4.3.2 BICM Pseudo-Capacity

The results from the previous section point out the fact that binary labeling strongly affects the BICM-GMI. Moreover, we have seen that the BICM-GMI curves for a given binary labeling intersect each other for different constellation sizes, which naturally leads to a quite general question: What is the best constellation, binary labeling, and input distribution for the BICM at a given SNR? Once this question is answered, approaching the fundamental limit will depend only on a good design of the binary encoder/decoder.

To formalize the previous question, and in analogy with (4.25), we define the BICM pseudo-capacity for a given constellation size as

4.64

where the optimization is over the constellation and its binary labeling, as well as over the vector of bitwise probabilities that induce a symbol-wise PMF (via (2.72)) satisfying the energy constraint . We use the name BICM pseudo-capacity because, in theory, the quantity in (4.64) might not be the largest achievable rate, which simply comes from the fact that the BICM-GMI is not the largest achievable rate for given . Nevertheless, the numerical results we present in this section will, in fact, show that the BICM pseudo-capacity can close the gap to the AWGN capacity, and thus, it is indeed a meaningful quantity to study.

Similarly to the CM capacity, the BICM pseudo-capacity in (4.64) represents an upper bound on the number of bits per symbol that can be reliably communicated using BICM, i.e., when for each SNR, the constellation, its labeling, and the bits' distribution are optimized.

In general, solutions of the problem in (4.64) are difficult to find, even when both the constellation and the labeling are fixed. This is because, unlike the CM-MI, the BICM-GMI is not a concave function of the input distribution. To clarify this, consider the following example.

Although not explicitly mentioned, the results in Fig. 4.13 are obtained by scaling the constellation for each so that . To formally define the problem in Example 4.17, we define the BICM constellation and labeling-constrained pseudo-capacity as

4.65

where the optimization is over all the and is chosen so that is satisfied.

For constellations with a small number of points with a fixed binary labeling, we may use an exhaustive search together with an efficient numerical implementation of the BICM-GMI (which we show in Section 4.5) to solve the problem in (4.65). While this approach has obvious complexity limitations, i.e., the complexity grows exponentially with the number of bits per symbol , it allows us to obtain an insight into the importance of the bit-level probabilistic shaping in BICM.

On the basis of the results in Examples 4.15–4.18, some concluding remarks about the BICM pseudo-capacity can be made. First of all, these results show the suboptimality of the BRGC in terms of maximizing the BICM-GMI for a given equally spaced constellation and uniform input distributions. For 8PAM and for low and medium SNR values, the NBC and the FBC give higher BICM capacities than the BRGC. Moreover, for asymptotically low rates, the optimum binary labeling for PAM constellations is the NBC. The results in these examples also show that a BICM system can be optimum for asymptotically low rates (i.e., it can achieve the SL). This can be obtained by using probabilistic shaping or by properly selecting the binary labeling.

4.3.3 Useful Relationships

In this section, we provide some useful relationships for the analysis of information-theoretic aspects of BICM transmission.

To show the usefulness of (4.66), consider (for simplicity) the AWGN channel. In this case, the conditional MI in (4.66) is given by (4.4) and can be expressed as

4.70

4.71

4.72

where

4.73

and where (4.72) follows from (2.75) and from the fact that , .

In (4.73), we use to denote a random variable with support and PMF given by (2.75), i.e., when the symbols are drawn from the subconstellation created by the binary labeling (see, e.g., Fig. 2.16). As the r.h.s. of (4.73) is an MI between and , we use the notation . Using (4.72) in (4.66) we obtain

4.74

The expression in (4.74) shows that the BICM-GMI can be expressed as a difference of CM-MIs. This interpretation is useful because it shows that a good characterization of MIs for arbitrary (sub)constellations leads to a good characterization of the BICM-GMI. This is particularly interesting as plenty of results already exist in the literature regarding CM-MIs, which, because of (4.74), can be “reused” to study the BICM-GMI. This approach has been taken, e.g., to study the low- and high-SNR behavior of BICM, where the asymptotic behavior of CM-MIs for arbitrary constellations is used to study the asymptotic behavior of the BICM-GMI.

We note that (4.76) generalizes the L-values defined in Section 3.3 allowing us to deal with arbitrary mismatched metrics . Theorem 5.1 demonstrates also that the distribution of the generalized L-values (4.76) suffices to calculate the BICM-GMI.

An immediate consequence of Theorem 5.1 is that when is given by (4.33), , , and . Thus,

4.81

In the case of uniformly distributed bits and for any generalized logarithmic-likelihood ratio (LLR), (4.75) becomes

4.82

Furthermore, if we assume the symmetry condition (3.74) is satisfied⁵ (i.e., , we obtain from (4.82)

4.83

4.84

Equivalently, we can express (4.83) as

4.85

where

4.86

4.87

and (4.87), inspired by (3.94), is a numerically stable⁶ version of (4.86).

Knowing the observations

4.88

and the transmitted bits , (4.85) can be transformed into a simple unbiased estimator of as

4.89

In the case of matched bit metrics with symmetric L-values and uniformly distributed bits, we obtain the estimate of the bit-level GMI via Monte Carlo integration

4.90

In a simple case of transmission over the scalar channel, i.e., when the model relating the channel outcome and the input is well known, we may calculate the GMI via the formulas provided in Section 4.5 with much better accuracy than the one obtained from (4.89) and (4.90). On the other hand, as it happens very often, the L-values are calculated using nonlinear expressions, and are affected by various random variables. Then, the direct integrals of Section 4.5 are too involved to derive and/or tedious to use. Moreover, because the PDFs or are also difficult to estimate (analytically or via histograms), the estimates (4.89) and (4.90) become very convenient.

4.4.1 Achievable Rates

Having analyzed CM and BICM, it is interesting to compare and relate them to multilevel coding (MLC), another popular coded modulation scheme we presented in Section 2.3.2. To this end, we start rewriting the CM-MI using the random variables , i.e.,

4.97

4.98

To obtain (4.97), we used the fact that the mapping is bijective, and to obtain (4.98), we used the chain rule of mutual information.

The quantity in (4.98) represents an MI between the bit and the output, conditioned on the knowledge of the bits . This bit-level MI represents the maximum rate that can be used at the th bit position, given a perfect knowledge of the previous bits.

MLC with multistage decoding (MSD) (see Fig. 2.5) is in fact a direct application of (4.98), i.e., parallel encoders send the encoded oddata using bits' positions , each of the encoders uses a coding rate . At the receiver side, the first bit level is decoded and the decisions are passed to the second decoder, which then passes the decisions to the third decoder, and so on, as shown in Fig. 2.6 (a).

Although different mapping rules produce different conditional MIs , the sum (4.98) remains constant. In other words, the sum of achievable rates using MLC encoding/decoding does not depend on the labeling and is equal to CM-MI. On the other hand, when MLC is based on parallel decoding of the individual levels (PDL), i.e., when no information is passed between the decoders (see Fig. 2.6b), BICM and MLC are equivalent in terms of achievable rates, although the main difference still remains: instead of encoder–decoder pairs, BICM uses only one pair.

Using (4.98) and (4.54), we can express the MI loss caused by BICM as

4.99

which clearly shows that the BICM-GMI will be approximately equal to the CM if the differences are small for all .

We have already demonstrated in Theorem 4.10 that the GMI is upper bounded by the MI, which indicates that the BICM-GMI is upper bounded by the CM-MI. This property can be also demonstrated analyzing the terms in the r.h.s. of (4.99). We note that, although often evoked in the literature, the data processing theorem cannot be used for the proof because the bits , the bit , and the observation do not form a Markov chain.

We are now in a position to establish the following inequalities

4.104

which hold for any , , , , and , and where and are related via (2.72). The second inequality in (4.104) is obvious because of constraining the input distribution to be discrete and unoptimized. The first inequality is due to Theorem 4.24 and is tight only if the bits conditioned on are independent, i.e.,

4.105

The expression in (4.105) holds, e.g., when using a 4QAM constellation with Gray labeling. In this case the bits and modulate the orthogonal dimensions of the constellation: and , respectively.

Similarly to (4.104), for fading channels, we obtain

4.106

Furthermore, because and are concave functions of the SNR, it follows from Jensen's inequality that

4.107

4.108

On the other hand, the BICM-GMI is not always a concave function of the SNR,⁷ and thus, we cannot guarantee an inequality similar to (4.107).

4.4.2 Robustness of the Encoding Scheme

While the rates achievable using MLC are never smaller than those attainable using BICM, it is important to note that the design of MLC, i.e., the choice of the coding rates for each of the coding levels, requires the knowledge of all the conditional MIs . This brings up an issue related to the robustness of the design with respect to lack of complete knowledge of the channel model.

To clarify this issue, we now assume that the MLC encoding/decoding scheme is designed for the AWGN channel for a coding rate . We also assume that the design is done so as to operate at the SNR decoding threshold , i.e., we design the encoders to ensure successful decoding at the SNR limit defined by the CM-MI, which we denote as

4.109

where is the inverse of the MI function defined in Section 4.1.

The decoding threshold of each of the MLC levels is given by

4.110

where is the inverse of the conditional MI .

We can thus define the SNR decoding threshold as the value of SNR, for which the message at all MLC levels can be decoded

4.111

In order to operate at the CM-MI limit with we need to satisfy the following:

4.112

that is, the SNR decoding thresholds for each of the decoders must be the same.

If a BICM scheme is designed for the same coding rate , the corresponding SNR decoding threshold is given by

4.113

where is the inverse function of the BICM-GMI.

To illustrate the previous definitions, we show in Fig. 4.15 the pairs and for a rate . The difference can be understood as the SNR penalty caused by using a BICM scheme for a given coding rate . For the BRGC, the penalty for is close to zero, however, for the BSGC it increases up to about .

We are now ready to see what happens if MLC and BICM are used when transmitting over a non-AWGN channel. In other words, we are interested in studying how the SNR decoding thresholds change if the design made for the AWGN channel is used in a different channel. This can be the case, e.g., when transmitting over a fading channel which was not foreseen during the design.

If the encoders with the rates with (designed so as to satisfy (4.112) for the AWGN channel) are used in a fading channel, the corresponding SNR decoding thresholds are given by

4.114

where is the inverse of the conditional MI .

Again, successful decoding at all decoding levels of MLC, is possible if is greater than all the individual SNR decoding thresholds in (4.114), i.e.,

4.115

Since the form of varies with , the values of the corresponding inverse functions will not be the same. Consequently, we cannot guarantee that the condition for the average SNR, similar to (4.112), will be satisfied, and, in general, we will have .

Therefore, the following relationship holds

4.116

and the difference should be interpreted as the SNR penalty because of a “code design mismatch”.

We emphasize that the mismatch happens because the coding rates were selected using the functions . If, on the other hand, we rather used , then the code designed would be matched to the fading channel. In such a case, the effect of the code design mismatch would appear if MLC were used over the AWGN channel. More generally, the code design mismatch phenomenon will be observed when the model of the channel used for the design is mismatched with respect to the actual channel.

Of course, in BICM transmission, the SNR decoding threshold is similar to (4.113), i.e.,

4.117

In many cases of practical interest, but in general, this relationship depends on the mapping as we discussed after (4.108). Of course, the relationship always holds, and we interpret the difference as the SNR gap because of the suboptimality of BICM. Unlike in the case of MLC, there is no effect of the code design mismatch: BICM remains “matched” to the channel independently of the underlying channel model.

4.5 Numerical Calculation of Mutual Information

In Sections 4.2 and 4.3, we have shown various examples of the CM-MI and the BICM-GMI calculated for the AWGN channel. While we cannot avoid the problem of enumerating the constellation points, we may efficiently calculate the multidimensional integrals (over ) appearing in the MI expressions using Gauss–Hermite (GH) quadratures. The methods we discussed in Section 4.3.3 may also be used but the numerical quadratures are much more accurate for small values of and, what is more important, they are much more reliable for being used in a numerical optimization.

For any function with bounded th derivative, the -points GH quadrature

4.118

is asymptotically () exact if we use the quadrature nodes as the th root of the Hermite polynomial

4.119

and the weights defined as

4.120

The values of and can be easily found for different values of , which determines the trade-off between the computation speed and the accuracy of the quadrature. The expression in (4.118) can be straightforwardly expanded to a quadrature over an -dimensional space as

4.121

where and . It is important to note that the expansion we use in (4.121) is not unique; however, it is a simple way of generalizing (4.118) to dimensions.

The CM-MI in (4.23) and in (4.24) can be expressed, respectively, as

4.122

4.123

where and are shown in Table 4.1. To obtain (4.122), we first used the substitution , then we used , and then split the logarithm of the quotient as a difference of logarithms. To obtain (4.123), we used in and split the logarithm of the multiplication as the sum of the logarithms.

Function	Expression

The BICM-GMI in (4.62) and (4.63) can be expressed, respectively, as

4.124

4.125

where and are shown in Table 4.1. To obtain (4.124), we first used (4.66), then the substitution , and then again split the logarithm of the quotient (in the conditional expectation (4.4)) as a difference of logarithms. The expression (4.125) can be obtained by repeating the procedure and using a uniform input distribution.

Using (4.121)–(4.123), we obtain the following ready-to-use expressions for the CM-MI

4.126

4.127

where (4.126) and (4.127) shall be used to evaluate the CM-MI with arbitrary and uniform input distributions, respectively.

Similarly, by using (4.121)–(4.125), we obtain the following ready-to-use expressions for the BICM-GMI

4.128

4.129

which shall be used to numerically evaluate the BICM-GMI with arbitrary and uniform input distributions, respectively.

All the results presented in this chapter were obtained with , which we found to provide an adequate accuracy in all numerical examples.

4.6 Bibliographical Notes

All the concepts in this chapter can be traced back to Shannon's works [1, 2]. For a detailed treatment of MIs, channel capacity, and so on, we refer the reader to standard textbook such as [3, 4].

The CM-MI has received different names in the literature. It is called joint capacity in [5], (constellation) constrained capacity in [6, 7], and coded modulation capacity in [8–12]. We use the name CM-MI to indicate that no optimization over the input distribution is performed.

The information-theoretical model for BICM was first introduced by Caire et al. using parallel memoryless BICO channels [8, Fig. 3]. Martinez et al. later refined this model in [10], where it was shown that the model of parallel channels leads to correct conclusions about achievable rates, but may yield incorrect error exponents/cutoff rates. Error exponents for BICM were also studied in [13].

The BICM-GMI has received different names in the literature. It is called parallel decoding capacity in [5], receiver constrained capacity in [6], and BICM capacity in [8–12]. Again, we refrain from using the word “capacity” to emphasize the lack of optimization over the input distribution. We adopted the name BICM-GMI, which fits well the finding of Martinez et al. [10] who recognized the BICM decoder as a mismatched decoder and showed that the BICM-GMI in (4.54) corresponds to an achievable rate of such a decoder. Theorem 4.11 is thus adopted from [10] and was also used in [14]. The results on the optimality of matched processing in Theorem 4.10 and Corollary 4.12 were inspired by [15].

As we indicated, the BICM-GMI is an achievable rate but it has not be proven to be the largest achievable rate. For example, a different achievable rate—the so-called LM rate—has been recently studied in [16, Part I]. Finding the largest achievable rate remains as an open research problem. Despite this cautionary statement, the BICM-GMI predicts well the performance of capacity-approaching codes, as shown in Example 4.13 (see also [17, Section 3], [18, Section IV]). The generator polynomials of the TENC used in Example 4.13 were chosen on the basis of the results in [19] and some of the puncturing patterns used were taken from [19, 20].

Changing the distribution of the constellation symbols—known as signal/constellation shaping—has been studied for a number years, see [21–24]. In the context of BICM, geometrical shaping (i.e., design of the constellation under the assumption of uniform distribution of the symbols) was studied in [5, 25, 26], and probabilistic shaping (i.e., varying the probabilities of the symbols and/or bits when using the fixed constellation) was proposed in [27, 28] and developed further in [16, 29–33]. Recently, a numerical algorithm to obtain optimal input distributions in BICM was proposed in [34]. As the BICM-GMI is not the largest achievable rate, probabilistic or geometric shaping in BICM yields what we call in Section 4.3.2 a “pseudo-capacity.”

The expression in (4.66) was first introduced in [9, Proposition 1], [7, eq. (65)] and later generalized in [35, Theorem 2] to multidimensional constellations and arbitrary input distributions. An alternative proof for Theorem 4.22 was presented in [16, Appendix B.1.1]. Achievable rates for 4D constellations for BICM in the context of coherent optical communications have been recently studied in [17].

The solution of (4.64) is in general unknown but results are available in the low-SNR regime. The low-SNR regime of the CM-MI has been studied in [36, 37], where conditions for constellations to achieve the SL are given. The fact that the SL is not always achieved for BICM was first noticed in [9] and later shown to be caused by the selection of the binary labeling [11]. The behavior of BICM in the low-SNR regime was studied in [9, 11, 35, 38, 39] as a function of the constellation and the binary labeling, assuming a uniform input distribution. Results for arbitrary input distributions under the indepence assumption of the bits have been recently presented in [40]. It is shown in [40] that probabilistic shaping offers no extra degrees of freedom in addition to what is provided by geometrical shaping for BICM in the low-SNR regime.

In terms of binary labelings, the NBC was shown to be optimal for PAM in the low-SNR regime [11, 35] and the FBC was analyzed in [41] for uncoded transmission. The 49 labelings that give different BICM-GMI in Fig. 4.12 correspond to the 49 classes in [42, Tables A.1 and A.2] and the 7 different classes of labelings that appear in Fig. 4.12 for high SNR correspond to those in [42, Table A.1]. The optimality of a Gray code was conjectured in [8, Section III-C] in a rather general setup, i.e., without specifying the Gray code, the constellation, or the SNR regime. This conjecture was later disproved in [43] where it is shown that for low and medium SNR, there exist other labelings that give a higher BICM-GMI (see also [12, Chapter 3]). The numerical results presented in [12, Chapter 3], [44] show that in the high-SNR regime, Gray codes are indeed optimal. A formal proof for the optimality of Gray codes in the high-SNR regime for arbitrary 1D constellations was recently given in [45]. A simple approximation for the BICM-GMI that can be used to optimize the binary labeling of the constellation was recently introduced in [18].

GH quadratures can be found in [46, Section 7.3.4] and tables with the coefficient and for different values of in [46, Appendix 7.3(b)]. Ready-to-use expressions for computing the CM-MI and BICM-GMI are given in [44].

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