5.4 Fading Channels

Up to now, we considered the case of transmission over fixed-gain channels. To address the problem of finding the distribution of the L-values in fading channels, we define the CDF in (5.31) explicitly conditioned on the SNR, i.e.,

5.141

Further, we average the CDF with respect to , and obtain the PDF of the L-values by differentiating with respect to , i.e.,

5.142

5.143

5.144

where, similarly to (5.141), we define

5.145

i.e., the dependence of on the is made explicit.

Having changed the order of differentiation and integration, the PDF of the L-values for fading channels in (5.144) and (5.145) is obtained by averaging over the distribution of the SNR. As we already derived for nonfading channels, we can thus reuse the results from the previous sections. Before presenting the result regarding the PDF of the L-values for fading channels, we introduce a useful lemma.

5.5 Gaussian Approximations

Although, as explained in the previous sections, we can find the PDF through the explicit derivation of the CDF, such an approach has two main disadvantages: (i) the expressions require analysis of the tessellation space and (ii) the resulting forms are piecewise functions. The former is critical when we have to deal with arbitrary constellations, particularly when the dimensionality of the signal space increases beyond . The latter complicates the analysis of the operations on the PDFs. For example, recall that the maximum likelihood (ML) decoderadds L-values, and thus, to characterize the performance of the decoder, we need to be able to find the CDF of the sum of L-values. This in turn requires finding the convolution of the PDFs and the integration of its tail,¹⁰ which becomes difficult for PDFs defined via piecewise functions.

The main objective of this section is to provide a model for the PDF of the L-values, such that (i) its parameters are easy to find, (ii) its form simplifies operations on the PDF of the L-values (such as convolution), and (iii) the integration over its tail accurately estimates the actual value of the probability of error.

Assuming that the L-values are Gaussian is convenient because it will, indeed, simplify the operations of convolution (necessary to find the PDF of the sum of the L-values) and integration (necessary to find the error probability). The Gaussian model is also interesting because the expression we derived for nonfading channels already contains Gaussian pieces whose parameters (mean and variance) depend in closed form on the constellation and labeling. The main idea is thus to take only the Gaussian piece from the piecewise model and to ignore windowing and weighting (correction) factors. The resulting simplicity is indeed appealing and, in such a case, the corresponding approximate PDF conditioned on a transmitted symbol will be

5.167

where the mean and variance are the parameters related to a Gaussian piece , indexed by and , i.e.,

5.168

5.169

where and are given by (5.43) and (5.44), respectively. The main problem consists then in finding the indices and such that (5.167) represents “well” the PDF .

We continue to keep in mind that the particular choice of and should be assessed using the criterion related to the objective we fixed, namely, an the accurate evaluation of the integration over the tails of the PDF. As the details on the performance evaluation of BICM receivers are mostly given in Chapter 6, here, we will only highlight the most important features of the two proposed heuristics with respect to their impact on the accuracy of evaluation of the BEP for uncoded transmission. While passing through Section 6.1 may be helpful, the following contents can be understood without such additional reading.

5.5.1 Consistent Model

For notational convenience, we will denote the transmitted symbol by and assume that it belongs to the subconstellation , i.e., is the value of the th bit in the label , (, see Section 2.5.2). The next definition introduces the first Gaussian model for the PDF of the L-values, namely, the so-called consistent model (CoM).

Definition 5.15 states that the tessellation region that provides the most “representative” Gaussian piece (i.e., the one whose parameters will be used for the approximation in (5.167)) is obtained by considering the transmitted symbol and the closest symbol from . From (5.43), we know that

5.172

5.173

5.174

and

5.175

Using (5.174) and (5.175) in (5.168) and (5.169), we conclude that for any ,

5.176

5.177

where

5.178

is the distance between and the closest symbol having a different bit at position . Therefore, (5.167) becomes

5.179

which has the same form we would obtain for a 2PAM constellation if we used and in (5.52).

The PDF in (5.179) also satisfies the consistency condition in Definition 3.8. This particular property explains the name we gave to the model that also corresponds to decomposing the BICM into “virtual” 2PAM transmissions, each characterized by the distance between the corresponding virtual 2PAM symbols.

Let us now verify how the obtained models affect the calculation of the BEP, which is an important parameter characterizing digital communications.¹¹ Here, the BEP for uncoded transmission in (3.110) can be obtained directly from the definition of the L-value.

5.187

5.188

where

5.189

is the BEP conditioned on the transmitted symbol .

For the case in Example 5.16, using the approximate PDF in (5.183), we obtain and , and from (5.184), we obtain . Thus,

5.190

5.191

We note that the expressions in (5.190) and (5.191) are similar to the exact error probability expressions given in (6.58) and (6.60), respectively. Although the dominant Q-function correctly appear in the expressions derived using the CoM, differences appears when other terms are compared. We also note that for sufficiently high SNR, . Thus, depending on the position , the transmitted bits are more (or less) prone to errors. This so-called unequal error protection (UEP) will be analyzed in Chapter 8.

For the case in Example 5.17, we obtain

5.192

5.193

The first (dominating) terms in (5.192) and (5.193) are again the same as the dominating terms in the exact expressions (6.65) and (6.66), and the differences appear only in the remaining terms.

Finally, it interesting to analyze the CoM at high SNR. When , is increasingly narrow and centered around . After the transformation , the L-values are most likely to lie in the vicinity of . In other words, for high SNR, is centered around and the components for or vanish. This is in fact what is observed in Figs. 5.22 and 5.23 for , i.e., the CoM approximates well the true PDF around its mean value. This observation has often been used as a justification to use the CoM to approximate the PDF of the L-values.

5.5.2 Zero-Crossing Model

For any , the conditional BEP in (5.189) can be expressed as

5.194

which shows that to have a good approximation of the conditional BEP, an accurate model of the tail of the PDF is necessary. When observing the results obtained with the CoM (see e.g., Fig. 5.22 (b)), we note that the (left) tail of the PDF is not well approximated. Consequently, the second terms in the BEP expressions obtained in Examples 5.16 and 5.17 do not match the terms that appear in the exact evaluation.

In this section, we introduce the zero-crossing model (ZcM) which aims at approximating the PDF well around , and thus, removing some of the discrepancies (in terms of BEP) observed when using the CoM. Before giving a formal definition, we introduce this model using an example. Consider an 8PSK constellation labeled by the BRGC, , , and . The exact PDF and the CoM approximation are shown in Fig. 5.24, where the shaded regions represent the integral in (5.194). This figure clearly shows that the CoM approximation fails to predict well the conditional BEP in (5.194). On the other hand, this figure also shows the ZcM defined below, which uses the Gaussian piece around . This results in a PDF that approximates much better the tail of the exact PDF, and thus, the conditional BEP in (5.194).

We can now compare the ZcM in Definition 5.18 with the CoM in Definition 5.15. First of all, we see that both ZcM and CoM require finding the symbol from which is closest to . This symbol (and the corresponding index) are thus common in both models. The difference is how to determine the complementary index (when ) or (when ). In Definition 5.15, this index is simply taken as equal to , while in Definition 5.18, we perform a search over to find closest to . In some cases, both definitions yield exactly the same results, namely, when the symbols and belong to the tessellation region they define, i.e., . In these cases, the PDF conditioned on or obtained from ZcM and CoM will be same.

Similarly to (5.190) and (5.191), now that we have the ZcM PDF of the L-values for 4PAM labeled by the BRGC in (5.197) and (5.198), we can calculate the BEP. As we have already stated in Example 5.19, the only difference with the CoM occurs for and , where we have and then we obtain

5.201

5.202

Again we compare the expressions in (5.201) and (5.202) with the exact expression in (6.58) and (6.60). Unlike in the CoM case, for the ZcM, the error probability for coincides with the exact calculation. For , the result is the same as in (5.191).

5.5.3 Fading Channels

The extension of the developed formulas to the case of the fading channel may be done by averaging the approximate Gaussian PDF in (5.167) over the distribution of the fading

5.208

In the particular case of Rayleigh fading, using the same approach as those leading to (5.160), we obtain

5.209

where and depend also on the Gaussian simplification strategy (CoM vs. ZcM).

5.5.4 QAM and PSK Constellations

Two of the most practically relevant constellations are the PAM and PSK constellations labeled by the BRGC. In these cases, it is possible to determine the forms of the approximate PDF without algorithmic steps. This is thanks to the structure of the binary labeling as well as to the regular geometry of the constellation. In other words, in these cases we are able to determine the parameters of the approximate PDF (i.e., the indices and of the representative tessellation regions ) as a function of the index of the transmitted symbol .

We start by considering the BRGC for 16PAM shown in Fig. 5.26. We note that for a given , the constellation may be split into groups of consecutive symbols. These groups are identified by rectangles on the left-hand side (l.h.s.) of Fig. 5.26. Furthermore, because of the symmetry of the labeling, it is enough to consider the PDF conditioned on , which implies that only the symbols are relevant for the analysis. For any symbol within each group, the closest symbol from lies also within the group. Because of the symmetries of the constellation, it is enough to analyze the first group, shown as shaded rectangles in Fig. 5.26.

To obtain the CoM, we note that for any (because we analyze , it follows that ), the closest symbol from is (e.g., we have for , for , etc.). The distance between the symbols is given , so we obtain

5.210

By using (5.210) in (5.179), we obtain the CoM PDF for an PAM constellation labeled by the BRGC, namely, for any (),

5.211

Similarly, in the case of the ZcM, it is enough to characterize the PDF of the symbols within the first group we identified because the closest “zero-crossing” tessellation region is also defined by the symbols from the same group. These regions are shown as thick lines in Fig. 5.26. Namely, for , we easily find and ; for , we obtain and ; and so on; then, we also find . More generally, for any , and . Thus, for any in the group and , we find from the relations (5.43) and (5.44) that

5.212

5.213

where (5.212) follows from using . Using (5.212) and (5.213) in (5.167) gives

5.214

The results in (5.211) and (5.214) are summarized in Table 5.1.

Gaussian Model
CoM
ZcM

The case of PSK shown in Fig. 5.27 may be treated in a similar way. The main difference is that the L-value and will have the same distribution because the groups of the symbols for and have the same form due to the circular symmetry of the constellation, i.e., the first and the last symbols ( and in Fig. 5.27) become adjacent. It is thus enough to calculate the parameters of the simplified form of the PDF for and consider only symbol , i.e., . Again, for the CoM, we have and , so from (5.135), we obtain . For the ZcM, we use and and thus and from (5.132), we get

5.215

These results are summarized in Table 5.2.

Gaussian Model
CoM
ZcM

5.6 Bibliographical Notes

The idea of the BICM channel can be tracked to [1], where the model from Fig. 5.2 corresponds to the model in [1, Fig. 3]. The model in Fig. 5.1 corresponds to the one introduced and formalized later in [2, Fig. 1]. The statistics of the L-values for the performance evaluation have been used in various works, e.g., in [1–4]. The formalism of using the PDF to evaluate the performance of the decoder in BICM transmissions was introduced in [5]. A probabilistic description of the L-values was also considered for the analysis of the decoding in turbo codes (TCs), low-density parity-check (LDPC) codes, or turbo-like processing in [6–8], respectively.

The probabilistic model for the L-values and 2PAM is well known [6 8, 9]. The explicit modeling of the PDF for BICM based on quadrature amplitude modulation (QAM) constellations appeared in [10–13], for phase shift keying (PSK) constellations labeled by the BRGC in [14], and for the case of arbitrary 2D constellations in [15 16]. The effect of fading on the PDF of L-values was shown for 2PAM in [2 6], for PAM in [17–19], and for 2D constellations in [20]. The probabilistic modeling of the L-values has also been considered in relay channels [21], and in multiple-input multiple-output (MIMO) transmission [22].

The Gaussian model of the L-values is well known and has been applied in [23]. The CoM defined in Section 5.5.1 has been used in [24] while the ZcM defined in Section 5.5.2 was proposed in [10 11], formalized in [12], and extended to the case of nonequidistant constellations in [25]. The ZcM has been recently used in [26] to study the asymptotic optimality of bitwise (BICM) decoders.

The space tessellation discussed in Section 5.3.1, based on solving the set of inequalities in (5.72), exploits the well-known duality between the line and points description [27, Chapter 8.2]. For details of finding a convex hull, we refer the reader to [27, Chapter 1.1], [28, Sec. 2.10]. The description of a method to find the vertices of the hull may be found in [28, Sec. 2.12].

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