This chapter describes different methods that can be used to evaluate the performance of bit-interleaved coded modulation (BICM) receivers. We focus on bit-error probability (BEP), symbol-error probability (SEP), and word-error probability (WEP) as performance metrics, as these are often considered relevant in practical systems. We pay special attention to techniques based on the knowledge of the probability density function (PDF) of the L-values we developed in Chapter 5, and in particular to the Gaussian models introduced in Section 5.5.
This chapter is organized as follows. In Section 6.1 we review methods to evaluate the performance for uncoded transmission and in Section 6.2 we study the performance of decoders (i.e., coded transmission). Section 6.3 is devoted to analyzing the pairwise-error probability (PEP), which appears as the key quantity when evaluating coded transmission. Section 6.4 studies the performance of BICM based on the Gaussian approximations of the PDF of the L-values.
By uncoded transmission we mean that the channel coding is absent (or is ignored), and thus, the decision on the transmitted symbols is made without taking into account the sequence of symbols (i.e., the constraints on the codewords). We are instead interested in deciding which symbol was transmitted at a given time instant using the channel output . These are “hard decisions” (briefly discussed in Section 3.4) which might eventually be fed to the channel decoder.
The criterion for making a hard decision depends on how it will be used. Probably the simplest and most intuitive approach is to minimize the SEP. Assuming an additive white Gaussian noise (AWGN) channel and equally likely symbols, we then obtain the expression we already showed in (3.100), i.e.,
Similarly, assuming equally likely bits, to minimize the BEP, the decision on the transmitted bits is
which can also be expressed as
or in terms of L-values, as
From (6.4) we conclude that a hard decision on the transmitted bits is equivalent to making a decision on the sign of the L-values in (3.50).1
The implementation of (6.3) has a complexity dominated by the calculation of the sum of exponential functions so simplifications may be sought. In particular, we may apply the max-log approximation, which used in (6.3) gives
The hard decision on the bits shown in (6.5) is also equivalent to determining the sign of the L-values calculated via the max-log approximation (i.e., when in (6.4) are max-log L-values). This was already shown in (3.106).
As we showed in Theorem 3.17, the decision in (6.5) is equivalent to the decision in (6.1), and thus, we can consider the vector of hard decisions on the bits
where . Owing to this equivalence we focus on the symbol-wise hard decisions in (6.1), or equivalently, on the bitwise hard decisions in (6.6) based on max-log L-values.
Since the hard decisions are equivalent to a one-bit (sign only) quantization of the L-values, we may continue to view them as a part of the BICM channel, which now becomes a binary-input binary-output channel. Methods for analyzing the performance of such a BICM channel in terms of SEP or BEP for uncoded transmission have been studied in the communication theory community over decades. The main reason for focusing on these quantities is that SEP/BEP for uncoded transmission are relatively easy to obtain compared to similar quantities in coded transmission. The underlying assumption is that systems with the same SEP/BEP will perform similarly when coding/decoding is added. As we will see in Chapter 8, this is, in general not true, as the design of the interleaver may change the performance of the coded transmission, while uncoded transmission is unaffected by interleaving.
The SEP is defined as
where
where the decision region of the symbol is defined using (6.1) as
and where is given by (3.54).
The set is the so-called Voronoi region of the symbol . In many practically relevant cases, the form of the Voronoi regions can be found by inspection. For example, when the constellation points are located on a rectangular grid (i.e., for -ary quadrature amplitude modulation (QAM) constellations) or on a circle (i.e., for -ary phase shift keying (PSK) constellations), finding the decision regions consists in determining the closest constellation points to the symbol along each dimension of the grid. In the case of QAM there are at most four such symbols, and in the case of PSK constellations there are only two of them. In the more general case of irregular constellations, the Voronoi regions may be found using numerical routines.
Let be the probability of detecting when was transmitted, i.e.,
From now on, we refer to as the transition probability (TP).
Using (6.12), the SEP (6.8) can then be expressed as
where to pass from (6.15) to (6.16) we used (6.9).
The BEP for the th bit was already defined in (3.110) for max-log metrics.2 For exact L-values, we have the following analogous definition
where are given by (6.3) and (6.4) and the “decision regions” for the th bit are thus
Similarly, in the case of max-log L-values, for the th bit in (3.110) with given by (6.5), the BEP is expressed as
where the decision regions in this case are
which follows from the fact that is the th bit of the label of the detected symbol via (6.1) (see (6.6)).
The calculation of in (6.18) requires integration over the decision region , i.e.,
This is in general quite difficult, mainly because is defined via nonlinear equations in (6.20) which, moreover, depend on (as we showed in Example 6.2).
On the other hand, with the max-log approximation, the decision region is the union of the Voronoi regions in (6.10), and thus, (i) it is independent of the SNR and (ii) it is defined via linear equations only. The BEP in this case is calculated as
where (6.24) follows from the law of total probability, (6.25) from (6.10) and (6.21), and where the is given by (6.12). The BEP averaged over the bits' positions is then defined as
where is the Hamming distance (HD) between the binary labels of and , and where to pass from (6.27) to (6.28) we simply reorganized the summations.
From (6.16) and (6.28) we see that once the Voronoi regions in (6.10) are found, the challenge of calculating the SEP and BEP is reduced to the calculation of the TPs in (6.12). The problem then boils down to efficiently calculating the integral (6.14). This is the focus of the following sections.
Before proceeding further, we express the Voronoi regions in (6.11) using the nonredundant inequalities defining , i.e.,
where the linear forms in this case are
We use an arbitrary variable to index the nonredundant inequalities defining the region, and thus, the definitions in (6.29) and (6.30) are similar to those we already used in (5.87) and (5.88).3
We start by analyzing the simplest possible case where only one nonredundant inequality exists (i.e., ). In this case, the problem is one-dimensional, and thus, the Voronoi region is a half-space. In this case, we express the TP in (6.14) as
where with slight abuse of notation we use in (without the argument ) to denote the parameters and defining the linear form in (6.30). The TP in (6.31) can be expressed as
where (6.32) follows from the fact that is a zero-mean, unit variance Gaussian random variable.
The result shown in Example 6.3 can also be obtained by simple inspection: because the symbol is at distance from the limit of the decision region , and the noise is Gaussian with variance , the calculation of yields (6.33). Nevertheless, we use the formalism of the decision region defined by the line for illustrative purposes.
The next step is to consider the region defined by two nonredundant inequalities and (i.e., ). In this case, the Voronoi region is a wedge, which we define in analogy to (5.95) as
For convenience, we also define a “complementary” wedge
The wedges and are schematically shown in Fig. 6.3. These simple forms of are important because they appear, e.g., in the case of PSK constellations (see Fig. 6.1). As we will see later, the analysis of a wedge also leads to expressions that can be used to study PAM, QAM, as well as arbitrary constellations.
For the case of the Voronoi region being a wedge, we denote the TP in (6.14) as
where in analogy to (5.96)
The TP in (6.37) can be expressed as
where
Owing to the normalization, and can be shown to be zero-mean, unit-variance Gaussian random variables with correlation
Using (6.41)–(6.44) (6.37) can be expressed as
where is given by (2.11).4
In the same way we defined a complementary wedge in (6.35), we define a “complementary” TP as
Following similar steps to those in (6.39)–(6.45), we find that the complementary TP in (6.46) is given by
The general expression (6.45) for the TP when the Voronoi region is a wedge particularizes to two interesting cases, which are shown schematically in Fig. 6.4.
which is valid for (see (2.12)), i.e., when the linear inequalities are not contradictory. The 2D integral is thus reduced to a 1D integration of a zero-mean Gaussian PDF along a line orthogonal to the stripe's limits.
With the expressions we have developed in (6.32) and (6.49), we are in a position to compute the SEP and BEP for any 1D constellation. In the following example we show how to do this for 4PAM and the binary reflected Gray code (BRGC).
As we showed in Example 6.4, the TP need not always be calculated for all pairs of the transmitted symbols. Instead, the regularity and the symmetry of the constellation can be exploited to simplify the calculations. This can also be done for 2D constellation as we show in the following example.
The expressions we derived in the previous example are entirely general; yet, particular cases of PSK constellations lead to simpler results that do not need bivariate Q-functions or, in fact, exploit the simplicity of the particular cases we analyzed in (6.48) and (6.49). We show this in the following example.
In Example 6.5 we were able to express for PSK constellation using bivariate Q-functions. Our objective now is to do the same in the case of an arbitrary 2D constellation. First, we assume that is a closed polygon defined by linear forms corresponding to the polygon's sides. We assume that the linear pieces are enumerated counter-clockwise, as shown in Fig. 6.7. In such a case, using the complementary wedge we defined in (6.35), the whole space can be expressed as a union of disjoint sets, i.e.,
The TP in (6.12) can then be expressed as
where is given by (6.46).
Similarly, if is an “infinite” polygon as shown in Fig. 6.8, we can write
and then,
We conclude then that the TPs for any 2D constellation can be calculated using only functions , which as we showed before, are expressed in terms of bivariate Q-function. These bivariate Q-functions now play the same role that the Q-function has when calculating for 1D constellations.
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