# 0.1 Digital multitone modulation

 Page 1 / 1

## Digital multitone modulation

We present the digital multitone modulation scheme and demonstrate its suitability for demodulation via FFAST.

## Digital multitone modulation scheme

Let a finite alphabet $\mathbb{A}=\left\{{a}_{1},{a}_{2},\cdots {a}_{|\mathbb{A}|}\right\}$ be given, where each symbol $a\in \mathbb{A}$ is associated with a unique sequence of $B$ ordered bits, ${b}_{B-1},\cdots ,{b}_{1},{b}_{0}$ , where $B=⌈{log}_{2}|\mathbb{A}|⌉$ and ${b}_{i}\in \left\{0,1\right\}$ for $i=0,\cdots ,B-1$ . For example, let $\mathbb{A}$ be the set of all lowercase letters in the English alphabet and associate each letter with its order in the alphabet. In this case, the binary sequence `01101' corresponds to the thirteenth letter, “m".

Generally speaking, Digital Multitone (DMT) Modulation is a “parallel communication scheme in which several carriers of different frequencies each carry narrowband signals simultaneously” [link] . These narrowband signals are usually sinusoids that encode the binary sequence associated with each symbol. If a bit is “high", then the corresponding sinusoid is expressed in the output signal; otherwise the bit is “low" and the sinusoid is not expressed. More precisely, given a symbol $a\in \mathbb{A}$ with the corresponding binary sequence ${b}_{B-1},\cdots ,{b}_{1},{b}_{0}$ , the message signal $m\left(t\right)$ is defined to be

$m\left(t\right)=\frac{1}{{\sum }_{k=0}^{B-1}{b}_{k}}\sum _{k=0}^{B-1}{b}_{k}cos\left(2\pi \left(k+1\right){f}_{0}t\right)$

for some fundamental carrier frequency, ${f}_{0}$ .

In our previous example where $\mathbb{A}$ is the English alphabet and the letter “m" corresponds to “01101”, the message signal $m\left(t\right)$ is the sum of the first, third, and fourth harmonics, as shown in the figure below:

In our computational experiments, we use digital multitone modulation to encode 8-bit Extended ASCII values. An Extended ASCII table can be found here . Below are several symbols and their digital multitone modulation signals.

## Sparsity in digital multitone modulation

Sparse FFT algorithms only achieve low runtime complexity if the input signal is sparse in its Fourier representation; that is, if for a length- $N$ signal, there are $k$ nonzero DFT coefficients with $k< . FFAST, the sparse FFT algorithm that we will be using, requires the sparsity constraint $k<{N}^{\frac{1}{3}}$ . Recall that the message signal, $m\left(t\right)$ , is defined as

$m\left(t\right)=\frac{1}{{\sum }_{k=0}^{B-1}{b}_{k}}\sum _{k=0}^{B-1}{b}_{k}cos\left(2\pi \left(k+1\right){f}_{0}t\right)$

so that the Nyquist frequency is $2B{f}_{0}$ . In order to ensure signal sparsity, the sampling frequency should be a multiple of the fundamental carrier frequency so that each of the sinusoidal components falls into a single frequency bin. This type of “on-the-grid" sampling may be expressed as

${f}_{s}=N{f}_{0}$

where $N$ is the length of the sampled signal. Note that in [link] each sinusoid contributes two DFT coefficients. Thus, $k=2B$ if all bits are high so that the sparsity condition may be expressed as $2B<{N}^{\frac{1}{3}}$ .

Sampling plays a large role in signal sparsification. There are many sampling methods that ensure sparsity and we present two different methods. The first method involves padding the input signal to achieve sparsity. Consider sampling at the Nyquist frequency so that ${f}_{s}=2B{f}_{0}$ . As stated, this sampled signal is not necessarily sparse – in fact, $k=N$ if all bits are high! However, periodizing the sampled signal sufficiently many times will result in higher frequency resolution by placing zero-value coefficients in between the nonzero coefficients, thus sparsifying the signal. This method results in a spectrum with nonzero coefficient few and far between. Second, consider sampling at a sufficiently high rate to satisfy the sparsity condition; that is, ${f}_{s}>8{B}^{3}{f}_{0}$ . First note that $8{B}^{3}{f}_{0}>2B{f}_{0}$ so that aliasing does not occur. This method results in a compact spectrum where only the first and last $B$ coefficients are nonzero. See the figure below for a spectra that are characteristic of these methods.

It should be noted that there are many sampling methods that ensure signal sparsity but for the purposes of this project, we care only that the signal is sparse. Sampling schemes are discussed further in [link] .

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
Do somebody tell me a best nano engineering book for beginners?
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
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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!