Properly Anti-Aliasing Window Transforms

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

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Downsampling with Anti-Aliasing

Hop Sizes for WOLA

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

Click for https://ccrma.stanford.edu/~jos/sasp/Critically_Sampled_Perfect_Reconstruction.html#30743

Click for https://ccrma.stanford.edu/~jos/sasp/Overlap_Add_Decomposition.html#21934

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

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

To downsample a signal by the factor N means to form a new signal consisting of every Nth sample of the original signal. — Click for https://ccrma.stanford.edu/~jos/mdft/Downsampling_Operator.html

Click for https://ccrma.stanford.edu/~jos/mdft/Alias_Operator.html

Click for https://ccrma.stanford.edu/~jos/sasp/Mathematical_Definition_STFT.html#19930

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

Click for https://ccrma.stanford.edu/~jos/sasp/Three_Term_Blackman_Harris_Window.html#10289

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

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Downsampling with Anti-Aliasing

Hop Sizes for WOLA

Properly Anti-Aliasing Window Transforms

For simplicity, define window-transform bandlimits at first zero-crossings about the main lobe. Given the first zero of $W(\omega)$ at $L \frac{2\pi}{M} \leq \frac{\pi}{R}$ , we obtain

$\displaystyle \zbox {R_{\hbox{max}}= \frac{M}{2L}}$

(10.27)

The following table gives maximum hop sizes for various window types in the Blackman-Harris family, where is both the number of constant-plus-cosine terms in the window definition (§3.3) and the half-main-lobe width in units of side-lobe widths $2\pi/M$ . Also shown in the table is the maximum COLA hop size we determined in Chapter 8.

L	Window Type (Length )	$R_{\hbox{max}}$	$R_{\hbox{\hbox{\sc Cola}}}$
1	Rectangular	M/2	M
2	Generalized Hamming	M/4	M/2
3	Blackman Family	M/6	M/3
L	-term Blackman-Harris	M/2L	M/L

In the table, any $R\leq R_{\hbox{max}}$ suppresses aliasing well.

It is interesting to note that the maximum COLA hop size is double the maximum downsampling factor which avoids aliasing of the main lobe of the window transform in FFT-bin signals $X_{\tilde m}(\omega_k)$ . Since the COLA constraint is a sufficient condition for perfect reconstruction, this aliasing is quite heavy (see Fig.9.21), yet it is all canceled in the reconstruction. The general theory of aliasing cancellation in perfect reconstruction filter banks will be taken up in Chapter 11.

**Figure 9.21:** Illustration of main-lobe aliasing intervals.
$\includegraphics[width=3in]{eps/WindowAliasingFD}$

It is important to realize that aliasing cancellation is disturbed by FBS spectral modifications.^10.5For robustness in the presence of spectral modifications, it is advisable to keep $R\leq R_{\hbox{max}}= M/(2L)$ . For compression, it is common to use $R = 2 R_{\hbox{max}}= R_{\hbox{\hbox{\sc Cola}}} = M/L$ together with a ``synthesis window'' in a weighted overlap-add (WOLA) scheme (§8.6).

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

``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

Properly Anti-Aliasing Window Transforms

``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