Impulse invariance  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, formula_1, is sampled with sampling period formula_2 to produce the discrete-time system's impulse response, formula_3. Thus, the frequency responses of the two systems are related by  If the continuous time filter is approximately band-limited (i.e. formula_6 when formula_7), 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/(2"T") 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 formula_10, 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. The zeros, if any, are not so simply mapped. 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. This method is known as the matched Z-transform method, or pole–zero mapping.