<< Chapter < Page Chapter >> Page >

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 A = { a 1 , a 2 , a | A | } be given, where each symbol a A is associated with a unique sequence of B ordered bits, b B - 1 , , b 1 , b 0 , where B = log 2 | A | and b i { 0 , 1 } for i = 0 , , B - 1 . For example, let 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 A with the corresponding binary sequence b B - 1 , , b 1 , b 0 , the message signal m ( t ) is defined to be

m ( t ) = 1 k = 0 B - 1 b k k = 0 B - 1 b k c o s ( 2 π ( k + 1 ) f 0 t )

for some fundamental carrier frequency, f 0 .

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

Decomposition of “m” in DMT Scheme

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.

Table of Extended ASCII Values
Different Symbols in DMT Scheme

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 < < N . FFAST, the sparse FFT algorithm that we will be using, requires the sparsity constraint k < N 1 3 . Recall that the message signal, m ( t ) , is defined as

m ( t ) = 1 k = 0 B - 1 b k k = 0 B - 1 b k c o s ( 2 π ( k + 1 ) f 0 t )

so that the Nyquist frequency is 2 B 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 = 2 B if all bits are high so that the sparsity condition may be expressed as 2 B < N 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 = 2 B 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 > 2 B 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.

Spectra for Different Sampling Schemes

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] .

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Using ffast to decrease computation time in digital multitone communication. OpenStax CNX. Dec 17, 2014 Download for free at http://legacy.cnx.org/content/col11731/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Using ffast to decrease computation time in digital multitone communication' conversation and receive update notifications?

Ask