Optimized Windows

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Spectrum Analysis Windows

Exact Discrete Gaussian Window

Optimal Windows for Audio Coding

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/

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

Spectrum analysis of sound is analogous to decomposing white light into its component colors by means of a prism — Click for https://ccrma.stanford.edu/~jos/mdft/Example_Applications_DFT.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 main lobe in the Fourier transform of a window function, such as the Chebyshev window, is the central local maximum in the transform magnitude (response about dc). Outside of the main lobe are usually side lobes (ripples in the magnitude spectrum). — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window_Side_Lobes.html

A side lobe in the Fourier transform of a window function, such as the Chebyshev window, is any local maximum in the transform magnitude outside of the main lobe (response about dc). — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window_Side_Lobes.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 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

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

Finite Impulse Response (FIR) digital filters may be designed by many methods. Perhaps the simplest is the window method. — Click for https://ccrma.stanford.edu/~jos/sasp/FIR_Digital_Filter_Design.html

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Spectrum Analysis Windows

Exact Discrete Gaussian Window

Optimal Windows for Audio Coding

Optimized Windows

We close this chapter with a general discussion of optimal windows in a wider sense. We generally desire

$\displaystyle W(\omega) \approx \delta(\omega),$

(4.59)

but the nature of this approximation is typically determined by characteristics of audio perception. Best results are usually obtained by formulating this as an FIR filter design problem (see Chapter 4). In general, both time-domain and frequency-domain specifications are needed. (Recall the potentially problematic impulses in the Dolph-Chebyshev window shown in Fig.3.33 when its length was long and ripple level was high). Equivalently, both magnitude and phase specifications are necessary in the frequency domain.

A window transform can generally be regarded as the frequency response of a lowpass filter having a stop band corresponding to the side lobes and a pass band corresponding to the main lobe (or central section of the main lobe). Optimal lowpass filters require a transition region from the pass band to the stop band. For spectrum analysis windows, it is natural to define the entire main lobe as ``transition region.'' That is, the pass-band width is zero. Alternatively, the pass-band could be allowed to have a finite width, allowing some amount of ``ripple'' in the pass band; in this case, the pass-band ripple will normally be maximum at the main-lobe midpoint ( $W(0)= 1+\delta$ , say), and at the pass-band edges ( $W(\epsilon) = W(-\epsilon) = 1-\delta$ ). By embedding the window design problem within the more general problem of FIR digital filter design, a plethora of optimal design techniques can be brought to bear [204,258,14,176,218].

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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

Optimized Windows

``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