As a familiar special case, set
where is the DFT matrix:
The inverse of this polyphase matrix is then simply the inverse DFT matrix:
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,
where , and
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.
Example: Polyphase Analysis of the STFT with 50% Overlap, Zero-Padding, and a Non-Rectangular Window
Necessary and Sufficient Conditions for Perfect Reconstruction