impulssivaste

Impulssivaste is the impulse response of a system, describing its output when driven by a brief input signal approaching an ideal Dirac delta. In continuous time, h(t) represents the system's response, while in discrete time it is h[n]. For linear time-invariant systems, the output is the convolution of input with h.

Equations: y(t) = x(t) h(t); y[n] = sum_k h[k] x[n-k]. The impulse response thus contains all information about the system's dynamics in LTI settings; knowing h allows predicting the output for any input via convolution.

Frequency-domain perspective: the Fourier transform of h(t) gives the frequency response H(ω); the Laplace transform yields H(s) for continuous time, the Z-transform H(z) for discrete time.

Properties and types: causality means h(t)=0 for t<0, stability requires the impulse response to be absolutely integrable (continuous) or summable (discrete). The impulse response of non-LTI systems is not well defined in the same sense.

Applications: analysis and design of filters, acoustic measurements of rooms or musical spaces, and system identification. In practice, impulse responses are measured by emitting a short pulse or a surrogate impulse (spike, exponential sweep, MLS) and recording the system's response.

Example: the impulse response of a simple RC circuit is h(t) = (1/RC) e^{-t/RC} u(t).