Wiener theorem

Now with bonus Bochner!

May 8, 2019 — May 8, 2019

functional analysis
Hilbert space
optimization
signal processing
stochastic processes
Figure 1

The special deterministic case of the Wiener-Khintchine theorem, written up with a slightly different notation for a slightly different project.

As seen in correlograms.

Consider an L2 signal f:RR. We overload notation and refer to a signal with implied free argument, say, t, so that f(rtξ),, for example, refers to denote the signal tf(rtξ). We write the inner product between signals tf(t) and tf(t) as f(t),f(t). Where it is not clear that the free argument is, e.g. t, we will annotate it f(t),f(t);t. Say that Ft{f(t)} is the Fourier transform of some f(t)L2, i.e. Ft{f(t)}(τ)=e2πitτf(t)dt=e2πitτ,f(t);t

and that Aξ{f(t)} is the autocorrelogram, i.e. At{f(t)}(τ)=f(t),f(tτ);t

What is the Fourier transform of Af? That is what the Wiener-Khintchine-relation () tells us. Assuming all terms are well-defined (which is non-trivial in general!), Fξ{Af(ξ)}(τ)=|Ft{f(t)}|2(τ).

To see this, assuming various terms are indeed well-defined, we use the list of properties of the Fourier transform from Wikipedia and grind out the identity… Fξ{Af(ξ)}(τ)=Fξ{f(t),f(tξ);t}(τ)=e2πiξτf(t),f(tξ);tdξ=e2πiξτf(t)f(tξ)dtdξ=f(t)e2πiξτf(tξ)dtdξ=f(t)e2πiξτf(tξ)dξdt=f(t)e2πiξτf(tξ)dξdt=f(t)Fξ{f(tξ)}(τ)dt=f(t)e2πitτFξ{f(ξ)}(τ)dt=f(t)e2πitτFξ{f(ξ)}(τ)dt=f(t)e2πitτFξ{f(ξ)}(τ)dtf is real=f(t)e2πitτdtFξ{f(ξ)}(τ)=Ft{f(t)}(τ)Fξ{f(ξ)}(τ)=Ft{f(t)}(τ)Ft{f(t)}(τ)=|Ft{f(t)}(τ)|2

Corollary: If we are interested in the power spectrum of the autocorrelogram, we note that it relates non-linearly to that of the source signal.

|Fξ{Af(ξ)}|2(τ)=|Ft{f(t)}|4(τ)

1 References

Wiener. 1930. Generalized Harmonic Analysis.” Acta Mathematica.
Yaglom. 1987a. Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References. Springer Series in Statistics.
———. 1987b. Correlation Theory of Stationary and Related Random Functions Volume I.