File:WOLA channelizer example.png

DescripcióWOLA channelizer example.png	English: Depiction of the implementation and operation of a Weighted Overlap Add (WOLA) channelizer (DTFT sampler). Wrap-around of a circular input buffer is used to offset phase discontinuities, caused by lack of a true time reference for the Fourier transform (DFT).[1][2]
Data	6 de març de 2020
Font	Treball propi
Autor	Bob K
Permís (Com reutilitzar aquest fitxer)	Jo, el titular dels drets d'autor d'aquest treball, el public sota la següent llicència:   L'ús d'aquest fitxer és regulat sota les condicions de Creative Commons de CC0 1.0 lliurament al domini públic universal. La persona que ha associat un treball amb aquest document ha dedicat l'obra domini públic, renunciant en tot el món a tots els seus drets de d'autor i a tots els drets legals relacionats que tenia en l'obra, en la mesura permesa per la llei. Podeu copiar, modificar, distribuir i modificar l'obra, fins i tot amb fins comercials, tot sense demanar permís. http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse
Altres versions	This was supposed to be a .svg file, but .svg did not support the yellow highlighting.
PNG genesis InfoField	Aquesta PNG imatge rasteritzada ha estat creada amb LibreOffice
Usage InfoField	Sampling the DTFT links to this image.
References InfoField	↑ Crochiere, R.E.; L.R.Rabiner (1983) «7.2» en Multirate Digital Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, p. 313–323 ISBN: 0136051626. ↑ Gumas, Charles Constantine (July 1997). "Window-presum FFT achieves high-dynamic range, resolution". Personal Engineering & Instrumentation News: 58–64. Archived from the original on 2001-02-10.   3. Sampling the DTFT

With no frequency or time decimation, the channelizer's DFT stage (fixed time-reference) calculation would be:

${\displaystyle {\begin{aligned}{\textrm {Chan}}_{f}(t)&=\sum _{n=0}^{11}y_{n}\cdot e^{-i2\pi {\frac {f}{12}}(t+n)};\quad f=0,1,2,...,11;\quad t=0,1,2,3,4,5,...\\&=e^{-i2\pi {\frac {f}{12}}t}\cdot \sum _{n=0}^{11}y_{n}\cdot e^{-i2\pi {\frac {f}{12}}n}\end{aligned}}}$

With frequency decimation by 4 ${\displaystyle (f=4k)}$ and time-decimation by 2, it becomes:

${\displaystyle {\begin{aligned}{\textrm {Chan}}_{k}(t)\ &=\ e^{-i2\pi {\frac {k}{3}}t}\cdot \sum _{n=0}^{11}y_{n}\cdot e^{-i2\pi {\frac {k}{3}}n};\quad k=0,1,2;\quad t=0,2,4,6,8,10,...\\&=\ e^{-i2\pi {\frac {k}{3}}t}\cdot \sum _{n=0}^{2}\left(\sum _{m=0}^{3}y_{n+3m}\cdot e^{-i2\pi {\frac {k}{3}}(n+3m)}\right)=e^{-i2\pi {\frac {k}{3}}t}\cdot \sum _{n=0}^{2}\underbrace {\left(\sum _{m=0}^{3}y_{n+3m}\right)} _{X_{n}}\cdot e^{-i2\pi {\frac {k}{3}}n}\\&=e^{-i2\pi {\frac {k}{3}}t}\cdot \underbrace {\sum _{n=0}^{2}X_{n}\cdot e^{-i2\pi {\frac {k}{3}}n}} _{\text{sliding time-reference}}\end{aligned}}}$

The phase-correcting sequence ${\displaystyle \left\{e^{-i2\pi {\frac {k}{3}}t};\ t=0,2,4,6,8,10,...\right\}}$ is equivalent to ${\displaystyle \left\{1,\ e^{+i2\pi {\frac {k}{3}}},\ e^{-i2\pi {\frac {k}{3}}},...\right\}}$ which repeats periodically.

The diagram shows that these corrections are produced automatically, by using the buffer's natural wrap-around characteristic as a time-reference. No buffer unwrapping logic or phase-correcting logic is required.  Or, in other words ( Multirate Digital Signal Processing, p. 321 ),

The modulation to convert the short-time spectrum from the sliding to the fixed-time references can be achieved without multiplication by recognizing that a modulo K shift of the (time-domain) sequence at the input of the DFT introduces a linear phase-shift of the DFT (frequency-domain).