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Introduction to Digital Filters
Elementary Audio Digital Filters
Elementary Filter Sections
Two-Zero
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Two-Zero
The signal flow graph for the general two-zero filter is given in Fig.B.7, and the derivation of frequency response is as follows:
![\fbox{
\begin{tabular}{rl}
Difference equation: & $y(n) = b_0 x(n) + b_1 x(n-1) ...
...+ b_1 \cos(\omega T) + b_2 \cos(2\omega T)}\right]$
\end{tabular}\vspace{10pt}
}](http://www.dsprelated.com/josimages/filters/img1387.png)
.
As discussed in §5.1,
the parameters 
and 
are called the numerator
coefficients, and they determine the two zeros. Using the
quadratic formula for finding the roots of a second-order polynomial,
we find that the zeros are located at
], then the two-zero case
reduces to two instances of our earlier analysis for the
one-zero. Assuming the zeros to be complex, we may express the zeros
in polar form as
and
, where
.
Forming a general two-zero transfer function in factored form gives
![\begin{eqnarray*}
H(z) &=& b_0 (1 - Re^{j\theta_c} z^{-1}) (1 - Re^{-j\theta_c} z^{-1})\\
&=& b_0 [1 - 2R\cos(\theta_c) z^{-1}+ R^2 z^{-2}]
\end{eqnarray*}](http://www.dsprelated.com/josimages/filters/img1395.png)
from which we identify
and
, so that
![$\displaystyle y(n) = b_0\{ x(n) - [2R \cos(\theta_c)]x(n - 1) + R^2 x(n - 2)\}
$](http://www.dsprelated.com/josimages/filters/img1398.png){kind=small}
, is now viewed as a notch
frequency, or antiresonance frequency. The closer R is to 1,
the narrower the notch centered at 
.
The approximate relation between bandwidth and 
given in
Eq.
(B.5) for the two-pole resonator now applies to the notch
width in the two-zero filter.
Figure B.8 gives some two-zero frequency responses obtained by
setting 
to 1 and varying . The value of 
, is again
. Note that the response is exactly analogous to the two-pole
resonator with notches replacing the resonant peaks. Since the plots
are on a linear magnitude scale, the two-zero amplitude response
appears as the reciprocal of a two-pole response. On a dB scale, the
two-zero response is an upside-down two-pole response.
 |
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Previous: Resonator Bandwidth in Terms of Pole RadiusNext: Complex Resonator
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.
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