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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 P ¯ l z can be defined in the time domain as p ¯ l m h ¯ m M l and p l m h m M l , where h n and 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 :

k th filterbank branch

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 i h i x n i 2 M k n i i h i 2 N k i x n i 2 M k n i h k i x n i 2 M k n
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 m M for m . For these outputs, the modulator has the value 2 M k m M 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 n 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 m M l h m M l 2 M k m M l h m M l 2 M k l p l m 2 M k l where p l m is the l th polyphase filter defined by the original (unmodulated) lowpass filter H z , we obtain
H k z l 0 M 1 m 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 :

k th filterbank branch containing M polyphase branches.

Because it is a linear operator, the downsampler can be moved through the adders and the (time-invariant) scalings 2 M k l . 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 l 0 M 1 v l m are identical for each filterbank branch, while the scalings 2 M k l l 0 M 1 once. Using these outputs we can compute the branch outputs via

y k m l 0 M 1 v l m 2 M k l
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 l 0 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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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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