Two-Zero

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Elementary Filter Sections

Resonator Bandwidth in Terms of Pole Radius

Complex Resonator

The phase response of an LTI filter is the angle of the (complex) frequency response — Click for https://ccrma.stanford.edu/~jos/filters/Phase_Response_I_I.html

Click for https://ccrma.stanford.edu/CCRMA/Courses/152/SPL.html

The amplitude response of an LTI filter is simply the magnitude of the (complex) frequency response — Click for https://ccrma.stanford.edu/~jos/filters/Amplitude_Response_I_I.html

A resonator is a recursive filter that boosts signal amplitude at a particular frequency. The simplest resonator is made using a two-pole filter. The impulse response of the two-pole resonator is an exponentially decaying sinusoidal oscillation, where the zero-crossing rate is (twice) the resonance frequency. — Click for https://ccrma.stanford.edu/~jos/filters/Two_Pole.html

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

The size of the passband, typically expressed in Hz. — Click for http://en.wikipedia.org/wiki/Bandpass

The difference equation is a recipe for computing samples of the output signal of a digital filter based on samples of the input signal and the filter coefficients. In an Infinite-Impulse-Response (IIR) digital filter, there are typically both feedforward and feedback coefficients in the difference equation. — Click for https://ccrma.stanford.edu/~jos/filters/Difference_Equation_I.html

The transfer function is defined for LTI filters as the z transform of the filter output signal, divided by the z transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html

The quadratic formula gives a closed-form solution for both roots of a second-order polynomial — Click for https://ccrma.stanford.edu/~jos/mdft/Quadratic_Formula.html

Click for https://ccrma.stanford.edu/~jos/filters/Signal_Flow_Graph_I.html

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

Click for https://ccrma.stanford.edu/~jos/filters/Signal_Flow_Graph.html

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Elementary Filter Sections

Resonator Bandwidth in Terms of Pole Radius

Complex Resonator

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: & $y(n) = b_0 x(n) + b_1 x(n-1) + b_2 x(n-2)$\\ [5pt] {\it z} transform: & $Y(z) = b_0 X(z) + b_1 z^{-1}X(z) + b_2 z^{-2}X(z)$\\ [5pt] Transfer function: & $H(z) = b_0+b_1z^{-1}+b_2z^{-2}$\\ [5pt] Frequency response: & $H(e^{j\omega T}) = b_0+b_1e^{-j\omega T}+b_2e^{-j2\omega T}$\\ [5pt] Amplitude response: & $G^2(\omega) = [b_0 + b_1 \cos(\omega T) + b_2 \cos(2\omega T)]^2$\\ &$\qquad\qquad + [-b_1 \sin(\omega T) - b_2 \sin(2\omega T)]^2$\\ [8pt] Phase response: & $\Theta(\omega) = \tan^{-1}\left[\displaystyle\frac{-b_1 \sin(\omega T) - b_2 \sin(2\omega T)}{b_0 + b_1 \cos(\omega T) + b_2 \cos(2\omega T)}\right]$ \end{tabular}\vspace{10pt} }$

**Figure B.7:** Signal flow graph for the general two-zero filter

.
$\includegraphics{eps/twozero}$

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

$\displaystyle z = \frac{-b_1 \pm \sqrt{b_1^2 - 4 b_0 b_2}}{2b_0}$

If the zeros are real [ $(b_1/2)^2 \geq b_2$ ], 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 $Re^{j\theta_c}$ and $Re^{-j\theta_c}$ , where $\theta_c = \omega_c T = 2\pi f_c T$ .

Forming a general two-zero transfer function in factored form gives

$\begin{eqnarray*} H(z) &=& b_0 (1 - Re^{j\theta_c} z^{-1}) (1 - Re^{-j\theta_c} z^{-1})\\ &=& b_0 [1 - 2R\cos(\theta_c) z^{-1}+ R^2 z^{-2}] \end{eqnarray*}$

from which we identify $b_1/b_0 = - 2R \cos(\theta_c)$ and , so that

$\displaystyle y(n) = b_0\{ x(n) - [2R \cos(\theta_c)]x(n - 1) + R^2 x(n - 2)\}$

is again the difference equation of the general two-zero filter with complex zeros. The frequency $\omega$ , is now viewed as a notch frequency, or antiresonance frequency. The closer R is to 1, the narrower the notch centered at $\omega_c$ .

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 $\theta _c$ , is again $\pi /4$ . 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.

**Figure B.8:** Frequency response of the two-zero filter

$y(n) = x(n) - 2R\cos (\theta _c) x(n - 1) + R^2 x(n - 2)$
with $\theta _c$ fixed at $\pi /4$ and for various values of . (a) Amplitude response. (b) Phase response.
$\includegraphics{eps/kfig2p25}$

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Two-Zero

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition) Copyright © 2024-09-03 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University