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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
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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