Summary

DescriptionWOLA 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]
Date	6 March 2020
Source	Own work
Author	Bob K
Permission (Reusing this file)	I, the copyright holder of this work, hereby publish it under the following license:   This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission. http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse
Other versions	This was supposed to be a .svg file, but .svg did not support the yellow highlighting.
PNG development InfoField	This PNG graphic was created with LibreOffice.
Usage InfoField	Sampling the DTFT links to this image.
References InfoField	↑ Crochiere, R.E.; L.R.Rabiner (1983) "7.2" in Multirate Digital Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, pp. 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

Continuous phase

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).