filterrespons

Filterresponse, typically written as filter response or transfer function in English, is a term used in signal processing to describe how a filter affects an input signal. It characterizes the filter’s effect across time and frequency, indicating which components are attenuated, amplified, or phase-shifted. The concept applies to both analog and digital filters.

For linear time-invariant systems, the filter response is fully described by the impulse response h(t) in continuous time or h[n] in discrete time. The output y of an input x is the convolution of x with the impulse response: y(t) = x(t) * h(t) for continuous signals, and y[n] = x[n] * h[n] for discrete signals. The frequency response, obtained by taking the Fourier transform of the impulse response, is H(ω) in continuous time or H(e^{jω}) in discrete time. This transfer function reveals how each frequency component is attenuated and phase-shifted by the filter. The magnitude |H(ω)| is called the amplitude response and ∠H(ω) the phase response. Designers often use Bode plots to visualize these properties.

Filters are commonly categorized as finite impulse response (FIR) or infinite impulse response (IIR). FIR filters have a finite, non-recursive impulse response and are inherently stable and linear-phase under suitable design. IIR filters use feedback and can achieve sharper effects with fewer coefficients but require careful stability analysis. Typical filter types include low-pass, high-pass, band-pass, and band-stop, each defined by its effect on the signal spectrum.

In practice, the filter response is measured or estimated through impulse or step tests, and visualized to guide design, verification, and implementation in audio, communications, image processing, and control systems.