impulssvastuse

Imulssvastuse is the output of a system when the input is an ideal impulse. In linear time-invariant (LTI) systems, the impulse response fully characterizes the system: the output for any input x(t) is the convolution y(t) = x(t) * h(t). In discrete time, y[n] = sum over k of x[k] h[n-k].

The impulse response h(t) (continuous time) or h[n] (discrete time) is the system’s reaction to a unit impulse δ(t) or δ[n]. The transfer function is obtained by transforming the impulse response: H(s) = Laplace{h(t)} for continuous time, H(jω) = Fourier{h(t)} for frequency response, or H(z) = Z{h[n]} for discrete time. The output spectra satisfy Y(ω) = H(ω) X(ω) in the frequency domain, and y(t) can be computed by convolution with h(t).

Properties: If h is absolutely summable, the system is BIBO stable. Causality implies h(t) = 0 for t < 0 in continuous time (and h[n] = 0 for n < 0 in discrete time). Finite impulse response (FIR) systems have h of finite length; infinite impulse response (IIR) systems have an infinite-length h.

Applications and measurement: Impulse response can be determined by applying an impulse or a very short pulse, or by deriving it from a step response. Knowing h enables efficient filtering via convolution or by multiplying spectra. It is central in audio processing, room acoustics (room impulse response), communication systems, and system identification.