Application Example

Considering the same system configurations as that used for Fig. 6.7(b), Fig. 6.8 shows the gain of using selective processing on the basis of the optimal execution order, i.e., semi- static scheduling. As we can observe from Fig. 6.8, the use of such a semi-static scheduling strategy yields a reduction on the computational energy consumption, while the decoding performance is maintained. Targeting FERs of interest, i.e., 1% _{∼ 0.1%, the achieved} reduction on the computational energy consumption is about20%.

5 6 7 8 9 10−4 10−3 10−2 10−1 100 SNR:(NtEs)/N0[dB] FER

Optimal execution order Semi-Static schedul.

(a) Decoding Performance

5 6 7 8 9 20 40 60 SNR:(NtEs)/N0[dB] Norm. Comput. Energy Comsum. (b) Decoding Complexity

Figure 6.8: Optimal static scheduling vs. semi-static scheduling with the selection rule parameterized byη[1] _{= 2 and η}[l]_{= 1.4· η}[l−1]_.

6.4

Summary

For an iterative receiver with two nested loops, the execution order of the inner and outer iterations needs to be decided. With respect to given channel statistics, the execution order that requires the minimal computational effort for achieving the minimal FER is of interest. This chapter initially has applied ACO for finding such execution order. As the ACO based method has no pre-assumption on the system parameters, it is applicable for general cases. In contrast, the EXIT-function based method in the literature has to rely on the assumption of asymptotically long codewords. Under practical constraints on the codeword length, the gain of using the ACO based method over the EXIT-function based method is pronounced. In order to further reduce the computational energy consumed by following the opti- mal execution order, we have subsequently introduced a semi-static scheduling strategy, which enables selective processing on the basis of the optimal execution order. Such semi- static scheduling strategy is able to reduce the computational energy consumption without degrading the decoding performance.

Finally, we have some comments on the possible extension of the ACO based method for solving the optimal execution order search problem in an iterative receiver with more than two nested loops. In principle, the ACO based method shall be straightforwardly extendable. We simply need to refine the tree graph of the execution order candidate set by adapting the out-degree of nodes in the tree to the number of nested iteration loops at the receiver. Based on the updated tree graph, the ACO algorithm labeled as MMAS is then applicable for finding the shortest path connected to the food source.

ML Decoding with

Incomplete Channel Information

In the previous chapters, the ML decoding problem was studied under the assumption that the instantaneous CSI is perfectly known by the receiver. Given the fact that in practice the knowledge of CSI is imperfect at the receiver, we extend our consideration to the ML decoding problem with incomplete channel information. Being more specific, we assume the knowledge of channel statistics is present at the receiver, but the instantaneous chan- nel realizations are unknown.

To accomplish the decoding task without knowing CSI completely, many typical re- ceivers have relied on the principle of synchronized detection [69], meaning that estimates of CSI are first calculated and then used as the actual CSI for coherent detection. For es- timating CSI, we have options between data-aided and non-data-aided algorithms. The estimation process of data-aided algorithms is based on pilot symbols. Non-data-aided al- gorithms do not require pilot symbols, but suffer from low estimation accuracy particularly at low SNRs [69]. In this chapter, pilot symbols are inserted into the data symbol sequence at the transmitter to support channel estimation.

After the empirical success of turbo decoding, iterative receiver concepts have been proposed, see, e.g., the editorials with the corresponding articles in [104, 105] and also the special issues in [9, 49]. Following the turbo principle, channel estimation should rely not only on pilot symbols, but also on the soft information acquired by the detector/decoder on data symbols. Using refined channel estimates, improvements on the decoding perfor- mance are permitted. The initial proposals for performing channel estimation and decod- ing in an iterative manner were heuristically obtained. In past years, systematic receiver designs have drawn many researchers’ attention. Being inline with the general approach of designing an optimum receiver in [69], we are particularly interested in a framework that allows us to systematically derive a receiver structure from the ML decoding problem. Without knowing instantaneous CSI, the likelihood function that the ML decoder tries to maximize typically has a complex form. In order to enable a real-time implementation of receiver in hardware, approximations are needed. One well-known high-SNR approxima- tion to the ML decoding problem is the joint MAP channel estimation and ML decoding problem. In [95], we have derived an approximate iterative solution to solve it.

In the literature, another general approach of designing receiver is to treat receiver de- 145

sign as attempting to solve a statistical inference problem. Following this approach, many iterative receiver algorithms, e.g., in [18, 45, 52, 59, 63, 76, 77, 125, 140], have been invented based on approximate inference algorithms, e.g., BP, expectation-maximization (EM), vari- ational message passing (VMP) and expectation propagation (EP). Here, we briefly note that the EM algorithm initially introduced to solve the ML estimation problem [24] can be viewed as a special instance of VMP, while VMP is a general purpose algorithm for variational Bayesian (VB) inference based on the mean field (MF) approximation [22, 121]. Using the EM approach for estimating a parameter, we obtain a hard estimate, namely, a particular value that the parameter can take on. In contrast, VMP yields a soft estimate, consisting of the hard estimate and also its reliability information. Furthermore, EP is a generalization of BP and it is obtained by forcing some beliefs to be members of a specific exponential family [70]. BP was applied in [125] for designing iterative receivers. How- ever, BP is not well suited to accomplish tasks that involve continuous random variables, e.g., channel estimation. Although the authors of [125] suggested to discretize the continu- ous random variables, the resulting algorithm still requires high complexity for sufficiently fine quantization. By projecting some beliefs into an exponential family, the computational intractability problem experienced by BP when dealing with continuous random variables can be efficiently resolved. Therefore, EP has been applied in [3] for channel estimation. However, the complexity required by EP is higher than that of VMP. And also, the authors of [3] encountered numerical issues while using EP. Therefore, VMP generally outperforms EP for channel estimation.

The algorithm derived in [95] to solve the joint MAP channel estimation and ML decod- ing problem can be identified as an outcome of combining EM with BP. Namely, the task of MAP channel estimation is based on the EM algorithm, while the iterative processing be- tween the detection and decoding units is an application of BP. The systematic derivation of such a hybrid message passing algorithm can be based on the generic message passing framework from [71] or based on the free energy approach based framework from [88]. Based on these frameworks, we can also combine BP and VMP, i.e., BP-VMP, which is an extension of BP-EM. The benefit of applying BP-VMP for designing iterative receivers has been empirically demonstrated in [99,100]. This motivates us to investigate its connection to the ML decoding problem.

In this chapter, we exemplarily consider a communication system as sketched in Fig. 7.1 and introduced in Section 7.1. Afterwards, the ML decoding problem with incomplete CSI is equivalently represented as a joint channel density estimation and ML decoding prob- lem in Section 7.2, where the joint MAP channel estimation and ML decoding problem can be identified as a special approximation of it, see Section 7.2.1. Following the approach introduced in the previous chapters, we further approximate the ML decoding problem by a constrained Bethe free energy minimization problem in Section 7.2.2. Attempting to find the ML solution by minimizing the constrained Bethe free energy, two approximate iterative solutions are presented in Section 7.3. Comparing them with the state-of-the-art hybrid message passing techniques, i.e., BP-EM and BP-VMP, the discovered connections on the one hand can help us understand the success of hybrid message passing algorithms in achieving near-ML decoding performance. On the other hand, they can confirm the universality of the Bethe free energy based approach to systematically derive efficient al-

Figure 7.1: A BICM based system under a frequency-flat Rayleigh fading SISO channel. gorithms for approximate ML decoding. Simulation results are presented and discussed in Section 7.4. The whole chapter is summarized in Section 7.5.

7.1

System Model

A BICM transmission system operating on a frequency-flat Rayleigh fading SISO channel is shown in Fig. 7.1. The information bit sequence m is first encoded by a convolutional encoder. The codebook of the employed CC is denoted as_{G. The output codeword c ∈ G} with length equal toNcis further interleaved and mapped onto the elements of a complex

modulation alphabet denoted as _{X . The resulting data symbols together with pilot sym-} bols are multiplexed into a single symbol sequence with lengthNs, i.e.,(sk)Nk=1s , where sk

represents the symbol transmitted at the time instantk. Let us group the time instants that are allocated for transmitting pilot symbols into a set and denote it as_Ipil. The remaining

time instants reserved for the data symbols are incorporated into the set_Idat ∆

={k}Ns

k=1\Ipil.

For a data symbolsk′ withk′ ∈ I_dat, it is mapped fromM_c = log∆ ₂(|X |) code bits in the

codeword c. Grouping the indices of their bit positions in the codeword into a set _Is,k′

and also describing the bits-to-symbol mapping rule asχ(_{·), we have s}k′ = χ([c]_I

s,k′). The

transmit energy per symbol, i.e.,E{|sk′|2}, is denoted as E_s.

At the receiver side, the received symbol at the time instantk equals

yk = hksk+ nk (7.1)

where hk ∈ C is the channel coefficient and nk ∈ C is additive white Gaussian noise.

For notational convenience, we introduce y, s, h and n as the vector representations of {yk}Nk=1s , {sk}Nk=1s , {hk}Nk=1s and {nk}Nk=1s , respectively. We assume {nk} is a temporally

white proper complex Gaussian noise sequence with zero-mean andE[_|nk|2] = N0. The

fading process_{hk} is stationary and zero-mean jointly proper Gaussian. The pdf of h is

given asp (h) =CN (h; 0Ns, Σh), where the mean vector 0Ns is a zero-valued vector with

lengthNsand Σh is the channel covariance matrix.