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Spectral Audio Signal Processing
    Multirate Polyphase Filter Banks
       Filter Banks Equivalent to STFTs
          Polyphase Analysis of Portnoff STFT

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Polyphase Analysis of Portnoff STFT

Consider the $ k$th filter-bank channel filter

$\displaystyle H_{k}(z) = H_{0}(zW_N^k), \; k=1,\ldots,N-1\,. $

The impulse-response $ h_0(n)$ can be any length $ L$. Denote the $ N$-channel polyphase components of $ H_0(z)$ by $ E_l(z)$, $ l=0,1,\ldots,N-1$. Then by the polyphase decomposition (§11.2.2), we have

\begin{eqnarray*} H_{0}(z) & = & \sum_{l=0}^{N-1} z^{-l}E_l(z^{N}) \\ [0.1in] H_... ... \\ [0.1in] & = & \sum_{l=0}^{N-1} z^{-l} E_l(z^{N}) W_N^{-kl}. \end{eqnarray*}

Consequently,

\begin{eqnarray*} H_{k}(z)X(z) & = & \sum_{l=0}^{N-1} z^{-l} E_l(z^{N})X(z) W_N^... ... \ldots \\ E_{N-1}(z^N) z^{-(N-1)} X(z) \end{array} \right] \end{eqnarray*}

If $ H_{0}(z)$ is a good $ N$th-band lowpass, the subband signals $ x_{k}(n)$ are bandlimited to a region of width $ 2\pi/N$. As a result, there is negligible aliasing when we downsample each of the subbands by $ N$. Commuting the downsamplers to get an efficient implementation gives Fig.11.30.

Figure: Polyphase implementation of Portnoff STFT filter bank with $ N=3$.
\includegraphics[width=\textwidth]{eps/polystft}

First note that if $ E_k(z) = 1$ for all $ k$, the system of Fig.11.30 reduces to a rectangularly windowed STFT in which the window length $ M$ equals the DFT length $ N=3$. The downsamplers ``hold off'' the DFT until the length 3 delay line fills with new input samples, then it ``fires'' to produce a spectral frame. A new spectral frame is produced after every third sample of input data is received.

In the more general case in which $ E_k(z)$ are nontrivial filters, such as $ E_k(z)=1+z^{-1}$, for example, they can be seen to compute the equivalent of a time aliased windowed input frame, such as $ x(n) + x(n-N)$. This follows because the filters operate on the downsampled input stream, so that the filter coefficients operate on signal samples separated by $ N$ samples. The linear combination of these samples by the filter implements the time-aliased windowed data frame in a Portnoff-windowed overlap-add STFT. Taken together, the polyphase filters $ E_k(z)$ compute the appropriately time-aliased data frame windowed by the $ w=\hbox{\sc Flip}(h_0)\longleftrightarrow H_{0}(z^{-1})$.

In the overlap-add interpretation of Fig.11.30, the window is hopped by $ N=3$ samples. While this was the entire window length in the rectangular window case ($ E_k(z) = 1$), it is only a portion of the effective frame length $ L$ when the analysis filters have order 1 or greater.

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written by 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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