Alternative Realizations

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Analysis of a Digital Comb Filter

Pole-Zero Analysis

First-Order Parallel Sections

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

The impulse response of a system is its output signal in response to the impulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.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

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

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

A comb filter is made from a delay line using a linear combination of its input and output signals. — Click for https://ccrma.stanford.edu/~jos/pasp/Comb_Filters.html

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

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

A digital filter can be visualized as a 'black box' that accepts a sequence of numbers and emits a new sequence of numbers. In digital audio signal processing applications, such number sequences usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects. — Click for https://ccrma.stanford.edu/~jos/filters/

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

Analysis of a Digital Comb Filter

Pole-Zero Analysis

First-Order Parallel Sections

Alternative Realizations

For actually implementing the example digital filter, we have only seen the difference equation

$\displaystyle y(n) = x(n) + g_1\, x(n-M_1) - g_2\, y(n-M_2)$

(from Eq.(3.1), diagrammed in Fig.3.1). While this structure, formally known as ``direct form I'', happens to be a good choice for digital comb filters, there are many other structures to consider in other situations. For example, it is often desirable, for numerical reasons, to implement low-pass, high-pass, and band-pass filters as series second-order sections. On the other hand, digital filters for simulating the vocal tract (for synthesized voice applications) are typically implemented as parallel second-order sections. (When the order is odd, there is one first-order section as well.) The coefficients of the first- and second-order filter sections may be calculated from the poles and zeros of the filter.

We will now illustrate the computation of a parallel second-order realization of our example filter . As discussed above in §3.11, this filter has five poles and three zeros. We can use the partial fraction expansion (PFE), described in §6.8, to expand the transfer function into a sum of five first-order terms:

$\begin{eqnarray*} H(z) &=& \frac{1 + 0.5^3 z^{-3}}{1 + 0.9^5 z^{-5}} \mathrel{=} \sum_{i=1}^5 \frac{r_i}{1-p_iz^{-1}}\\ &\approx& \frac{0.1657}{1 + 0.9z^{-1}} + \frac{ 0.1894 - j 0.0326 }{1 - ( 0.7281 + j 0.5290 )z^{-1}} + \frac{ 0.1894 + j 0.0326 }{1 - ( 0.7281 - j 0.5290 )z^{-1}} \\ && \qquad\qquad \mathrel{+} \frac{ 0.2277 + j 0.0202 }{1 - ( -0.2781 + j 0.8560 )z^{-1}} + \frac{ 0.2277 - j 0.0202 }{1 - ( -0.2781 - j 0.8560 )z^{-1}}\\ [5pt] &=& \frac{0.1657}{1+0.9000z^{-1}} + \frac{0.3788 -0.2413z^{-1}}{1 - 1.4562z^{-1}+ 0.8100z^{-2}}\\ && \qquad\qquad\qquad\; \mathrel{+} \frac{0.4555 + 0.0922z^{-1}}{1 + 0.5562z^{-1}+ 0.8100z^{-2}}, \end{eqnarray*}$

where, in the last step, complex-conjugate one-pole sections are combined into real second-order sections. Also, numerical values are given to four decimal places (so ` ' is replaced by ` $\approx$ ' in the second line). In the following subsections, we will plot the impulse responses and frequency responses of the first- and second-order filter sections above.

Subsections

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``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

Alternative Realizations

``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