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Figure B.3: Signal flow graph for the general one-pole filter
$ y(n) = b_0 x(n) - a_1 y(n - 1).$

Fig.B.3 gives the signal flow graph for the general one-pole filter. The road to the frequency response goes as follows:

Figure B.4: Family of frequency responses of the one-pole filter
$ y(n) = x(n) - a_1 y(n - 1)$
for various real values of $ a_1$ . (a) Amplitude response. (b) Phase response.

Difference equation: & $y(n) = b_0 x(n) - a_1 y(n-1)$\\ [5pt]
{\it z} transform: & $Y(z) = b_0 X(z) - a_1 z^{-1}Y(z)$\\ [5pt]
Transfer function: & $H(z) = \displaystyle\frac{b_0}{1+a_1z^{-1}}$\\ [5pt]
Frequency response: & $H(e^{j\omega T}) = \displaystyle\frac{b_0}{1+a_1e^{-j\omega T}}$

The one-pole filter has a transfer function (hence frequency response) which is the reciprocal of that of a one-zero. The analysis is thus quite analogous. The frequency response in polar form is given by

G(\omega) &=& \frac{\vert b_0\vert}{\sqrt{[1 + a_1 \cos(\omega T)]^2 + [-a_1 \sin(\omega T)]^2}}\\
&=& \frac{\vert b_0\vert}{\sqrt{1 + a_1^2 + 2a_1 \cos(\omega T)}}\\ [10pt]
\Theta(\omega) &=&
-\tan^{-1}\left[\frac{-a_1 \sin(\omega T)}{1 + a_1 \cos(\omega T)}\right], & b_0>0 \\ [5pt]
\pi-\tan^{-1}\left[\frac{-a_1 \sin(\omega T)}{1 + a_1 \cos(\omega T)}\right], & b_0<0 \\
\end{array} \right..

A plot of the frequency response in polar form for $ b_0 = 1$ and various values of $ a_1$ is given in Fig.B.4.

The filter has a pole at $ z = -a_1$ , in the $ z$ plane (and a zero at $ z$ = 0). Notice that the one-pole exhibits either a lowpass or a highpass frequency response, like the one-zero. The lowpass character occurs when the pole is near the point $ z = 1$ (dc), which happens when $ a_1$ approaches $ -1$ . Conversely, the highpass nature occurs when $ a_1$ is positive.

The one-pole filter section can achieve much more drastic differences between the gain at high frequencies and the gain at low frequencies than can the one-zero filter. This difference is achieved in the one-pole by gain boost in the passband rather than attenuation in the stopband; thus it is usually desirable when using a one-pole filter to set $ b_0$ to a small value, such as $ 1 -
\left\vert a_1\right\vert$ , so that the peak gain is 1 or so. When the peak gain is 1, the filter is unlikely to overflow.B.1

Finally, note that the one-pole filter is stable if and only if $ \left\vert a_1\right\vert < 1$ .

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University