# 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] .

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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