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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

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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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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