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Polyphase Analysis of Portnoff STFT

Consider the $ k$ th filter-bank channel filter

$\displaystyle H_{k}(z) \eqsp H_{0}(zW_N^k), \; k=1,\ldots,N-1\,.$ (12.96)

The impulse-response $ h_0(n)$ can be any length $ L$ sequence. 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}) \\ [5pt] H_{k}(z) & = & \sum_{l=0}^{N-1} (zW_N^k)^{-l} E_l[(z W_N^k)^{N}] \\ [5pt] & = & \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^{-kl} \\ \left[\begin{array}{c} H_{0}(z) \\ \ldots \\ H_{N-1}(z) \end{array} \right] & = & \left[\begin{array}{ccc} & & \\ & W_N^{-kl} & \\ & & \end{array} \right] \left[\begin{array}{c} E_0(z^N) z^{-0} X(z) \\ \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.11.29.

\begin{psfrags} % latex2html id marker 30545\psfrag{X(z)}{{\large $X(z)$}}\psfrag{E0(z^3)}{{\large $E_0(z^3)$}}\psfrag{E1(z^3)}{{\large $E_1(z^3)$}}\psfrag{E2(z^3)}{{\large $E_2(z^3)$}}\psfrag{E0(z)}{{\large $E_0(z)$}}\psfrag{E1(z)}{{\large $E_1(z)$}}\psfrag{E2(z)}{{\large $E_2(z)$}}\psfrag{X0(z)}{{\large $X_0(z)$}}\psfrag{X1(z)}{{\large $X_1(z)$}}\psfrag{X2(z)}{{\large $X_2(z)$}}\begin{figure}[htbp] \includegraphics[width=\twidth]{eps/polystft} \caption{Polyphase implementation of Portnoff STFT filter bank for $N=3$.} \end{figure} \end{psfrags}

First note that if $ E_k(z) = 1$ for all $ k$ , the system of Fig.11.29 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 $ w=\hbox{\sc Flip}(h_0)\;\leftrightarrow\;H_{0}(z^{-1})$ .

In the overlap-add interpretation of Fig.11.29, 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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