impulseresponse

An impulse response of a system is the output produced when the input is an impulse, an extremely short and broadband signal. In continuous time, the impulse is the Dirac delta δ(t); in discrete time, the Kronecker delta δ[n]. For linear time-invariant systems, the impulse response h(t) (continuous) or h[n] (discrete) completely characterizes the system: the output to any input x(t) is the convolution y(t) = ∫ h(τ) x(t−τ) dτ, or y[n] = ∑_k h[k] x[n−k].

Because LTI systems are linear and time-invariant, the response to any input can be constructed as a weighted sum of shifted impulses, making h the fundamental descriptor of the dynamics. Properties include causality, where h(t)=0 for t<0 if the system cannot respond before input is applied; and BIBO stability, which is guaranteed if ∑_n |h[n]| < ∞ (discrete) or ∫ |h(t)| dt < ∞ (continuous).

Obtaining the impulse response can be done by applying an impulse input, or, in practice, by using a close approximation such as a very short pulse. In experiments, h can also be identified by applying known excitation signals (chirps, pseudorandom sequences) and performing deconvolution or system identification to estimate h.

In the frequency domain, the Fourier transform H(ω) of h(t) (or the DTFT H(e^{jω}) of h[n]) is the system transfer function, describing gain and phase shift as a function of frequency. The impulse response also relates to other responses: the step response is the integral of h, and the impulse response is the derivative of the step response for continuous-time LTI systems.

Applications include filter design, signal analysis, control systems, and audio processing, where h enables prediction of outputs for arbitrary inputs via convolution.