# 2.6 Signals and systems problems

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## Complex number arithmetic

Find the real part, imaginary part, the magnitude and angle of the complex numbers given by the following expressions.

1. $-1$
2. $\frac{1+\sqrt{3}i}{2}$
3. $1+i+e^{i\frac{\pi }{2}}$
4. $e^{i\frac{\pi }{3}}+e^{i\pi }+e^{-(i\frac{\pi }{3})}$

## Discovering roots

Complex numbers expose all the roots of real (and complex) numbers. For example, there should be two square-roots, three cube-roots, etc. of any number.Find the following roots.

1. What are the cube-roots of 27? In other words, what is $27^{\left(\frac{1}{3}\right)}$ ?
2. What are the fifth roots of 3 ( $3^{\left(\frac{1}{5}\right)}$ )?
3. What are the fourth roots of one?

## Cool exponentials

Simplify the following (cool) expressions.

1. $i^{i}$
2. $i^{(2i)}$
3. $i^{i^{i}}$

## Complex-valued signals

Complex numbers and phasors play a very important role in electrical engineering. Solving systems for complexexponentials is much easier than for sinusoids, and linear systems analysis is particularly easy.

1. Find the phasor representation for each, andre-express each as the real and imaginary parts of a complex exponential. What is the frequency (in Hz)of each? In general, are your answers unique? If so, prove it; if not, find an alternative answer forthe complex exponential representation.
1. $3\sin (24t)$
2. $\sqrt{2}\cos (2\pi \times 60t+\frac{\pi }{4})$
3. $2\cos (t+\frac{\pi }{6})+4\sin (t-\frac{\pi }{3})$
2. Show that for linear systems having real-valued outputs for real inputs, that when the input is thereal part of a complex exponential, the output is the real part of the system's output to the complexexponential (see [link] ). $S(\Re (Ae^{i\times 2\pi ft}))=\Re (S(Ae^{i\times 2\pi ft}))$

For each of the indicated voltages, write it as the real part of a complex exponential( $v(t)=\Re (Ve^{st})$ ).Explicitly indicate the value of the complex amplitude $V$ and the complex frequency $s$ . Represent each complex amplitude as a vector in the $V$ -plane, and indicate the location of the frequencies in the complex $s$ -plane.

1. $v(t)=\cos (5t)$
2. $v(t)=\sin (8t+\frac{\pi }{4})$
3. $v(t)=e^{-t}$
4. $v(t)=e^{-(3t)}\sin (4t+\frac{3\pi }{4})$
5. $v(t)=5e^{(2t)()}\sin (8t+2\pi )$
6. $v(t)=-2$
7. $v(t)=4\sin (2t)+3\cos (2t)$
8. $v(t)=2\cos (100\pi t+\frac{\pi }{6})-\sqrt{3}\sin (100\pi t+\frac{\pi }{2})$

Express each of the following signals as a linear combination of delayed and weighted step functions andramps (the integral of a step).

## Linear, time-invariant systems

When the input to a linear, time-invariant system is the signal $x(t)$ , the output is the signal $y(t)$ ( [link] ).

1. Find and sketch this system's output when the input is the depicted signal .
2. Find and sketch this system's output when the input is a unit step.

## Linear systems

The depicted input $x(t)$ to a linear, time-invariant system yields the output $y(t)$ .

1. What is the system's output to a unit step input $u(t)$ ?
2. What will the output be when the input is the depicted square wave ?

## Communication channel

A particularly interesting communication channel can be modeled as a linear, time-invariant system. When thetransmitted signal $x(t)$ is a pulse, the received signal $r(t)$ is as shown .

1. What will be the received signal when the transmitter sends the pulse sequence ${x}_{1}(t)$ ?
2. What will be the received signal when the transmitter sends the pulse signal ${x}_{2}(t)$ that has half the duration as the original?

## Analog computers

So-called analog computers use circuits to solve mathematical problems, particularly when they involve differential equations. Suppose we are given the following differential equation to solve. $\frac{d y(t)}{d t}}+ay(t)=x(t)$ In this equation, $a$ is a constant.

1. When the input is a unit step ( $x(t)=u(t)$ ), the output is given by $y(t)=(1-e^{-(at)})u(t)$ . What is the total energy expended by the input?
2. Instead of a unit step, suppose the input is a unit pulse (unit-amplitude, unit-duration) delivered to the circuit at time $t=10$ . What is the output voltage in this case?Sketch the waveform.

what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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