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[apcalculus] Digital analog converter (DAC) making a discrete continuous?
Posted:
Sep 17, 2012 9:45 PM


This article caught in the New York Times:
http://tinyurl.com/9z9bmkc
caught my attention because at this time of year I assume you like me are teaching continuity and the classic how can we make (x^2  9)/( x  9) continuous at x = 9? Yeah, I like this problem. You get to review difference of squares, you get to look at its surprising graph, you get to review limits (the left and right are equal), you get to analyze this numerically by looking at the table of values (I definitely am a proponent of the rule of 4) and you get to review the definition of continuity and with a judicious selection of the proper yvalue you get to fill in the "hole" and repair the discontinuity. I like all of that but the back of my mind I kept thinking "so what? aka when are we ever going to use this stuff?"
Does anyone know the inner workings of a digital analog converter? Specifically, I know that "normal" sounds are analog and look like sine waves. To make say a CD or an MP3 these analog waves are "sampled" and made into a discrete series of 1 and 0s which I would consider a discontinuous set of data. To reconstitute the music these 1 and 0s need to be reassembled into a analog sound wave.
So am I correct in assuming from the article:
" Later, your sound system will play a sophisticated game of connectthedots to turn that choppy data back into a wave. That takes math. How authentic the playback sounds depends largely on how well the system turns digital dots back into the original wave.
That dedigitizing is done by a computer chip and software combination called a digitaltoanalog converter (or DAC) ? and here is a little secret of the audio industry ? to keep prices down, manufacturers often scrimp on that part. That means mediocre math."
that the mediocre math the article is referring to is how to accurately fill in the hole as in the classic (x^2  9)/( x  9) example? Any calculus/audiophiles out there that would know whether my supposition is correct? If so I am really excited because it is hard to come up with everyday examples of discontinuous functions other than taxi cabs and making copies and Kinkos :)
PS A fun piece of trivia: most people think a standard CD is solid and opaque. But if you hold a CD up to strong light like sunlight coming through a window you can see through it. Alas, your students are getting to the point where they don't even know what a CD is let alone a 45 (other than a gun) or an LP. Sigh.
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