# 0.5 Sampling with automatic gain control  (Page 14/19)

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Consider performing an iterative maximization of

$J\left(x\right)=8-6|x|+6cos\left(6x\right)$

via [link] with the sign on the update reversed (so that the algorithm will maximize ratherthan minimize). Suppose the initialization is $x\left[0\right]=0.7$ .

1. Assuming the use of a suitably small stepsize $\mu$ , determine the convergent value of $x$ .
2. Is the convergent value of $x$ in part (a) the global maximum of $J\left(x\right)$ ? Justify your answer by sketching the error surface.

Suppose that a unimodal single-variable performance function has only one point with zeroderivative and that all points have a positive second derivative. TRUE or FALSE: A gradient descent methodwill converge to the global minimum from any initialization.

Consider the modulated signal

$r\left(t\right)=w\left(t\right)cos\left(2\pi {f}_{c}t+\Phi \right)$

where the absolute bandwidth of the baseband message waveform $w\left(t\right)$ is less than ${f}_{c}/2$ . The signals $x$ and $y$ are generated via

$\begin{array}{cc}\hfill x\left(t\right)& =\text{LPF}\left\{r\left(t\right)cos\left(2\pi {f}_{c}t+\theta \right)\right\}\hfill \\ \hfill y\left(t\right)& =\text{LPF}\left\{r\left(t\right)sin\left(2\pi {f}_{c}t+\theta \right)\right\}\hfill \end{array}$

where the LPF cutoff frequency is ${f}_{c}/2$ .

1. Determine $x\left(t\right)$ in terms of $w\left(t\right)$ , ${f}_{c}$ , $\Phi$ , and $\theta$ .
2. Show that
$\frac{\partial }{\partial \theta }\left\{\frac{1}{2}{x}^{2}\left(t\right)\right\}=-x\left(t\right)y\left(t\right)$
using the fact that derivatives and filters commute as in [link] .
3. Determine the values of $\theta$ maximizing ${x}^{2}\left(t\right)$ .

Consider the function

$J\left(x\right)={\left(1,-,|,x,-,2,|\right)}^{2}.$
1. Sketch $J\left(x\right)$ for $-5\le x\le 5$ .
2. Analytically determine all local minima and maxima of $J\left(x\right)$ for $-5\le x\le 5$ . Hint: $\frac{d\phantom{\rule{4pt}{0ex}}|f\left(b\right)|}{db}=\mathrm{sign}\left(f\left(b\right)\right)\frac{d\phantom{\rule{3.33333pt}{0ex}}f\left(b\right)}{db}$ where $\mathrm{sign}\left(a\right)$ is defined in [link] .
3. Is $J\left(x\right)$ unimodal as a function of $x$ ? Explain your answer.
4. Develop an iterative gradient descent algorithm for updating $x$ to minimize $J$ .
5. For an initial estimate of $x=1.2$ , what is the convergent value of $x$ determined by an iterative gradient descent algorithm with a satisfactorily small stepsize.
6. Compute the direction (either increasing $x$ or decreasing $x$ ) of the update from (d) for $x=1.2$ .
7. Does the direction determined in part (f) point from $x=1.2$ toward the convergent value of part (e)? Should it (for a correct answer to (e))? Explain your answer.

## Automatic gain control

Any receiver is designed to handle signals of a certain average magnitude most effectively. The goal of an AGC is toamplify weak signals and to attenuate strong signals so that they remain (as much as possible) within the normaloperating range of the receiver. Typically, the rate at which the gain varies is slow compared with the data rate, though itmay be fast by human standards.

The power in a received signal depends on many things: the strength of the broadcast, the distance from the transmitter to the receiver, thedirection in which the antenna is pointed, and whether there are any geographic features such as mountains (or tall buildings) that block,reflect, or absorb the signal. While more power is generally better from the point of view of trying to decipher the transmitted message,there are always limits to the power handling capabilities of the receiver. Hence if the received signal is too large (on average), itmust be attenuated. Similarly, if the received signal is weak (on average), then it must be amplified.

[link] shows the two extremes that the AGC is designed to avoid. In part (a), the signal is much larger than the levels of thesampling device (indicated by the horizontal lines). The gain must be made smaller. In part (b), the signal ismuch too small to be captured effectively, and the gain must increased.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
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
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