Polyphase Analysis of Portnoff STFT

• Recall that the analysis filter in an STFT can be expressed as a frequency-shift of a prototype lowpass filter:
  $$H_{k}(z) = H_{0}(zW_N^k), \; k=1,\ldots,N-1$$
  In principle, the lowpass filter's impulse-response 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,
  $$\begin{eqnarray*} H_{0}(z) & = & \sum_{l=0}^{N-1} z^{-l}E_l(z^{N}) \\ [0.1in] H_{k}(z) & = & \sum_{l=0}^{N-1} (zW_N^k)^{-l} E_l((z W_N^k)^{N}) \\ [0.1in] & = & \sum_{l=0}^{N-1} z^{-l} E_l(z^{N}) W_N^{-kl} \end{eqnarray*}$$
  

• Consequently,
  $$\begin{eqnarray*} H_{k}(z)X(z) & = & \sum_{l=0}^{N-1} z^{-l} E_l(z^{N})X(z) W_N^{-kl} \\ \left[\begin{array}{c} H_{0}(z) \\ \ldots \\ H_{N-1}(z) \end{array} \right] & = & \left[\begin{array}{ccc} & & \\ & W_N^{-kl} & \\ & & \end{array} \right] \left[\begin{array}{c} E_0(z^N) z^{-0} X(z) \\ \ldots \\ E_{N-1}(z^N) z^{-(N-1)} X(z) \end{array} \right] \end{eqnarray*}$$
  

• 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 is straightforward:
  
  

• We see that the polyphase filters compute the appropriate time-aliases of the flipped window $H_{0}(z^{-1})$ .

• The window is hopped by $N$ samples, but recall that it is operating on input date time aliased by the factor $L/N$ , where $L$ is the filter length. Thus, the hop size is only a fraction of the anti-aliasing filter impulse-response length.

Question: What familiar case do we get when $E_k(z) = 1$ for all $k$ ?