# 0.2 Derivation of the equations for a basic fdm-tdm transmux

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Two intuitively reasonable approaches to developing the equations for the FDM-TDM transmultiplexer are presented in this section. The first emulates [link] . We first develop the equations for a digital counterpart of the analog tuners used in the filter bank and then observe that significant computational improvements can be obtained when the tuning frequencies are linked together in a simple way. The second subsection starts from a different point, that of using the discrete Fourier transform as a spectral channelizer. We ultimately find out that these two approaches yield essentially the same analytical results.

## Fundamental equations for a single-channel digital tuner

The input FDM signal is assumed to be the continuous-time waveform ${x}_{c}\left(t\right)$ . The analog-to-digital converter shown in [link] samples this waveform at the uniform rate of f s samples per second, producing the discrete-time sequence $x\left(k\right)$ , where $x\left(k\right)\equiv {x}_{c}\left(t=kT\right)$ , the integer k is the time index, and T is the sampling interval given by $T=\frac{1}{{f}_{s}}$ . The spectrum of this sequence is shifted down in frequency by multiplying it by a complex exponential of the form ${e}^{-j2\pi {f}_{0}kT}$ , where f 0 is the desired amount of the frequency downconversion. The product of $x\left(k\right)$ and this exponential is then filtered in discrete time by using the pulse response $h\left(k\right)$ . The duration of the pulse response $h\left(k\right)$ is assumed to be finite and in particular of length no greater than L , an integer. The filter output $\overline{y}\left(k\right)$ is then decimated by a factor of M , yielding the sequence $y\left(r\right)$ , where the integer r is the decimated time index.

These processing steps are shown in graphical form in [link] . Both sides of the two-sided spectrum of the sampled input signal are seen in [link] (a). For the moment, the input signal is assumed to be real-valued and therefore the spectrum is symmetrical around 0 Hz Even though real-valued inputs are assumed here, all of the ensuing analysis applies to complex-valued signals as well. . A channel of interest in this spectrum has been shaded and its center frequency is noted to be f 0 . Multiplying the input signal by ${e}^{-j2\pi {f}_{0}kT}$ has the effect of shifting the spectrum to the left (assuming $0\le {f}_{0}\le \frac{{f}_{s}}{2}$ ) and centering the desired channel at 0 Hz. The downconverted signal is now complex-valued, and therefore spectral symmetry around 0 Hz is neither required nor expected. The transfer function of the lowpass filter appears in [link] (c). The filter pulse response $h\left(k\right)$ is chosen to attain the desired spectral characteristics. In particular, the filter needs to pass the channel of interest without degradation and suppress all others sufficiently. How to design such a pulse response is discussed in Appendix A. In general, the quality of the filter grows with the value of the parameter L . The filter shown here is symmetrical around 0 Hz and its pulse response $h\left(k\right)$ can therefore be real-valued. This is not required however.

After the application of the shifted signal $\rho \left(k\right)$ to the filter, the spectrum shown in [link] (d) results. The desired channel is isolated from all others. It is sampled, however, at a rate far faster than required by the Nyquist sampling theorem. The filter output is then decimated by the factor M , resulting in the spectrum shown in [link] (e). The channel's bandwidth is the same as before but now its percentage bandwidth, that is, its bandwidth compared to its final sampling rate, is much higher. In a good digital tuner the percentage bandwidth after decimation usually ranges between 0.5 and 0.9, where unity is the theoretical limit imposed by the sampling theorem.

what is variations in raman spectra for nanomaterials
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
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
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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
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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
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
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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 did you get the value of 2000N.What calculations are needed to arrive at it
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