<< Chapter < Page Chapter >> Page >

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.

Periodic 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 = { 0 if  k = 0 ,   k  even 1 j k π [ cos ( 2 k π 3 ) cos ( k π 3 ) ] if  k  odd MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@64FA@
    C k ( 1 ) = 1 j π = 0 + j 0.3183 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaWgaaWcbaGaam4AaaqabaGccaGGOaGaaGymaiaacMcacqGH9aqpdaWcbaWcbaGaeyOeI0IaaGymaaqaaiaadQgacqaHapaCaaGccqGH9aqpcaaIWaGaey4kaSIaamOAaiaaicdacaGGUaGaaG4maiaaigdacaaI4aGaaG4maaaa@4758@ . Plot the magnitude and phase of thecoefficients C k for k { 10 , 9 , , 9 , 10 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgacqGHiiIZdaGadaqaaiabgkHiTiaaigdacaaIWaGaaiilaiabgkHiTiaaiMdacaGGSaGaeSOjGSKaaiilaiaaiMdacaGGSaGaaGymaiaaicdaaiaawUhacaGL9baaaaa@44B3@ . 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 k K max K max C k e jk ω 0 t k 0 K max 2 C k cos k ω 0 t 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 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEhadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIXaaaaa@398F@ .
  • Produce plots of x t for t 5 5 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 .

Questions & Answers

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
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
characteristics of micro business
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 ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
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
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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
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, Continuous time linear systems laboratory (ee 235). OpenStax CNX. Sep 28, 2007 Download for free at http://cnx.org/content/col10374/1.8
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Continuous time linear systems laboratory (ee 235)' conversation and receive update notifications?