6.3 PEP Evaluation

From the results in Sections 6.2.4 and 6.2.5, we know that finding the PEP in (6.166) and (6.226) is instrumental to evaluate the performance of the ML and BICM decoders, respectively. In the case of the BICM decoder we need to evaluate

where

and c06-math-1015 is the average PDF given in (6.227). Alternatively, we might want to calculate

where

and carry out the summation (6.233) over c06-math-1018. Note that, to simplify the notation we removed the subindexing with c06-math-1019 we used in (6.210).

Although we used the same notation in (6.283) and (6.285), the random variables c06-math-1020 described by these PDFs are not the same. In the first case, to evaluate the PEP, we have to consider only i.i.d. random variables. In the case of (6.285) the random variables are non-i.i.d.; we treat this case in Section 6.3.4.

6.3.1 Numerical Integration

In general, we will have to deal with distributions different from the Gaussian case we showed in Example 6.24. The direct way to obtain the tail integral of the multiple convolution is via direct and inverse Laplace-type transforms, i.e., from the relationship linking the MGF (see (2.6)) of the L-values c06-math-1021 and their PDF c06-math-1022 given in (6.227).13 More specifically,

6.287 equation

and

6.288 equation

where c06-math-1026 is taken from the domain of the MGF.

Expressing (6.282) as

6.289 equation
6.290 equation

where c06-math-1029 and c06-math-1030 is the inverted step function with MGF c06-math-1031. The PEP can be then calculated by inverting the MGF of c06-math-1032, i.e.,

Finding exact analytical solutions of the integral (6.291) is often impossible so numerical integration is then used. Before studying this approach, we show a useful lemma.

In what follows, we show how to numerically calculate (6.291). To this end, we use a Gauss-Chebyshev (GCh) quadrature, which states that for any function c06-math-1050 that can be represented by polynomials for c06-math-1051,

where c06-math-1053 and c06-math-1054.

Using the substitution c06-math-1055, (6.291) becomes

6.295 equation
6.296 equation

where (6.297) follows from the fact that the second part of the integrand in (6.297) is odd. This is due to Lemma 6.25, which shows that c06-math-1059 is real and even.

After the change of variable c06-math-1060, (6.297) becomes

where

6.299 equation

Thus, using (6.294) in (6.298) yields

6.300 equation

We then use c06-math-1064 and take advantage of the symmetry c06-math-1065 and c06-math-1066. This yields the final expression for the PEP

where c06-math-1068 and c06-math-1069 is the MGF of the random variable c06-math-1070 with PDF given by (6.227).

To evaluate the PEP in (6.301), one would ideally use a large value of c06-math-1078. However, a good tradeoff between accuracy and implementation complexity is typically obtained for relatively small values of c06-math-1079. Most of the examples in this chapter were calculated using c06-math-1080 or c06-math-1081.

c06f018

Figure 6.18 The real (a) and imaginary (b) parts of the MGF c06-math-1071 in (6.303) for c06-math-1072 and c06-math-1073. The points along the integration line c06-math-1074 (shown with circles) correspond to the GCh quadrature nodes with c06-math-1075. As expected (see Lemma 6.25), c06-math-1076 for c06-math-1077

We conclude this section by making some remarks about the PEP expression in (6.301):

  • The numerical complexity grows with the number of terms required by (6.243), as the integration (6.301) must be carried out for each value of c06-math-1097. This brings implementation burden when we need to evaluate c06-math-1098 for many different values of c06-math-1099.
  • The MGF c06-math-1100 must be known for c06-math-1101 values of c06-math-1102 required in (6.301). While in some cases we may be able to analytically calculate the MGF from the analytical form of the PDF c06-math-1103, in others, it is too cumbersome or even impossible. In fact, it may be difficult to derive tractable expressions for the PDF c06-math-1104 in the first place, e.g., when the L-values depend on many random parameters.
  • In the cases where the PDF is not known, we may replace (6.286) by Monte Carlo integration. To this end, we use a sum over realizations c06-math-1105 of c06-math-1106, conditioned on the inputs of the BICM channel in Fig. 5.1, i.e.,
    6.309 equation
    6.310 equation

    The complexity of the PEP evaluation is then dominated by (6.311), which has to be evaluated c06-math-1110 times for different values of c06-math-1111 as required by (6.301).

  • The expression (6.301) does not explicitly relate to the parameters of BICM (e.g., constellation and labeling), and thus, it does not provide insights for the design.

6.3.2 Saddlepoint Approximation

To go around the first two difficulties mentioned above, the so-called saddlepoint approximation (SPA) may be used to efficiently approximate the PEP. The SPA is a tool known in statistical analysis to efficiently calculate cumulative distribution functions (CDFs) and PDFs of a sum of random variables using the cumulant generating function (CGF)

For a sum of i.i.d. random variables such as c06-math-1113, we obtain

and thus, (6.291) becomes

A formal derivation of the SPA may be found in the textbooks we reference in Section 6.5. Here, we provide an intuitive explanation for why SPA “work”. We start by approximating the CGF in (6.312) via a truncated Taylor series around c06-math-1116, i.e.,

where c06-math-1118 and c06-math-1119 are the first and second derivatives of the CGF given by

6.316 equation
6.317 equation

The Taylor expansion in (6.315) is done around the so-called saddlepoint c06-math-1122, chosen to satisfy c06-math-1123, which used in (6.314) gives the SPA

6.318 equation
6.319 equation

After the change of variables c06-math-1126 we obtain

where

6.321 equation

and where (6.320) results from c06-math-1129.

Using c06-math-1130 transforms (6.321) into

Alternatively, if the (tighter) approximation c06-math-1132 is used, we obtain

We can also transform the expression in (6.322) and (6.323) back into the MGF domain via (6.312). For example, (6.323) becomes

6.324 equation

We have obtained expressions that depend solely on the MGF or CGF evaluated at c06-math-1135. This is an important simplification when compared to (6.301), as once c06-math-1136 and c06-math-1137 are known, we can evaluate c06-math-1138 for any c06-math-1139. The caveat is that now, c06-math-1140 must be found. This is usually the most difficult part of the SPA method because solving nonlinear saddlepoint equation c06-math-1141, or equivalently

is, in general, not trivial. However, for the cases we study here, it is in fact quite simple.

We note that to solve (6.325) we need to choose c06-math-1143 to satisfy

From (3.85) we know that the function c06-math-1145 is odd, which allows us to conclude that for

the integrand in (6.326) is also odd, and thus, (6.327) solves (6.325). To show that this solution is unique, we need to demonstrate that c06-math-1147 is convex, i.e.,

6.328 equation

which can be recognized as Hölder's inequality, and therefore, holds independently of the distribution of the random variable c06-math-1149.

c06f019

Figure 6.19 CGF c06-math-1172 (solid lines) and its second-order approximation c06-math-1173 (dashed lines) are shown together with the corresponding MGF c06-math-1174 (solid lines) and its approximation c06-math-1175 (dashed lines): (a) c06-math-1176, (b) c06-math-1177, (c) c06-math-1178, and (d) c06-math-1179. The circles indicate the position of the c06-math-1180 quadrature nodes necessary to implement the solution from Section 6.3.1. Here c06-math-1181

Thanks to the SPA, once we know c06-math-1182 and c06-math-1183 for one real argument c06-math-1184, we may calculate c06-math-1185 for any value of c06-math-1186. This stands in contrast to the numerical integration approach, where we have to evaluate the MGF for c06-math-1187 points (see (6.301)), and next carry out the summations for each c06-math-1188. Thus, not only does the SPA allow us to obtain analytical solutions in some cases, but it also offers a clear advantage over numerical integration when the MGF is difficult to acquire.

6.3.3 Chernoff Bound

Another useful approximation of the PEP relies on using upper bounding techniques, as shown in the following theorem.

The proof of Theorem 6.29 shows that c06-math-1204 for any c06-math-1205. In Theorem 6.29 we use c06-math-1206 as this value of c06-math-1207 minimizes c06-math-1208, and thus, tightens the bound (6.340).

For the particular case of 2PAM transmission, from (6.329) we have c06-math-1209, so the Chernoff bound in (6.336) is given by

Using c06-math-1211, we bound the true PEP in (6.258) as

6.343 equation

which shows that the Chernoff bound in (6.342) goes to zero (as c06-math-1213) slower than the actual value of the PEP. The next example shows PEP calculation for 2PAM in fading channels.

c06f020

Figure 6.20 Comparison of the PEP estimation methods for a 2PAM constellation in Rayleigh fading channel. To obtain c06-math-1214 we used (6.308) with c06-math-1215 quadrature points

6.3.4 Nonidentically Distributed L-values

We can now revisit the assumption that the L-values c06-math-1247 entering the metric c06-math-1248 are identically distributed. Such an assumption is sufficient when using the model (6.227), but in the most general case (6.218), we need to calculate c06-math-1249 in (6.284). In this case, c06-math-1250 is a sum of c06-math-1251 random variables, c06-math-1252 with distribution c06-math-1253, c06-math-1254 with a distribution c06-math-1255, and so on, where c06-math-1256. We denote the MGF of the c06-math-1257th distribution by c06-math-1258, and its CGF by c06-math-1259. The extension of the previously obtained formulas thus requires replacing the MGF c06-math-1260 with the product c06-math-1261.

The numerical integration (6.301) is generalized as

6.353 equation

The SPA-based solution (6.323) becomes

6.354 equation

and the upper bound is generalized as

where

6.357 equation

and thus c06-math-1267.

We will exploit the bound (6.355) in Lemma 7.13 and use it in the following example to find the PEP of trellis-coded modulation (TCM) receivers.

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