Convolution

4 min read#fourier

The sliding-overlap operation that blends two signals — and which, magically, becomes simple multiplication in the frequency domain.

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Convolution

Convolution is the operation that slides one signal across another, multiplying and accumulating the overlap at every shift. For two signals f and g:

(f * g)(\tau) = \int_{-\infty}^{\infty} f(u)\,g(\tau - u)\,du.

Intuitively, g is a little kernel — a stencil — that you drag along f; at each position the output is the area of their overlap. Smear a signal with a blob-shaped kernel and you blur it; smear it with a kernel that has positive and negative parts and you sharpen or differentiate it. Every linear filter — every blur, echo, equalizer, and edge detector — is a convolution.

The kernel sliding across the signal

Below, a bump-shaped kernel (the moving highlight) slides left to right across a two-peak signal (top). At each position the running output (bottom) records how much the kernel currently overlaps the signal. Where the kernel sits on a peak, the output rises; between peaks, it dips.

Top: a fixed signal with a sliding kernel. Bottom: the convolution — the accumulated overlap at each position.

The convolution theorem

Convolution looks expensive — a sliding integral at every point — yet the Fourier Transform collapses it into something trivial. Convolution in time is multiplication in frequency, and vice versa:

\widehat{f * g} = \hat f \cdot \hat g.

This is the engine of practical signal processing. To filter a signal, you do not laboriously slide a kernel across it; you transform to the Frequency Domain, multiply by the filter's response, and transform back. A blur becomes "attenuate the high frequencies"; an echo becomes "boost a comb of frequencies." The hard operation in one domain is the easy one in the other.

See also

Related topics

  • Fourier TransformThe generalization of the Fourier series to non-periodic signals — an integral that resolves any signal into a continuous band of frequencies.
  • Frequency DomainThe second of the two equivalent views of a signal — describing it by which frequencies it contains rather than by how it varies in time.
  • Gibbs PhenomenonThe persistent ~9% overshoot that a truncated Fourier sum makes near a jump discontinuity — and that never goes away no matter how many terms you add.
  • Fourier AnalysisAny signal is a sum of pure sinusoids — and reading it as such turns the tangled language of time into the clean language of frequency.
  • Fourier SeriesA periodic signal is an exact sum of harmonics — sinusoids at integer multiples of one fundamental frequency.
  • SpectrumThe magnitude of a signal's frequency content — the map of where its energy lives, read as a landscape of peaks.

Referenced by