# 0.2 Derivation of the equations for a basic fdm-tdm transmux  (Page 7/10)

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${W}_{m}\left(\omega \right)=\sum _{p=0}^{N-1}{\overline{w}}_{m}\left(p\right)·{e}^{-j\omega pT},\phantom{\rule{4pt}{0ex}}\frac{-\pi }{T}\le \omega \le \frac{\pi }{T}.$

Suppose that we first choose the simple case with uniform weighting, that is, $w\left(p\right)=1$ for $0\le p\le N-1$ and, therefore, ${\overline{w}}_{m}\left(p\right)={e}^{j2\pi \frac{mp}{N}}$ . In this case, ${W}_{m}\left(w\right)$ is given by

${W}_{m}\left(\omega \right)={e}^{-j\frac{\pi m}{N}}·{e}^{-j\frac{\left(N-1\right)\omega T}{2}}·\frac{sin\frac{N\omega T}{2}}{sin\left(\frac{\pi m}{N}-\frac{\omega T}{2}\right)}$

The magnitude of this transfer function is plotted in [link] . From this plot we can conclude that the pulse response ${\overline{w}}_{m}\left(p\right)$ has what might be generally considered to be the frequency response of a bandpass filter. The filter is centered on bin m and its bandwidth is nominally $\frac{{f}_{s}}{N}$ . While it might be characterized as a bandpass filter, we also note that the passband is quite rounded and the stopband rejection is relatively poor. The first sidelobes are only 13 dB lower than the peak of the passband response.

We've now shown that the path from the input to the m-th bin can be described as a finite impulse response (FIR) filtering operation and that the transfer function of that filtering operation has a fairly sloppy bandpass characteristic, at least when the data weighting is uniform. What happens for other values of m then? The answer is "the same thing." For each value of m between 0 and $N-1$ , the pulse response ${\overline{w}}_{m}\left(p\right)$ is computed, leading to the transfer function ${W}_{m}\left(\omega \right)$ . An overlay of these bandpass responses is shown in the lower portion of [link] . From this we can conclude that the block diagram shown in [link] describes a single-input, N-output bank of filters. The filter center frequencies are spaced uniformly in increments of $\frac{{f}_{s}}{N}$ Hz. All N outputs are sampled in time as frequently as the input. When the weighting function $w\left(p\right)$ is uniform, then the bandpass filters have the form of [link] , shown in [link] .

## The implications of attaining the desired bandpass characteristic

We've just shown that the DFT of delayed versions of the input sequence $x\left(k\right)$ has the general properties of a bank of regularly-spaced bandpass filters. Two considerations leave us short of our goals. The first is that the shape of the transfer function for each bandpass filter is not good enough for most applications and must be improved. The second is that of reducing the amount of computation required. We address the first one in this section, and temporarily defer the computation issue.

[link] shows two transfer functions. The one pointed to from the left is exactly the same as that shown in [link] and defined in [link] . The one pointed to from the right is representative of the type needed for demultiplexing FDM multichannel telephone signals. It offers essentially flat response for most of the passband, has very sharp transition bands, and suppresses all energy outside of the transition bands by 55 dB or more. While other applications may require different transfer functions, as a rule they will be much more stringent than unweighted transfer function shown in [link] .

How then do we attain different transfer function characteristics? In fact, we use some of the remaining degrees of freedom, the weighting function $w\left(p\right)$ . By allowing $w\left(p\right)$ to be non-uniform we can now alter the shape of the transfer function of each bandpass filter. By using well-known FIR filter design techniques (see An Introduction to the FDM-TDM Digital Transmultiplexer: Appendix A ) it is possible to attain virtually any shape. It is not, however, possible to always attain the desired shape and the desired bandwidth while keeping the duration of the pulse response constant. In fact, as discussed in An Introduction to the FDM-TDM Digital Transmultiplexer: Appendix A , for a constant bandwidth, the pulse response duration must grow as the transition bandwidth is forced to be smaller and as the stopband suppression in increased. The chain of events described in [link] then unfolds.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
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
has a lot of application modern world
Kamaluddeen
yes
narayan
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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