### Polyphase View of the STFT

As a familiar special case, set

(12.66) |

where is the

*DFT matrix:*

(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,

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