<< 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

what is the role of psychology in the development of therapies for disorders of a circadian rhythm?
wazir Reply
To study Abnormal Psychology means, to study mainly the nature of
Baddo Reply
abnormal psychology? can you explain please?
la
abnormal, this branch of psychology typically deals with behavior in a clinical context.
pakki
yo
Baddo
To study Abnormal Psychology means, to study mainly the nature of 1). Conscious Mind 2). Unconscious Mind 3). Subconscious Mind 4). Normal Mind
pakki
yo
Kylee
abnormal psychology- is the scientific study of Psychological disorders and their treatments
utkarsh
Utkarsh Verma can you help me pass out my exams? tomorrom
Baddo
10 - 11 MCQ test can you all or some help? the only subject I'm weak in is psychology
Baddo
which branch of psychology is the test on?
pakki
How Abnormal Behavior is Defined While it may seem that the definition of “abnormal” indicates a behavior that is considered outside of the human norm, it is actually a much slipperier slope. Therefore, abnormal psychology takes on the task of scientifically studying and classifying human behaviors
Anye
Yo
Monica
What is the role of psychology in the development of therapies for circadian rhythm disorders?
wazir
please I wish to find out how difficult/easy it is, for someone with a bachelor degree in philosophy to continue with masters and doctoral in psychology, specifically, I-O psychology.
Anye
do the individuals know that they are being researched on? and why does anyone speak up regarding the matter? is it "don't ask, don't tell" policy?
Stepphun Reply
examine any 5 perceptual principals of organization
kundai Reply
Is it possible for a child to be schizophrenic?
Esther Reply
yes. early onset of schizophrenia starts at 13 to 18. very rare but possible.
Kristel
Good question
Mulder
It's uncommon for children to be diagnosed with schizophrenia.
pakki
what is the role of psychology in the development of therapies for circadian rhythm disorders?
wazir
is there a difference between age and romantic attachment style?
Alysha Reply
I don't think so. Well, not always. Certain things should be timeless.
Know
I'm 48 & he's 26 ...so we shall see?
Tanya
in my opinion if you both are comfortable with each other then definitely age is nothing but a number but then also there will be little difference in the cognition due to age
utkarsh
yes Utkarsh.I agree with you.We call it as generation gap. But there can be exceptions also...
Shailaja
yes generation gap can affect the relationship but this totally depends on the couple that how they settle up with each other
utkarsh
that is true, balance is needed. I spent a long time with an older man and I was more his child than his partner. power struggles are hard in relationships with generational gaps
Ashley
I think all it all depends on your comfort or braveness and attitude toward life, then age is just numbers.
Sujeet
new perspective n ones mindset towards life plays an important role... if the change is for gud.. go-ahead
Smitha
Yes romantic attachment styles change with age if that's what you are asking. here is where you can read more about it ***google.com/amp/s/qz.com/1206940/attachment-style-changes-with-age/amp/
krish
can empirical research be performed on people being diagnosed or people who have borderline personality disorder?
Stepphun Reply
yes it can
Angela
***pubmed.ncbi.nlm.nih.gov/3711874/
Angela
thank you for the feedback
Stepphun
👍
Angela
yes?
Stepphun
discuss the three models of positional coding in working memory citing examples.Please help me with points to this question
jane Reply
can you clear me up a doubt that for to become a psychriatist what process are needed
Mimi Reply
What are the vocational choices for women in pakistan also discuss its aims and objectives.
Malik Reply
I want to research on Egocentrism and Autistic thinking on myself.
RAVI Reply
I want to research on Egocentrism and Autistic thinking on myself. Have you any suggestion for me?
RAVI
I want to research on Egocentrism and Autistic thinking on myself. Have you any suggestion for me?
RAVI
why do you think they correlate?
James
what is the role of psychology in the development of therapies for circadian rhythm?
wazir
what is Egocentrism
Ishfaq
External goal that have the capacity to motivate behaviour
Haider Reply
Reinforcements and Punishments
Khyzar
indeed! but positive punishment if I may add
Angela
hi. i am a student of B.A. Psychology (major), i wanted to enquire as to which textbook would be most informative on 'Biopsychology' ?
Bhargavi
Hey, I'm currently a BS Psychology major and I'd like to recommend the 10th Edition Biological Psychology book by Kalat
Ayessa
thanks, it really means a lot😀
Bhargavi
You're welcome.
Ayessa
what is logical atomism
Koshivani
what is HMC found in UK and US in 1915
Samu Reply
List five branches of psychology
Faith Reply
clinical psychology social psychology educational psychology experimental psychology neurological psychology behavioural psychology criminal psychology industrial psychology
Nazish
I am a older retired person who has always found psychology interesting I am teaching myself with information that I find online textbooks etc is there any advice that you can give me to help me thank you
Brenda
can anyone explain Shiffrin and Schneider experiment of controlled and automatic human information?
AKSHATA Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.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