# 0.12 Linear equalization  (Page 5/17)

 Page 5 / 17

## The least-squares solution

Define the matrix

$\begin{array}{ccc}\hfill \Psi & =& {\left[F-{\left({R}^{T}R\right)}^{-1}{R}^{T}S\right]}^{T}\left({R}^{T}R\right)\left[F-{\left({R}^{T}R\right)}^{-1}{R}^{T}S\right]\hfill \\ & =& {F}^{T}\left({R}^{T}R\right)F-{S}^{T}RF-{F}^{T}{R}^{T}S+{S}^{T}R{\left({R}^{T}R\right)}^{-1}{R}^{T}S.\hfill \end{array}$

The purpose of this definition is to rewrite [link] in terms of $\Psi$ :

$\begin{array}{ccc}\hfill {J}_{LS}& =& \Psi +{S}^{T}S-{S}^{T}R{\left({R}^{T}R\right)}^{-1}{R}^{T}S\hfill \\ & =& \Psi +{S}^{T}\left[I-R{\left({R}^{T}R\right)}^{-1}{R}^{T}\right]S.\hfill \end{array}$

Since ${S}^{T}\left[I-R{\left({R}^{T}R\right)}^{-1}{R}^{T}\right]S$ is not a function of $F$ , the minimum of ${J}_{LS}$ occurs at the $F$ that minimizes $\Psi$ . This occurs when

${F}^{†}={\left({R}^{T}R\right)}^{-1}{R}^{T}S,$

assuming that ${\left({R}^{T}R\right)}^{-1}$ 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 ${J}_{LS}$ at $F={F}^{†}$ is the summed squared delayed source recovery error. This is the remaining term in [link] ; that is,

${J}_{LS}^{\mathrm{min}}={S}^{T}\left[I-R{\left({R}^{T}R\right)}^{-1}{R}^{T}\right]S.$

The formulas for the optimum $F$ in [link] and the associated minimum achievable ${J}_{LS}$ in [link] are for a specific $\delta$ . To complete the design task, it is also necessary to findthe optimal delay $\delta$ . The most straightforward approach is to set up a series of $S=RF$ calculations, one for each possible $\delta$ , to compute the associated values of ${J}_{LS}^{\mathrm{min}}$ , 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

Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
what is nano technology
what is system testing?
Got questions? Join the online conversation and get instant answers!
The content of the passage is focused on the mathematical derivation and explanation of the least-squares solution in the context of linear equalization, particularly in signal processing. It involves matrix operations and optimization to find the best linear equalizer for a given channel and delay.
no_question

the passage does not contain a direct mathematical question or a request for a calculation. it is a detailed explanation of a mathematical concept and its application.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 0.12 Linear equalization  (Page 5/17)

 Page 5 / 17

## The least-squares solution

Define the matrix

$\begin{array}{ccc}\hfill \Psi & =& {\left[F-{\left({R}^{T}R\right)}^{-1}{R}^{T}S\right]}^{T}\left({R}^{T}R\right)\left[F-{\left({R}^{T}R\right)}^{-1}{R}^{T}S\right]\hfill \\ & =& {F}^{T}\left({R}^{T}R\right)F-{S}^{T}RF-{F}^{T}{R}^{T}S+{S}^{T}R{\left({R}^{T}R\right)}^{-1}{R}^{T}S.\hfill \end{array}$

The purpose of this definition is to rewrite [link] in terms of $\Psi$ :

$\begin{array}{ccc}\hfill {J}_{LS}& =& \Psi +{S}^{T}S-{S}^{T}R{\left({R}^{T}R\right)}^{-1}{R}^{T}S\hfill \\ & =& \Psi +{S}^{T}\left[I-R{\left({R}^{T}R\right)}^{-1}{R}^{T}\right]S.\hfill \end{array}$

Since ${S}^{T}\left[I-R{\left({R}^{T}R\right)}^{-1}{R}^{T}\right]S$ is not a function of $F$ , the minimum of ${J}_{LS}$ occurs at the $F$ that minimizes $\Psi$ . This occurs when

${F}^{†}={\left({R}^{T}R\right)}^{-1}{R}^{T}S,$

assuming that ${\left({R}^{T}R\right)}^{-1}$ 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 ${J}_{LS}$ at $F={F}^{†}$ is the summed squared delayed source recovery error. This is the remaining term in [link] ; that is,

${J}_{LS}^{\mathrm{min}}={S}^{T}\left[I-R{\left({R}^{T}R\right)}^{-1}{R}^{T}\right]S.$

The formulas for the optimum $F$ in [link] and the associated minimum achievable ${J}_{LS}$ in [link] are for a specific $\delta$ . To complete the design task, it is also necessary to findthe optimal delay $\delta$ . The most straightforward approach is to set up a series of $S=RF$ calculations, one for each possible $\delta$ , to compute the associated values of ${J}_{LS}^{\mathrm{min}}$ , 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

Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
what is nano technology
what is system testing?
Got questions? Join the online conversation and get instant answers!
user
The content of the passage is focused on the mathematical derivation and explanation of the least-squares solution in the context of linear equalization, particularly in signal processing. It involves matrix operations and optimization to find the best linear equalizer for a given channel and delay.

TYPE: NO_QUESTION

The passage does not contain a direct mathematical question or a request for a calculation. It is a detailed explanation of a mathematical concept and its application.