Impulse invariance is a technique for designing discrete-time infinite-impulse-response (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discrete-time system. The frequency response of the discrete-time system will be a sum of shifted copies of the frequency response of the continuous-time system; if the continuous-time system is approximately band-limited to a frequency less than the Nyquist frequency of the sampling, then the frequency response of the discrete-time system will be approximately equal to it for frequencies below the Nyquist frequency.

The continuous-time system's impulse response, ${\displaystyle h_{c}(t)}$, is sampled with sampling period ${\displaystyle T}$ to produce the discrete-time system's impulse response, ${\displaystyle h[n]}$.

Thus, the frequency responses of the two systems are related by

If the continuous time filter is approximately band-limited (i.e. ${\displaystyle H_{c}(j\Omega )<\delta }$ when ${\displaystyle |\Omega |\geq \pi /T}$), then the frequency response of the discrete-time system will be approximately the continuous-time system's frequency response for frequencies below π radians per sample (below the Nyquist frequency 1/(2T) Hz):

Note that aliasing will occur, including aliasing below the Nyquist frequency to the extent that the continuous-time filter's response is nonzero above that frequency. The bilinear transform is an alternative to impulse invariance that uses a different mapping that maps the continuous-time system's frequency response, out to infinite frequency, into the range of frequencies up to the Nyquist frequency in the discrete-time case, as opposed to mapping frequencies linearly with circular overlap as impulse invariance does.

If the continuous poles at ${\displaystyle s=s_{k}}$, the system function can be written in partial fraction expansion as

Thus, using the inverse Laplace transform, the impulse response is

The corresponding discrete-time system's impulse response is then defined as the following