# 0.2 Derivation of the equations for a basic fdm-tdm transmux  (Page 2/10)

 Page 2 / 10

In principle, the parameters f s (and hence T ), f 0 , L , and M can be chosen arbitrarily. In fact, significant simplications to the implementation of the tuner occur if they are carefully chosen. To do this we must first develop a general equation for the decimated tuner output $y\left(r\right)$ . Spectral Description of Each Step in the Digital Tuning of a Single Channel

The undecimated filter output $\overline{y}\left(k\right)$ can be written as the convolutional sum of $\rho \left(k\right)$ and the filter pulse response $h\left(k\right)$ :

$\overline{y}\left(k\right)=\sum _{l=0}^{L-1}h\left(l\right)\rho \left(k-l\right).$

Substituting the expression for $\rho \left(k\right)$ yields

$\overline{y}\left(k\right)=\sum _{l=0}^{L-1}h\left(l\right)x\left(k\phantom{\rule{4pt}{0ex}}-l\right){e}^{-j2\pi {f}_{0}T\left(k-l\right)}.$

Separating the two terms in the exponential produces the next expression:

$\overline{y}\left(\mathrm{k}\right)={e}^{-j2\pi {f}_{0}Tk}·\sum _{l=0}^{L-1}h\left(l\right)x\left(k-l\right){e}^{j2\pi {f}_{0}Tl}.$

Decimation by the factor M is introduced by evaluating $\overline{y}\left(k\right)$ only at the values of k where $k=rM$ . We denote the decimated output as $y\left(r\right)$ , given by

$y\left(r\right)\equiv \overline{y}\left(k=rM\right)={e}^{-j2\pi {j}_{0}TrM}·\sum _{l=0}^{L-1}h\left(l\right)x\left(rM-l\right){e}^{j2\pi {f}_{0}Tl}$

## Choosing various system parameters to simplify the general equation for the tuner output

Equation 4 holds for arbitrary choice of L , M , f 0 , and f s . To obtain the equations for the basic FDM-TDM transmultiplexer, we must first simplify the general equation for the output of the digital tuner. We do this by making the three key assumptions:

1. We assume that the sampling rate f s and the tuning frequency f 0 are integer multiples of the same frequency step $\Delta f$ . In the case of FDM multichannel telephone systems for example, $\Delta f$ is typically 4 kHz. We define the integer parameters N and n with the expressions ${f}_{s}\equiv N·\Delta f$ and ${f}_{0}\equiv n·\Delta f$ .
2. We next assume that the pulse response duration L is an integer multiple of the factor N defined above. We define the positive integer parameter Q where $L\equiv Q·N$ . This is a nonrestrictive assumption since Q can be chosen large enough to make it true for any value of L . If $QN$ exceeds the minimum required value of L , then $h\left(k\right)$ can be made artificially longer by padding it with zero values. The factor Q turns out to be an important design parameter. The parameters Q and N are determined separately and the resulting value of L follows from their choice.
3. We also assume that the decimation factor M is chosen to be closely related to the parameter N . Typical values are $M=N$ and $M=\frac{N}{2}$

We can now examine the effects of these assumptions. First, the relationship between f s , f 0 , and $\Delta f$ allows $y\left(r\right)$ to be written as

${y}_{n}\left(r\right)={e}^{-j2\pi \frac{nrM}{N}}·\sum _{l=0}^{L-1}h\left(l\right)x\left(rM-l\right){e}^{j2\pi \frac{nl}{N}}.$

We subscript the decimated output $y\left(r\right)$ by the parameter n to indicate that it depends on the tuning frequency ${f}_{0}=n·\Delta f$ .

The second assumption, the definition of the parameter Q , permits the single sum to be split into a nested double sum. To do this, define the new integer indices q and p by the expressions

$l\equiv qN+p,\phantom{\rule{4pt}{0ex}}where\phantom{\rule{4pt}{0ex}}0\le q\le Q-1\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}0\le p\le N-1.$

Examination of [link] shows that the pulse response running index l has a unique value in the range from 0 to $L-1$ for each permissible value of p and q . This permits the single convolutional sum over the index l to be replaced (for reasons to be shown) with a double sum over the indices p and q . In particular,

$\begin{array}{ccc}{y}_{n}\left(r\right)\hfill & =\hfill & {e}^{-j2\pi \frac{nrM}{N}}·\sum _{l=0}^{L-1}h\left(l\right)x\left(rM-l\right){e}^{j2\pi \frac{nl}{N}}\hfill \\ & =\hfill & {e}^{-j2\pi \frac{nrM}{N}}·\sum _{p=0}^{N-1}\sum _{q=0}^{Q-1}h\left(qN+p\right)x\left(rM-qN-p\right){e}^{j2\pi \frac{n\left(qN+p\right)}{N}}\hfill \\ & =\hfill & {e}^{-j2\pi \frac{nrM}{N}}·\sum _{p=0}^{N-1}\sum _{q=0}^{Q-1}h\left(qN+p\right)x\left(rM-qN-p\right){e}^{j2\pi \frac{nqN}{N}}{e}^{j2\pi \frac{np}{N}}\hfill \\ & =\hfill & {e}^{-j2\pi \frac{nrM}{N}}·\sum _{p=0}^{N-1}{e}^{j2\pi \frac{np}{N}}\left[\sum _{q=0}^{Q-1}h\left(qN+p\right)x\left(rM-qN-p\right)\right].\hfill \end{array}$

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
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
characteristics of micro business
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Got questions? Join the online conversation and get instant answers! By Gerr Zen      By Rhodes By  By