3.1 Uniformally modulated (dft) filterbank

The uniform modulated filterbank can be implemented using polyphase filterbanks and DFTs, resulting in huge computationalsavings. below illustrates the equivalent polyphase/DFT structures for analysis andsynthesis. The impulse responses of the polyphase filters $P_l(z)$ and ${\overline{P}}_l(z)$ can be defined in the time domain as ${\overline{p}}_l(m)=\overline{h}(mM+l)$ and $p_l(m)=h(mM+l)$ , where $h(n)$ and $\overline{h}(n)$ denote the impulse responses of the analysis and synthesis lowpass filters, respectively.

Recall that the standard implementation performs modulation, filtering, and downsampling, in that order. The polyphase/DFTimplementation reverses the order of these operations; it performs downsampling, then filtering, then modulation (if weinterpret the DFT as a two-dimensional bank of "modulators"). We derive the polyphase/DFT implementation below, startingwith the standard implementation and exchanging the order of modulation, filtering, and downsampling.

Polyphase/dft implementation derivation

We start by analyzing the $k$ th filterbank branch, analyzed in :

The first step is to reverse the modulation and filtering operations. To do this, we define a "modulated filter" $H_k(z)$ :

Notice that the only modulator outputs not discarded by the downsampler are those with time index $n=mM$ for $m\in \mathbb{Z}$ . For these outputs, the modulator has the value $e^{i\frac{2\pi }{M}kmM}=1$ , and thus it can be ignored. The resulting system is portrayedby:

Next we would like to reverse the order of filtering and downsampling. To apply the Noble identity, we must decompose $H_k(z)$ into a bank of upsampled polyphase filters. The techniqueused to derive polyphase decimation can be employed here:

Because it is a linear operator, the downsampler can be moved through the adders and the (time-invariant) scalings $e^{-i\frac{2\pi }{M}kl}$ . Finally, the Noble identity is employed to exchange the filtering and downsampling. The $k$ th filterbank branch becomes:

Observe that the polyphase outputs $\{v_l(m)\colon l=\{0, \dots , M-1\}\}$ are identical for each filterbank branch, while the scalings $\{e^{-i\frac{2\pi }{M}kl}\colon l=\{0, \dots , M-1\}\}$ once. Using these outputs we can compute the branch outputs via

The polyphase/DFT synthesis bank can be derived in a similar manner.