Next, we re-derive the maximum-likelihood decoding metric for use of priority-first search decoding algorithm. Continuing the derivation from (4.1) based on BTkBk = Gθ,k for 1≤ k ≤ M and 1≤ θ ≤ Θ, we can establish in terms of similar procedure as in Section 3.3.1 that:
ˆb = arg min As it turns out, the recursive on-the-fly metric for the priority-first search decoding algorithm
3Under the assumption that Q ≥ P , the ith diagonal element of the target Gθ,1 is given by Q− i + 1, and the diagonal elements of the target Gθ,k are equal to Q for 2 ≤ k < M; hence, their inverse matrices exist. However, when P ≤ N − (M − 1)Q, Gθ,M has no inverse. In such case, we re-define Dθ,M as:
Dθ,M , 0[N −(M −1)Q]×[N −(M −1)Q]⊕ G−1θ,M(N− (M − 1)Q + 1),
where Gθ,M(j) is a (P− j + 1) × (P − j + 1) matrix that contains the jth to P th rows and the jth to P th columns of Gθ,M.
is: 1≤ k ≤ M. In addition, the low-complexity heuristic function is given by:
ϕ2(b(`)),
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 4.1: Word error rates (BERs) for the codes of Double-28, SA-14, Single-28(Q=15) and Double-28(Q=15) over the quasi-static channel with Qchan= 15.
with initial values vm,k(θ)(b(k−1)Q−P +2) = 0 and β0,k(θ) =PQ+P −1
m=1 α(θ)m,k.
It is worth mentioning that if the single-tree code is adopted, ϕ2(·) can be further reduced to: low-complexity on-the-fly decoding can indeed be conducted under the single code tree condition.
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−28 SA−14 Single−28(Q=15) Double−28(Q=15)
Figure 4.2: Bit error rates (BERs) for the codes of Double-28, SA-14, Single-28(Q=15) and Double-28(Q=15) over the quasi-static channel with Qchan= 15.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
10−3 10−2 10−1 100
SNR (dB)
WER
Double−28 SA−14 Single−28(Q=15) Double−28(Q=15)
Figure 4.3: Word error rates (BERs) for the codes of Double-28, SA-14, Single-28(Q=15) and Double-28(Q=15) over the quasi-static channel with Qchan≥ L.
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−28 SA−14 Single−28(Q=15) Double−28(Q=15)
Figure 4.4: Bit error rates (BERs) for the codes of Double-28, SA-14, Single-28(Q=15) and Double-28(Q=15) over the quasi-static channel with Qchan≥ L.
Table 4.1: Average numbers of node expansions per information bit for the codes of Single-28(Q=15) and Double-Single-28(Q=15) using the priority-first search decoding guided by either evaluation function f1or evaluation function f2over the quasi-static channel with Qchan= 15.
SNR 3dB 4dB 5dB 6dB 7dB 8dB 9dB
Double-28(Q=15)-f1 2860 2440 2076 1790 1564 1359 1200 Double-28(Q=15)-f2 1271 1029 877 685 582 484 413 ratio of f1/ f2 2.3 2.4 2.4 2.6 2.7 2.8 2.9 Single-28(Q=15)-f1 1658 1367 1074 899 701 593 488 Single-28(Q=15)-f2 766 625 482 392 321 254 219 ratio of f1/f2 2.2 2.2 2.2 2.3 2.2 2.3 2.2
SNR 10dB 11dB 12dB 13dB 14dB 15dB
Double-28(Q=15)-f1 1040 958 899 811 780 723 Double-28(Q=15)-f2 353 312 277 250 229 207 ratio of f1/ f2 2.9 3.1 3.2 3.2 3.4 3.5 Single-28(Q=15)-f1 448 356 309 277 244 232 Single-28(Q=15)-f2 177 149 133 121 104 92 ratio of f1/f2 2.5 2.4 2.3 2.3 2.3 2.5
4.4 Simulation Results
Figures 4.1 and 4.2 compare four codes over fast-fading channels whose channel coefficients vary in every 15-symbol period. Notably, we will use Qchan to denote the varying period of the channel coefficients h, and retain Q as the design parameter for the nonlinear codes.
In notations, “Double-28” and “SA-14” denote the codes defined in the previous sections, and “Single-28(Q=15)” and “Double-28(Q=15)” are the codes constructed based on the rule introduced in this section under the design parameter Q = 15. Again, the mapping between the bit patterns and codewords for the SA-14 code is defined by simulated annealing.
Both Figs. 4.1 and 4.2 show that the Double-28 code seriously degrades when the channel coefficients unexpectedly vary in an intra-codeword fashion. This hints that the assumption that the channel coefficients remain constant in a coding block is critical in the code design in Section 3.2. Figures 4.3 and 4.4 then indicate that the codes taking into considerations the varying nature of the channel coefficients within a codeword is robust in its performance when being applied to channels with constant coefficients. Thus, we may conclude that for a channel whose coefficients vary more often than a coding block, it is advantageous to use the code design for a fast-fading environment considered in the section.
A more striking result from Fig. 4.1 is that even if the codeword length of the Single-28(Q=15) and the Double-Single-28(Q=15) codes is twice of the SA-14 code, their word error rates are still markedly superior at medium-to-high SNRs. Note that the SA-14 code is the computer-optimized code specifically for Qchan = 15 channel. This hints that when the channel memory order is known, performance gain can be obtained by considering the inter-subblock correlation, and favors a longer code design. The gain can be regarded as obtaining from a time diversity due to varying channels.
The decoding complexity, measured in terms of average number of node expansions per in-formation bit, for codes of Single-28(Q=15) and Double-28(Q=15) are illustrated in Tab. 4.1.
Similar observation is attained that the decoding metric f2 yields less decoding complexity
than the on-the-fly decoding one f1; however, the saving in complexity reduces when channels with fast-fading are considered.
4.5 Summary
An extension of the code design for combined channel estimation and error correction to channels with independently varying fading subblocks is established in this chapter. This design can directly construct a code of any desired code length and code rate, of which the performance is shown to be comparable to the best computer-searched code for the channels simulated. Although we only derive the coding scheme and its decoding metric for a fixed Q, further extension to the situation that the channel coefficients h vary nonstationarily as the periods Q1, Q2, . . ., QM are not equal is straightforward. Such design may be suitable for, e.g., the frequency-hopping scheme of Global System for Mobile communications (GSM) and Universal Mobile Telecommunications System (UMTS), or the time-hopping scheme in IS-54, in which cases the channel coefficients change (or hop) at protocol-aware scheduled time [19].
The performance of our constructed code can be further (slightly) improved if the code-words are selected uniformly from all feasible (c1, c2,· · · , cM) ∈ {−1, 0, 1}M. For exam-ple, select only half (i.e., 213) of the codewords according to c1 = 0 and c2 = −1 for the (28, 14)(Q = 15) code, and pick the remaining half of the codewords from those binary sequences satisfying (4.3) with c1 = 0 and c2 = 1. This however will slightly increase the decoding complexity. The trade-off between selecting codewords from fixed (c1, . . . , cM) or multiple (c1, . . . , cM)’s is thus evident.
Chapter 5
Code Designs for
Frequency-Nonselective Varying Fading Channels
The error correcting code design that jointly considers channel estimation is especially useful in situation when either the fading is rapid enough to preclude a good estimate of channel taps or the cost of implementing the channel estimators is high. One example is the reliable delivery of often short-in-length control signal such as channel quality indicator (CQI) in a highly mobile environment.
At this background, Xu et al. proposed a novel nonlinear coding scheme suitable for blind noncoherent detection of the transmitted control signal to the 802.16m standard body [39].
In the proposal, the uplink CQI information is encoded using a (12, 6) code. The codeword will then be repeatedly transmitted three times (perhaps through different OFDM channels) in order to further benefit from diversity gain (which can be equivalently regarded as a (36, 6) coding scheme).
Since most of the existing blind-detectable noncoherent codes are designed with the help of computer search, they exhibit no apparent structure for efficient decoding. The operation-intensive exhaustive search therefore becomes the only decoding option, of which the dramatically increasing decoding complexity prevents its practical use for codes of long
codeword length or high code rate.
In this chapter, we take a different approach in such code design. Based on self-orthogonality framework, we propose a systematic (N, K) coding scheme that can deal with any given N and K for channels with possibly varying channel coefficients in a coding block. It is an extension of our previous work that targets the blind detection over channels with static (i.e., constant) channel coefficients during the transceiving of a codeword [35]. Simulations show that our constructed (36, 6) code has almost the same performance as Xu’s three-times-repetitive (12, 6) code when the channel independently varies its coefficients three times in a coding block. In case the channel remains constant during the entire coding block, our constructed code has 0.7 dB performance improvement over Xu’s code.
Xu’s code is specifically designed for a frequency-nonselective OFDM system, while our systematic code construction scheme can also be applied in a frequency selective environment.
Our simulation results indicate that with a proper design, a blind-detectable noncohrent code can be made robust for channels whose taps may vary more often than a coding block.
A side advantage of our code construction scheme is that its systematic structure makes it maximum-likelihoodly decodable by the priority-first search algorithm. Thus, when be-ing compared with the operation-intensive exhaustive decoder, the decodbe-ing complexity is greatly reduced especially when codes of longer code length is adopted.
This chapter includes the following sections. Section 5.1 describes the system model we consider. Section 5.2 mentions our systematical codeword-selection procedure to construct codes for joint channel and data estimation for frequency-nonselective fading channels. Sim-ulations are discussed in Section 5.3. Finally, Section 5.4 summarizes the chapter.
5.1 System Model
Suppose that a codeword b = [b1 · · · bN]T is transmitted over a block fading channel of memory order 0, of which channel coefficients may vary in every Q symbols, where bi ∈ {±1}
and Q > P . Then, the channel model we consider in this chapter actually is a special case of the model in Chapter 4 with P = 1. It can be derived that the joint maximum-likelihood decoder [4, 30] upon the reception of y is given by (2.2):
ˆb = arg min
b∈C min
h ky − Bhk2
= arg max
b∈C
XM k=1
ykyHk − PBk
2 (5.1)
= arg max
b∈C
XM k=1
kyHkbkk2,
where yk , [y(k−1)Q+1 y(k−1)Q+2 · · · ykQ] is the output portion affected by bk, and PBk , Bk(BTkBk)−1BT
k. In the above derivation, we assume that the receiver, although it knows nothing about h, has perfect knowledge about the values of P and Q.
5.2 Code Design
Now, as far as the code design for frequency nonselective channels is concerned,A0(b1, . . . , b`) is simply the set of all binary ±1-sequences of length N, whose first ` bits are assigned as the arguments indicate, and which at the same time satisfies that
BT
kBk= Q for 1≤ k < M = dL/Qe BT
MBM = N − (M − 1)Q. (5.2)
Again, the next step is to determine the integer ∆ that satisfies (4.4). We however found that letting ∆ be the largest integer satisfying (4.4) as we did in Section 4.2 may not generate the alphabetically uniform-pick code with the best error performance. In certain cases, the second largest integer satisfying (4.4) is indeed a better choice. Further investigation that
follows along this direction suggests that a better choice of ∆ will yield a code with larger minimum pairwise distance in the sense of PM
k=1kPB¯k− PBkk2, where{ ¯Bk}Mk=1 and{Bk}Mk=1
respectively correspond to codewords ¯b and b.
It may not be practical to examine the minimum pairwise distance for all 2K codewords for the determination of the best ∆. Instead, we choose K codewords as representatives.
These representative codewords correspond to ρ = 2j∆ for 0≤ j ≤ K − 1. We then adopt the ∆ that minimizes the pairwise distance among these K codewords subject to (4.4).
When N > K + 4 and P = 0, the proposed process of determining ∆ is indeed equivalent to that the ∆-th codeword must be of the form
[−1 · · · − 1
| {z }
K+1
1 1 u 1],
where u is a maximum-length shift-register sequence. In other words, the first K + 3 bits are fixed as [−1 · · · − 1 1 1], and the last bit is always equal to 1. This is because under P = 0, all binary ±1-sequences satisfy (5.2), which results in that (2j+1∆)-th codeword is exactly the logical left-shift of (2j∆)-th codeword.
We close this section by pointing out that the size of set AP(b1, . . . , b`) for P = 1 has explicit formula as
|AP(b1, . . . , b`)| = 2N −`.
5.3 Simulation Results
In our simulations, the channel parameters follow those in [30], where h is zero-mean complex-Gaussian distributed with E[hhH] = (1/(P + 1))IP +1.
We first compare our constructed (36, 6) code with Xu’s three-times-repetitive (12, 6) code over frequency nonselective channels. As shown in Figure 5.1, the two codes has comparable performance when channel coefficients vary independently in every 12 symbols. In case the
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
Eb/N0 (dB)
WER
Proposed (36,6) Xu et al. (12,6)
Figure 5.1: Word error rates (WERs) for the constructed (36, 6) code and Xu’s three-times-repetitive (12, 6) code over flat fading channel with coefficients varying independently in every 12 symbols.
channel coefficients remain constant over the entire coding block, the proposed (36, 6) code performs 0.7 dB better than Xu’s code as shown in Figure 5.2. It should be emphasized that when P = 0, AP(b1, . . . , b`) is irrelevant to the design parameter Q; hence, the (36, 6) code in Figure 5.1 is identical to the one used in Figure 5.2. This indicates that the proposed (36, 6) code can adapt more robustly to the two simulated scenarios than Xu’s code.
5.4 Summary
In this chapter, we propose a novel systematic code construction scheme for joint channel estimation and error correction for channels with independently varying fading subblocks.
Unlike the existing noncoherent codes that are designed with the help of computer search, a code of desired code length and code rate can be directly generated with our coding scheme. We then compare our codes with the three-times-repetitive (12, 6) code proposed by Xu et al. for use of channel quality indicator (CQI) in uplink control for IEEE 802.16m.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10−2
10−1 100
Eb/N0 (dB)
WER
Proposed (36,6) Xu et al. (12,6)
Figure 5.2: Word error rates (WERs) for the constructed (36, 6) code and Xu’s three-times-repetitive (12, 6) code over flat fading channel with coefficients unchanged during the trans-mission of a codeword.
Simulations show that our constructed (36, 6) code has comparable performance to Xu’s code when channel coefficients changes randomly in every 12 symbols. If the channel taps remain constant in the entire coding block of length 36, our code outperforms Xu’s code by 0.7 dB.
This indicates that the new constructed code adapts more robustly to the two simulated scenarios.
Chapter 6
A Systematic Space-Time Code Design
Coding and transmission schemes for noncoherent receivers used in input multiple-output (MIMO) flat-fading channels can be roughly classified into two categories.1 Schemes in the first category devise the space-time constellations for a given noncoherent receiver structure using computer search [1, 2, 16], while schemes in the second category couple the well-known space-time block codes with blind detection [21,22,31]. A brief summary of these schemes is as follows.
In [1], Beko et al. propose a two-phase code design approach, where the first phase produces a rough space-time code constellation that is subsequently refined in the second phase through a search-based geodesic descent optimization algorithm (GDA). In [2], Borran et al. uses the Kullback-Leibler distance as a design criterion to partition the signal space into several subsets, resulting in a reduction of number of parameters to be computer-searched.
The authors in [16] construct unitary space-time signals by random search upon a Fourier-based structure, which only requires optimizing L− 1 parameters instead of L(L − 1)/2 in
1There are some notable papers that deal with similar problems, but cannot be classified into the two categories. For example, both [9] and [10] consider the so-called training codes that incorporate training symbols into their codewords. As anticipated, the receiver estimates the channel coefficients via training symbols. Such designs are very different from ours, which combines channel estimation and error correction by adopting joint maximum-likelihood decoding at the receiver. In [5], a noncoherent code is constructed through a mapping from coherent code. The code structure however only allows for a suboptimal efficient decoder.
the correlation matrix, where L is the number of space-time signals.
On the other hand, [21], [22] and [31] incorporate blind detection to existing space-time block codes. Based on the semidefinite relaxation (SDR) approach, an efficient suboptimal blind detection scheme is also suggested by Ma et al. in [21]. Later in [22], Ma further addresses the necessary properties for the family of orthogonal space-time block codes that can well co-work with blind detection.
Two main problems of designing codes or signal constellations based on unconstrained computer-search are that the design complexity is in general high, especially for codes of long block length, and the codes often need to be redesigned when design assumptions change.
Moreover, their decoding depends mostly on operationally intensive exhaustive search, which further prevents their practical use in the case of long block lengths. Obviously, these problems can be solved by realizing a systematic code construction and its respective low-complexity decoder. Such an approach designed under two-transmit-antenna and half-rate condition is presented in this paper.
Furthermore, one main difference between our work and the existing works on combining known space-time block codes with blind detection, is that we aim at achieving a coding gain in contrast to targeting only improved diversity gains at maximum rate.
The chapter is organized in the following fashion. Section 6.1 introduces the system model. Section 6.2 presents our code design scheme that is devised based on the unitary and full-rank properties. Section 6.3 derives the maximum-likelihood metric that can be used by priority-first search decoding. Simulations are summarized and discussed in Section 6.4.
6.1 System Model
We consider an MIMO system with AT transmit antennas and AR receive antennas. The N × AR complex received matrix Y = [y1 y2 . . . yAR] is then given by
Y= BH + N,
where B = [b1 b2 . . . bAT] is the N×ATtransmitted code matrix, and N = [n1 n2 . . . nAR] is an N×ARzero-mean complex Gaussian matrix with independent and identically distributed elements and covariance matrix
E[ninHi ] = σ2
1 0 · · · 0 0 1 · · · 0 ... ... ... ...
0 0 · · · 1
N ×N
.
Also, bi = [b1,i b2,i . . . bN,i]T is the bipolar codeword transmitted by antenna i with each bi,n∈ {±1/√
AT}. Likewise, yj = [y1,j y2,j . . . yN,j]T is the received vector at the jth receive antenna.
Because H is assumed an unknown constant matrix, the Gaussian assumption on the additive noise matrix N immediately gives that the maximum-likelihood (ML) decision about the transmitted codeword should be made based on the generalized likelihood ratio test (GLRT) as
ˆ
B = arg min
B min
H kY − BHk2
= arg min
B kY − B ˆHk2
= arg min
B k(IN − PB)Yk2, (6.1)
where ˆH, (BTB)−1BTY is the least-square estimate of H with respect to codeword B and received matrix Y, and
PB , B(BTB)−1BT
is a function of the codeword B. Here, IN denotes an N× N identity matrix.
6.2 Code Design
6.2.1 Criteria for Good Codes
Several criteria for good codes have been proposed in the literature [1, 12, 13, 36]. We will in particular center on two of them: unitary and pairwise full-rank.
Firstly, it has been derived in [36] that unitary codewords, i.e., BTB= (N/AT)· IAT, can maximize the average signal-to-noise ratio (SNR) regardless of the statistics on H. It has also been shown that when H is zero-mean complex Gaussian distributed, a unitary signal maximizes the capacity [24] and minimizes the union bound of word error rate (WER) [3] at high SNR. These results suggest that a good code can perhaps be constructed by collecting unitary codewords.
Secondly, it is better to have full-rank codeword pairs, where a pair of codewords, B(i) and B(j), is said to be pair-wisely full-rank if
rank
B(i) B(j)
= 2AT,
subject to N ≥ 2AT. This is because at fairly high SNR, the average error probability is well approximated by the sum of pair-wise word error rates, namely, the union bound [1].
Also at fairly high SNR, the pair-wise word error is in turn well approximated by Pr
Bˆ = B(j)
B(i) transmitted
≈ Q 1
√2kHk q
λmin(Lij)
where
Lij , IAR ⊗ B(i)T IN − PB(j) B(i)
,
and λmin(Lij) is the smallest eigenvalue of Lij. Here, “⊗” indicates the Kronecker product, and Q(x) , √12πR∞
x e−t2/2dt is the area under the tail of a standard Gaussian probability density function. Hence, if [B(i) B(j)] do not achieve full column rank, we can obtain by [13]
that
det
B(i)T IN − PB(j) B(i)
= 0.
This subsequently implies that λmin(Lij) = 0, and (6.2) will be close to 1/2 at fairly high SNR, which is a situation that a good code should avoid.
Therefore, a code that satisfies both the above criteria should guarantee a good pairwise-error-based union bound (which in turn hints to have a good performance). This viewpoint will be confirmed by the subsequent simulations.
6.2.2 The Proposed Code Design
Denote the information sequence by k = [k1 k2 . . . kK]T, where ki ∈ {±1}. The correspond-ing codeword is then proposed to be
B= 1
√AT
k k s
−k s k
where “ ” denotes the Hadamard product, and
s= It remains to show that the code just introduced satisfies pair-wise full-rank criterion.
Let Ai,j , B(i)TB(j). Then for the validity of the pair-wise full-rank criterion, it suffices to prove that
for 1 ≤ i, j ≤ 2K with i6= j. By denoting respectively the kth eigenvalue and kth eigenvector of Ai,jAT λ2 6= (N/AT)2 by differentiating the following two cases.
Case 1: si = sj = s.
We end this section by commenting that our design can be viewed as a high-dimensional variation of Alamouti codes. Hence, the unitary property is satisfied simply by the Alamouti code structure. By properly introducing the additional Hadamard product, our code can further fulfill the pairwise full-rank property.