Summary

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

The Rectangular Window

Side Lobes

Generalized Hamming Window Family

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

A causal filter is any filter whose impulse response is zero prior to time zero. — Click for https://ccrma.stanford.edu/~jos/filters/Causal_Recursive_Filters.html

Click for https://ccrma.stanford.edu/CCRMA/Courses/152/SPL.html

A Fourier Series (FS) expansion is the appropriate Fourier transform for periodic signals. — Click for https://ccrma.stanford.edu/~jos/mdft/Fourier_Series_FS_Relation.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

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

The Discrete Time Fourier Transform (DTFT) is the appropriate Fourier transform for discrete-time signals of arbitrary length. It can be obtained as the limit of a Discrete Fourier Transform (DFT) as its length goes to infinity. — Click for https://ccrma.stanford.edu/~jos/mdft/Discrete_Time_Fourier_Transform.html

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

The Rectangular Window

Side Lobes

Generalized Hamming Window Family

Rectangular Window Summary

The rectangular window was discussed in Chapter 5 (§3.1). Here we summarize the results of that discussion.

Definition ( odd):

$\displaystyle w_R(n) \isdef \left\{\begin{array}{ll} 1, & \left\vert n\right\vert\leq\frac{M-1}{2} \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right.$

(4.12)

Transform:

$\displaystyle W_R(\omega) = M\cdot \hbox{asinc}_M(\omega) \isdef \frac{\sin\left(M\frac{\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)}$

(4.13)

The DTFT of a rectangular window is shown in Fig.3.7.

**Figure 3.7:** Rectangular window discrete-time Fourier transform.
$\includegraphics[width=\twidth]{eps/Rect}$

Properties:

Zero crossings at integer multiples of

$\displaystyle \Omega_M \isdef \frac{2\pi}{M} = \hbox{frequency sampling interval for a length $M$\ DFT.}$ (4.14)
Main lobe width is $2 \Omega_M = \frac{4\pi}{M}$ .
As increases, the main lobe narrows (better frequency resolution).
has no effect on the height of the side lobes (same as the ``Gibbs phenomenon'' for truncated Fourier series expansions).
First side lobe only 13 dB down from the main-lobe peak.
Side lobes roll off at approximately 6dB per octave.
A phase term arises when we shift the window to make it causal, while the window transform is real in the zero-phase case (i.e., centered about time 0).

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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

Rectangular Window Summary

``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