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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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characteristics of micro business
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Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
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CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
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s. Reply
of graphene you mean?
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in general
s.
Graphene has a hexagonal structure
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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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