# X and f z z integraldisplay f x x f y xz x dx which

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x ) , and f z ( z ) = integraldisplay −∞ f x ( x ) f y ( x,z x ) dx which is the convolution of the individual PDF. This result can be extended for z being the sum of any number of PDFs, for which the total PDF is the convolution of all the individual PDFs. Regardless of the shapes of the individual PDFs, the tendency upon multiple convolutions is to tend toward a net PDF with a Gaussian shape. This can easily be verified by successive convolution of uniform PDFs, for example, which have the form of gate functions. By the third of fourth convolution, it is already difficult to distinguish the result from a Bell curve. Random processes A random or stochastic process is an extension of a random variable, the former being the latter but also being a function of an independent variable, usually time or space. The outcome of a random process is therefore not just a single value but rather a curve (a waveform). The randomness in a random process refers to the uncertainty regarding which curve will be realized. The collection of all possible waveforms is called the ensemble, and a member of the ensemble is a sample waveform. Consider a random process that is a function of time. For each time, a PDF for the random variable can be determined from the frequency distribution. The PDF for at time t is denoted f x ( x ; t ) , emphasizing that the function can be different at different times. However, the PDF f x ( x ; t ) does not completely specify the random process, lacking information about how the random variables representing different times are correlated. That information is contained in the autocorrelation function, itself derivable from the JDF. In the case of a random process with zero mean and unity variance, we have ρ ( t 1 ,t 2 ) = E [ x ( t 1 ) x ( t 2 )] = E ( x 1 ,x 2 ) = integraldisplay −∞ integraldisplay −∞ x 1 x 2 f x 1 ,x 2 ( x 1 ,x 2 ) dx 1 dx 2 23

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(For a complex random process, the autocorrelation function is defined as E ( x 1 ,x 2 ) .) Consequently, not only the PDF but also the second-order or JDF must be known to specify a random process. In fact, the n -th order PDF f x 1 ,x 2 , ··· ,x n ( x 1 ,x 2 , · · · ,x n ; t 1 ,t 2 , · · · ,t n ) must be known to completely specify the random process with n sample times. Fortunately, this near impossibility is avoided when dealing with Gaussian random processes and linear systems, for which it is sufficient to deal with first and second order statistics only. Furthermore, lower-order PDFs are always derivable from a higher-order PDF, e.g. f x 1 ( x 1 ) = integraldisplay −∞ f x 1 ,x 2 ( x 1 ,x 2 ) dx 2 so one need only specify the second-order statistics to completely specify a Gaussian random process. Among random processes is a class for which the statistics are invariant under a time shift. In other words, there is no preferred time for these processes, and the statistics remain the same regardless of where the origin of the time axis is located. Such processes are said to be stationary. True stationarity requires invariance of the n -th order PDF.
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