frequencyresponse

Frequency response describes how a system responds to inputs over frequency. For linear time-invariant systems, the output to any input equals the convolution of the input with the system's impulse response. In the frequency domain, the output spectrum is the product of the input spectrum and the system's frequency response H(ω) (continuous time) or H(e^{jΩ}) (discrete time). The frequency response is a complex function H(ω) = |H(ω)| e^{j∠H(ω)}; the magnitude |H(ω)| indicates amplification or attenuation vs frequency, and the phase ∠H(ω) indicates the phase shift. Representations include the magnitude response and phase response, often shown on Bode plots (log magnitude vs log frequency, and phase vs log frequency).

For continuous-time systems, H(s) = Y(s)/X(s) with s = σ + jω; for steady-state sinusoidal inputs, s = jω and H(jω) describes the response. The frequency response is fully determined by the impulse response h(t) (or h[n] in discrete time) via Fourier transform. It is central in designing filters, audio gear, communications systems, and measurement equipment. A flat magnitude response with linear phase is ideal in some designs, but causality and stability impose trade-offs. Causality implies certain phase behavior and realizability; stability requires the impulse response to be absolutely summable (discrete) or absolutely integrable (continuous).

Measuring frequency response can be done with swept-sine, logarithmic sweeps, or white-noise stimuli, from which H(ω) is estimated.