2.5 Chapter Conclusions
3.1.1 Iterative Detection and Decoding Fundamentals
3.1.1.1 System Model
Before introducing the channel coding blocks in our MIMO system model, let us briefly review the mathematical model of a SDMA system supportingU users and having N receive antennas at the
BS, which is formulated as:
y=Hs+n, (3.1)
wherey, H, s, n are the (N×1)-element received signal column vector, the (N×U)-element FDCTF matrix, the(U×1)-element transmitted signal column vector, and the(N×1)-element
3.1.1. Iterative Detection and Decoding Fundamentals 53 Deinterleaver −1 Interleaver Binary Source Hard Decision LP2 LA2 LA1 LE1 Interleaver LE2 LP2,i LP1
...
n AWGN Sink Binary H Modulator Channel Channel Encoder code rate R s y MS BS Decoder Detector MAPFigure 3.1: Schematic diagram of iterative detection and decoding
AWGN column vector, respectively. Each element sm of the transmitted signal vector s can be further represented assu = map(xu), u = 1, 2, ..., U, where map(·)represents a specific bit- to-symbol mapping scheme and x<u>
is a(log2Mc×1)block of raw bits. In other words, each element of the transmitted signal vectors, i.e. a constellation symbol, containslog2Mcnumber of information bits.
When no channel encoder is employed at the transmitter, the estimates of the transmitted sig- nal s can be obtained by the low-complexity near-ML detectors of Chapter 2. Note that all the low-complexity near-ML SDs we encountered so far in Chapter 2 are HIHO detectors.
Thanks to the employment of channel coding, the SNR required for achieving a desirable BER may be further reduced. Hence in Figure 3.1 a MIMO system employing a channel encoder and an iterative receiver is portrayed. The interleaver and deinterleaver pair seen at the receiver side of Figure 3.1 divides the receiver into two parts, namely, the inner MAP detector and the outer decoder. Note that in Figure 3.1, the subscript ‘1’ denotes variables associated with the inner de- tector, while the subscript ‘2’ represents variables associated with the outer channel decoder. It was detailed throughout [104] and [105] that the iterative exchange of extrinsic information be- tween these serially concatenated receiver blocks results in substantial performance improvements. In this treatise we assume familiarity with the classic turbo detection principles [104]. Natually, the inner MIMO detector has to be capable of processing the soft-bit information provided by the soft-output channel decoder. On the other hand, the outer channel decoder also has to be capable of processing the soft reliability information provided by the soft-output inner MIMO detector. The resultant soft bit information is iteratively exchanged between the inner MIMO detector and the outer channel decoder.
3.1.1. Iterative Detection and Decoding Fundamentals 54
3.1.1.2 MAP Bit Detection
In contrast to the conventional HIHO detector, which outputs hard symbol decisions, and hence results in hard bit decisions also at the output of the demodulator, the inner MIMO detector of Figure 3.1 has to be capable of providing soft bit reliability information for further processing by the outer channel decoder. The advantage of providing soft bit information is that the channel decoder benefits from exploiting the reliability information provided by the detector and returns to the detector its improved-confidence soft-information in the interest of iteratively improving the resultant A Posteriori Probability (APP). Hence, the probability of bit errors is minimized. This SISO scheme may be referred to as a MAP detector. Conventionally, the APP is quantified in terms of the Log-Likelihood Ratio (LLR) as [104]:
LD(xk|y) =lnP[xk = +1|y]
P[xk = −1|y], (3.2)
wherey is the received symbol vector, andxk, k=0, 1, ..., U·log2Mc−1 is the kth element of the corresponding transmitted bit vectorx. Since the bits in the vector x have been channel encoded and scrambled by the interleaver, we may assume that the bits of the vector x are statistically independent of each other. With the aid of Bayes’ theorem, the LLRs of Eq.(3.2) can be rewritten as [51] [106]: LD(xk|y) = ln p(y|xk = +1)P[xk = +1]/p(y) p(y|xk =−1)P[xk =−1]/p(y) (3.3) = lnP[xk = +1] P[xk =−1] +ln p(y|xk = +1) p(y|xk = −1) (3.4) = LA(xk) +ln ∑x∈Xk,+1 p(y|x)·exp ∑j∈Jk,xLA(xj) ∑x∈Xk, −1 p(y|x)·exp ∑j∈Jk,xLA(xj) | {z } LE(xk|y) , (3.5)
where Xk,+1 represents the set of M
U c
2 number of legitimate transmitted bit vectors x associated withxk = +1, and similarly, Xk,−1is defined as the set corresponding toxk = −1. Specifically, we have:
Xk,+1={x|xk = +1}, Xk,−1={x|xk =−1}. (3.6) Note here that the value of xk = −1 represents a logical value of 0, while xk = 1 represents a
logical value of1. Furthermore Jk,xis the set of indices j, which is defined as:
Jk,x={j|j=0, 1, ..., U·log2Mc−1, j6=k}. (3.7) The a priori LLR valueLAdefined for thejth bit is given by [104]:
LA(xj) =lnP[xj = +1]
P[xj = −1]. (3.8)
According to [51], following a number of manipulations, the a posteriori LLR value can be ex- pressed with the aid of the a priori LLRs as:
LD(xk|y) = LA(xk) +ln∑x∈Xk,+1 p(y|x)·exp( 1 2x[Tk]·LA,[k]) ∑x∈Xk,−1 p(y|x)·exp(1 2x[Tk]·LA,[k]) | {z } LE(xk|y) , (3.9)
3.1.2. Chapter Contributions and Outline 55
where the subscript[k]denotes the exclusion of thekth element of a vector. Hence, x[k]represents a
specific sub-vector of the bit vectorx obtained by omitting thekth component and retaining the rest
of them. Similarly,LA,[k]represents the specific sub-vector of the a priori LLR vectorLAobtained
by excluding thekth element, where LAis the vector containing the a priori LLR value of all the bits inx.
Observe from Eq.(3.9) that the a posteriori LLR is equal to the sum of the a priori LLR and the so-called extrinsic LLR, which is the second component in the equation. Note that although the above derivation of the soft reliability information is valid for the bit vectorx1which is associated with the inner MIMO detector, the subscript ‘1’ is omitted, since Eq.(3.9) also holds for the bit vectorx2associated with the outer channel code. Assuming that an AWGN channel is encountered, the conditional probability of receiving the MIMO output signaly, provided that x was transmitted , namelyp(y|x), can be computed as:
p(y|s=map(x)) = exp[− 1 2σ2 w · ||y−Hs|| 2] (2πσ2 w)N , (3.10)
where the denominator is a constant when the noise variance 2σw2 is constant, hence it can be
omitted in the calculation of the LLR values. In order to reduce the computational complexity imposed, the Jacobian logarithm [104] may be employed to approximate the extrinsic LLRs as follows:
jac ln(a1, a2) = ln(ea1+ea2), (3.11)
= max(a1, a2) +ln(1+e−|a1−a2|), (3.12) where the second term may be omitted in order to further approximate the original log value, sinceln(1+e−|a1−a2|)can be regarded as a refinement of the coarse approximation provided by the
maximum. Consequently, when using the above-mentioned Jacobian approximation, the extrinsic LLR, i.e. the second term of Eq.(3.9) can be rewritten as:
Le(xk|y) = 1 2xmax∈Xk,+1{− 1 σ2 w|| y−Hs||2+x[Tk]·LA,[k]} − 12 max x∈Xk,−1{− 1 σ2 w|| y−Hs||2+xT[k]·LA,[k]}, (3.13) which represents the information exchanged between the inner MIMO detector and the outer chan- nel decoder, as seen in Figure 3.1.
3.1.2 Chapter Contributions and Outline
Even with the aid of the Jacobian approximation of Eq. (3.12), the calculation of the extrinsic LLR value using Eq. (3.13) may still impose an excessive computational complexity, depending on the number of users U and on the constellation size Mcof the modulation scheme employed, since a brute-force full-search has to be carried out by the MAP detector in order to find the joint maximimum of the two terms of Eq. (3.13). From our discourse on the SD scheme provided in Section 2.2 as well as in the light of the corresponding complexity reduction techniques of Section 2.3, we may argue that the HIHO SD constitutes a computationally efficient solution to
3.1.2. Chapter Contributions and Outline 56
the ML detection problem in uncoded MIMO systems. For the sake of approaching the channel capacity at a low complexity, the SISO SD algorithm was contrived by Hochwald and ten Brink in [51], where a list of the best hypothesized transmitted MIMO symbol candidates was generated, which was representative of the entire lattice in computing the soft bit information, resulting in the concept of the LSD of Section 3.2.1. However, in order to achieve a good performance, when the LSD is employed in an iterative detection aided channel coded system, the list size has to remain sufficiently large, resulting in a potentially excessive complexity. Hence, for the sake of further reducing the complexity imposed by the LSD of Section 3.2.1, we proposed various solutions to the problem of how to maintain a near-MAP performance with the aid of a small candidate list size. More specifically, the novel contributions of this chapter are as follows:
• Our discovery is that in contrast to the conventional SD, it is plausible to set the search center of the SD to a point which is typically closer to the real ML solution than the conventional LS or MMSE solution. Commencing the search from a more accurate search center may be considered as a process of search-complexity reduction.
• A generic center-shifting SD scheme is proposed for channel coded iterative receivers based on the above-mentioned perception, which substantially reduces the detection complexity by decomposing it into two stages, namely the iterative search-center-update phase and the reduced-complexity search around it. Three search-center-update algorithms are devised in order to iteratively shift the search center to a point closer to the true ML point with the aid of the soft-bit-information delivered by the outer channel decoder.
• We propose a novel complexity-reduction scheme, referred to as the Apriori-LLR-Threshold (ALT) based technique for the LSD, which is also based on the exploitation of the soft-bit- information, namely, the a priori LLRs provided by the outer channel decoder in the context of iterative detection aided channel coded systems.
• We significantly improve the performance of the conventional two-stage SD-aided turbo receiver by intrinsically amalgamating our proposed center-shifting-assisted SD with the decoder of a Unity-Rate-Code (URC) having an Infinite Impulse Response (IIR), both of which are embedded in a channel-coded SDMA/OFDM transceiver, hence creating a pow- erful three-stage serially concatenated scheme. Moreover, for the sake of achieving a near- capacity performance, Irregular Convolutional Codes (IrCCs) are used as the outer code for the proposed iterative center-shifting SD aided three-stage system.
• The convergence characteristics of the proposed schemes are visualized and analyzed with the aid of EXIT charts. Furthermore, performance versus complexity comparsions are car- ried out amongst the above-mentioned novel schemes.
The remainder of this chapter is organized as follows. The fundamentals of the conventional LSD are briefly reviewed in Section 3.2.1, followed by a discussion on the center-shifting the- ory in the context of the SD in Section 3.2.2, which partitions the SD into two parts, i.e. the search-center-update phase and the search around it. Then, three search-center-update algorithms are contrived in Section 3.2.3 in order to iteratively update the search center to a point, which is expected to be increasingly closer to the true ML MIMO symbol point. This search-center-update