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Describes the role of cyclic prefixing.

Why the prefix?

In an ideal world, we wouldn't have to worry about cyclic prefixes; the channel would be a wire and everything we send would be received in exactly the same form. However, implementing our DMT system in an ideal world would be pretty useless, as all the cool benefits from DMT would go to waste. Thus, we implemented a channel with an ugly (but not too ugly) frequency response, which added two forms of interference: inter-symbol-interference (ISI) and inter-channel-interference (ICI), which we will discuss next.

Inter-symbol-interference (isi)

ISI arises from the fact that the channel performs a linear convolution of its impulse response with the time-domain waveform. By this time, the blocks (now known as symbols ) have been mirrored, IFFT'd and concatenated. At the intersection of adjacent symbols, the linear convolution of the signal with the impulse response (whose support is assumed to be less than the symbol length but greater than unity) overlaps parts of both symbols. This means that independent symbols affect each other; one symbol"bleeds"into another. The addition of a prefix provides a buffer between symbols that prevends this.


Inter-symbol-interference (ISI): This indicates the linear convolution of the channel impulse response h with the time-domain input to the channel x . Notice the overlap at the intersection of the two symbols.

Inter-channel-interference (ici)

ICI comes from the fact that the carrier frequencies for DMT lose their orthogonality due to the frequency response of the channel. If we looked at the FFT of a block at the input to the channel, we'd see a sinc function at each carrier frequency since the IFFT modulates each carrier with a rectangular pulse. The DFT basis is orthonormal, so each of the basis vectors (sinusoids) are orthogonal to all the rest; this means each of the sincs are orthogonal (zero inner product) as well. The frequency response of the channel has the effect of attenuating certain frequencies more than others, so each of the sincs is changed by a different amount. Since the inner product is a measure of the similarity of two vectors, two previously"completely dissimilar"sinc functions now have at least some degree of similarity; i.e they are no longer orthogonal. Without orthogonal carriers, the FFT cannot exactly recover the correctspectral coefficients. Cyclic padding solves this problem by turning the linear convolution of the channel inpulse response with the signal into a cyclic convolution.


Inter-channel-interference (ICI): This represents two carrier frequencies as sincs in the frequency-domain. The frequency response of the channel causes the carrier frequencies for the transmitted signal to lose orthogonality.

Cyclic prefix

The addition of a cyclic prefix to each symbol solves both ISI and ICI. In our system, we assume the channel impulse response has a known length L . The prefix consists simply of copying the last L -1 values from each symbol and appending them in the same order to the front of the symbol. By having this buffer of essentially junk data in the front, the convolution of the impulse response with the signal at the end of a symbol does not affect any of the actual data at the beginning of the next symbol. In addition, by repeating the last elements at the beginning, the first real"data"elements of each symbol experience overlap with the"end"of the symbol, just as in cyclic convolution. This means the linear convolution of the channel impulse response with the concatenated symbols becomes concatenated cyclic convolutions of the impulse response with the individual symbols. Since cyclic convolution directly corresponds to multiplication in the frequency domain, this has great import with respect to equalization, as we will see later. After the time-domain signal passes through the channel, it is broken back into the parallel symbols and the prefix is simply discarded.

Cyclic prefix

This is one time-domain symbol with the cyclic prefix and last L elements shown in red (This is a slightly exaggerated pad length).

Our related MATLAB functions: cyclicpad.m , decyclicpad.m

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, Ece 301 projects fall 2003. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10223/1.5
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