# 0.5 Sampling with automatic gain control  (Page 15/19)

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There are two basic approaches to an AGC. The traditional approach uses analog circuitry to adjust the gain before the sampling. The more modern approach uses the output of the sampler to adjust thegain. The advantage of the analog method is that the two blocks (the gain and the sampling) are separate and do not interact.The advantage of the digital adjustment is that less additional hardware is required since the DSP processing is already presentfor other tasks.

A simple digital system for AGC gain adjustment is shown in [link] . The input $r\left(t\right)$ is multiplied by the gain $a$ to give the normalized signal $s\left(t\right)$ . This is then sampled to give the output $s\left[k\right]$ . The assessment block measures $s\left[k\right]$ and determines whether $a$ must be increased or decreased.

The goal is to choose $a$ so that the power (or average energy) of $s\left(t\right)$ is approximately equal to some specified ${\mathbf{s}}^{2}$ . Since

${a}^{2}{\left(\text{avg},\left\{,{r}^{2},\left(t\right),\right\}|}_{t=kT}\approx \text{avg}\left\{{s}^{2}\left(kT\right)\right\}\approx \text{avg}\left\{{s}^{2}\left[k\right]\right\},$

it would be ideal to choose

${a}^{2}\approx \frac{{\mathbf{s}}^{2}}{\text{avg}\left\{{r}^{2}\left(kT\right)\right\}},$

because this would imply that $\text{avg}\left\{{s}^{2}\left(kT\right)\right\}\approx {\mathbf{s}}^{2}$ . The averaging operation (in this case a movingaverage over a block of data of size $N$ ) is defined by

$\text{avg}\left\{x\left[k\right]\right\}=\frac{1}{N}\sum _{i=k-N+1}^{k}x\left[i\right]$

Unfortunately, neither the analog input $r\left(t\right)$ nor its power are directly available to the assessment blockin the DSP portion of the receiver, so it is not possible to directly implement [link] .

Is there an adaptive element that can accomplish this task? As suggested in the beginning of "Iteration and Optimization" , there are three steps to the creation of a viableoptimization approach: setting a goal, choosing a solution method, and testing. As in any real life engineering task, a propermathematical statement of the goal can be tricky, and this section proposes two (slightly different) possibilitiesfor the AGC. By comparing the resulting algorithms (essentially, alternativeforms for the AGC design), it may be possible to trade off among various design considerations.

One sensible goal is to try to minimize a simple function of the difference between the power of the sampled signal $s\left[k\right]$ and the desired power ${\mathbf{s}}^{2}$ . For instance, the averaged squared error in the powers of $s$ and $\mathbf{s}$ ,

$\begin{array}{cc}\hfill {J}_{LS}\left(a\right)& =\text{avg}\left\{\frac{1}{4},{\left({s}^{2}\left[k\right]-{\mathbf{s}}^{2}\right)}^{2}\right\}\hfill \\ & =\frac{1}{4}\text{avg}\left\{{\left({a}^{2}{r}^{2}\left(kT\right)-{\mathbf{s}}^{2}\right)}^{2}\right\},\hfill \end{array}$

penalizes values of $a$ which cause ${s}^{2}\left[k\right]$ to deviate from ${\mathbf{s}}^{2}$ . This formally mimics the parabolic form of the objective [link] in the polynomial minimization example of the previous section.Applying the steepest descent strategy yields

$a\left[k+1\right]=a\left[k\right]-\mu {\left(\frac{d{J}_{LS}\left(a\right)}{da}|}_{a=a\left[k\right]},$

which is the same as [link] , except that the name of the parameter has changed from $x$ to $a$ . To find the exact form of [link] requires the derivative of ${J}_{LS}\left(a\right)$ with respect to the unknown parameter $a$ . This can be approximated by swapping the derivative and the averaging operations to give

$\begin{array}{c}\\ \hfill \frac{d{J}_{LS}\left(a\right)}{da}& =& \frac{1}{4}\frac{d\text{avg}\left\{{\left({a}^{2}{r}^{2}\left(kT\right)-{\mathbf{s}}^{2}\right)}^{2}\right\}}{da}\hfill \\ & \approx & \frac{1}{4}\text{avg}\left\{\frac{d{\left({a}^{2}{r}^{2}\left(kT\right)-{\mathbf{s}}^{2}\right)}^{2}}{da}\right\}\hfill \\ & =& \text{avg}\left\{\left({a}^{2}{r}^{2}\left(kT\right)-{\mathbf{s}}^{2}\right)a{r}^{2}\left(kT\right)\right\}.\hfill \end{array}$

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
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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
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Abigail
for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
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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.
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what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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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
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many many of nanotubes
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what is the function of carbon nanotubes?
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I'm interested in nanotube
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what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
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