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The first portion of the exponential term in the sum vanishes since its argument is always an integer multiple of 2 π . Moving the terms of the summation in the last step is possible since the remaining term of the exponential does not depend on the running index q . It is useful to give a short name to the terms in brackets in the last equation. Noting that it is a function of the decimated time index r and the running index p , we define the variable v ( r , p ) by the expression

v ( r , p ) q = 0 Q - 1 h ( q N + p ) · x ( r M - q N - p ) .

Notice that v ( r , p ) is a function of the input data x ( k ) , the filter pulse response h ( l ) , and the system parameters Q , M , and N , but it is not a function of the selected conversion frequency f 0 , represented in the equation for y n ( r ) by the integer n .

Substituting v ( r , p ) into the equation for the decimated output y n ( r ) of the tuner tuned to frequency f 0 = n · Δ f yields

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

Notice that the frequency dependency of the tuner shows up only in the exponential terms.

Before discussing this result in detail it remains to examine the effects of the third assumption. To do this, define the decimation factor M by the expression M N K , where K = 1 , 2, or 4. Look first at the exponential terms preceding the sum. It can now be written as

e - j 2 π n r M N = e - j 2 π n r K = [ e - j 2 π K ] n r = [ - j 4 K ] n r .

With K defined this way, the most general expression for y n ( r ) is

y n ( r ) = [ - j 4 K ] n r · p = 0 N - 1 e j 2 π n p N v ( r , p ) , w h e r e
v ( r , p ) = q = 0 Q - 1 h ( q N + p ) · x ( r N K - q N - p ) .

It can be verified that for K = 1 , 2, or 4, the factor multiplying the sum is at most a negation or a swapping from imaginary to real or vice versa. Thus no actual multiplication is needed. By far the cleanest case is the one in which the other system parameters (for example, N , Q , and h ( k ) ) are selected so that the decimation factor M exactly equals N , or equivalently that K = 1 . In this case, the exponential preceding the sum collapses to unity, yielding what will be termed in this technical note as the basic FDM-TDM transmux equation Because of the K = 1 assumption, this equation is the simplest of all those seen to this point and will be referred to as the basic equation . Many applications require M to be chosen differently however (see Section 4 for example) and in these cases equation 12 should be used. :

y n ( r ) = p = 0 N - 1 e j 2 π n p N v ( r , p ) , w h e r e
v ( r , p ) = q = 0 Q - 1 h ( q N + p ) · x ( ( r - q ) N - p ) .

Interpretation of the basic tuner equation in terms of the discrete fourier transform

Examination of [link] shows that each sample of the tuner output, when tuned to frequency f 0 = n Δ f , is the N-point inverse discrete Fourier transform (DFT) of the preprocessed data { v ( r , p ) } , evaluated at frequency index n . The signal flow described by the equation is shown in [link] . The sampled input data x ( k ) passes into a digital tapped delay line of length Q N at the sampling rate f s . Every M-th sample, the complete contents of the delay line, all Q N samples, are used to compute { v ( r , p ) } . Thus the v ( r , p ) are computed at the decimated rate f s M . Each of the N elements of v ( r , p ) is computed by weighting Q of the delayed input samples by the appropriate coefficient from the pulse response vector h ( k ) and summing them together. Notice that at each decimated sampling interval all of the delayed data and all of the pulse response coefficients are used to compute the v ( r , p ) . Notice also that since Q N is usually much greater than M , each input sample is used in the production of the { v ( r , p ) } over several consecutive values of the decimated sampling index r .

Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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