Momentum invariance transformation

The impulse invariance transformation (impulse invariant transformation, IIR) is a mathematical process (a system response invariant transformation) and is used to synthesize time-discrete , mainly digital filters .

Explanation

To this end, which is the impulse response of an analog filter by equidistant sampling in the time-discrete pulse response with transferred. The impulse response of the time-discrete filter thus corresponds to the impulse response of the analog filter at the sampling times . ${\displaystyle g_{a}(t)}$ ${\displaystyle g(k)}$ ${\displaystyle g(k)=g_{a}(kT)}$ ${\displaystyle kT}$

To carry out the momentum-invariant transformation, proceed as follows. Using an inverse Laplace transformation , the impulse response is obtained from the transfer function of the analog filter: ${\displaystyle g_{a}(t)}$ ${\displaystyle G(s)}$

${\displaystyle g_{a}(t)=L^{-1}\left\{G(s)\right\}}$

In order to "scan" the impulse response, substituting by in . Here is the sampling period. The z-transfer function can now be obtained from the sampled impulse response with the help of the z-transformation . In summary, the momentum-invariant transformation can be called ${\displaystyle t}$ ${\displaystyle kT}$ ${\displaystyle g_{a}(t)}$ ${\displaystyle T}$

${\displaystyle G(z)=T\cdot Z\left\{\left.L^{-1}\left\{G(s)\right\}\right|_{t=k\,T}\right\}}$

write. The multiplication by reduces the prefactor of the spectrum of the sampled signal, so that the spectrum of the sampled signal (apart from aliasing) becomes independent of the sampling period. ${\displaystyle T}$ ${\displaystyle {\frac {1}{T}}}$

In this way, a time-discrete filter can be designed which has the same impulse response at the sampling times as a corresponding analog filter. This is hardly noticeable with a suitably high sampling in the frequency range. The time-discrete filter thus approximates the frequency response of the analog filter. ${\displaystyle kT}$

With the transformation

${\displaystyle G(z)=\left(1-z^{-1}\right)\cdot Z\left\{\left.L^{-1}\left\{{\frac {G(s)}{s}}\right\}\right|_{t=k\,T}\right\}}$

a z-transfer function would be obtained which has the same step response at the sampling times . ${\displaystyle kT}$

example

Comparison of the impulse or step response of the filter

Consider an analog filter with the following transfer function:

${\displaystyle G(s)={\frac {6}{s^{2}+4s+5}}}$

The impulse response of the filter is:

${\displaystyle L^{-1}\left\{G(s)\right\}=6\mathrm {e} ^{-2t}\sin t}$

We now substitute by what we ${\displaystyle t}$ ${\displaystyle kT}$

${\displaystyle 6\mathrm {e} ^{-2kT}\sin kT}$

receive. The z-transform of is . The prefactor can be written as ; using the damping theorem of the z-transformation, which reads there , one thus obtains ${\displaystyle \sin kT}$ ${\displaystyle {\frac {z\sin T}{z^{2}-2z\cos T+1}}}$ ${\displaystyle \mathrm {e} ^{-2kT}}$ ${\displaystyle a^{-k}=\left(\mathrm {e} ^{2T}\right)^{-k}}$ ${\displaystyle Z\left\{a^{-k}x[k]\right\}=X(az)}$

${\displaystyle G(z)=6T\cdot {\frac {az\sin T}{a^{2}z^{2}-2az\cos T+1}}}$

for the discretized transfer function of the filter. For a comparison of the impulse response or the step response of the analog and the discretized filter, see the picture opposite.

literature

• Hermann Götz: Introduction to digital signal processing . 3rd revised and expanded edition. Teubner, Stuttgart et al. 1998, ISBN 3-519-20117-8 , ( Teubner study scripts 117 electrical engineering ).
• Alan V. Oppenheim, Ronald W. Schafer: Discrete-time signal processing . 3rd revised edition. Oldenbourg, Munich et al. 1999, ISBN 3-486-22948-6 .