Yes I’m godawfully lazy, I love to sit at home on Monday mornings sipping on cocoa, I love going out on a wild romp through bookstores while they conduct mid-terms, I also love to not write stuff in exams that they tell me to write because the “government” is trying to steal my ideas. But that does not mean you give me “assignments” that involve me twisting myself into knots to find out what the aperture effect is! I’ll tell you what the aperture effect is. This effect is characterized by the presence of a hole. And that’s all that the aperture effect is, folks.
(Side-effects include sudden onset of nausea, clinical depression on a monthly cycle, PTSD, proclivity towards irrationality and a constant feeling on insecurity.)
Isn’t it obvious I’m watching Kung-Fu Panda! I couldn’t be bluer if I were the sky behind the clouds in the background of the peach tree when Po is stuffing his face with peaches. I wish I had peaches right now. They would give me inner peace, maybe.
Ok so when you sample a signal you use pulses of some finite width, coz f*** you, Dirac’s delta functions have goddamn irrational width and by irrational I mean infinitesimally small and NOT the square root of pi. Aperture effect and the Dirac delta function should get drunk. Bang, Bang Bangity Bang, I said Bang, Bang Bangity Bang!
Aarrghh Oogway, I can control when the fruit will fall, and I can control where to plant the seed. That is no illusion! And a peach cannot defeat this assignment!
I need your help Master! I have to believe, *deep breathe. Sniff, he’s gone. Master Oogway has moved on. I’m going to take a moment here.
You know the way the panda is running full tilt when he hears Tai-Lung is here? Yea.. I get that feeling sometimes. Especially when they ask me to convolve signals! I mean what?! Aren’t we trying to make life simple? Now you add an asterisk instead of a multiplication sign and suddenly make everything more complicated than Secret Ingredient noodle soup! (I have to believe!) *Deep breathes.
AWWWWESOME! THE POOL OF SACRED TEARS! AWWW MAN BIRTHPLACE OF KUNG-FU!
CONVOLVING ANYTHING WITH AN IMPULSE FUNCTION SIMPLY SHIFTS THE ORIGINAL FUNCTION IN FREQUENCY ! AWWYEA! I FINALLY GET IT!
Oh oh oh and the “sifting” (not shifting, you moronic mudpies) property of the delta function, which is the same I was coarsely abusing above, makes the convolving easy to visualize! Indeed it does! Wow! I’ve learned the secrets of harmony and focus….more or less.
Ok so now, we need to use this newfound knowledge to generate flat-top sampled signals which means that your signals should look like this
And not this
So we already have the Burj and now we need to make it into the Sears. How would an engineer do it? He would shave of the top. Yes, literally. Sometimes I feel if maths hadn’t been discovered/invented, engineers would have become terrorists and Jacks the Rippers. Slash, cut, shave, make bitstreams outta analog waveforms, slit.
So the Burj is described by the function,
So now we take a random pulse train with finite width (hell yea) and convolve it (HELL YEA) with the Burj (HELL THE F*** YEA).
And after a little impulse function magic (the x(delta)(t) thingy turns out to be super-helpful) and some Fourier’s Baby Formula (coz we all know the answer to any problem is baby formula) we get this!
Aww right! So now we get a flat-top sampled signal by using the sample-and-hold operation. You know that one, where you taste the water in a new hotel, keep its in your mouth, swiggle it around like wine trying to determine exactly which poison it might have. So yes, the consequence of the hold operation is that there is a significant attenuation of the higher-frequency spectral replicates. Oh you meant the hotel hold operation, yes, if its poisoned you’ll die.
And now, here’s the what the last 21,697 seconds have been all about. That higher frequency attenuation IS THE APERTURE EFFECT.
WTF!!!! THAT’S IT?! HIGHER FREQUENCIES ARE ATTENUATED?! THAT WAS THE ANSWER THE WHOLE TIME! AND I HAVE TO WRITE AN ENTIRE PAGE ABOUT IT!
SCREW MATHS, I’M GOING TO TURN INTO TAI-LUNG! Atleast he’s cool enough to open locks using feathers.
D.O.O.F. 