# 0.2 Derivation of the equations for a basic fdm-tdm transmux  (Page 9/10)

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We now develop some equations that describe the steps just traversed. Starting with [link] we replace N with ${N}^{\text{'}}=QN$ , obtaining an expression for the time sequence seen at the m-th DFT bin.

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

Suppose, as discussed above, that we eliminate the filter overlapping by evaluating only every Q-th DFT bin. Thus we compute ${X}_{m}\left(k\right)$ only for those values of m that are integer multiples of Q . Specifically, if n is assumed to be an integer, then we only compute ${X}_{m}\left(k\right)$ for values of m given by $m=Qn$ . This is leads to

${X}_{m}\left(k\right)={X}_{Qn}\left(k\right)\equiv {X}_{n}\left(k\right)=\sum _{p=0}^{QN-1}x\left(k-p\right)w\left(p\right){e}^{j2\pi \frac{np}{N}}.$

Since we have achieved the goal of constructing spectrally concentrated bandpass filters (albeit at the cost of expanding the size of all steps preceding the final DFT computation), we can now consider decimating the filter outputs. Since the filter bandwidths are nominally $\frac{{f}_{s}}{N}$ Hz, decimation by up to N is possible without violating the sampling theorem. Suppose we decimate by the factor M , where $0 . This means evaluating the integer time index k only at integer multiples of M . If we allow the integer to be the decimated time index, the decimated version of the n-th DFT bin output is

${X}_{n}\left(k=rM\right)={X}_{n}\left(r\right)=\sum _{p=0}^{QN-1}x\left(rM-p\right)w\left(p\right){e}^{j2\pi \frac{np}{N}}.$

At this point we can start making comparisons. [link] closely resembles [link] and [link] closely resembles [link] . In fact, if we use the definition of $v\left(r,p\right)$ developed earlier, then [link] becomes

${X}_{n}\left(r\right)=\sum _{p=0}^{N-1}v\left(r,p\right)·{e}^{j2\pi \frac{np}{N}}$

which differs from the equation for ${y}_{n}\left(r\right)$ developed in [link] only in the absence of a residual carrier term. If, for example, we want to compute ${y}_{n}\left(r\right)$ , we can do it by selecting the right DFT output bin ( n in this case) and multiplying it by the residual carrier term, if any. Thus for all practical purposes, the bank of tuners viewpoint and the DFT-based filter bank viewpoint yield the same structure and same results.

## The effect of bin decimation on an fft

More insight into the relationship between the DFT-based filter bank and the basic FDM-to-TDM transmultiplexer shown in [link] can be gained by considering the common situation where an FFT is used to compute the DFT. In the preceding section, it was shown that the DFT filter implicitly uses a QN-point DFT but in fact only N output bins are computed. Consider now the FFT flow graph shown in [link] (a). The input is $QN$ (8, in this case) weighted input samples $\overline{x}\left(p\right)$ and the output is $QN$ bins. Suppose now that all we want is the odd numbered output bins. Careful examination of the flow graph shows that more than just the output points can be deleted. Look at $x\left(0\right)$ , for example. It is computed using numbers from the previous stage which are only used to compute undesired outputs . Thus these intermediate terms need not be computed either. This process can continue until the point where the intermediate points are needed. To see how this works, examine [link] (b). Removing all unneeded nodes reveals something very interesting. The FFT processing naturally breaks into two sections. The second section, the QN-point FFT pruned of all unneeded nodes, is recognized to have the flow graph of an N-point FFT. In fact, if the bin decimation is not offset from bin 0, then the twiddle factors are exactly those of an N-point FFT as well. The section preceding the N-point FFT can be written as N Q-point sums of weighted, delayed input data samples. These sums can be recognized as the $\left\{v\left(r,p\right)\right\}$ . Thus by pruning out the unneeded nodes in a QN-point FFT taken over the weighted input data, the computation of the filter bank gracefully separates into the cascade of a preprocessor that computes the $\left\{v\left(r,p\right)\right\}$ and an N-point FFT. The resulting block diagram is exactly the same as that shown in [link] .

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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
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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?
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What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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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
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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
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