# 0.2 Derivation of the equations for a basic fdm-tdm transmux  (Page 6/10)

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The computational efficiency of the transmultiplexer can therefore be traced to two key items:

1. Separation of the tuning computation into two segments, one of which (the $\left\{v\left(r,p\right)\right\}$ ) need be computed only once
2. The use of the FFT algorithm to compute the inverse DFT

The first accrues from strategic choices of the sampling and tuning frequencies, while the second depends on N being chosen to be a highly composite integer.

## The transmux as a dft-based filter bank

We have just developed an FDM-TDM transmultiplexer by first writing the equations for a single, decimated digital tuner. The equations for a bank of tuners come from then assuming that (1) they all use the same filter pulse response and (2) their center frequencies are all integer multiples of some basic frequency step. In this section, we develop an alternate view, which happens to yield the same equations. It produces a different set of insights, however, making its presentation worthwhile.

## Using the dft as a filter bank

Instead of building a bank of tuners and then constraining their tuning frequencies to be regularly spaced, suppose we start with a structure known to provide equally-spaced spectral measurements and then manipulate it to obtain the desired performance.

Consider the structure shown in [link] . The sampled input signal $x\left(k\right)$ enters a tapped delay line of length N . At every sampling instant, all N current and delayed samples are weighted by constant coefficients $w\left(i\right)$ (where $w\left(i\right)$ scales $x\left(k-i\right)$ , for i between 0 and $N-1$ ), and then applied to an inverse discrete Fourier transform Whether or not it is implemented with an FFT is irrelevant at this point. Also, we happen to use the inverse DFT to produce a result consistent with that found in the proceeding subsction, but the forward DFT could also be used. . The complete N-point DFT is computed for every value of k and produces N outputs. The output sample stream from the m-th bin of the DFT is denoted as ${X}_{m}\left(k\right)$ .

Since DFTs are often associated with spectrum analysis, it may seem counterintuitive to consider the output bins as time samples. It is strictly legal from an analytical point of view, however, since the DFT is merely an N-input, N-output, memoryless, linear transformation. Even so, the relationship of this scheme and digital spectrum analysis will be commented upon later. We continue by first examining the path from the input to a specific output bin, the m-th one, say. For every input sample $x\left(k\right)$ there is an output sample ${X}_{m}\left(k\right)$ . By inspection we can write an equation relating the input and chosen output:

${X}_{m}\left(k\right)=\sum _{p=0}^{N-1}x\left(k-p\right)w\left(p\right){e}^{j2\pi \frac{mp}{N}}$

the m-th bin of an N-point DFT of the weighted, delayed data. We can look at this equation another way by defining ${\overline{w}}_{m}\left(p\right)$ by the expression

${\overline{w}}_{m}\left(p\right)\equiv w\left(p\right)·{e}^{j2\pi \frac{np}{N}}$

and observing that [link] can be written as

${X}_{m}\left(k\right)=\sum _{p=0}^{N-1}x\left(k-p\right)·{\overline{w}}_{m}\left(p\right).$

From this equation it is clear ${X}_{m}\left(k\right)$ is the output of the FIR digital filter that has $x\left(k\right)$ as its input and ${\overline{w}}_{m}\left(p\right)$ as its pulse response. Since the pulse response does not depend on the time index k , the filtering is linear and shift-invariant. For such a filter we can compute its transfer function, using the expression

where we get a research paper on Nano chemistry....?
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
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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
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
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Alexandre
what is the stm
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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
how did you get the value of 2000N.What calculations are needed to arrive at it
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