 Introduction
 Spectral Analysis of Nonperiodic Functions
 Statistical Methods
 References
3.3 STATISTICAL METHODS
We now turn our attention away from the time and frequency domain and toward the probability domain where statistical methods of analysis are employed. As indicated in Section 3.1, such methods are required because of the uncertainty resulting from the introduction of noise and other factors during transmission.
3.3.1 The Cumulative Distribution Function and the Probability Density Function
A random variable X^{[1],[2]} is a function that associates a unique numerical value X(&lamda;_{i}) with every outcome &lamda;_{i} of an event that produces random results. The value of a random variable will vary from event to event, and depending on the nature of the event will be either discrete or continuous. An example of a discrete random variable X_{d} is the number of heads that occur when a coin is tossed four times. As X_{d} can only have the values 0, 1, 2, 3, and 4, it is discrete. An example of a continuous random variable X_{c} is the distance of a shooter's bullet hole from the bull's eye. As this distance can take any value, X_{c} is continuous.
Two important functions of a random variable are the cumulative distribution function (CDF) and the probability density function (PDF).
The cumulative distribution function, F(x), of a random variable X is given by
where P[X(&lamda;) ≤ x] is the probability that the value X(&lamda;) taken by the random variable X is less than or equal to the quantity x.
The cumulative distribution function F(x) has the following properties:

0 ≤ F(x) ≤ 1

F(x_{1}) ≤ F(x_{2}) if x_{1} ≤ x_{2}

F(–∞) = 0

F(+∞) = 1
The probability density function f(x) of a random variable X is the derivative of F(x) and thus is given by
The probability density function f(x) has the following properties:
Further, from Eqs. (3.22) and (3.23), we have
The function within the integral is not shown as a function of x because, as per Eq. (3.22), x is defined here as a fixed quantity. It has been arbitrarily shown as a function of z, where z has the same dimension as x, f(z) being the same PDF as f(x). Some texts, however, show it equivalently as a function of x, with the understanding that x is used in the generalized sense.
The following example will help in clarifying the concepts behind the PDF, f(x), and the CDF, F(x). In Fig. 3.4(a) a fourlevel pulse amplitude modulated signal is shown. The amplitude of each pulse is random and equally likely to occupy any of the four levels. Thus, if a random variable X is defined as the signal level v, and P(v = x) is the probability that v = x, then
Figure 3.4 A fourlevel PAM signal and its associated CDF and PDF.
With this probability information we can determine the associated CDF, F_{4L}(v). For example, for v = –1
In a similar fashion, F_{4}_{L}(v) for other values of v may be determined. A plot of F_{4}_{L}(v) versus v is shown in Fig. 3.4(b).
The PDF f_{4}_{L}(v) corresponding to F_{4}_{L}(v) can be found by differentiating F_{4}_{L}(v) with respect to v. The derivative of a step of amplitude V is a pulse of value V. Thus, since the steps of F_{4}_{L}(v) are of value 0.25,
A plot of f_{4}_{L}(v) versus v is shown in Fig. 3.4(c).
3.3.2 The Average Value, the Mean Squared Value, and the Variance of a Random Variable
The average value or mean, m, of a random variable X, also called the expectation of X, is also denoted either by or E(x). For a discrete random variable, X_{d}, where n is the total number of possible outcomes of values x_{1}, x_{2}, . . . , x_{n}, and where the probabilities of the outcomes are P(x_{1}), P(x_{2}), . . . , P(x_{n}) it can be shown that
For a continuous random variable X_{c}, with PDF f_{c}(x), it can be shown that
and that the mean square value, is given by
Figure 3.5 shows an arbitrary PDF of a continuous random variable. A useful number to help in evaluating a continuous random variable is one that gives a measure of how widely spread its values are around its mean m. Such a number is the root mean square (rms) value of (X – m) and is called the standard deviation σ of X.
Figure 3.5 A Probability Distribution Function (PDF) of a continuous random variable.
The square of the standard deviation, σ^{2}, is called the variance of X and is given by
The relationship between the variance σ^{2} and the mean square value E (X^{2}) is given by
We note that for the average value m = 0, the variance σ^{2} = E (X^{2}).
3.3.3 The Gaussian Probability Density Function
The Gaussian or, as it's sometimes called, the normal PDF^{[1]}^{,}^{[2]} is very important to the study of wireless transmission and is the function most often used to describe thermal noise. Thermal noise is the result of thermal motions of electrons in the atmosphere, resistors, transistors, and so on and is thus unavoidable in communication systems. The Gaussian probability density function, f(x), is given by
where m is as defined in Eq. (3.28) and σ as defined in Eq. (3.31). When m = 0 and σ = 1 the normalized Gaussian probability density function is obtained. A graph of the Gaussian PDF is shown in Fig. 3.6(a).
Figure 3.6 The Gaussian random variable.
The CDF corresponding to the Gaussian PDF is given by
When m = 0, the normalized Gaussian cumulative distribution function is obtained and is given by
A graph of the Gaussian cumulative distribution function is shown in Fig. 3.6(b). In practice, since the integral in Eq. (3.35) is not easily determined, it is normally evaluated by relating it to the wellknown and numerically computed function, the error function. The error function of v is defined by
and it can be shown that erf(0) = 0 and erf(∞) = 1.
The function [1 – erf(v)] is referred to as the complementary error function, erfc(v). Noting that , we have
Tabulated values of erfc(v) are only available for positive values of v.
Using the substitution , it can be shown^{[1]} that the Gaussian CDF F(x) of Eq. (3.35) may be expressed in terms of the complementary error function of Eq. (3.37) as follows:
3.3.4 The Rayleigh Probability Density Function
The propagation of wireless signals through the atmosphere is often subject to multipath fading. Such fading will be described in detail in Chapter 5. Multipath fading is best characterized by the Rayleigh PDF.^{[1]} Other phenomena in wireless transmission are also characterized by the Rayleigh PDF, making it an important tool in wireless analysis. The Rayleigh probability density function f(r) is defined by
and hence the corresponding CDF is given by
A graph of f(r) as a function of r is shown in Fig. 3.7. It has a maximum value of , which occurs at r = α. It has a mean value , a meansquare value , and hence, by Eq. (3.32), a variance σ^{2} given by
Figure 3.7 The Rayleigh probability density function. (From Taub, H., and Schilling, D., Principles of Communication Systems, McGrawHill, 1971, and reproduced with the permission of the McGrawHill Companies.)
A graph of F(r) versus 10 log_{10} (r^{2} / 2α^{2}), which is from Feher,^{[3]} is shown in Fig. 3.8. If the amplitude envelope variation of a radio signal is represented by the Rayleigh random variable R, then the envelope has a meansquare value of , and hence the signal has an average power of . Thus, 10 log_{10} (r^{2} / 2α^{2}), which equals 10 log_{10} (r^{2} / 2) – 10 log_{10} (α^{2}), represents the decibel difference between the signal power level when its amplitude is r and its average power. From Fig. 3.8 it will be noted that for signal power less than the average power by 10 dB or more, the distribution function F(r) decreases by a factor of 10 for every 10dB decrease in signal power. As a result, when fading radio signals exhibit this behavior, the fading is described as Rayleigh fading.
Figure 3.8 The Rayleigh cumulative distribution function. (By permission from Ref. 3.)
3.3.5 Thermal Noise
White noise^{[1]} is defined as a random signal whose power spectral density is constant (i.e., independent of frequency). True white noise is not physically realizable since constant power spectral density over an infinite frequency range implies infinite power. However, thermal noise, which as indicated earlier has a Gaussian PDF, has a power spectral density that is relatively uniform up to frequencies of about 1000 GHz at room temperature (290K), and up to about 100 GHz at 29K.^{[4]} Thus, for the purpose of practical communications analysis, it is regarded as white. A simple model for thermal noise is one where the twosided power spectral density G_{n}(f) is given by
where N_{0} is a constant.
In a typical wireless communications receiver, the incoming signal and accompanying thermal noise is normally passed through a symmetrical bandpass filter centered at the carrier frequency f_{c} to minimize interference and noise. The width of the bandpass filter, W, is normally small compared to the carrier frequency. When this is the case the filtered noise can be characterized via its socalled narrowband representation.^{[1]} In this representation, the filtered noise voltage, n_{nb}(t), is given by
where n_{c}(t) and n_{s}(t) are Gaussian random processes of zero mean value, of equal variance and, further, independent of each other. Their power spectral densities, and , extend only over the range –W/2 to W/2 and are related to G_{n} (f) as follows:
The relationship between these power spectral densities is shown in Fig. 3.9. This narrowband noise representation will be found to be very useful when we study carrier modulation methods.
Figure 3.9 Spectral density relationships associated with narrowband representation of noise.
3.3.6 Noise Filtering and Noise Bandwidth
In a receiver, a received signal contaminated with thermal noise is normally filtered to minimize the noise power relative to the signal power prior to demodulation. If, as shown in Fig. 3.10, the input twosided noise spectral density is N_{0} / 2, the transfer function of the real filter is H_{r}(f), and the output noise spectral density is G_{no}(f), then, by Eq. (3.21), we have
Figure 3.10 Filtering of white noise.
and thus the normalized noise power at the filter output, P_{o}, is given by
A useful quantity to compare the amount of noise passed by one receiver filter versus another is the filter noise bandwidth.^{[1]} The noise bandwidth of a filter is defined as the width of an ideal brickwall (rectangular) filter that passes the same average power from a white noise source as does the real filter. In the case of a real low pass filter, it is assumed that the absolute values of the transfer functions of both the real and brickwall filters are normalized to one at zero frequency. In the case of a real bandpass filter, it is assumed that the brickwall filter has the same center frequency as the real filter, f_{c} say, and that the absolute values of the transfer functions of both the real and brickwall fitlers are normalized to one at f_{c}.
For an ideal brickwall low pass filter of twosided bandwidth B_{n} and H_{bw} (f) = 1 from –B_{n}/2 to +B_{n}/2
Thus, from Eqs. (3.46) and (3.47) we determine that
Figure 3.11 shows the transfer function H_{bw}(f) of a low pass brickwall filter of twosided noise bandwidth B_{n} superimposed on the twosided transfer function H_{r}(f) of a real filter.
Figure 3.11 Low pass filter twosided noise bandwidth, Bn.