The Wiener-Khintchine Theorem

An important method of treating such signals that do not decay at infinity, due independently to Wiener in 1930 and Khintchine in 1934, starts from the form of the convolution theorem given in the previous blog [equation 15 reproduced again:

FT \int_{-\infty}^{\infty} f(\tau)f(t+\tau)d\tau = |F(\omega)|^{2} ]

The inverse Fourier transform of this equation is

\int_{-\infty}^{\infty}f(\tau)f^{*}(t+\tau)d\tau=(1/2\pi)\int_{-\infty}^{\infty}|F(\omega)|^{2}e^{i\omega t}d\omega equation 16

For many signals of interest, such as sinusoids, step functions, and random noise of fixed statistical properties, the autocorrelation integral on the left does not converge. But, if we define a truncated version of f(t) by

f_{T}(t)= f(t) if -T<t<T and f_{T}(t)=0 otherwise

then we can write

\int_{-\infty}^{\infty}f_{T}(\tau)f_{T}(t+\tau)d\tau which equals

(1/2\pi)\int_{-\infty}^{\infty}|F_{T}(\omega)|^{2}e^{i\omega t} d\omega equation 17

where F_{T}(\omega) is the Fourier transform of f_{T}(t). Dividing by the time interval 2T and taking the limit, equation 17 becomes

\lim_{T \rightarrow \infty} (1/2\pi) \int_{-T}^{T}f(\tau)f(t+\tau)d\tau=(1/2\pi)\int_{-\infty}^{\infty}\lim_{T \rightarrow \infty} (|F_{T}(\omega)|^{2})(e^{i\omega t})/(2T)d\omega

Wiener (1949) was able to show that, under the condition that the limit on the left exists, the limit inside of the right hand integral converges to a function:

P(\omega)=\lim_{T \rightarrow \infty} (|F_{t}(\omega)|^{2})/(2T) equation 18

which we call the power spectrum density of f. Using these revised definitions of autocorrelation,

\phi (t)=\lim_{T \rightarrow \infty} (1/2T) \int_{-T}^{T} f(\tau)f(\tau +t)d\tau equation 19

and power spectrum, our result now reads

P(\omega)=(1/2\pi)\int_{-\infty}^{\infty}P(\omega)e^{i \omega t}d\omega equation 20 

which is called the Wiener-Khintchine theorem, the autocorrelation is the inverse Fourier transform of  the power spectrum. This is a very significant result, not a simple restatement of our starting point, equation 16. In Equation 16, both f(t) and F(\omega) must be square integrable,  that is, they must contain finite energy over all time and frequency. In equation 20, f(t) must only be sufficiently well behaved so that

\lim_{T \rightarrow \infty} (1/2T)\int_{-\infty}^{\infty} |f(t)|^{2}dt < \infty

That is to say, f(t) is only required to have finite power (signal squared per unit time) to have a power spectrum, but f(t) must have a finite energy (that is, square integrable or, with additional restrictions, be only absolutely integrable0 to possess a Fourier transform. Two classes of functions of interest, periodic functions and some types of random noise, satisfy the first condition, but not the second.

We will exploit the Wiener-Khintchine theorem further when we discuss Power Spectral Estimation. Here, we have introduced it to show how Fourier integral theory can be generalized to include functions with infinite energy but finite power. Having presented this vignette of the theory of generalized Fourier integrals, we now feel free to abandon further convergence questions in our heuristic discussion of Fourier transform pairs.

More later…

Nalin Pithwa

 

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