All right reserved.

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)
*

*
where
*

*
[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.
*

*
(Continues...)
*

*
*

Excerpted fromDigital Communication over Fading ChannelsbyMarvin K. Simon Mohamed-Slim AlouiniCopyright © 2005 by John Wiley & Sons, Inc.. Excerpted by permission.

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