# 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.

## Discussion

The continuous-time system's impulse response, , is sampled with sampling period 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. when ), 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):

for

### Comparison to the bilinear transform

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.

### Effect on poles in system function

If the continuous poles at , 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.

### Poles and zeros

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. In the case of all-pole filters, the methods are equivalent.

### Stability and causality

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.

### Corrected formula

When a causal continuous-time impulse response has a discontinuity at , the expressions above are not consistent.[1] This is because should really only contribute half its value to .

Making this correction gives

Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function