One-Zero

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

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

A lowpass filter rejects high frequencies and does not affect low frequencies. — Click for http://ccrma.stanford.edu/~jos/filters/Simplest_Lowpass_Filter_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 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

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

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

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

One-Zero

**Figure B.1:** Signal flow graph for the general one-zero filter

.
$\includegraphics{eps/kfig2p17}$

Figure B.1 gives the signal flow graph for the general one-zero filter. The frequency response for the one-zero filter may be found by the following steps:

$\fbox{ \begin{tabular}{rl} Difference equation: & $y(n) = b_0x(n) + b_1x(n - 1)$\ \\ [5pt] {\it z} transform: & $Y(z) = b_0X(z) + b_1z^{-1}X(z)$\\ [5pt] Transfer function: & $H(z) = b_0 + b_1z^{-1}$\\ [5pt] Frequency response: & $H(e^{j\omega T}) = b_0 + b_1e^{-j\omega T}$ \end{tabular}}$

By factoring out $e^{-j\omega T/2}$ from the frequency response, to balance the exponents of

, we can get this closer to polar form as follows:

$\begin{eqnarray*} H(e^{j\omega T}) &=& b_0 + b_1 e^{-j\omega T}\\ &=& (b_0 - b_1) + b_1 + b_1e^{-j\omega T}\\ &=& (b_0 - b_1) + e^{-j\omega T/2}(b_1e^{j\omega T/2}+ b_1e^{-j\omega T/2})\\ &=& (b_0 - b_1) + e^{-j\omega T/2}2b_1\cos(\omega T/2)\\ &=& (b_0 - b_1) + e^{-j\pi f T} 2b_1\cos(\pi f T) \end{eqnarray*}$

**Figure B.2:** Family of frequency responses of the one-zero filter

for various values of . (a) Amplitude response. (b) Phase response.
$\includegraphics{eps/kfig2p19}$

We now apply the general equations given in Chapter 7 for filter gain $G(\omega)$ and filter phase $\Theta(\omega)$ as a function of frequency:

$\begin{eqnarray*} H(e^{j\omega T}) &=& b_0 + b_1e^{-j\omega T}\\ &=& b_0 + b_1 \cos(\omega T) - jb_1 \sin(\omega T)\\ [10pt] G(\omega) &=& \sqrt{[b_0 + b_1 \cos(\omega T)]^2 + [-b_1 \sin(\omega T)]^2}\\ &=& \sqrt{b_0^2 + b_1^2 + 2b_0b_1 \cos(\omega T)}\\ [10pt] \Theta(\omega) &=& \tan^{-1}\left[\frac{-b_1 \sin(\omega T)}{b_0 + b_1 \cos(\omega T)}\right] \end{eqnarray*}$

A plot of $G(\omega)$ and $\Theta(\omega)$ for

and various real values of

, is given in Fig.B.2. The filter has a zero at

in the

plane, which is always on the real axis. When a point on the unit circle comes close to the zero of the transfer function the filter gain at that frequency is low. Notice that one real zero can basically make either a highpass (

) or a lowpass filter (

). For the phase response calculation using the graphical method, it is necessary to include the pole at

One-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