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
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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

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).