Generalized Window Method

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

FIR Digital Filter Design

Conclusions

Minimum-Phase Filter Design

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

An Finite Impulse Response (FIR) digital filter has an impulse response that reaches zero in a finite number of samples. Such filters cannot have any feedback loops. FIR filters are also called nonrecursive. The transfer function of an FIR filter is a polynomial. — Click for https://ccrma.stanford.edu/~jos/filters/FIR_Digital_Filters.html

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Fast Fourier Transforms (FFT) are fast algorithms for computing the Discrete Fourier Transform (DFT) — Click for https://ccrma.stanford.edu/~jos/mdft/Fast_Fourier_Transform_FFT.html

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The impulse signal is the shortest pulse 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). The impulse is typically denoted by the Greek delta symbol δ and is often called the 'Dirac delta function', after Paul Dirac who pioneered its use (in addition to being a lead contributor in the development of Quantum Mechanics). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.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 Fourier Transform (FT) gives the (complex) spectrum of a continuous-time signal of any length. — Click for https://ccrma.stanford.edu/~jos/mdft/Fourier_Transform_FT_Inverse.html

Minimum-phase polynomials have all their zeros inside the unit circle of the complex plane. — Click for https://ccrma.stanford.edu/~jos/filters/Minimum_Phase_Polynomials.html

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Forward-backward filtering is a technique for implementing any filter as a zero-phase filter, provided that the data to be filtered are available offline so that the data can be filtered both forward and backward in time. — Click for https://ccrma.stanford.edu/~jos/filters/Forward_Backward_Filtering.html

An audio equalizer is a filter used to adjust the spectrum of an audio signal, usually in some number of frequency bands. High quality audio equalizers provide one or more bands of adjustment per critical band of human hearing. — Click for https://github.com/bdejong/musicdsp/blob/master/source/Filters/197-rbj-audio-eq-cookbook.rst

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

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JOS Home Page

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Index of terms in JOS Website

Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

FIR Digital Filter Design

Conclusions

Minimum-Phase Filter Design

Generalized Window Method

Reiterating and expanding on points made in §4.6.3, often we need a filter with a frequency response that is not analytically known. An example is a graphic equalizer in which a user may manipulate sliders in a graphical user interface to control the gain in each of several frequency bands. From the foregoing, the following procedure, based in spirit on the window method (§4.5), can yield good results:

Synthesize the desired frequency response as the smoothest possible interpolation of the desired frequency-response points. For example, in a graphic equalizer, cubic splines [286] could be used to connect the desired band gains.^5.12
If the desired frequency response is real (as in simple band gains), either plan for a zero-phase filter in the end, or synthesize a desired phase, such as linear phase or minimum phase (see §4.8 below).
Perform the inverse Fourier transform of the (sampled) desired frequency response to obtain the desired impulse response.
Plot an overlay of the desired impulse response and the window to be applied, ensuring that the great majority of the signal energy in the desired impulse response lies under the window to be used.
Multiply by the window.
Take an FFT (now with zero padding introduced by the window).
Plot an overlay of the original desired response and the response retained after time-domain windowing, and verify that the specifications are within an acceptable range.

In summary, FIR filters can be designed nonparametrically, directly in the frequency domain, followed by a final smoothing (windowing in the time domain) which guarantees that the FIR length will be precisely limited. As we'll discuss in Chapter 8, it is necessary to precisely limit the FIR filter length to avoid time-aliasing in an FFT-convolution implementation.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Generalized Window Method

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University