### Choice of Output Signal and Initial Conditions

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*initial condition*corresponds to modal initial state

*i.e.*, it is in

*phase quadrature*with respect to ). Phase-quadrature outputs are often useful in practice,

*e.g.*, for generating complex sinusoids.

#### Butterworth Lowpass Poles and Zeros

When the maximally flat optimality criterion is applied to the general (analog) squared amplitude response , a surprisingly simple result is obtained [64]: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.1}Also, 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

with

#### Example: Second-Order Butterworth Lowpass

In the second-order case, we have, for the analog prototype,To convert this to digital form, we apply the bilinear transform

(I.4) | |||

(I.5) | |||

(I.6) | |||

(I.7) |

Note that the numerator is , as predicted earlier. As a check, we can verify that the dc gain is 1:

*i.e.*, that there is a (double) notch at half the sampling rate. 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

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Bilinear Transformation

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Finding the Eigenstructure of A