Complex exponentials are eigensignals of filters (time-invariant and linear systems) . In the discrete case, the response to the input \(e_\nu [n] = \exp(i 2\pi \nu n)\) is \(H(\nu) e_\nu[n]\), a complex exponential with identical frequency, multiplied by the frequency response of the filter \(H(\nu)\), given by the discrete time Fourier transform of the impulse response of the filter: \[H(\nu) = \sum_{n\in\mathbf Z} h[n] \exp(-i 2\pi \nu n) .\]

We plot here the real part of the complex exponentials, for which the filter acts as a gain and a delay.

Drag the coefficients of the filters to watch the effect on the output and the frequency response.

Filtered sound

Predefined filters :

Input frequency :