Correlograms

This material is revised and expanded from the appendix of draft versions of a recent conference submission, for my own reference. I used (deterministic) correlograms a lot in that, and it was startlingly hard to find a decent summary of their properties anywhere. Nothing new here, but… see the material about doing this in a probabilistic way via Wiener-Khintchine representation and covariance kernels which lead to a natural probabilistic spectral analysis.

Consider an $L_2$ signal $f:\mathbb{R}\to \mathbb{R}.$ We frequently overload notation and refer to a signal with free argument $t$, so that $f(rt-\xi ),$ for example, refers to 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 what the free argument is, e.g. $t$, we annotate it $⟨f(t),f^{\prime }(t);t⟩$.

The correlogram $\mathcal{A}:L_2(\mathbb{R})\to L_2(\mathbb{R})$ maps signals to signals. Specifically, $\mathcal{A}\{f\}$ is a signal $\mathbb{R}\to \mathbb{R}$ such that

$$\mathcal{A}\{f\}:=\xi \mapsto ⟨f(t),f(t-\xi );t⟩$$ This is the covariance between $f(t)$ and $f(t-\xi ).$ (Note that we here discuss the covariance between given deterministic signals, not between two stochastic sources; covariance of stochastic processes is a broader, let alone inferring the covariance of stochastic processes.) Note also that this is what I would call an autocovariance not an auto-correlation, since it’s not normalized, but I’ll stick with the latter for now for reasons of convention.

We derive the properties of this transform.

Multiplication by a constant. Consider a constant $c\in \mathbb{R}.$

$$\begin{array}{rl}\mathcal{A}\{cf\}(\xi )& =⟨cf(t),cf(t-\xi )⟩\\ & =c^2⟨f(t),f(t-\xi );t⟩\\ & =c^2\mathcal{A}\{f\}(\xi ).\end{array}$$

Time scaling:

$$\begin{array}{rl}\mathcal{A}\{f(rt)\}(\xi )& =⟨f(rt),f(rt-\xi );t⟩\\ & =\int f(rt)f(rt-\xi )\mathrm{d}t\\ & =\frac{1}{r}\int f(t)f(t-\frac{\xi }{r})\mathrm{d}t\\ & =\frac{1}{r}\mathcal{A}\{f\}\left(\frac{\xi }{r}\right)\end{array}$$

Addition:

$$\begin{array}{rl}\mathcal{A}\{f+f^{\prime }\}(\xi )& =⟨f(t)+f^{\prime }(t),f(t-\xi )+f^{\prime }(t-\xi );t⟩\\ & =⟨f(t),f(t-\xi )⟩+⟨f(t),f^{\prime }(t-\xi );t⟩+⟨f^{\prime }(t),f(t-\xi )⟩+⟨f^{\prime }(t),f^{\prime }(t-\xi );t⟩\\ & =\mathcal{A}\{f\}(\xi )+⟨f^{\prime }(t),f(t-\xi );t⟩+⟨f(t),f^{\prime }(t-\xi );t⟩+\mathcal{A}\{f^{\prime }\}(\xi ).\\ & =\mathcal{A}\{f\}(\xi )+⟨f^{\prime }(t),f(t-\xi );t⟩+⟨f(t+\xi ),f^{\prime }(t);t⟩+\mathcal{A}\{f^{\prime }\}(\xi ).\\ & =\mathcal{A}\{f\}(\xi )+⟨f^{\prime }(t),f(t-\xi );t⟩+⟨f^{\prime }(t),f(t+\xi );t⟩+\mathcal{A}\{f^{\prime }\}(\xi ).\end{array}$$

We can say little about the term $⟨f^{\prime }(t),f(t-\xi );+⟩⟨f^{\prime }(t),f(t+\xi );t⟩$ without more information about the signals in question. However, we can solve a randomized version. Suppose $S_i,i\in \mathbb{N}$ are i.i.d. Rademacher variables, i.e. that they assume a value in $\{+1,-1\}$ with equal probability. Then, we can introduce the following property:

Randomised addition:

$$\begin{array}{rl}\mathbb{E}[\mathcal{A}\{S_{1}f+S_2{f}^{\prime }\}(\xi )& =\mathbb{E}[\mathcal{A}\{S_{1}f\}(\xi )+⟨S_2{f}^{\prime }(t),S_{1}f(t-\xi );t⟩+⟨S_2{f}^{\prime }(t),S_{1}f(t+\xi );t⟩+\mathcal{A}\{S_2{f}^{\prime }\}(\xi )]\\ & =\mathbb{E}[\mathcal{A}\{S_{1}f\}(\xi )]+\mathbb{E}⟨S_2{f}^{\prime }(t),S_{1}f(t-\xi );t⟩+\mathbb{E}⟨S_2{f}^{\prime }(t),S_{1}f(t+\xi );t⟩+\mathbb{E}[\mathcal{A}\{S_2{f}^{\prime }\}(\xi )]\\ & =\mathcal{A}\{f\}(\xi )+\mathbb{E}[S_1{S}_2]⟨f^{\prime }(t),f(t-\xi );t⟩+\mathbb{E}[S_1{S}_2]⟨f^{\prime }(t),f(t+\xi );t⟩+\mathcal{A}\{f^{\prime }\}(\xi )\\ & =\mathcal{A}\{f\}(\xi )+\mathcal{A}\{f^{\prime }\}(\xi )\end{array}$$