Skip to content Skip to navigation

Connexions

You are here: Home » Content » Uniformally Modulated (DFT) Filterbank

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the author

Recently Viewed

This feature requires Javascript to be enabled.

Uniformally Modulated (DFT) Filterbank

Module by: Phil Schniter

Summary: This module covers the Uniformally Modulated Filterbanks.

The uniform modulated filterbank can be implemented using polyphase filterbanks and DFTs, resulting in huge computational savings. Figure 1 below illustrates the equivalent polyphase/DFT structures for analysis and synthesis. The impulse responses of the polyphase filters P l z P l z and P ¯ l z P ¯ l z can be defined in the time domain as p ¯ l m= h ¯ mM+l p ¯ l m h ¯ m M l and p l m=hmM+l p l m h m M l , where hn h n and h ¯ n h ¯ n denote the impulse responses of the analysis and synthesis lowpass filters, respectively.

Figure 1
Figure 1 (poly_f1.png)

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 kkth filterbank branch, analyzed in Figure 2:

Figure 2: kkth filterbank branch
Figure 2 (poly_f2.png)

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

vkn=∑ihixn-iⅇⅈ2πMkn-i= ∑ihiⅇ-ⅈ2πNkixn-i ⅇⅈ2πMkn= ∑ihkixn-i ⅇⅈ2πMkn 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 (1)
The equation above indicated that xn xn is convolved with the modulated filter and that the filter output is modulated. This is illustrated in Figure 3:

Figure 3
Figure 3 (poly_f3.png)

Notice that the only modulator outputs not discarded by the downsampler are those with time index n=mM n m M for m∈ℤ m . For these outputs, the modulator has the value ⅇⅈ2πMkmM=1 2 M k m M 1 , and thus it can be ignored. The resulting system is portrayed by:

Figure 4
Figure 4 (poly_f4.png)

Next we would like to reverse the order of filtering and downsampling. To apply the Noble identity, we must decompose H k z 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 nz-n=∑l=0M-1∑m=-∞∞ h k mM+lz-mM+l Hk z n hk n z n l 0 M 1 m h k m M l z m M l (2)
Noting the fact that the llth polyphase filter has impulse response: h k mM+l=hmM+lⅇ-ⅈ2πMkmM+l=hmM+lⅇ-ⅈ2πMkl= p l mⅇ-ⅈ2πMkl 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 p l m is the llth polyphase filter defined by the original (unmodulated) lowpass filter Hz Hz , we obtain
Hk z=∑l=0M-1∑m=-∞∞ p l mⅇ-ⅈ2πMklz-mM-l=∑l=0M-1ⅇ-ⅈ2πMklz-l∑m=-∞∞ p l mzM-m=∑l=0M-1ⅇ-ⅈ2πMklz-l P l zM 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 (3)
The kkth filterbank branch (now containing MM polyphase branches) is in Figure 5:

Figure 5: kkth filterbank branch containing MM polyphase branches.
Figure 5 (poly_f5.png)

Because it is a linear operator, the downsampler can be moved through the adders and the (time-invariant) scalings ⅇ-ⅈ2πMkl 2 M k l . Finally, the Noble identity is employed to exchange the filtering and downsampling. The kkth filterbank branch becomes:

Figure 6
Figure 6 (poly_f6.png)

Observe that the polyphase outputs { v l m|l=0…M-1} v l m l 0 … M 1 v l m are identical for each filterbank branch, while the scalings {ⅇ-ⅈ2πMkl|l=0…M-1} 2 M k l l 0 … M 1 once. Using these outputs we can compute the branch outputs via

y k m=∑l=0M-1 v l mⅇ-ⅈ2πMkl y k m l 0 M 1 v l m 2 M k l (4)
From the previous equation it is clear that y k m y k m corresponds to the kkth DFT output given the MM-point input sequence { v l m|l=0…M-1} v l m l 0 … M 1 . Thus the MM filterbank branches can be computed in parallel by taking an MM-point DFT of the MM polyphase outputs (see Figure 7).

Figure 7
Figure 7 (poly_f7.png)

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

Comments, questions, feedback, criticisms?

Send feedback