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

X m ( k ) = p = 0 Q N - 1 x ( k - p ) ω ( p ) e j 2 π m p Q N .

Suppose, as discussed above, that we eliminate the filter overlapping by evaluating only every Q-th DFT bin. Thus we compute X m ( k ) 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 ( k ) for values of m given by m = Q n . This is leads to

X m ( k ) = X Q n ( k ) X n ( k ) = p = 0 Q N - 1 x ( k - p ) w ( p ) e j 2 π n p 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 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 < M N . 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 ( k = r M ) = X n ( r ) = p = 0 Q N - 1 x ( r M - p ) w ( p ) e j 2 π n p 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 ( r , p ) developed earlier, then [link] becomes

X n ( r ) = p = 0 N - 1 v ( r , p ) · e j 2 π n p N

which differs from the equation for y n ( r ) developed in [link] only in the absence of a residual carrier term. If, for example, we want to compute y n ( r ) , 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 Q N (8, in this case) weighted input samples x ¯ ( p ) and the output is Q N 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 ( 0 ) , 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 { v ( r , p ) } . 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 { v ( r , p ) } and an N-point FFT. The resulting block diagram is exactly the same as that shown in [link] .

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
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Lale Reply
no can't
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where we get a research paper on Nano chemistry....?
Maira Reply
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what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
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Jyoti Reply
ya I also want to know the raman spectra
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Crow Reply
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
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what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
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scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Source:  OpenStax, An introduction to the fdm-tdm digital transmultiplexer. OpenStax CNX. Nov 16, 2010 Download for free at http://cnx.org/content/col11165/1.2
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