# Fourier series and gibbs phenomenon  (Page 2/2)

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## That funny phase

You probably noticed in the last problem that even though the wave forms looked fairly different, the sound was similar. Let's look into this a bit deeper with a simpler sound.

• Create a script file called phasefun.m to put your code in for this problem.
• Pick two harmonic frequencies and generate a signal from two cosines at these frequencies added together and call it sig1 . Use Fs = 8000 (remember that you can reproduce only frequencies that are less than Fs/2 ).
• Now generate a second signal called sig2 exactly the same as the first one, except time delay the second cosine by a half cycle (half of its period).
• Use subplot to show a few periods of both signals, do they look different? Save the plot as phasesigs.tif . What did the time delay do to the phase?
• Play each signal with soundsc , do they sound different?
• Redo sig2 with a few different delays and compare the sound to the first signal.
• Create a sig3 that is one cosine at some frequency. Now add sig3 with a timed delayed version of itself and call it sig4 . Use a quarter cycle delay.
• Use subplot and plot a few periods of sig3 and sig4 . Play them with soundsc , do they sound different to you?
• What is suggested about our hearing capabilities from this experiment?
• You will need to show the TA the following files: phasefun.m phasesigs.tif

## Truncated fourier series

In this section, we'll reconstruct the periodic function x(t) , shown in Figure 1, by synthesizing a periodic signal from a variable number of Fourier Series coefficients, and observe similarities and differences in the synthesized signal.

## Gibbs phenomena

• Create a script file called gibbs.m to put your code in for this problem.
• Click here to download the MATLAB function Ck.m . Take a look at the contents of the function. This function takes one argument $k$ , and creates the $k$ th Fourier series coefficient for the squarewave above:
${C}_{k}\left(1\right)=\frac{-1}{j\pi }=0+j0.3183$ . Plot the magnitude and phase of thecoefficients ${C}_{k}$ for $k\in \left\{-10,-9,\dots ,9,10\right\}$ . The magnitude and phase should be plotted separately using the subplot command, with the magnitude plotted in the top half of the window and the phase in the bottom half. Place frequency $w$ on the x axis. Use the MATLAB command stem instead of plot to emphasize that the coefficients are a function of integer-valued (not continuous) $k$ . Label your plots.
• Save the graph as Coeff.tif .
• Write whatever script/function files you need to implement the calculation of the signal $x(t)$ with a truncated Fourier series:
$x(t)=\sum_{k=-{K}_{\mathrm{max}}}^{{K}_{\mathrm{max}}} {C}_{k}e^{(\mathrm{jk}{\omega }_{0}t)}=\sum_{k=0}^{{K}_{\mathrm{max}}} 2\left|{C}_{k}\right|cos(k{\omega }_{0}t+{\angle C}_{k})$
for a given ${K}_{max}$
You can avoid numerical problems and ensure a real answer if you use the cosine form. For this example, ${w}_{0}=1$ .
• Produce plots of $x(t)$ for $t\in \left[-5 , 5\right]$ for each of the following cases: ${K}_{max}$ = 5; 15; and 30. For all the plots, use as your time values the MATLAB vector t=-5:.01:5 . Stack the three plots in a single figure using the subplot command and include your name in the title of the figure. Save the figure as FourTrunc.tif
• Add clear comments describing what the files do. You will need to show the TA the following files: gibbs.m Coeff.tifFourTrunc.tif

As you add more cosines you'll note that you do get closer to the square wave (in terms of squared error), but that at the edges there is some undershoot and overshoot that becomes shorter in time, but the magnitude of the undershoot and overshoot stay large. This persistent undershoot and overshoot at edges is called Gibbs Phenomenon.

In general, this kind of "ringing" occurs at discontinuities if you try to synthesize a sharp edge out of too few low frequencies. Or, if you start with a real signal and filter out its higher frequencies, it is "as if" you had synthesized the signal from low frequencies. Thus, low-pass filtering (a filter that only passes low-frequencies) will also cause this kind of ringing.

For example, when compressing an audio signal, higher frequencies are usually removed (that is, the audio signal is low-pass filtered). Then, if there is an impulse edge or "attack" in the music, ringing will occur. However, the ringing (called "pre-echo" in audio) can be heard only before the attack, because the attack masks the ringing that comes after it (this masking effect happens in your head). High-quality MP3 systems put a lot of effort into detecting attacks and processing the signals to avoid pre-echo.

## What to show the ta

Show the TA ALL m-files that you created or edited and the files below. gibbs.m Coeff.tifFourTrunc.tif sigsynth.maddcosines.m synthwaves.tifphasefun.m phasesigs.tifany wav files created

An applet here provides a great interface for listening to sinusoids and their harmonics. There are some well-known auditory illusions associated with the perception of pitch here .

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
Do somebody tell me a best nano engineering book for beginners?
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
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
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
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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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