Impulse invariance

The Impulsinvarianz transform ( Impulsinvariante transform, IIR) is a mathematical method ( a systemantwortinvariante transformations ) and is used for the synthesis of discrete-time, mainly digital filters.

Explanation

For this, the impulse response of the analog filter is transferred through equidistant sampling in the time-discrete pulse response with. The impulse response of the discrete-time filter thus corresponds to the sampling time points with the impulse response of the analog filter.

To carry out the impulsinvariante transformation, proceed as follows. Using an inverse Laplace transform to obtain the impulse response of the transfer function of the analog filter is:

In order to " scan " the impulse response, is substituted by at. Here is the sample period. The z- transfer function is now obtained from the sampled impulse response by using the z- transformation. In summary, the transformation can impulsinvariante so as

. Write

As a result, a discrete-time filters are designed, which at the sampling instants has the same impulse response as a corresponding analog filter. This is no obvious advantage in suitably high sampling in the frequency domain. The time-discrete filters thereby approximates the frequency response of the analog filter.

The transformation

One would obtain a z- transfer function, which has the same step response at the sampling instants.

Example

Given an analog filter with the following transfer function:

The pulse response of the filter is:

We now substitute by what we

Receive. The z-transform of reads. The pre-factor can be defined as write; applying the attenuation rate of the z-transform, which is obtained as thus

For the discretized transfer function of the filter. For comparison, the impulse response or the step response of the analog and the discretized filter, see picture on the left.