Convolution: Introduction¶

Interpreted by Filtering¶

\(f * h\) is (at least) as good as the best properties of \(f\) and \(h\) separately (as long as \(h\) is not a delta function)

Filtering a signal (function) is to eliminate some of its frequency and let the others go through, then check the consequences in the time domain.
Filter is a system that convolves an input (which can vary) with a fixed signal (function), a.k.a. impulse response
In the frequency domain, filtering is just multiplication: \(G = F \cdot H\)
In the time domain, filtering is synonymous with convolution: \(g = f * h\) where \(h\) is the fixed fixed impulse response, or the inverse Fourier transform of the transform function \(H\).
It is easy to understand filtering (convolution) in terms of frequency, not so easy in the time domain
low-pass filter works as a smoother: removing the short-term fluctuations and leaving the longer-term trend.
high-pass filter: e.g. edge detection
To design a filter is to design \(H\)

If \(f\) is differentiable, convolution \(f * g\) is differentiable:

\[(f * g)' = f' * g\]