2.7 The Interleaver

The interleaver c02-math-0861 bijectively maps the binary codewords c02-math-0862 into the codewords c02-math-0863. The interleaver is determined by the interleaving vector c02-math-0864 which defines the mapping between c02-math-0865 and c02-math-0866 as follows:

2.99 equation

where c02-math-0868 (see (2.80)) and

2.100 equation

The sequences c02-math-0870 and c02-math-0871 are obtained, respectively, by reading columnwise the elements of the sequences of binary vectors c02-math-0872 and c02-math-0873. Very often, we will skip the notation which includes the interleaving vector c02-math-0874, i.e., we use c02-math-0875.

We now define two simple and well-known interleavers.

As the interleaving transforms the code c02-math-0888 into the code c02-math-0889, considerations in the domain of the codewords c02-math-0890 may help clarify the relationships between modulation and coding. In what follows, we “reuse” the DS-related definitions from Section 2.6.

Further, we adapt Definition 2.23 as follows.

We note that the interleaving does not change the HW of the codewords, and thus, we have c02-math-0903. On the other hand, in general, c02-math-0904. In fact, the GHW c02-math-0905 used in c02-math-0906 and in c02-math-0907 has, in general, different dimensions: c02-math-0908 for c02-math-0909 and c02-math-0910 for c02-math-0911.

At this point, it is also useful to define the generalized input-output distance spectrum (GIODS) of the code c02-math-0912, which relates the input sequences c02-math-0913 and the corresponding codewords c02-math-0914.

It is then convenient to introduce the input-dependent distance spectrum (IDS) and generalized input-dependent distance spectrum (GIDS) of the encoder.

As the IDS and GIDS defined above relate the codewords c02-math-0926 and the input sequences c02-math-0927, they depend on the encoding operation. They are thus different from the DS or GDS in Definitions 2.21 and 2.23 which are defined solely for the codewords, i.e., in abstraction of how the encoding is done.

The following relationships are obtained via marginalization of the GIODS

2.113 equation
2.114 equation

The codewords c02-math-0930 and c02-math-0931 are related to each other via the bijective operation of interleaving, which owes its presence to considerations similar to those we evoked in Section 2.6 to motivate the introduction of the DS. We repeat here the assertion we made that the “protection” against decoding errors can be related to the HW of the codewords (or the HD between pairs of codewords). Then,in order to simplify the decoder's task of distinguishing between the binary codewords, we postulate, not only to have a large distance between all pairs of codewords, but also to transmit every bit of the binary difference between the codewords, using different symbols c02-math-0932. In such a case, we decrease the probability that the respective observations c02-math-0933 are simultaneously affected by the channel in an adverse manner. These considerations motivate the following definition.

Clearly, the codeword diversity c02-math-0939 is bounded as

2.116 equation

We thus say that the codeword c02-math-0941 achieves its maximum diversity if c02-math-0942, which depends on how the interleaver allocates all the nonzero elements in c02-math-0943 to different labels c02-math-0944. In addition, if c02-math-0945, the codeword cannot achieve its maximum diversity. From now on, we only consider the cases c02-math-0946.

The interleaver diversity efficiency is bounded as c02-math-0955, where interleavers that achieve the upper bound are desirable. This is justified by the assertions we made above, namely that it is desirable to transmit bits in which the codewords differ using independent symbols. As a yardstick to compare the diversity efficiency of different interleavers, we define the random interleaver.

We note that the codewords c02-math-0961 are generated in a deterministic manner, but for a random interleaver, the mapping between c02-math-0962 and c02-math-0963 is random. Therefore, both the codeword diversity c02-math-0964 and the interleaver diversity efficiency c02-math-0965 are random. To evaluate them meaningfully, we then use expectations, i.e., we consider their average with respect to the distribution of the interleaving vector c02-math-0966 given in (2.119).10

For large c02-math-0992, treating c02-math-0993 as a continuous variable, we may apply a first-order Taylor-series approximation of c02-math-0994 around c02-math-0995, to obtain

From now, we will refer to c02-math-0997 as the average complementary interleaver diversity efficiency.

In Fig. 2.23, we show the expression (2.121) as well as the approximation in (2.127). We observe that the average complementary interleaver diversity efficiency is strongly affected by an increase in c02-math-0998. For example, when c02-math-0999, to keep the average complementary interleaver diversity efficiency at c02-math-1000, c02-math-1001 is necessary for c02-math-1002 but we need c02-math-1003 when c02-math-1004. It is also interesting to note from Fig. 2.23 that a random interleaver does not guarantee an average interleaver diversity c02-math-1005, even for large c02-math-1006.

c02f023

Figure 2.23 Average complementary interleaver diversity efficiency (solid lines) and its large-c02-math-1007 approximation (2.127) (dashed lines) for c02-math-1008 (hollow markers, e.g., c02-math-1009) and for c02-math-1010 (filled markers, e.g., •)

Another property of the code c02-math-1011 we are interested in is related to the way the codewords' weights are distributed across the bit positions after interleaving, which we formalize in the following.

Up to now, we have considered two effects of the interleaving of the code c02-math-1048 independently: the GDS in Definition 2.36 deals with the assignment of c02-math-1049 nonzero bits from c02-math-1050 into different positions c02-math-1051 at the modulator's input, while the interleaver diversity efficiency in Definition 2.40 deals with the assignment of these c02-math-1052 bits into labels at different time instants c02-math-1053. We may also jointly consider these two “dimensions” of the interleaving. Namely, for all c02-math-1054 with c02-math-1055, we have

i.e., the ratio between the number of codewords in the set c02-math-1057 which achieve their maximum diversity and the number of codewords in the set c02-math-1058 depends solely on c02-math-1059. The proof of (2.140) can be made along the lines of the proofs of Theorems 2.43 and 2.45.

Theorem 2.45 takes advantage of the enumeration over all possible interleavers c02-math-1060, which makes the enumeration over all codewords unnecessary. As we can use any codeword c02-math-1061, the assignment of the bits c02-math-1062 to the position c02-math-1063 in the label c02-math-1064 becomes independent of the codeword c02-math-1065. Thus, we can simply treat the assigned position c02-math-1066 as a random variable c02-math-1067. The assignment is not “biased” toward any position c02-math-1068, so c02-math-1069 has a uniform distribution on the set c02-math-1070, which leads to (2.139). Such a model is depicted in Fig. 2.24.

c02f024

Figure 2.24 Model of the random interleaver: the bits c02-math-1071 are mapped to the bit positions c02-math-1072 within the label c02-math-1073, where c02-math-1074 is a uniformly distributed random variable, i.e., c02-math-1075

The concept of random interleavers leads to the random position-assignment model from Fig. 2.24. Of course, in practice, a fixed interleaving vector c02-math-1076 is used. However, we will be able to apply the simple model depicted in Fig. 2.24 to analyze also a fixed interleaver, provided that it inherits some of the properties of the random interleaver. In such a case, we talk about quasirandom interleavers which we define in the following.

As we refer to properties of the interleavers for c02-math-1085, when discussing their effects, we have in mind a particular family of interleavers which defines how to obtain the interleaving vector c02-math-1086 for each value of c02-math-1087. Therefore, strictly speaking, the definition above applies to a family of interleavers and not to a particular interleaver with interleaving vector c02-math-1088.

The essential difference between the random interleaving and quasirandom interleaving lies in the analysis of the DS or GDS. In the case of random interleaving, instead of enumerating all possible codewords c02-math-1089, we fix one of them and average the spectrum over all possible realizations of the interleaver. In the case of a fixed (quasirandom) interleaving, we enumerate the codewords c02-math-1090 produced by interleaving all the codewords c02-math-1091 from c02-math-1092. The property of quasirandomness assumes that the enumeration over the codewords will produces the same average (spectrum) as the enumeration over the interleavers.

Of course, quasirandomness is a property which depends on the code and how the family of interleavers is defined. In the following example, we analyze a CC and two particular interleavers in the light of the conditions of quasirandomness.

c02f025

Figure 2.25 Complementary interleaver diversity efficiency c02-math-1125 for the CENC with c02-math-1126 and two types of interleavers: (a) rectangular and (b) pseudorandom

c02f026

Figure 2.26 Values of c02-math-1127 for c02-math-1128 when using a rectangular interleaver for different values of c02-math-1129: (a) c02-math-1130, (b) c02-math-1131, (c) c02-math-1132, and (d) c02-math-1133. The thick solid lines are the distributions of the random interleaver c02-math-1134

c02f027

Figure 2.27 Values of c02-math-1135 for c02-math-1136 when using a pseudorandom interleaver for different values of c02-math-1137: (a) c02-math-1138, (b) c02-math-1139, (c) c02-math-1140, and (d) c02-math-1141. The thick solid lines are the distributions of the random interleaver c02-math-1142

2.8 Bibliographical Notes

TCM was originally proposed at ISIT 1976 [1] and then developed in [2–4]. TCM quickly became a very popular research topic and improved TCM paradigms were soon proposed: rotationally invariant TCM [5, 6], multidimensional TCM [3, 7–9], TCM based on cosets and lattices [10, 11], TCM with nonequally spaced symbols [12–15], etc. TCM went also quickly from research to practice; it was introduced in the modem standards in the early 1990s (V.32 [16] and V.32bis [17]) increasing the transmission rates up to 14.4 kbps. TCM is a well-studied topic and extensive information about it can be found, e.g., in [18, 19, Section 8.12], [20, Chapter 4], [21, Section 8.2], [22, Chapter 14], [23, Chapter 18]. TCM for fading channels is studied in [20, Chapter 5].

MLC was proposed by Imai and Hirakawa in [24, 25]. MLC with MSD as well as the design rules for selecting the c02-math-1143 rates of the encoders were analyzed in detail in [26, 27]. MLC for fading channels, which includes bit interleavers in each level, has been proposed in [28], and MLC using capacity-approaching (turbo) codes was proposed in [29].

BICM was introduced in [30] and later analyzed from an information-theoretic point of view in [31, 32]. BICM-ID was introduced in [33–35] where BICM was recognized as a serial concatenation of encoders (the encoder and the mapper) and further studied in [36–40]. For relevant references about the topics related to BICM treated in this book, we refer the reader to the end of each chapter.

The discrete-time AWGN model and the detection of signals in a continuous-time AWGN channel is a well-studied topic in the literature, see, e.g., [19, Chapter 3], [41, Chapter 2], [42, Chapter 5], [43, Section 2.5], [44, Chapters 26 and 28]. For more details about models for fading channels, we refer the reader to [19, Chapter 13] or to [42, Chapter 3]. In particular, more details on the Nakagami fading distribution [45] are given in [42, Section 3.2.2].

The BRGC was introduced in [46] and studied for uncoded transmission in [47, 48] where its asymptotic optimality for PAM, PSK, and QAM constellations was proved. For more details about Gray labelings, we refer the reader also to [49]. The expansion used to define the BRGC was introduced in [47]. An alternative construction that can be used is based on reflections, which is detailed in [47, Section IV]. The FBC was analyzed in [50] for uncoded transmission and the BSGC was recently introduced in [51].

For 8PSK and in the context of BICM-ID, other labelings have been proposed; see [52] and references therein. For example, the SSP labeling was proposed in [40, Fig. 2 (c)], (later found via an algorithmic search in [38, Fig. 2 (a)], and called M8), or the MSP labeling [53, Fig. 2 (b)]. These two labelings are shown in Fig. 2.17 (b) and (c). The M16 labeling used in BICM-ID for 16QAM was first proposed in [38, Fig. 2 (b)].

The design of bit labelings for improving the performance of BICM-ID has been studied in [37, 38, 54–59] and references therein. Most of the works consider one-to-one mappers (i.e., a bijective mapping from labels to constellation symbols); however, when signal shaping is considered, a many-to-one mapping (c02-math-1144) may be useful, as shown in [59, Section 6.2].

The channel capacity defined in 1948 by Shannon [60] spurred a great deal of activity and approaching Shannon's limit became one of the most important problems among researchers for about 45 years. The complexity of encoding/decoding was always an issue but these limits have been continuously pushed by the development of integrated circuits. CENCs provided a low-complexity encoding strategy and the Viterbi algorithm [61] gave a clever and relatively low-complexity decoding method. Introduced by Elias [62] in 1955, CENCs were studied extensively and their detailed description can be found in popular textbooks, e.g., [23, Chapter 11], [63, Chapter 5], [64]. The name for CENCs with ODS was coined by Frenger et al. in [65], however, most of the newly reported spectra in [65] had already been presented in [66, Tables III–V], [67, Tables II–IV], as later clarified in [68].

TCs were invented by Berrou et al. [69] and surprised the coding community with their performance maintaining relatively simple encoding and decoding via iterative processing. Since then, they have been analyzed in detail in many works and textbooks, e.g., [63, Chapter 8.2; 23, Chapter 16]; TCs are also used in many communication standards such as 3G and 4G telephony [70, Section 16.5.3], digital video broadcasting (DVB) standards [71], as well as in deep space communications [72, 73].

Most often, the BICM literature assumes that random interleaving with infinite length is used [31], which leads to the simple random multiplexing model we have shown in Fig. 2.24. The formulas describing the diversity efficiency of the finite-length interleaver can be found in [32, Chapter 4.3].

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