Given two real-valued analog signals \(f(t), g(t)\), there are two very similar but distinct bilinear signal processing operations that are commonly performed in the field of analog signal processing on \(f\) and \(g\) to obtain a new function. The first is to take the convolution of \(f\) and \(g\):
\[(f\ast g)(t):=\int_{-\infty}^{\infty}f(t’)g(t-t’)dt’\]
while, replace that minus sign with a plus sign yields the cross-correlation of \(f\) and \(g\):
\[(f\star g)(t):=\int_{-\infty}^{\infty}f(t’)g(t+t’)dt’\]
At an intuitive level, to understand the convolution, it helps to artificially think of \(f(t)\) for instance as being the main signal of interest (even though convolution is commutative \(f\ast g=g\ast f\)), and \(g(t)\) as being a filter/kernel function that modifies \(f(t)\). For example, if \(g(t)=1\) for \(t\in[-1/2,1/2]\) and \(g(t)=0\) elsewhere (i.e. a bump function), then the convolution \((f\ast g)(t)=\int_{t-1/2}^{t+1/2}f(t’)dt’\) has the simple interpretation of a moving average, a smoothed/blurred version of the original signal \(f\). Or one can be more refined about how one implements this moving average, using a Gaussian filter \(g(t)=e^{-t^2/2\sigma^2}/\sigma\sqrt{2\pi}\) instead, to give a more gentle smoothing (in fact, convolution with a Gaussian filter is sometimes called taking the Weierstrass transform of \(f(t)\)). In addition, as \(\sigma\to 0\), the Gaussian would approach a Dirac delta distribution \(g(t)=\delta(t)\) which provides (by the sifting property) an identity for convolution \(f\ast \delta =\delta\ast f=f\). This fact also underlies the impulse response technique for solving linear time-invariant systems of the form \(\mathcal Ly=f\); simply find a Green’s function \(\hat y(t)\) that solves \(\mathcal L\hat y=\delta\) for a “point source” \(\delta(t)\) first, then it follows that \(f\ast\mathcal L\hat y=f\ast\delta\) and linearity of \(\mathcal L\) means that \(\mathcal L(f\ast\hat y)=f\) so \(y=f\ast\hat y=\hat y\ast f\) solves the differential equation. In line with the remarks earlier, the intuitive way to conceptualize this formula is to think of \(\hat y(t)\) as the “main signal” and the source function \(f(t)\) as a filter that weights \(\hat y(t)\) appropriately at each moment \(t\in\textbf R\) in time, to produce the correct signal \(y(t)\). Slogan:
\[\text{solution}=\text{Green’s function}\ast\text{source}\]
There is also an important convolution theorem which asserts that if one takes the Fourier transform of the above slogan, then the convolution interchanges with multiplication (where the Fourier transform of the Green’s function is called the transfer function).
By contrast, the cross-correlation of two real-valued analog signals provides a way to measure the “delay” between \(f(t)\) and \(g(t)\). Suppose hypothetically that \(f\) and \(g\) were basically the exact same signal, just that \(f\) occurred in time by an amount \(\Delta t\) earlier than \(g\) did; mathematically, \(f(t)=g(t+\Delta t)\). Then the cross-correlation of \(f\) with \(g\) would be:
\[(f\star g)(t)=\int_{-\infty}^{\infty}g(t’+\Delta t)g(t+t’)dt’\]
Then in general finding the time \(t\) at which the cross-correlation \((f\star g)(t)\) is maximized will provide a good proxy for \(\Delta t\), the delay between the two signals.
Of course, in the field of digital signal processing, the integrals would be replaced by series, but other than that the interpretation remains identical.