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The term a 2 r 2 ( k T ) inside the parentheses is equal to s 2 [ k ] . The term a r 2 ( k T ) outside the parentheses is not directly available to the assessment mechanism, though it can reasonably beapproximated by s 2 [ k ] a . Substituting the derivative into [link] and evaluating at a = a [ k ] gives the algorithm

a [ k + 1 ] = a [ k ] - μ avg ( s 2 [ k ] - s 2 ) s 2 [ k ] a [ k ] .

Care must be taken when implementing [link] that a [ k ] does not approach zero.

Of course, J L S ( a ) of [link] is not the only possible goal for the AGC problem.What is important is not the exact form of the performance function, but where the performance function has its optimal points.Another performance function that has a similar error surface (peek ahead to [link] ) is

J N ( a ) = avg { | a | ( s 2 [ k ] 3 - s 2 ) } = avg { | a | ( a 2 r 2 ( k T ) 3 - s 2 ) } .

Taking the derivative gives

d J N ( a ) d a = d avg { | a | ( a 2 r 2 ( k T ) 3 - s 2 ) } d a avg { d | a | ( a 2 r 2 ( k T ) 3 - s 2 ) d a } = avg { sgn ( a [ k ] ) ( s 2 [ k ] - s 2 ) } ,

where the approximation arises from swapping the order of the differentiation and the averagingand where the derivative of | · | is the signum or sign function, which holds as long as the argument is nonzero.Evaluating this at a = a [ k ] and substituting into [link] gives another AGC algorithm

a [ k + 1 ] = a [ k ] - μ avg { sgn ( a [ k ] ) ( s 2 [ k ] - s 2 ) } .

Consider the “logic” of this algorithm. Suppose that a is positive. Since s is fixed,

avg { sgn ( a [ k ] ) ( s 2 [ k ] - s 2 ) } = avg { ( s 2 [ k ] - s 2 ) } = avg { s 2 [ k ] } - s 2 .

Thus, if the average energy in s [ k ] exceeds s 2 , a is decreased. If the average energy in s [ k ] is less than s 2 , a is increased. The update ceases when avg { s 2 [ k ] } s 2 , that is, where a 2 s 2 r 2 , as desired. (An analogous logic applies when a is negative.)

The two performance functions [link] and [link] define the updates for the two adaptive elements in [link] and [link] . J L S ( a ) minimizes the square of the deviation of the power in s [ k ] from the desired power s 2 . This is a kind of “least square” performance function(hence the subscript LS). Such squared-error objectives are common, and will reappear in phase trackingalgorithms in Chapter  [link] , in clock recovery algorithms in Chapter  [link] , and in equalization algorithms in Chapter  [link] . On the other hand, the algorithm resulting from J N ( a ) has a clear logical interpretation (the N stands for `naive'), and the update is simpler, since [link] has fewer terms and no divisions.

To experiment concretely with these algorithms, agcgrad.m provides an implementation in M atlab . It is easy to control the rate at which a [ k ] changes by choice of stepsize: a larger μ allows a [ k ] to change faster, while a smaller μ allows greater smoothing. Thus, μ can be chosen by the system designer to trade off the bandwidth of a [ k ] (the speed at which a [ k ] can track variations in the energy levels of the incoming signal) versus theamount of jitter or noise. Similarly, the length over which the averaging is done (specified by the parameter lenavg ) will also affect the speed of adaptation;longer averages imply slower moving, smoother estimates while shorter averages imply faster moving, more jittery estimates.

n=10000;                           % number of steps in simulation vr=1.0;                            % power of the inputr=sqrt(vr)*randn(n,1);             % generate random inputs ds=0.15;                           % desired power of outputmu=0.001;                          % algorithm stepsize lenavg=10;                         % length over which to averagea=zeros(n,1); a(1)=1;              % initialize AGC parameter s=zeros(n,1);                      % initialize outputsavec=zeros(1,lenavg);              % vector to store terms for averaging for k=1:n-1  s(k)=a(k)*r(k);                  % normalize by a(k)   avec=[sign(a(k))*(s(k)^2-ds),avec(1:lenavg-1)];  % incorporate new update into avec  a(k+1)=a(k)-mu*mean(avec);       % average adaptive update of a(k) end
agcgrad.m minimize the performance function J ( a ) = avg { | a | ( ( 1 / 3 ) a 2 r 2 - d s ) } by choice of a (download file)

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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.
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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
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
how can I make nanorobot?
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
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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