Chapter 8
Interleaver Design
In binary modulation all the bits receive the same “treatment” when transmitted over the channel, i.e., the L-values have the same conditional probability density function (PDF). The error probability is also the same for all bits, i.e., each bit is equally “protected” against errors. High-order modulations, on the other hand, introduce unequal error protection (UEP). This is an inherent property of bit-interleaved coded modulation (BICM) and depends on the labeling and the form of the constellation. While previously we considered the use of quasi-random interleavers (which average out the UEP), in this chapter we want to take advantage of the UEP. To this end, we propose to design interleavers for BICM, which requires the analytical tools developed in the previous chapters to be refined.
This chapter is organized as follows. In Section 8.1 we introduce the idea of UEP and study a particular interleaver that takes into account the presence of the UEP: the so-called multiple-input interleaver (M-interleaver). In Section 8.1.1 we study the performance of BICM with M-interleavers generalizing the results from Chapter 6. In Section 8.1.1 we also study the problem of a joint interleaver and code design for such BICM transceivers.
8.1 UEP in BICM and M-interleavers
UEP may be easily explained using the communication-theoretic tools of Chapter 6 or the information-theoretic tools of Chapter 4. Let us revisit these concepts through a simple example.
Figure 8.1 UEP for 8PAM constellation labeled by the BRGC over the AWGN channel: (a) BEP for uncoded transmission in (6.26) and (b) normalized BICM-GMI and the bit-wise MIs 
Example 8.1 illustrates the concept of UEP caused by the binary labeling. In what follows we focus on its communication-theoretic analysis. Namely, we characterize the performance of the BICM decoder by explicitly taking into account the fact that the L-values calculated by the demapper for different bit positions have different distributions.
8.1.1 Preserving UEP in BICM
Let us consider a model for the BICM transmission, which generalizes the BICM channel in Fig. 5.1. This is done in Fig. 8.2 where all the elements appearing after the encoder and before the decoder are treated as a BICM channel with a vectorial input 
and a vectorial output 
. In this way, the code bits from the 
different classes are explicitly shown in Fig. 8.2 as the inputs to the BICM channel.
Figure 8.2 Refined model of the BICM channel: the 
classes of code bits are identified at the inputs of the BICM channel
Figure 8.3 Zehavi's BICM channel: (a) each class of the code bits is individually interleaved and mapped to a given input of the mapper (multiple interleavers) and (b) equivalent parallel channel model defined by the conditional PDFs 
, 
(parallel channel model)
As discussed in Section 6.2.4, if the interleaver complies with the conditions of quasirandomness (see Definition 2.46), the L-values 
may be considered independent. Using similar considerations, for 
, we can replace all the elements of the generalized BICM channel by a set of parallel channels. This is shown in Fig. 8.3 (b), where each channel yields L-values having distribution 
.
Let us now compare the conventional single-input interleaver (S-interleaver) 
with the M-interleaver shown in Fig. 8.3. The S-interleaver is the one we studied in the previous chapters and operates without differentiating the classes of code bits. For the S-interleaver, according to the considerations in Section 6.2 leading to (6.227), we have
For the M-interleaver with 
, the 
th L-values 
is the deinterleaved version of 
, so we obtain
Thus, using the S-interleaver we average the PDFs of the L-values and remove the effect of UEP. Eliminating UEP was, in fact, considered useful in some works on BICM (as it simplifies the analysis). On the other hand, the M-interleaver considered above allows us to preserve the UEP introduced by the binary labeling. While the idea of using M-interleavers was already present in the early works on BICM, this concept was not thoroughly studied in the literature.
We will show that the M-interleaver we propose are a generalization of S-interleavers, and thus, with an appropriate design, we can only obtain performance gains. This is shown in Section 8.1.4. From now, we refer to BICM transmission with S-interleavers as BICM-S, and BICM with M-interleaver will be called BICM-M. Further, in order to exploit the presence of UEP, we will generalize the M-interleaver to deal with the case of 
and derive performance metrics that will allow us to design such M-interleavers.
The order in which we enumerate the classes of bits is arbitrary and so is the numbering of the mapper's inputs. Instead of changing the order of the encoder's output or the order of the mapper's inputs, we may modify the M-interleaver shown in Example 8.2. For example, we may pass the bits from the first encoder's output to the second mapper's input (and not the first one as we did in Example 8.2), those from the second encoder's output to the third mapper's input, and those from the third encoder's output to the first mapper's input. In general, there are 
possible permutations, each changing the structure of the transceiver. One of the questions we will answer in this chapter is whether such a permutation changes the performance of the receiver for a given encoder. Choosing the most appropriate permutation becomes a part of interleaver design.
8.1.2 The M-Interleaver
In order to deal with the more general case 
, we propose a particular interleaver shown in Fig. 8.4, where we use 
interleavers at the input, the deterministic bit-reorganizing multiplexer (MUX) and 
interleavers at the output of the MUX.
Figure 8.4 The M-interleaver: the input sequences 
are passed through 
interleavers 
and the resulting sequences 
are reorganized by a deterministic MUX into sequences 
, which are next interleaved into output sequences 
The role of the MUX is to reorganize the interleaved bits' sequences 
. These sequences 
are first divided into subsequences as follows:
where each subsequence 
contains 
bits and 
is constrained to be such that 
. Next, the subsequences are concatenated into output sequences
each of them composed of 
bits, as required to generate 
labels 
.
Figure 8.5 Example of interleaving via the M-interleaver with 
, 
. The values defining the MUX are 
, 
, 
, 
, and 
The M-interleaver is then defined by a matrix 
with entries 
, which have to satisfy two obvious constraints. The first one is that all the bits 
must be assigned to one of the mapper's input, i.e.,
which translates into
The second constraint is that each mapper's input is equally “loaded” with bits from the encoder's output
or equivalently,
The matrix 
can then be written as
where the last row and the last column of 
take into account the constraints imposed on 
. Consequently, when designing 
, only 
for 
and 
may be freely set (considering also 
).
The motivation behind the structure of the M-interleaver should be now clear: we would like to assign the code bits to the protected positions so as to improve the performance of the receiver. However, we cannot do it arbitrarily. For example, we cannot assign all the bits 
to the most protected positions (in Example 8.1 this would be the position 
) as this will violate the constraints defined in (8.8).
We note here that the particular case of 
, i.e., when the matrix 
is square and each row/column contains only one nonzero element, the double set of interleavers—input (
) and output (
)—is not necessary. When 
we may use just one set of interleavers as is shown already in Fig. 8.3. Having two sets of interleavers, however, is necessary for the more general case 
.
8.1.3 Modeling the M-Interleaver
We recall that the S-interleaver in Fig. 2.24 is modeled as a random MUX assigning the code bits belonging to the class 
to the 
th position of the mapper. The randomness comes from the fact that the interleavers are generated randomly, and then, the position 
is random as well, which we denote by 
. For S-interleavers we considered up to now, this variable was uniformly distributed.
To analyze the M-interleaver and take advantage of the considerations in Section 2.7 and Section 6.2, we model the interleaving vectors 
and 
as random variables drawn with equal probability from the set of all possible permutations. Then,
where 
means that 
belongs to the subsequence 
, i.e., it is assigned to the mapper's output 
. Similarly,
i.e., for any position 
, the bit 
is obtained from the 
th code bits' class with probability 
.
To adapt the model Fig. 2.24 to the case of the M-interleaver, we first need to consider a random switch indexed by 
, i.e., we use 
instead of 
. We also replace the values of the assignment probabilities as shown in Fig. 8.6. Now 
has a meaning of the probability that the bit belonging to the class 
is mapped to the position 
of the mapper, i.e., 
.
Figure 8.6 Probabilistic model of the M-interleaver defined in Section 8.1.2: the bits 
are multiplexed to the position 
within the label 
, where 
is a random variable with distribution 
The immediate consequence of the random assignment of the bits 
to the positions 
is that the corresponding L-value 
has a distribution that changes with 
. Thus the PDFs for 
are given by
Using the PDFs of the L-values in (8.15) we transform the BICM channel in Fig. 8.2 to a set of parallel BICO channels with channel transition probabilities 
, which is shown in Fig. 8.3 (b).
In the case of Zehavi's interleaving in Example 8.2, using (8.10) and (8.15) we recover the formulas we already had in (8.2). In the case of the S-interleaver, the matrix 
should be chosen as
so using (8.15) we obtain the PDF
which coincides, as expected, with (8.1).
We emphasize that in order to obtain the formula (8.15) and use it further to evaluate the performance of the decoder, we need to satisfy the conditions of quasi-randomness, similar to those we explained in Section 2.7. We do not dwell on this anymore and simply assume that such conditions are fulfilled.
8.1.4 Performance Evaluation
We have established a parallel channel model (see Fig. 8.3 (b)), which relies solely on the knowledge of the PDF of the L-values 
associated with the classes of code bits. To study the performance of the decoders, we should distinguish amongst the codewords according to the input they provide to the parallel channels. Assuming that the joint symmetry condition in (3.75) is satisfied, this input depends on the generalized Hamming weight (GHW) of the codeword 
. Namely,for each codeword 
with 
the metric 
of the BICM decoder is given by the sum of 
L-values with PDF 
, 
L-values with PDF 
, and so on. This is what we showed already in (6.211), and consequently, the PDF of the metric 
is given by
Then using the same approximation strategies as in Section 6.2.4 we obtain an approximation for the word-error probability (WEP):
where 
is the generalized distance spectrum (GDS) of the code 
in Definition 2.23, and
In analogy to Example 6.22, the following example shows the performance of BICM with convolutional encoders (CENCs), but now using M-interleaver.
In the following example we show how to compute the GIWD 
and the GWD 
for a CENC, which follow as a generalization of the procedure in Example 6.23.
Figure 8.7 Generalized state machine for 
used in the analysis of BICM-M
By using a vector of dummy variables 
, the procedure shown in Example 8.6 (
) can be extended to any value of 
, and in general, the GWD and GIWD can be expressed in terms of an 
dimensional vector 
. Once the GIWD of the encoder is calculated, in order to predict the BEP performance of BICM-M via (8.22), we only need to compute 
in (8.20). This is done in the following section by exploiting the Gaussian simplifications we introduced in Section 6.4.
8.2 Exploiting UEP in 
Constellations
In Section 8.1 we defined the models to characterize the performance of BICM-M. In order to efficiently optimize the transceiver, we need closed-form expressions that relate the performance of the decoder to the multiplexing matrix 
and the channel model. In this section we turn our attention to the relatively simple case of 
PAM constellations for which we obtained simplified forms for the PDFs of the L-values (see Section 5.5).
8.2.1 The Generalized BICM Channel
For 
PAM constellations labeled by the BRGC, the results in Section 6.4.2 tell us that the conditional PDF of the max-log L-values can be approximated via a Gaussian function. After marginalization over all transmitted symbols (assumed equiprobable), we obtain
where 
is given by (6.386), 
by (6.387), and 
by (6.388).
As the L-values 
in (8.35) are modeled as a mixture of Gaussian random variables, and the L-values 
are a mixture of L-values 
(see (8.15)), the L-values 
are also a Gaussian mixture. The way the Gaussian PDFs are mixed is defined by 
in (8.9) and by the constellation bit-wise Euclidean distance spectrum (CBEDS) matrix 
, which gathers all parameters 
, as defined in (6.374). More specifically, if we define the matrix 
of dimensions 
the PDF of the L-values at the 
th decoder's input is given by
From (8.37), 
can be interpreted as the probability of observing the 
th Gaussian PDF at the 
th parallel channel in Fig. 8.3 (b).
8.2.2 Performance Evaluation
The expression in (8.37) gives an analytical approximation for the PDFs of the L-values for BICM-M. In this section we study the performance of such BICM-M transceivers. From now on, we limit our considerations to the parameters defined by the zero-crossing model (ZcM) approximation (see Section 6.4.2), which proved to yield more accurate results than those obtained from the consistent model (CoM) (see Section 6.4.4). The next theorem shows how the pairwise-error probability (PEP) in (8.20) can be approximated.
Figure 8.8 BEP approximation in (8.22) based on Theorem 8.8 (lines) and simulated BEP (markers) for BICM-M, 
, 4PAM (
) and the three interleavers defined in (8.51). We used the ODS CENC 
from Table 2.1 (
) and the TENC in Example 2.31. The interleaver size of the TENC 
is 
and 10 iterations are performed by the turbo decoder
We note that all combinations in (8.49) are in general tedious to evaluate (specially for large values of 
and/or 
), and thus, we propose further approximations. One straightforward simplification is to consider, for each 
, only the Gaussian density with the smallest mean-to-standard deviation ratio (e.g., 
). The intuition behind this approximation is that the error terms related to other Gaussian functions decrease quickly with 
. This approximation yields the following approximation on the BEP:
The expression in (8.52) is quite simple to evaluate compared to the original expression based on (8.45), and it still takes into account the optimization parameters (MUX and encoder).
We can also obtain the following asymptotic approximation
which we expect to be tight as 
. This result provides us with a new criterion to select the optimum encoder and interleaver (see Section 8.2.3).
Figure 8.9 BEP approximation in (8.22) based on Theorem 8.8 (lines) and simulated BEP (markers) for BICM-M for the rate 
ODS CENC 
from Table 2.1 (
) and for the TENC in Example 2.31 for 
and 8PAM (
, 
). The interleaver size of the TENC 
is 
and 10 iterations are performed by the turbo decoder. The asymptotic bounds based on (8.52) for the TENC and on (8.53) for the CENC are also shown
Analyzing the approximation of the BEP expression (8.22) together with (8.45), we can appreciate three terms: 
, which depends only on the encoder; 
, which depends only on the channel (see (8.46)); and 
, which depends on the interleaver (see (8.36)). Assuming that the constellation and the labeling are fixed, the optimum performance of the system will be achieved by a joint design of the interleaver and the encoder. We study this in the following section.
8.2.3 Joint Encoder and Interleaver Design
It is well known that ODS CENCs in Table 2.1 are the optimum CENCs for binary transmission. However, according to (8.53), when UEP is introduced by the channel, the optimization criterion is different, i.e., a joint optimization of the MUX (i.e., how the code bits are assigned to different bit positions in the constellation) and the CENC should be done. More specifically, the expression in (8.53) shows that we first need to optimize the free Hamming distance (FHD) of the encoder and then we need to optimize the GIWD of the encoder and the MUX matrix 
. To formally define the optimum design for a given constraint length 
and code rate 
, we define 
as the set of encoders in 
that give MFHD, where 
is the CENC universe defined in Section 2.6.1.
Note that for given values of 
and 
, the problem of selecting the optimum interleaver configuration (selection of 
) is a continuous multidimensional optimization problem. For simplicity, however, the optimization is performed over only a limited number of points. Using Definition 8.11, an exhaustive search for pairs 
with constraint length up to 
was performed. Three different configurations were tested: code rate 
(
, 
) with 4PAM (
), 8PAM (
), or 16PAM (
). The (simplified) optimization space for 
in these cases was 
for 
, 
for 
, and 
for 
. The results are presented in Table 8.1, where we highlight CENCs found that are different (in terms of their IWD) from the ODS CENCs in Table 2.1. Among the 24 combinations studied, 6 resulted in new optimal encoders.
Table 8.1 Optimum CENC and multiplexers (MUXs) for 
wit 4PAM, 8PAM, and 16PAM. New encoders found, better than the ODS CENCs in Table 2.1, are highlighted
4PAM ( ) |
8PAM ( ) |
16PAM ( ) |
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| 2 | 5 |  |
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| 3 | 6 |  |
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| 4 | 7 |  |
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| 5 | 8 |  |
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| 6 | 10 |  |
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| 7 | 10 |  |
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| 8 | 12 |  |
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| 9 | 12 |  |
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The cost function in (8.55) is shown in Fig. 8.10 as a function of the interleaver parameter 
for 
, 4PAM, and 
. The ODS CENC 
is shown with a thick black line. Analyzing this curve, it is clear that the performance of this encoder can be optimized by setting 
, and that the curve has a maximum for 
, which will result in the worst interleaver design for this particular encoder. The cost function obtained for the encoder 
(thick dashed line) attains the smallest value among all other encoders (including the ODS one). Consequently, if the MUX is adequately designed setting 
(best BICM-M), this encoder is the optimal encoder for this particular configuration. However, if the interleaver is not optimized, e.g., setting 
(BICM-S), the new encoder is not optimal anymore.
Figure 8.10 Cost function in (8.55) for all possible encoders in 
for 
(
, 
), 4PAM, and 
as a function of the MUX parameter 
. The thick solid line represents the ODS CENC 
, and the thick dashed line the optimum BICM-M design based on 
We conclude by showing in Fig. 8.11 the performance of the optimum design 
compared with all encoders with 
and 
(enumerated using the variable 
) using the best and the worst interleaver design (
and 
). The lines represent the range of variation between the best and the worst interleaver design, i.e., any other interleaver configuration will have a coefficient between the corresponding pair of markers. We note that the optimum design may significantly outperform other encoders, e.g., 16PAM and 
in Fig. 8.11. The improvement with respect to ODS CENCs is less evident but clear. Thus, the results presented in this section indicate that optimizing the interleaver and encoder should be a mandatory step in the design of BICM-M.
Figure 8.11 Cost function in (8.55) for all the 21 possible encoders in 
, for the best and worst interleaver design: 
(‘
’), 
(‘
’), and 
(‘
’). The lines represent the range of variation between the best and the worst interleaver design
8.3 Bibliographical Notes
UEP in terms of BEP for uncoded transmission with 
PSK and 
QAM constellations labeled by the BRGC has been studied in [1] and for 16QAM and 64QAM in [2, Figs. 5.2 and 5.4]. Unequal power allocation for systematic/parity bits to impose UEP is an idea first used for turbo-encoded BICM in [3] and later analyzed in [4–8]. UEP for turbo-encoded schemes has been studied in [3 4, 6 8–10]. The influence of the block length and code rate for optimal power allocation was analyzed in [4 11] and interleaver design aiming to assign the code bits to different bit positions for high-order modulation schemes was studied in [12].
The BICM transceivers with M-interleaver we analyzed in this chapter in fact correspond to the original model introduced by Zehavi in [13] for BICM (and also for BICM with iterative demapping (BICM-ID) in [14]), where the application of parallel interleavers was postulated. Over the years, different names have been given to these interleavers: e.g., “in-line” 15, “intralevel” [16], “M” [17], “dual” [18], or “modular” [19]. BICM-M has been studied in [17 20], BICM-ID with M-interleavers in [14], BICM for serially concatenated systems in [15], and orthogonal frequency-division multiplexing (OFDM)-based BICM in [16] (see also [21]). BICM-M has also been proposed in the third-generation partnership project (3GPP) standard [18 22]. Nevertheless, most of the existing literature on BICM and BICM-ID still follows the framework set in [23] and assumes the use of one singe-input interleaver (BICM-S). This simplifies the analysis of the resulting system, but leads to suboptimality, as we have shown in this chapter.
The generalized transfer function of the encoder was briefly introduced in the original paper of Zehavi [13, eq. ((4.8))] and the GIWD in [19, Section IV-A]. As the PEP computation for BICM-M is not straightforward, the application of the GIWD of the encoder was not very popular. This problem was solved by the analytical expressions for the PDF of the L-values (and their Gaussian approximations) we introduced in Chapter 5. This was first done in [24] for Gaussian channels and later in [25] for fading channels. BICM-ID with M-interleavers was shown to outperform BICM-ID with a single interleaver in [26] (see also [27, Chapter 4]).
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