When the maximally flat optimality criterion is applied to the general
(analog) squared amplitude response
, a surprisingly simple
result is obtained [64]:
![$\displaystyle G_a^2(\omega_a) = \frac{1}{1+\omega_a^{2N}} \protect$](http://www.dsprelated.com/josimages_new/filters/img2331.png) |
(I.1) |
where
![$ N$](http://www.dsprelated.com/josimages_new/filters/img278.png)
is the desired order (number of
poles). This simple result
is obtained when the response is taken to be maximally flat at
![$ \omega_a=\infty$](http://www.dsprelated.com/josimages_new/filters/img2332.png)
as well as
dc (
i.e., when both
![$ G_a^2(\omega_a)$](http://www.dsprelated.com/josimages_new/filters/img2333.png)
and
![$ G_a^2(1/\omega_a)$](http://www.dsprelated.com/josimages_new/filters/img2334.png)
are maximally flat at dc).
I.1Also, an arbitrary scale factor for
![$ \omega_a$](http://www.dsprelated.com/josimages_new/filters/img2339.png)
has been set such that
the cut-off frequency (-3dB frequency) is
![$ \omega_c = 1$](http://www.dsprelated.com/josimages_new/filters/img2340.png)
rad/sec.
The analytic continuation
(§D.2)
of
to the whole
-plane may be obtained by substituting
to obtain
The
![$ 2N$](http://www.dsprelated.com/josimages_new/filters/img1192.png)
poles of this expression are simply the
roots of unity when
![$ N$](http://www.dsprelated.com/josimages_new/filters/img278.png)
is odd, and the roots of
![$ -1$](http://www.dsprelated.com/josimages_new/filters/img1211.png)
when
![$ N$](http://www.dsprelated.com/josimages_new/filters/img278.png)
is even. Half of these
poles
![$ s_k$](http://www.dsprelated.com/josimages_new/filters/img2343.png)
are in the left-half
![$ s$](http://www.dsprelated.com/josimages_new/filters/img1471.png)
-plane (
re
![$ \left\{s_k\right\}<0$](http://www.dsprelated.com/josimages_new/filters/img2344.png)
) and
thus belong to
![$ H_a(s)$](http://www.dsprelated.com/josimages_new/filters/img2327.png)
(which must be stable). The other half belong
to
![$ H_a(-s)$](http://www.dsprelated.com/josimages_new/filters/img2345.png)
. In summary, the poles of an
![$ N$](http://www.dsprelated.com/josimages_new/filters/img278.png)
th-order Butterworth
lowpass prototype are located in the
![$ s$](http://www.dsprelated.com/josimages_new/filters/img1471.png)
-plane at
![$ s_k = \sigma_k +
j\omega_k = e^{-j\theta_k}$](http://www.dsprelated.com/josimages_new/filters/img2346.png)
, where [
64, p. 168]
![\begin{displaymath}\begin{array}{rcrl} \sigma_k &=&-\!&\sin(\theta_k)\\ \omega_k &=&&\cos(\theta_k) \end{array} \protect\end{displaymath}](http://www.dsprelated.com/josimages_new/filters/img2347.png) |
(I.2) |
with
for
![$ k=0,1,2,\dots,N-1$](http://www.dsprelated.com/josimages_new/filters/img2349.png)
. These poles may be quickly found graphically
by placing
![$ 2N$](http://www.dsprelated.com/josimages_new/filters/img1192.png)
poles uniformly distributed around the unit circle (in
the
![$ s$](http://www.dsprelated.com/josimages_new/filters/img1471.png)
plane, not the
![$ z$](http://www.dsprelated.com/josimages_new/filters/img45.png)
plane--this is not a frequency axis) in
such a way that each complex pole has a complex-conjugate counterpart.
A Butterworth lowpass filter additionally has
zeros at
.
Under the bilinear transform
, these all map to the
point
, which determines the numerator of the digital filter as
.
Given the poles and zeros of the analog prototype, it is straightforward
to convert to digital form by means of the bilinear transformation.
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