# 3.1 Uniformally modulated (dft) filterbank

 Page 1 / 1
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 ${\overline{P}}_{l}(z)$ can be defined in the time domain as ${\overline{p}}_{l}(m)=\overline{h}(mM+l)$ and ${p}_{l}(m)=h(mM+l)$ , where $h(n)$ and $\overline{h}(n)$ 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 :

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

${v}_{k}(n)=\sum h(i)x(n-i)e^{i\frac{2\pi }{M}k(n-i)}=\left(\sum h(i)e^{-i\frac{2\pi }{N}ki}x(n-i)\right)e^{i\frac{2\pi }{M}kn}=\left(\sum {h}_{k}(i)x(n-i)\right)e^{i\frac{2\pi }{M}kn}$
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 $n=mM$ for $m\in \mathbb{Z}$ . For these outputs, the modulator has the value $e^{i\frac{2\pi }{M}kmM}=1$ , 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:

${H}_{k}(z)=\sum$ 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: ${h}_{k}(mM+l)=h(mM+l)e^{-i\frac{2\pi }{M}k(mM+l)}=h(mM+l)e^{-i\frac{2\pi }{M}kl}={p}_{l}(m)e^{-i\frac{2\pi }{M}kl}$ where ${p}_{l}(m)$ is the $l$ th polyphase filter defined by the original (unmodulated) lowpass filter $H(z)$ , we obtain
${H}_{k}(z)=\sum_{l=0}^{M-1} \sum$ 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 :

Because it is a linear operator, the downsampler can be moved through the adders and the (time-invariant) scalings $e^{-i\frac{2\pi }{M}kl}$ . Finally, the Noble identity is employed to exchange the filtering and downsampling. The $k$ th filterbank branch becomes:

Observe that the polyphase outputs $\{{v}_{l}(m)\colon l=\{0, \dots , M-1\}\}$ are identical for each filterbank branch, while the scalings $\{e^{-i\frac{2\pi }{M}kl}\colon l=\{0, \dots , M-1\}\}$ once. Using these outputs we can compute the branch outputs via

${y}_{k}(m)=\sum_{l=0}^{M-1} {v}_{l}(m)e^{-i\frac{2\pi }{M}kl}$
From the previous equation it is clear that ${y}_{k}(m)$ corresponds to the $k$ th DFT output given the $M$ -point input sequence $\{{v}_{l}(m)\colon l=\{0, \dots , M-1\}\}$ . 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.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!