Digital Differentiator Design

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

The Fourier transform of a signal is a frequency-domain function — Click for https://ccrma.stanford.edu/~jos/mdft/

A filter is said to be stable if its impulse response decays to zero as time goes to infinity — Click for https://ccrma.stanford.edu/~jos/filters/Stability_Revisited.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

Sampling is the process of converting a continuous-time signal into a discrete-time signal. — Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

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

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

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 invfreqz function (in matlab) computes the coefficients of a recursive digital filter based on specified frequency-response samples. — Click for http://www.dsprelated.com/showcode/20.php

Matlab is a high-level scripting language for scientific computing — Click for https://ccrma.stanford.edu/~jos/matlab/

In digital signal processing, FIR is an acronym for Finite Impulse Response — Click for https://ccrma.stanford.edu/~jos/filters/FIR_Digital_Filters.html

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.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

Digital Differentiator Design

We saw the ideal digital differentiator frequency response in Fig.8.1, where it was noted that the discontinuity in the response at $\omega=\pm\pi$ made an ideal design unrealizable (infinite order). Fortunately, such a design is not even needed in practice, since there is invariably a guard band between the highest supported frequency $f_{\mbox{\tiny max}}$ and half the sampling rate

**Figure 8.2:** FIR differentiator designed by the matlab function `invfreqz` (Octave). Top: Overlay of the ideal amplitude response ( $\vert j2\pi f\vert$ ), fitted filter amplitude response, and guard-band limit (at 20 kHz). Bottom: Overlay of ideal phase response ( $\pi /2$ radians), fitted filter phase response, and guard-band limit (20 kHz).
$\includegraphics[width=\twidth]{eps/iirdiff-mag-phs-N0-M10}$

Figure 8.2 illustrates a more practical design specification for the digital differentiator as well as the performance of a tenth-order FIR fit using invfreqz (which minimizes equation error) in Octave.^9.12 The weight function passed to invfreqz was 1 from 0 to 20 kHz, and zero from 20 kHz to half the sampling rate (24 kHz). Notice how, as a result, the amplitude response follows that of the ideal differentiator until 20 kHz, after which it rolls down to a gain of 0 at 24 kHz, as it must (see Fig.8.1). Higher order fits yield better results. Using poles can further improve the results, but the filter should be checked for stability since invfreqz designs filters in the frequency domain and does not enforce stability.^9.13

Digital Differentiator Design

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University