Infinite impulse response filter 2nd order lowpass Impulse Invariant approximation.

Cuthbert Nyack
The sampled "equivalent" of a second order Butterworth filter derived by use of the Impulse Invariant approximation is shown in the applets below. The Butterworth filter has an upper 3dB frequency of 1rad/s. The first applet shows the filter response on a log-log scale. The vertical scale for amplitude is in dB and is shown in the scrollbar label as dB/Vdiv. Horizontal scale is in rad/s and goes to a maximum of 2xsampling frequency. Only the range up tp ½ of the sampling frequency is important but the rest is included to show the periodicity of the response. Because of the log scale, the higher periods are greatly "compressed". If the /Hdiv is "a" then the horizontal scale goes from a-1 to a3 rad/s. For phase the vertical scale is from -pi to pi/3.

In both applets red shows the sampled filter response, orange shows the analog filter response, green shows the sampled filter phase and cyan shows the analog filter phase. magenta shows the ideal filter response. The applet below shows the response on a linear scale which gives a better view of what happens at low sampling frequencies. In applet below vertical scale is from 0 to 4/3 and horizontal scale is from 0 to 4rad/s.

The response at low sampling frequencies is not good and this approach is only used when it is necessary to have equivalence between the analog and sampled impulse response. As the sampling time is increased the gain at zero frequencies become less than one. This is because the T factor is no longer adequate to make the area under the analog nd sampled responses identical.
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COPYRIGHT © 1999 Cuthbert A. Nyack.