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The least-squares solution

Define the matrix

Ψ = [ F - ( R T R ) - 1 R T S ] T ( R T R ) [ F - ( R T R ) - 1 R T S ] = F T ( R T R ) F - S T R F - F T R T S + S T R ( R T R ) - 1 R T S .

The purpose of this definition is to rewrite [link] in terms of Ψ :

J L S = Ψ + S T S - S T R ( R T R ) - 1 R T S = Ψ + S T [ I - R ( R T R ) - 1 R T ] S .

Since S T [ I - R ( R T R ) - 1 R T ] S is not a function of F , the minimum of J L S occurs at the F that minimizes Ψ . This occurs when

F = ( R T R ) - 1 R T S ,

assuming that ( R T R ) - 1 exists. A matrix is invertible as long as it has noeigenvalues equal to zero. Since R T R is a quadratic form it has no negative eigenvalues. Thus, all eigenvalues must be positivein order for it to be invertible. The corresponding minimum achievable by J L S at F = F is the summed squared delayed source recovery error. This is the remaining term in [link] ; that is,

J L S min = S T [ I - R ( R T R ) - 1 R T ] S .

The formulas for the optimum F in [link] and the associated minimum achievable J L S in [link] are for a specific δ . To complete the design task, it is also necessary to findthe optimal delay δ . The most straightforward approach is to set up a series of S = R F calculations, one for each possible δ , to compute the associated values of J L S min , and pick the delay associated with the smallest one.

This procedure is straightforward to implement in M atlab , and the program LSequalizer.m allows you to play with the various parameters to get a feel for theireffect. Much of this program will be familiar from openclosed.m. The first three lines define a channel, create a binary source, and thentransmit the source through the channel using the filter command. At the receiver, the data are put through a quantizer, and then the erroris calculated for a range of delays. The new part is in the middle.

b=[0.5 1 -0.6];                   % define channelm=1000; s=sign(randn(1,m));       % binary source of length m r=filter(b,1,s);                  % output of channeln=3;                              % length of equalizer - 1 delta=3;                          % use delay <=n p=length(r)-delta;R=toeplitz(r(n+1:p),r(n+1:-1:1));  % build matrix R S=s(n+1-delta:p-delta)';          % and vector Sf=inv(R'*R)*R'*S                  % calculate equalizer f Jmin=S'*S-S'*R*inv(R'*R)*R'*S     % Jmin for this f and deltay=filter(f,1,r);                  % equalizer is a filter dec=sign(y);                      % quantize and count errorserr=0.5*sum(abs(dec(delta+1:end)...                   -s(1:end-delta)))
LSequalizer.m find a LS equalizer f for the channel b (download file)

The variable n defines the length of the equalizer, and delta defines the delay that will be used in constructing the vector S defined in [link] (observe that delta must be positive and less than or equal to n ). The Toeplitz matrix R is defined in [link] and [link] , and the equalizer coefficients f are computed as in [link] . The value of minimum achievable performance is Jmin , which is calculated as in [link] . To demonstrate the effect of the equalizer, thereceived signal r is filtered by the equalizer coefficients, and the output is then quantized.If the equalizer has done its job (i.e., if the eye is open), then there should be some shift sh at which no errors occur.

For example, using the default channel b= [0.5 1 -0.6],and length 4 equalizer ( n=3 ), four values of the delay delta give

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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