# 3.1 Uniformally modulated (dft) filterbank

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This module covers the Uniformally Modulated Filterbanks.

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" ${H}_{k}(z)$ :

${v}_{k}(n)=\sum h(i)x(n-i)e^{i\frac{2\pi }{M}k(n-i)}=\left(\sum h(i)e^{-i\frac{2\pi }{N}ki}x(n-i)\right)e^{i\frac{2\pi }{M}kn}=\left(\sum {h}_{k}(i)x(n-i)\right)e^{i\frac{2\pi }{M}kn}$
The equation above indicated that $x(n)$ is convolved with the modulated filter and that the filter output is modulated. This is illustrated in :

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:

${H}_{k}(z)=\sum$ h k n z n l 0 M 1 m h k m M l z m M l
Noting the fact that the $l$ th polyphase filter has impulse response: ${h}_{k}(mM+l)=h(mM+l)e^{-i\frac{2\pi }{M}k(mM+l)}=h(mM+l)e^{-i\frac{2\pi }{M}kl}={p}_{l}(m)e^{-i\frac{2\pi }{M}kl}$ where ${p}_{l}(m)$ is the $l$ th polyphase filter defined by the original (unmodulated) lowpass filter $H(z)$ , we obtain
${H}_{k}(z)=\sum_{l=0}^{M-1} \sum$ p l m 2 M k l z m M l l 0 M 1 2 M k l z l m p l m z M m l 0 M 1 2 M k l z l P l z M
The $k$ th filterbank branch (now containing $M$ polyphase branches) is in :

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

${y}_{k}(m)=\sum_{l=0}^{M-1} {v}_{l}(m)e^{-i\frac{2\pi }{M}kl}$
From the previous equation it is clear that ${y}_{k}(m)$ corresponds to the $k$ th DFT output given the $M$ -point input sequence $\{{v}_{l}(m)\colon l=\{0, \dots , M-1\}\}$ . Thus the $M$ filterbank branches can be computed in parallel by taking an $M$ -point DFT of the $M$ polyphase outputs (see ).

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

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