In practice this can be very difficult to ascertain

Info icon This preview shows pages 25–27. Sign up to view the full content.

In practice this can be very difficult to ascertain. Stationarity is often asserted not on the basis of the sample basis but instead in terms of the known properties of the generating mechanism. A less restrictive requirement is satisfied by processes which are wide-sense or weakly stationary. Wide-sense sta- tionarity requires only that the PDF be invariant under temporal translation through second order. For such processes, expectations are constant, and the autocorrelation function is a function of lag only, i.e. ρ ( t 1 ,t 2 ) = ρ ( t 2 t 1 ) . Even this less restrictive characterization of a random variable is an idealization and cannot be met by a process that occurs over a finite span of times. In practice, the term is meant to apply to a process approximately over some finite time span of interest. Demonstrating stationarity or even wide-sense stationarity may be difficult in practice since if the PDF is not known a priori since one generally does not have access to the ensemble and so cannot perform the ensemble average necessary to compute, for example, the correlation function. However, a special class of stationary random processes called ergodic processes lend themselves to experimental interrogation. For these processes, all of the sample waveforms in the ensemble are realized over the course of time and an ensemble average can therefore be replicated by a time average (sample average). E ( x ( t )) lim T →∞ 1 T integraldisplay T/ 2 T/ 2 x ( t ) dt ρ ( τ ) lim T →∞ 1 T integraldisplay T/ 2 T/ 2 x ( t ) x ( t + τ ) dt An ergodic process obviously cannot depend on time and therefore must be stationary. Gaussian random processes The properties of random variables were stated in section 1.5.5. A Gaussian random process is one for which the random variables x ( t 1 ) , x ( t 2 ) , · · · , x ( t n ) are jointly Gaussian according to (1.14) for every n and for every set ( t 1 ,t 2 , · · · ,t n ) . All of the simplifying properties of jointly normal random variables therefore apply to Gaussian random processes. In particular, since the complete statistics of Gaussian processes are determined by their second- order statistics (autocorrelation functions), a Gaussian random process that is wide-sense stationary is automatically stationary in the strict sense. Moreover, the response of any linear system to a Gaussian random process will also be a Gaussian random process. This fact is crucial for evaluating the propagation of stochastic signals and noise through the components of communication and radar systems. Power spectral density of a random process Communication and radar engineers are often concerned with the frequency content of a signal. If the signal in question is a stationary random variable, at least in the wide sense, then it must exist over an infinite time domain and 24
Image of page 25

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

therefore would have to be a power signal. But what is the PSD for such a signal? The sample functions of the process that occupy the ensemble are potentially all different, with different power spectral densities, and no one of them is preferred to another.
Image of page 26
Image of page 27
This is the end of the preview. Sign up to access the rest of the document.
  • Spring '13
  • HYSELL
  • The Land, power density, Solid angle

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern