Polyphase View of the STFT
As a familiar special case, set
![]() |
(12.66) |
where

![]() |
(12.67) |
The inverse of this polyphase matrix is then simply the inverse DFT matrix:
![]() |
(12.68) |
Thus, the STFT (with rectangular window) is the simple special case of a perfect reconstruction filter bank for which the polyphase matrix is constant. It is also unitary; therefore, the STFT is an orthogonal filter bank.
The channel analysis and synthesis filters are, respectively,
![\begin{eqnarray*}
H_k(z) &=& H_0(zW_N^k)\\ [5pt]
F_k(z) &=& F_0(zW_N^{-k})
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2142.png)
where
, and
![]() |
(12.69) |
corresponding to the rectangular window.
Looking again at the polyphase representation of the
-channel
filter bank with hop size
,
,
,
dividing
, we have the system shown in Fig.11.25.
Following the same analysis as in §11.4.1 leads to the following
conclusion:

Our analysis showed that the STFT using a rectangular window is
a perfect reconstruction filter bank for all
integer hop sizes in the set
.
The same type of analysis can be applied to the STFT using the other
windows we've studied, including Portnoff windows.
Next Section:
Example: Polyphase Analysis of the STFT with 50% Overlap, Zero-Padding, and a Non-Rectangular Window
Previous Section:
Necessary and Sufficient Conditions for Perfect Reconstruction