Impulse response

An impulse response records the output produced by an idealized input concentrated at one instant. In continuous time, that input is the Dirac delta, written δ(t); in discrete time, it is the unit impulse δ[n].¹ Although an ideal impulse is a mathematical abstraction, a sufficiently short pulse can approximate one when its duration is small relative to the system's characteristic response time.

The response itself may be a voltage, pressure, displacement, sound waveform, image, or any other measurable output. Its shape reveals delay, amplification, decay, oscillation, resonant frequencies, and possible instability. A causal system has an impulse response that is zero before the impulse arrives, while a noncausal mathematical model may have nonzero values earlier in time.

Impulse responses are used in acoustics, electronics, mechanics, optics, control engineering, seismology, and digital signal processing. They provide a common language for comparing systems whose physical inputs and outputs differ substantially.

For a linear time-invariant system, the impulse response completely determines the response to any suitable input. The output is the convolution of the input with the impulse response: y(t) = x(t) * h(t) in continuous time, or y[n] = x[n] * h[n] in discrete time.¹

Linearity means that responses to scaled and added inputs can be scaled and added; time invariance means that delaying the input delays the output without changing its shape. Decomposing an input into shifted impulses therefore lets the system's output be built by adding shifted, weighted copies of h. This result is the basis of finite impulse response and infinite impulse response filters.

In the frequency domain, the Fourier transform of the impulse response is the system's frequency response, often written H(f) or H(ω).² Poles, zeros, phase, and magnitude can thus be studied either through h or through the corresponding transfer function. The equivalence fails for nonlinear or time-varying systems, where one impulse response cannot describe every operating condition.

An impulse response is usually estimated by exciting a real system with a short pulse, a swept sine, a pseudorandom sequence, or broadband noise, then recording the output. The measured waveform is deconvolved with the known excitation to reduce the influence of the test signal itself.

Digital systems often obtain h[n] directly by feeding a unit sample into a simulation or filter implementation. In laboratory measurements, finite pulse width, sensor bandwidth, background noise, clipping, clock mismatch, and reflections can distort the result. Windowing and averaging can improve the estimate, but they also alter time or frequency resolution.

In room acoustics, the measured impulse response separates direct sound from early reflections and late reverberation, supporting estimates such as reverberation time and enabling auralization through convolution. In imaging and instrumentation, a related impulse response is called a point-spread function when the input is localized in space rather than time.

Impulse responses can expose properties that a frequency-response plot hides. The location of the peak indicates latency, while a long tail may reveal energy storage, reverberation, dispersion, or slow thermal and mechanical relaxation. A response can also have a large transient peak even when its steady-state gain is modest.

The distinction between a system's impulse response and its inverse is practically important. Deconvolution attempts to recover an input or undo a system, but it can strongly amplify noise at frequencies where the response is small. Equalizers therefore use regularization, limits on gain, or minimum-phase designs rather than a mathematically exact inverse in every case.³

For a stable continuous-time system, absolute integrability of h(t) is a sufficient condition for bounded-input, bounded-output stability; discrete-time tests use the corresponding sum over |h[n]|.⁴ In nonlinear systems, higher-order or amplitude-dependent impulse-response methods are needed, so the ordinary single waveform is only a local or approximate description.

Dirac delta

An idealized impulse with zero duration and unit area, used to probe continuous-time systems.

Convolution

An operation that combines an input with a system's impulse response to calculate the output of an LTI system.

Frequency response

The complex gain of a system as a function of frequency; it is the Fourier transform of the impulse response under suitable conditions.

Point-spread function

The spatial analogue of an impulse response, describing how an imaging system represents a point source.