Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books



Chapters

Introduction to Digital Filters
    Elementary Audio Digital Filters
       Elementary Filter Sections
          One-Pole

Chapter Contents:

Search Introduction to Digital Filters

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

One-Pole

Figure B.3: Signal flow graph for the general one-pole filter
$ y(n) = b_0 x(n) - a_1 y(n - 1).$
\begin{figure}\input fig/kfig2p20.pstex_t \end{figure}

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.
\begin{figure}\input fig/kfig2p21.pstex_t \end{figure}

\fbox{ \begin{tabular}{rl} Difference equation: & $y(n) = b_0 x(n) - a_1 y(n-1)... ...$H(e^{j\omega T}) = \displaystyle\frac{b_0}{1+a_1e^{-j\omega T}}$ \end{tabular}}

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

\begin{eqnarray*} G(\omega) &=& \frac{\vert b_0\vert}{\sqrt{[1 + a_1 \cos(\omega... ... + a_1 \cos(\omega T)}\right], & b_0<0 \\ \end{array} \right.. \end{eqnarray*}

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$.

Order a Hardcopy of Introduction to Digital Filters

Previous: One-Zero
Next: Two-Pole
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )