# 1.4 Introduction problems

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Problems for introduction to signals and systems.

## Rms values

The rms (root-mean-square) value of a periodic signal is defined to be $s=\sqrt{\frac{1}{T}\int_{0}^{T} s(t)^{2}\,d t}$ where $T$ is defined to be the signal's period : the smallest positive number such that $s(t)=s(t+T)$ .

1. What is the period of $s(t)=A\sin (2\pi {f}_{0}t+\phi )$ ?
2. What is the rms value of this signal? How is it related to the peak value?
3. What is the period and rms value of the depicted square wave , generically denoted by $\mathrm{sq}(t)$ ?
4. By inspecting any device you plug into a wall socket, you'll see that it is labeled "110 volts AC". What isthe expression for the voltage provided by a wall socket? What is its rms value?

## Modems

The word "modem" is short for "modulator-demodulator." Modems are used not only for connecting computers totelephone lines, but also for connecting digital (discrete-valued) sources to generic channels. In thisproblem, we explore a simple kind of modem, in which binary information is represented by the presence orabsence of a sinusoid (presence representing a "1" and absence a "0"). Consequently, the modem's transmittedsignal that represents a single bit has the form $x(t)=A\sin (2\pi {f}_{0}t)\text{,}0\le t\le T$ Within each bit interval $T$ , the amplitude is either $A$ or zero.

1. What is the smallest transmission interval that makes sense with the frequency ${f}_{0}$ ?
2. Assuming that ten cycles of the sinusoid comprise a single bit's transmission interval, what is thedatarate of this transmission scheme?
3. Now suppose instead of using "on-off" signaling, we allow one of several different values for the amplitude during any transmission interval. If $N$ amplitude values are used, what is the resulting datarate?
4. The classic communications block diagram applies to the modem. Discuss how the transmitter mustinterface with the message source since the source is producing letters of the alphabet, not bits.

## Advanced modems

To transmit symbols, such as letters of the alphabet, RUcomputer modems use two frequencies (1600 and 1800 Hz) and several amplitude levels. A transmission is sentfor a period of time $T$ (known as the transmission or baud interval) and equals the sum of two amplitude-weighted carriers. $x(t)={A}_{1}\sin (2\pi {f}_{1}t)+{A}_{2}\sin (2\pi {f}_{2}t)\text{,}0\le t\le T$ We send successive symbols by choosing an appropriate frequency and amplitude combination, and sending themone after another.

1. What is the smallest transmission interval that makes sense to use with the frequencies given above?In other words, what should $T$ be so that an integer number of cycles of the carrier occurs?
2. Sketch (using Matlab) the signal that modem produces over several transmission intervals. Make sure youaxes are labeled.
3. Using your signal transmission interval, how many amplitude levels are needed to transmit ASCIIcharacters at a datarate of 3,200 bits/s? Assume use of the extended (8-bit) ASCII code.
We use a discrete set of values for ${A}_{1}$ and ${A}_{2}$ . If we have ${N}_{1}$ values for amplitude ${A}_{1}$ , and ${N}_{2}$ values for ${A}_{2}$ , we have ${N}_{1}{N}_{2}$ possible symbols that can be sent during each $T$ second interval. To convert this number into bits (the fundamental unit ofinformation engineers use to qualify things), compute $\log_{2}({N}_{1}{N}_{2})$ .

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
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?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Good
Answers please
Nikki Reply

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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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