As we continue to step forward into the new millennium with wireless technologies leading the way in which we communicate, it becomes increasingly clear that the dominant consideration in the design of systems employing such technologies will be their ability to perform with adequate margin over a channel perturbed by a host of impairments, not the least of which is multipath fading. This is not to imply that multipath fading channels are something new to be reckoned with; indeed, they have plagued many a system designer for well over 40 years, but rather to serve as a motivation for their ever-increasing significance in the years to come. At the same time, we do not in any way wish to diminish the importance of the fading channel scenarios that occurred well prior to the wireless revolution since indeed many of them still exist and will continue to exist in the future. In fact, it is safe to say that whatever means are developed for dealing with the more sophisticated wireless applications will no doubt also be useful for dealing with the less complicated fading environments of the past.
With the above in mind, what better opportunity is there than now to write a comprehensive book that will provide simple and intuitive solutions to problems dealing with communication system performance evaluation over fading channels? Indeed, as mentioned in the preface, the primary goal of this book is to present a unified method for arriving at a set of tools that will allow the system designer to compute the performance of a host of different digital communication systems characterized by a variety of modulation/detection types and fading channel models. By "set of tools" we mean a compendium of analytical results that not only allow easy yet accurate performance evaluation but at the same time provide insight into the manner in which this performance depends on the key system parameters. To emphasize what was stated above, the set of tools that will be developed in this book are useful not only for the wireless applications that are rapidly filling our current technical journals but also to a host of others involving satellite, terrestrial, and maritime communications.
Our repetitive use of the word "performance" thus far brings us to the purpose of this introductory chapter, namely, to provide several measures of performance related to practical communication system design and to begin exploring the analytical methods by which they may be evaluated. While the deeper meaning of these measures will be truly understood only after their more formal definitions are presented in the chapters that follow, the introduction of these terms here serves to illustrate the various possibilities that exist depending on both need and relative ease of evaluation.
1.1 SYSTEM PERFORMANCE MEASURES
1.1.1 Average Signal-to-Noise Ratio (SNR)
Probably the most common and well understood performance measure characteristic of a digital communication system is signal-to-noise ratio (SNR). Most often this is measured at the output of the receiver and is thus directly related to the data detection process itself. Of the several possible performance measures that exist, it is typically the easiest to evaluate and most often serves as an excellent indicator of the overall fidelity of the system. While traditionally the term "noise" in signal-to-noise ratio refers to the ever-present thermal noise at the input to the receiver, in the context of a communication system subject to fading impairment, the more appropriate performance measure is average SNR, where the term "average" refers to statistical averaging over the probability distribution of the fading. In simple mathematical terms, if [gamma] denotes the instantaneous SNR [a random variable (RV)] at the receiver output that includes the effect of fading, then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
is the average SNR, where [p.sub.[gamma]] ([gamma]) denotes the probability density function (PDF) of [gamma]. In order to begin to get a feel for what we will shortly describe as a unified approach to performance evaluation, we first rewrite (1.1) in terms of the moment generating function (MGF) associated with [gamma]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)
Taking the first derivative of (1.2) with respect to s and evaluating the result at s = 0, we immediately see from (1.1) that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)
that is, the ability to evaluate the MGF of the instantaneous SNR (perhaps in closed form) allows immediate evaluation of the average SNR via a simple mathematical operation, namely, differentiation.
To gain further insight into the power of the statement above, we note that in many systems, particularly those dealing with a form of diversity (multichannel) reception known as maximal-ratio combining (MRC) (to be discussed in great detail in Chapter 9), the output SNR, [gamma], is expressed as a sum (combination) of the individual branch (channel) SNRs, namely, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where L denotes the number of channels combined. In addition, it is often reasonable in practice to assume that the channels are independent of each other, that is, that the RVs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are themselves independent. In such instances, the MGF [M.sub.[gamma]] (s) can be expressed as the product of the MGFs associated with each channel [i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which as we shall later on in the text can, for a large variety of fading channel statistical models, be computed in closed form. By contrast, even with the assumption of channel independence, the computation of the PDF [p.sub.[gamma]] ([gamma]), which requires convolutional of the various PDFs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that characterize the L channels, can still be a monumental task. Even in the case where these individual channel PDFs are of the same functional form but are characterized by different average SNRs, [bar][gamma].sub.l], the evaluation of [p.sub.[gamma]] ([gamma]) can still be quite tedious. Such is the power of the MGF-based approach; namely, it circumvents the need for finding the first-order PDF of the output SNR, provided that one is interested in a performance measure that can be expressed in terms of the MGF. Of course, for the case of average SNR, the solution is extremely simple, namely, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] regardless of whether the channels are independent, and in fact, one never needs to find the MGF at all. However, for other performance measures and also the average SNR of other combining statistics, such as the sum of an ordered set of random variables typical of generalized selection combining (GSC) (to be discussed in Chapter 9), matters are not quite this simple and the points made above for justifying an MGF-based approach are, as we shall see, especially significant.
1.1.2 Outage Probability
Another standard performance criterion characteristic of diversity systems operating over fading channels is the so-called outage probability-denoted by [P.sub.out] and defined as the probability that the instantaneous error probability exceeds a specified value or equivalently the probability that the output SNR, [gamma], falls below a certain specified threshold, [gamma]th. Mathematically speaking, we have
[P.sub.out] = [[infinity].sup.[gamma]th.sub.0] [p.sub.[gamma]([gamma])[d.sub.[gamma]] (1.4)
which is the cumulative distribution function (CDF) of [gamma], namely, [P.sub.[gamma]] ([gamma]), evaluated at [gamma] = [gamma]th. Since the PDF and the CDF are related by [p.sub.[gamma]] ([gamma]) = [d]P.sub.[gamma]]([gamma]) /[d.sub.[gamma]] and since [P.sub.[gamma]] (0) = 0, then the Laplace transforms of these two functions are related by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)
Furthermore, since the MGF is just the Laplace transform of the PDF with argument reversed in sign [i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]], then the outage probability can be found from the inverse Laplace transform of the ratio [M.sub.[gamma]] (-s) /s evaluated at [gamma] = [gamma]th
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)
where [omega] is chosen in the region of convergence of the integral in the complex s plane. Methods for evaluating inverse Laplace transforms have received widespread attention in the literature. (A good summary of these can be found in the paper by Abate and Whitt). One such numerical technique that is particularly useful for CDFs of positive RVs (such as instantaneous SNR) is discussed in Appendix 9B and applied therein in Chapter 9. For our purpose here, it is sufficient to recognize once again that the evaluation of outage probability can be performed based entirely on the knowledge of the MGF of the output SNR without ever having to compute its PDF.
1.1.3 Average Bit Error Probability (BEP)
The third performance criterion and undoubtedly the most difficult of the three to compute is average bit error probability (BEP). On the other hand, it is the one that is most revealing about the nature of the system behavior and the one most often illustrated in documents containing system performance evaluations; thus, it is of primary interest to have a method for its evaluation that reduces the degree of difficulty as much as possible.
The primary reason for the difficulty in evaluating average BEP lies in the fact that the conditional (on the fading) BEP is, in general, a nonlinear function of the instantaneous SNR, as the nature of the nonlinearity is a function of the modulation/detection scheme employed by the system. Thus, for example, in the multichannel case, the average of the conditional BEP over the fading statistics is not a simple average of the per channel performance measure as was true for average SNR. Nevertheless, we shall see momentarily that an MGF-based approach is still quite useful in simplifying the analysis and in a large variety of cases allows unification under a common framework.
Suppose first that the conditional BEP is of the form
[P.sub.b](E|[gamma]) = [ITLITL.sub.1] exp ([-a.sub.1[gamma]]) (1.7)
such as would be the case for differentially coherent detection of phase-shift-keying (PSK) or noncoherent detection of orthogonal frequency-shift-keying (FSK) (see Chapter 8). Then, the average BEP can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)
where again [M.sub.[gamma]] (s) is the MGF of the instantaneous fading SNR and depends only on the fading channel model assumed.
Suppose next that the nonlinear functional relationship between [P.sub.b] (E |[gamma]) and [gamma] is such that it can be expressed as an integral whose integrand has an exponential dependence on [gamma] in the form of (1.7),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)
where for our purpose here h ([xi]) and g ([xi]) are arbitrary functions of the integration variable and typically both [[xi].sub.1] and [[xi].sub.2] are finite (although this is not an absolute requirement for what follows). While not at all obvious at this point, suffice it to say that a relationship of the form in (1.9) can result from employing alternative forms of such classic nonlinear functions as the Gaussian Q-function and Marcum Q-function (see Chapter 4), which are characteristic of the relationship between [P.sub.b] (E |[gamma]) and [gamma] corresponding to, for example, coherent detection of PSK and noncoherent detection of quadriphase-shift-keying (QPSK), respectively. Still another possibility is that the nonlinear functional relationship between [P.sub.b] (E |[gamma]) and [gamma] is inherently in the form of (1.9); thus, no alternative representation need be employed. An example of such occurs for the conditional symbol error probability (SEP) associated with coherent and differentially coherent detection of M-ary PSK (M-PSK) (see Chapter 8). Regardless of the particular case at hand, once again averaging (1.9) over the fading gives (after interchanging the order of integration)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)
As we shall see later on in the text, integrals of the form in (1.10) can, for many special cases, be obtained in closed form. At the very worst, with rare exception, the resulting expression will be a single integral with finite limits and an integrand composed of elementary functions. Since (1.8) and (1.10) cover a wide variety of different modulation/detection types and fading channel models, we refer to this approach for evaluating average error probability as the unified MGF-based approach and the associated forms of the conditional error probability as the desired forms. The first notion of such a unified approach was discussed in Ref. 2 and laid the groundwork for much of the material that follows in this text.
It goes without saying that not every fading channel communication problem fits this description; thus, alternative, but still simple and accurate, techniques are desirable for evaluating system error probability in such circumstances. One class of problems for which a different form of MGF-based approach is possible relates to communication with symmetric binary modulations wherein the decision mechanism constitutes a comparison of a decision variable with a zero threshold. Aside from the obvious uncoded applications, the above-mentioned class also includes the evaluation of pairwise error probability in error-correction-coded systems as discussed in Chapter 12. In mathematical terms, letting D |[gamma] denote the decision variable, then the corresponding conditional BEP is of the form (assuming arbitrarily that a positive data bit was transmitted)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)
where [p.sub.D|[gamma] (D) and P.sub.D|[gamma] (D) are, respectively, the PDF and CDF of this variable. Aside from the fact that the decision variable D |[gamma] can, in general, take on both positive and negative values whereas the instantaneous fading SNR, [gamma], is restricted to only positive values, there is a strong resemblance between the binary probability of error in (1.11) and the outage probability in (1.4). Thus, by analogy with (1.6), the conditional BEP of (1.11) can be expressed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)
where M.sub.D|[gamma] (-s) now denotes the MGF of the decision variable D |[gamma], that is, the bilateral Laplace transform of p.sub.D|[gamma] (D) with argument reversed.
To see how M.sub.D|[gamma] (-s) might explicitly depend on [gamma], we now consider the subclass of problems where the conditional decision variable D |[gamma] corresponds to a quadratic form of independent complex Gaussian RVs, such as a sum of the squared magnitudes of, say, L independent complex Gaussian RVs-a chi-square RV with 2L degrees of freedom. Such a form occurs for multiple-(L)-channel reception of binary modulations with differentially coherent or noncoherent detection (see Chapter 9). In this instance, the MGF [M.sub.D|[gamma]] (s) happens to be exponential in [gamma] and has the generic form
[M.sub.D|[gamma]] (s) = ƒ1(s)exp([gamma)ƒ2(s)) (1.13)
If as before we let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then substituting (1.13) into (1.12) and averaging over the fading results in the average BEP
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.15)
is the unconditional MGF of the decision variable, which also has the product form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16)
Finally, by virtue of the fact that the MGF of the decision variable can be expressed in terms of the MGF of the fading variable (SNR) as in (1.15) [or (1.16)], then, analogous to (1.10), we are once again able to evaluate the average BEP solely on the basis of knowledge of the latter MGF.
Excerpted from Digital Communication over Fading Channels by Marvin K. Simon Mohamed-Slim Alouini Copyright © 2005 by John Wiley & Sons, Inc.. Excerpted by permission.
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