Wiener theorem

Now with bonus Bochner!


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

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As seen in correlograms.

Consider an \(L_2\) signal \(f: \bb{R}\to\bb{R}.\) We will overload notation and refer to a signal with implied free argument, say, \(t\), so that \(f(rt-\xi),\) would, for example, refers to denote the signal \(t\mapsto f(rt-\xi).\) We write the inner product between signals \(t\mapsto f(t)\) and \(t\mapsto f'(t)\) as \(\inner{f(t)}{f'(t)}\). Where it is not clear that the free argument is, e.g. \(t\), we will annotate it \(\finner{f(t)}{f'(t)}{t}\). Say that \(\ftrn{F}{t}\{f(t)\}\) is the Fourier transform of some \(f(t)\in L_2\), i.e.

\[\begin{aligned} \ftrn{F}{t}\{f(t)\}(\tau)&=\int e^{2\pi i t \tau}f(t)\dd t\\ &=\finner{e^{2\pi i t \tau}}{f(t)}{t} \end{aligned}\]

and that \(\ftrn{A}{\xi}\{f(t)\}\) is the autocorrelogram, i.e.

\[\ftrn{A}{t}\{f(t)\}(\tau)=\finner{f(t)}{f(t-\tau)}{t}\]

What is the Fourier transform of \(\trn{A}f\)? That is what the Wiener-Khintchine-relation (Wiener 1930) tells us. Assuming all terms are well-defined (which is non-trivial in general!),

\[\ftrn{F}{\xi}\{\trn{A}f(\xi)\}(\tau)=|\ftrn{F}{t}\{f(t)\}|^2(\tau).\]

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…

\[\begin{aligned} \ftrn{F}{\xi}\{\trn{A}f(\xi)\}(\tau) &=\ftrn{F}{\xi}\{\finner{f(t)}{f(t-\xi)}{t}\}(\tau) & \\ &=\int e^{2\pi i \xi \tau} \finner{f(t)}{f(t-\xi)}{t}\dd \xi & \\ &=\int e^{2\pi i \xi \tau} \int f(t)f(t-\xi)\dd t \dd \xi & \\ &=\int \int f(t) e^{2\pi i \xi \tau}f(t-\xi)\dd t \dd \xi & \\ &=\int \int f(t) e^{2\pi i \xi \tau}f(t-\xi)\dd \xi\dd t & \\ &=\int f(t) \int e^{2\pi i \xi \tau}f(t-\xi)\dd \xi\dd t & \\ &=\int f(t) \ftrn{F}{\xi}\{f(t-\xi)\}(\tau)\dd t & \\ &=\int f(t) e^{2\pi i t \tau} \ftrn{F}{\xi}\{f(-\xi)\}(\tau)\dd t & \\ &=\int f(t) e^{2\pi i t \tau} \ftrn{F}{\xi}\{f(\xi)\}(\tau)\dd t & \\ &=\int f(t) e^{2\pi i t \tau} \overline{\ftrn{F}{\xi}\{f(\xi)\}(\tau)}\dd t & f \text{ is real} \\ &=\int f(t) e^{2\pi i t \tau} \dd t \overline{\ftrn{F}{\xi}\{f(\xi)\}(\tau)} & \\ &=\ftrn{F}{t}\{f(t)\}(\tau) \overline{\ftrn{F}{\xi}\{f(\xi)\}(\tau)} & \\ &=\ftrn{F}{t}\{f(t)\}(\tau) \overline{\ftrn{F}{t}\{f(t)\}(\tau)} & \\ &=|\ftrn{F}{t}\{f(t)\}(\tau)|^2 & \end{aligned}\]

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.

\[|\ftrn{F}{\xi}\{\trn{A}f(\xi)\}|^2(\tau)=|\ftrn{F}{t}\{f(t)\}|^4(\tau)\]

References

Wiener, Norbert. 1930. “Generalized Harmonic Analysis.” Acta Mathematica 55: 117–258. https://doi.org/10.1007/BF02546511.

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