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Also covariances

This material is revised and expanded from the appendix of draft versions of a recent conference submission, for my own reference. I used correlograms a lot in that, but it was startling that despite being simple and, to my mind, non-controversial, it is hard to find a decent summary of their properties anywhere.

Credit to Ning Ma for the following image:

autocorrelogram of a processed audio signal
autocorrelogram of a processed audio signal

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Consider an \(L_2\) signal \(f: \bb{R}\to\bb{R}.\) We will frequently overload notation and refer to as 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 \(t\mapsto f'(t)\) as \(\inner{f(t)}{f'(t)}\). Where it is not clear that the free argument is, e.g. \(t\), we will annotate it \(\finner{f(t)}{f'(t)}{t}\).

The autocorrelation transform or correlogram \(\cc{A}:L_2(\bb{R}) \to L_2(\bb{R})\) maps signals to signals. Specifically, \(\mathcal{A}\{f\}\) is a signal \(\bb{R}\to\bb{R}\) such that \[\mathcal{A}\{f\}:=\xi \mapsto \finner{ f(t) }{ f(t-\xi) }{t}\] 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; inferring the covariance of stochastic processes is a different problem. Note also that this is what I would call an autocovariance not an auto-correlation, since it’s not normalized. We seem to be stuck with this unfortunate terminology for historical reasons.

We derive the properties of this transform.

Multiplication by a constant. Consider a constant \(c\in \bb{R}.\) \[\begin{aligned}\mathcal{A}\{cf\}(\xi)&= \inner{ cf(t) }{ cf(t-\xi) }\\ &= c^2\finner{ f(t) }{ f(t-\xi) }{t}\\ &= c^2\mathcal{A}\{f\}(\xi).\\ \end{aligned}\]

Time scaling: \[\begin{aligned}\mathcal{A}\{f(r t)\}(\xi) &=\finner{ f(r t) }{ f(r t-\xi) }{t}\\ &= \int f(r t)f(r t-\xi)\dd t\\ &= \frac{1}{r }\int f(t)f(t-\frac{\xi}{r})\dd t\\ &= \frac{1}{r} \mathcal{A}\{f\}\left(\frac{\xi}{r}\right)\\ \end{aligned}\]

Addition: \[\begin{aligned}\mathcal{A}\{f+f'\}(\xi) &=\finner{ f(t)+f'(t) }{ f(t-\xi)+f'(t-\xi) }{t}\\ &=\finner{ f(t) }{ f(t-\xi)\rangle+\langle f(t),f'(t-\xi) }{t} +\finner{ f'(t) }{ f(t-\xi)\rangle+\langle f'(t),f'(t-\xi) }{t}\\ &= \mathcal{A}\{f\}(\xi)+ \finner{ f'(t) }{ f(t-\xi)}{t} +\finner{f(t)}{f'(t-\xi) }{t} +\mathcal{A}\{f'\}(\xi).\\ &= \mathcal{A}\{f\}(\xi)+ \finner{ f'(t) }{ f(t-\xi)}{t} +\finner{f(t+\xi)}{f'(t) }{t} +\mathcal{A}\{f'\}(\xi).\\ &= \mathcal{A}\{f\}(\xi)+ \finner{ f'(t) }{ f(t-\xi)}{t} +\finner{f'(t) }{f(t+\xi)}{t} +\mathcal{A}\{f'\}(\xi).\\ \end{aligned}\]

We can say little about the term \(\finner{ f'(t) }{ f(t-\xi)}+\finner{f'(t) }{f(t+\xi)}{t}\) without more information about the signals in question. However, we can solve a randomized version. Suppose \(S_i, \, i\in\bb{N}\) 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: \[\begin{aligned} \bb{E}[ \mathcal{A}\{S_1f + S_2f'\}(\xi) &=\bb{E}[ \mathcal{A}\{S_1f\}(\xi) + \finner{ S_2 f'(t) }{ S_1 f(t-\xi)}{t} +\finner{S_2f'(t) }{S_1 f(t+\xi)}{t} +\mathcal{A}\{S_2f'\}(\xi)]\\ &=\bb{E}[ \mathcal{A}\{S_1f\}(\xi)] + \bb{E}\finner{ S_2 f'(t) }{ S_1 f(t-\xi)}{t} + \bb{E}\finner{S_2f'(t) }{S_1 f(t+\xi)}{t} +\bb{E}[ \mathcal{A}\{S_2f'\}(\xi)]\\ &=\mathcal{A}\{f\}(\xi)+ \bb{E}[ S_1S_2]\finner{ f'(t) }{ f(t-\xi) }{t} + \bb{E}[ S_1S_2]\finner{ f'(t) }{ f(t+\xi) }{t}+\mathcal{A}\{f'\}(\xi)\\ &=\mathcal{A}\{f\}(\xi)+ \mathcal{A}\{f'\}(\xi)\\ \end{aligned}\]