Convolution in the time domain equals multiplication in the frequency domain, and vice versa

The convolution theorem is a cornerstone of signal processing: the convolution of two time-domain signals produces the same result as multiplying their frequency-domain spectra, and conversely, multiplying two time-domain signals (as in ring modulation) is equivalent to convolving their spectra. This duality has practical consequences for audio: 1) Filtering (convolution with an impulse response) in the time domain corresponds to multiplying by the filter’s frequency response in the frequency domain—the basis of the FFT-based fast convolution. 2) Ring modulation (multiplication) creates sideband products (spectral convolution). 3) Time-domain shaping (enveloping) produces corresponding spectral transformations. Understanding this duality allows choosing the more computationally efficient domain for any operation.

Examples

Convolving a dry vocal with a reverb impulse response applies the room acoustics: equivalent to multiplying the vocal spectrum by the room’s frequency response. Fast convolution exploits this by FFT-transforming both signals, multiplying, then inverse-FFTing—orders of magnitude faster for long impulse responses.

Assessment

A reverb impulse response is 2 seconds long. Explain why fast convolution (via FFT) is more efficient than direct convolution for applying this reverb, and identify which mathematical property makes it possible.