Polyphase View of the STFT

As a familiar special case, set

$\displaystyle \bold{E}(z) = \bold{W}_N^\ast$

where $\bold{W}_N^\ast$ is the DFT matrix:

$\displaystyle \bold{W}_N^\ast[kn] = \left[e^{-j2\pi kn/N}\right]$

The inverse of this polyphase matrix is then simply the inverse DFT matrix:

$\displaystyle \bold{R}(z) = \frac{1}{N}\bold{W}_N$

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,

$\begin{eqnarray*} H_k(z) &=& H_0(zW_N^k)\\ [0.1in] F_k(z) &=& F_0(zW_N^{-k}) \end{eqnarray*}$

where $W_N\isdef e^{-j2\pi/N}$ , and

$\displaystyle F_0(z)=H_0(z)=\sum_{n=0}^{N-1}z^{-n}\;\longleftrightarrow\;[1,1,\ldots,1]$

corresponding to the rectangular window.

figure[htbp] $\includegraphics{eps/polyNchanSTFT}$

Looking again at the polyphase representation of the -channel filter bank with hop size , $\bold{E}(z)=\bold{W}_N^\ast$ , $\bold{R}(z)=\bold{W}_N$ , dividing , we have the system shown in Fig.10.26. Following the same analysis as in §10.4.1 leads to the following conclusion:

$\displaystyle \zbox {\hbox{The polyphase representation is an \emph{overlap-add} representation.}}$

Our analysis showed that the STFT using a rectangular window is a perfect reconstruction filter bank for all integer hop sizes in the set $R\in\{N,N/2,N/3,\ldots,N/N\}$ . The same type of analysis can be applied to the STFT using the other windows we've studied, including Portnoff windows.

Previous: Necessary and Sufficient Conditions for Perfect Reconstruction
Next: Example: Polyphase Analysis of the STFT with 50% Overlap, Zero-Padding, and a Non-Rectangular Window

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
    Multirate Filter Banks
       Perfect Reconstruction Filter Banks
          Polyphase View of the STFT