<< Chapter < Page Chapter >> Page >
Problems dealing with Fourier Series.

Simple fourier series

Find the complex Fourier series representations of the following signals without explicitly calculating Fourier integrals. What is the signal's period in each case?

  1. s t t
  2. s t t 2
  3. s t t 2 2 t
  4. s t 2 t t
  5. s t 10 t 6 1 2 t
  6. s t given by the depicted waveform .

Fourier series

Find the Fourier series representation for the following periodic signals . For the third signal, find the complex Fourier series for the triangle wave without performing the usual Fourier integrals. Hint: How is this signal related toone for which you already have the series?

Phase distortion

We can learn about phase distortion by returning to circuits and investigate the following circuit .

  1. Find this filter's transfer function.
  2. Find the magnitude and phase of this transfer function. How would you characterize thiscircuit?
  3. Let v in t be a square-wave of period T . What is the Fourier series for the output voltage?
  4. Use Matlab to find the output's waveform for the cases T 0.01 and T 2 . What value of T delineates the two kinds of results you found? The software in fourier2.m might be useful.
  5. Instead of the depicted circuit, the square wave is passed through a system that delays its input, whichapplies a linear phase shift to the signal's spectrum. Let the delay τ be T 4 . Use the transfer function of a delay to compute usingMatlab the Fourier series of the output. Show that the square wave is indeed delayed.

Approximating periodic signals

Often, we want to approximate a reference signal by a somewhat simpler signal. To assess the quality of anapproximation, the most frequently used error measure is the mean-squared error. For a periodic signal s t , ε 2 1 T t 0 T s t s ˜ t 2 where s t is the reference signal and s ˜ t its approximation. One convenient way of findingapproximations for periodic signals is to truncate their Fourier series. s ˜ t k K K c k 2 k T t The point of this problem is to analyze whether thisapproach is the best ( i.e. , always minimizes the mean-squared error).

  1. Find a frequency-domain expression for the approximation error when we use the truncatedFourier series as the approximation.
  2. Instead of truncating the series, let's generalize the nature of the approximation to including any setof 2 K 1 terms: We'll always include the c 0 and the negative indexed term corresponding to c k . What selection of terms minimizes the mean-squarederror? Find an expression for the mean-squared error resulting from your choice.
  3. Find the Fourier series for the depicted signal . Use Matlab to find the truncated approximation and bestapproximation involving two terms. Plot the mean-squared error as a function of K for both approximations.

Long, hot days

The daily temperature is a consequence of several effects, one of them being the sun's heating. If thiswere the dominant effect, then daily temperatures would be proportional to the number of daylight hours. The plot shows that the average daily high temperature does not behave that way.

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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
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
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Pdf generation problem modules. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10514/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Pdf generation problem modules' conversation and receive update notifications?

Ask