impulseresponses

Impulse responses describe how a dynamic system responds to a Dirac delta input. In continuous time, this is written as h(t); in discrete time, h[n]. For linear time-invariant LTI systems, the output y(t) is the convolution of the input x(t) with the impulse response: y(t) = ∫ h(τ) x(t−τ) dτ, or y[n] = ∑ h[k] x[n−k]. Because the system is LTI, the impulse response fully characterizes its behavior and can be used to predict any output for any input.

In the frequency domain, the Fourier transform H(ω) of h(t) (or the Z-transform H(z) of h[n]) is the system’s transfer function, relating input and output spectra. This makes convolution in time equivalent to multiplication in frequency, simplifying analysis and filter design. Impulse responses also enable deconvolution, which retrieves inputs from known outputs when the system is known.

Practically, impulse responses are measured by applying a short, ideally instantaneous impulse. In many real-world cases, approximate impulses or specially designed signals such as exponential sweeps or pseudo-random sequences are used, followed by deconvolution to recover h. For causal, stable systems, h(t) = 0 for t < 0, and the energy of h is finite.

Common examples include the impulse response of an RC circuit, h(t) = (1/RC) e^{−t/(RC)} u(t) for t ≥ 0, and a discrete-time first-order system with h[n] = a^n u[n], 0 < a < 1. Impulse responses underpin applications across audio, room acoustics, communications, and control, enabling filtering, equalization, reverberation modeling, and system identification. See also step response, transfer function, and convolution.