 17 14 12 10 9 6
ratio of f1/ f2 9.8 10.7 12.0 13.6 14.4 16.0 16.1 16.8 17.0 17.8 17.3 9.0 Double-26-f1 2361 2006 1695 1416 1189 981 813 677 523 499 392 105
Double-26-f2 175 130 94 69 53 39 29 23 18 15 13 6
ratio of f1/ f2 13.5 15.4 18.0 20.5 22.4 25.2 28.0 29.4 29.1 33.3 30.2 17.5 Double-30-f1 8455 7073 5760 5133 3759 3430 2644 1996 1765 1368 1081 192
Double-30-f2 459 332 232 166 119 86 60 44 33 25 20 7
ratio of f1/ f2 18.4 21.3 24.8 30.9 31.6 39.9 44.1 45.4 53.4 54.7 54.1 27.4
Single-22-f1 460 371 308 250 200 163 130 105 85 69 57 12
Single-22-f2 45 33 26 20 15 12 10 8 7 6 5 4
ratio of f1/ f2 10.2 11.2 11.8 12.5 13.3 13.5 13.0 13.1 12.1 11.5 11.4 3.0
Single-26-f1 1635 1328 1061 839 666 522 403 312 244 191 152 21
Single-26-f2 112 79 57 42 31 23 17 13 11 9 7 4
ratio of f1/ f2 14.6 16.8 18.6 20.0 21.5 22.7 23.7 23.9 22.2 21.2 21.7 5.3
Single-30-f1 5871 4695 3857 2924 2335 1813 1328 884 805 572 416 39
Single-30-f2 284 199 144 101 72 51 35 26 18 14 11 4
ratio of f1/ f2 20.6 23.6 26.8 29.0 32.4 35.5 38.0 34.0 44.7 40.9 37.8 9.8
Table 3.2: The attained diversity levels of codes, which are least-square-approximated based on WER performance curves within 8–15 dBs.
Diversity N = 8 N = 10 N = 12 N = 14 N = 16 N = 18 N = 22 N = 26 N = 30
SA-N 1.84 1.84 1.88 1.87 1.89 1.87 1.91 — —
Single-N — — — — — — 1.89 1.90 1.87
Double-N 1.67 1.80 1.79 1.85 1.88 1.87 1.87 1.89 1.87
We close this section by commenting on the attained diversity level d of the simulated codes. The diversity level d serves as approximation of the word error probability at high SNR, i.e., Pe ≈ SNR−d. From Table 3.2, we observe that the attained diversities of codes of length 22 are around 1.9, which is close to the anticipated value of P = 2. The tables also suggest that the diversities degrade at small N , and the computer-searched codes have somewhat higher diversities within the considered SNR range. We conclude that under the
constraint of the self-orthogonal structure, the simulated codes can turn the second delayed channel path into another diversity. This results in a blind detection performance of diversity level close to P .