Uniformally Modulated (DFT) Filterbank 2.13 2001/12/31 2005/10/18 16:17:10.955 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Phil Schniter schniter@ee.eng.ohio-state.edu DFT filterbank Noble identities polyphase This module covers the Uniformally Modulated Filterbanks. The uniform modulated filterbank can be implemented using polyphase filterbanks and DFTs, resulting in huge computational savings. below illustrates the equivalent polyphase/DFT structures for analysis and synthesis. 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/DFT implementation reverses the order of these operations; it performs downsampling, then filtering, then modulation (if we interpret the DFT as a two-dimensional bank of "modulators"). We derive the polyphase/DFT implementation below, starting with the standard implementation and exchanging the order of modulation, filtering, and downsampling.

Polyphase/DFT Implementation Derivation We start by analyzing the kth filterbank branch, analyzed in :

kth filterbank branch

The first step is to reverse the modulation and filtering operations. To do this, we define a "modulated filter" H k z : vk 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 hk i x n i 2 M k n The equation above indicated that xn 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 portrayed by:

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 technique used to derive polyphase decimation can be employed here: Hk z n hk n z n l 0 M 1 m h k m M l z m M l Noting the fact that the lth 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 lth polyphase filter defined by the original (unmodulated) lowpass filter Hz , we obtain Hk 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 kth filterbank branch (now containing M polyphase branches) is in :

kth 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 kth 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 kth 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.