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 continuous-time system's impulse response,
to produce the discrete-time system's impulse response,
Thus, the frequency responses of the two systems are related by If the continuous time filter is approximately band-limited (i.e.
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.
, 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 Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function Thus the poles from the continuous-time system function are translated to poles at z = eskT.
[clarification needed] If the system function has zeros as well as poles, they can be mapped the same way, but the result is no longer an impulse invariance result: the discrete-time impulse response is not equal simply to samples of the continuous-time impulse response.
Since poles in the continuous-time system at s = sk transform to poles in the discrete-time system at z = exp(skT), poles in the left half of the s-plane map to inside the unit circle in the z-plane; so if the continuous-time filter is causal and stable, then the discrete-time filter will be causal and stable as well.
When a causal continuous-time impulse response has a discontinuity at
has different right and left limits, and should really only contribute their average, half its right value
Making this correction gives Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function The second sum is zero for filters without a discontinuity, which is why ignoring it is often safe.