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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.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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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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