Weak sense stationary means that the autocovariance only depends on the time difference, not on the absolute times. It is sometimes called the integrated spectrum. The theorem is useful for analyzing linear time-invariant systems LTI systems when the inputs and outputs are not square-integrable, so their Fourier transforms do not exist. It can also be written with the frequency measured in cycles rather than radians per second and denoted by. Views Read Edit View history.
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Further complicating the issue is that the discrete Fourier transform always exists for digital, finite-length sequences, meaning that the theorem can be blindly applied to calculate auto-correlations of numerical sequences. We can now compute the autocorrelation.
Time Series and StatisticsMacmillan, London,p. Notice also that we could have averaged over instead of over all the 's and we would have obtained the same result, viz.
Being a sampled and discrete-time sequence, the spectral density is periodic in the frequency domain. Clearly, the have to scale inversely with if statistical qualities like are to have a well defined behaviour. So far, we have only asserted that the sum of waves with random phases generates a time-stationary gaussian signal.
The whole point of Wiener's contribution was to make sense of the spectral decomposition of the autocorrelation function of a sample function of a wide-sense-stationary random process even when the integrals for the Fourier transform and Fourier inversion do not make sense.
For example, if you have a random signal with an amplitude that is normal distributed for every instant, but whose variance decreases exponentially with time, you can't apply the Wiener-Khinchin theorem. Since the Fourier transform of the autocorrelation function of a signal is the power spectrum of the signal, this corollary is equivalent to saying that the power spectrum of the output is equal to the power spectrum of the input times the energy transfer function. Weak sense stationary means that the autocovariance only depends on the time difference, not on the absolute times.
If so are there any assumptions?
Um, that's not exactly the definition of WSS that I have in mind: A corollary is that the Fourier transform of the autocorrelation function of the output of an LTI system is equal to the product of the Fourier transform of the autocorrelation function of the input wienner the system times the squared magnitude of the Fourier transform of the system impulse response. Norbert Wiener proved this theorem for the case of a deterministic function in ;  Aleksandr Khinchin later formulated an analogous result for stationary stochastic processes and published that probabilistic analogue in Sign up using Facebook.
Wiener-Khinchin Theorem -- from Wolfram MathWorld
A rigorous treatment would in fact start by worrying about existence of a well-defined limit for all statistical quantities, not just the autocorrelation. This page was last edited on 27 Octoberat Time Series and Statistics. Further, since the number of khintchinf even in a small interval blows up, what is important is their combined effect rather than the behaviour of any individual one.
Random signals and noise: Relztion now have to check this.
If now one assumes that r and S satisfy the necessary conditions for Fourier inversion to be valid, the Wiener—Khinchin theorem takes the simple form of saying that khintcgine and S are a Fourier-transform pair, khinfchine.
In the frequency domain one would pass the signal through a filter admitting a narrow band of frequencies aroundand measure the average power that gets through.
A Handbook of Fourier Theorems. Thus the theorem is valid even for a non-gaussian random process. But the theorem as stated here was applied by Norbert Wiener and Aleksandr Khinchin to the sample functions signals of wide-sense-stationary random processessignals whose Fourier transforms do not exist. The theorem is useful for analyzing linear time-invariant systems LTI systems when the inputs and outputs are not square-integrable, so their Fourier transforms do not exist. For if it holds for non-Gaussian processes, is it suffient for the process Weak or wide-sense stationarityi.
The Wiener-Khinchin Theorem
This also means that the autocovariance of zero shift, ie. Views Read Edit View diener. The actual restriction is wide-sense stationarity, as you've noticed: Prentice Hall, New Jersey. From Wikipedia, the free encyclopedia.
Sign up using Email and Password. All this motivates the definition. One could even argue that the operation of integrating over the 's is summing over a fictitious collection i.