Recalling that , the output signal from any diagonal state-space model is a linear combination of the modal signals. The two immediate outputs and in Fig.G.3 are given in terms of the modal signals and as
The output signal from the first state variable is
The initial condition corresponds to modal initial state
where is the desired order (number of poles). This simple result is obtained when the response is taken to be maximally flat at as well as dc (i.e., when both and are maximally flat at dc).I.1Also, an arbitrary scale factor for has been set such that the cut-off frequency (-3dB frequency) is rad/sec.
The analytic continuation (§D.2) of to the whole -plane may be obtained by substituting to obtain
In the second-order case, we have, for the analog prototype,
To convert this to digital form, we apply the bilinear transform
Note that the numerator is , as predicted earlier. As a check, we can verify that the dc gain is 1:
In the analog prototype, the cut-off frequency is rad/sec, where, from Eq.(I.1), the amplitude response is . Since we mapped the cut-off frequency precisely under the bilinear transform, we expect the digital filter to have precisely this gain. The digital frequency response at one-fourth the sampling rate is
and dB as expected.
Note from Eq.(I.8) that the phase at cut-off is exactly -90 degrees in the digital filter. This can be verified against the pole-zero diagram in the plane, which has two zeros at , each contributing +45 degrees, and two poles at , each contributing -90 degrees. Thus, the calculated phase-response at the cut-off frequency agrees with what we expect from the digital pole-zero diagram.
In the plane, it is not as easy to use the pole-zero diagram to calculate the phase at , but using Eq.(I.3), we quickly obtain
A related example appears in §9.2.4.
Finding the Eigenstructure of A