Wiener theorem

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 $L_2$ signal $f:\mathbb{R}\to \mathbb{R}.$ We overload notation and refer to a signal with implied free argument, say, $t$, so that $f(rt-\xi ),$, for example, refers to denote the signal $t\mapsto f(rt-\xi ).$ We write the inner product between signals $t\mapsto f(t)$ and $t\mapsto f^{\prime }(t)$ as $⟨f(t),f^{\prime }(t)⟩$. Where it is not clear that the free argument is, e.g. $t$, we will annotate it $⟨f(t),f^{\prime }(t);t⟩$. Say that ${\mathcal{F}}_t\{f(t)\}$ is the Fourier transform of some $f(t)\in L_2$, i.e. $$\begin{array}{rl}{\mathcal{F}}_t\{f(t)\}(\tau )& =\int e^{2\pi it\tau }f(t)\mathrm{d}t\\ & =⟨e^{2\pi it\tau },f(t);t⟩\end{array}$$

and that ${\mathcal{A}}_{\xi }\{f(t)\}$ is the autocorrelogram, i.e. $${\mathcal{A}}_t\{f(t)\}(\tau )=⟨f(t),f(t-\tau );t⟩$$

What is the Fourier transform of $\mathcal{A}f$? That is what the Wiener-Khintchine-relation (Wiener 1930) tells us. Assuming all terms are well-defined (which is non-trivial in general!), $${\mathcal{F}}_{\xi }\{\mathcal{A}f(\xi )\}(\tau )=|{\mathcal{F}}_t\{f(t)\}{|}^2(\tau ).$$

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… $$\begin{array}{rlr}{\mathcal{F}}_{\xi }\{\mathcal{A}f(\xi )\}(\tau )& ={\mathcal{F}}_{\xi }\{⟨f(t),f(t-\xi );t⟩\}(\tau )& \\ & =\int e^{2\pi i\xi \tau }⟨f(t),f(t-\xi );t⟩\mathrm{d}\xi & \\ & =\int e^{2\pi i\xi \tau }\int f(t)f(t-\xi )\mathrm{d}t\mathrm{d}\xi & \\ & =\int \int f(t)e^{2\pi i\xi \tau }f(t-\xi )\mathrm{d}t\mathrm{d}\xi & \\ & =\int \int f(t)e^{2\pi i\xi \tau }f(t-\xi )\mathrm{d}\xi \mathrm{d}t& \\ & =\int f(t)\int e^{2\pi i\xi \tau }f(t-\xi )\mathrm{d}\xi \mathrm{d}t& \\ & =\int f(t){\mathcal{F}}_{\xi }\{f(t-\xi )\}(\tau )\mathrm{d}t& \\ & =\int f(t)e^{2\pi it\tau }{\mathcal{F}}_{\xi }\{f(-\xi )\}(\tau )\mathrm{d}t& \\ & =\int f(t)e^{2\pi it\tau }{\mathcal{F}}_{\xi }\{f(\xi )\}(\tau )\mathrm{d}t& \\ & =\int f(t)e^{2\pi it\tau }\overline{{\mathcal{F}}_{\xi }\{f(\xi )\}(\tau )}\mathrm{d}t& f\text{ is real}\\ & =\int f(t)e^{2\pi it\tau }\mathrm{d}t\overline{{\mathcal{F}}_{\xi }\{f(\xi )\}(\tau )}& \\ & ={\mathcal{F}}_t\{f(t)\}(\tau )\overline{{\mathcal{F}}_{\xi }\{f(\xi )\}(\tau )}& \\ & ={\mathcal{F}}_t\{f(t)\}(\tau )\overline{{\mathcal{F}}_t\{f(t)\}(\tau )}& \\ & =|{\mathcal{F}}_t\{f(t)\}(\tau ){|}^2& \end{array}$$

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.

$$|{\mathcal{F}}_{\xi }\{\mathcal{A}f(\xi )\}{|}^2(\tau )=|{\mathcal{F}}_t\{f(t)\}{|}^4(\tau )$$