[music-dsp] RE: Semi OT: How Much Math in Practical DSP?
William DeMeo
williamdemeo at yahoo.com
Sun Sep 5 18:13:54 EDT 2004
Hello,
I also thought your question was a good one, so I'll share my point of view, for better of for
worse...
In my humble opinion, in many fields of research -- digital audio in particular -- the more math
you have, the deeper your understanding of and analytical capacity for the subject. Often it seems
that the more mathematically inclined researchers are more likely to make novel, significant
contributions to the field (though this is probably a misconception, resulting from bias).
I've found that the time spent learning abstract math has paid greater (though perhaps less
tangible, obvious, and immediate) dividends than, say, learning how to write a software synth, or
a Linux driver. Obviously, the latter has more obvious and practical benefit, but it might be a
short, quickly forgotten chapter of your life. The math, on the other hand, will stick with you
forever, whether you realize it or not (and whether you like it or not!)
Specifically regarding differential equations: I don't mean to imply that I have a vast and
comprehensive knowledge of the digital audio literature, but it seems to me that diff eqs come up
only rarely. Far more important for analysis and synthesis, Fourier/Gabor transforms, etc. are
basic vector space methods. A good second semester linear algebra course -- something along the
lines of "Linear Algebra" by Peter Lax -- is excellent preparation.
you wrote:
>> However, I've heard that truely understanding say FFT
>> would require years of study and I'm probably not
>> into math that much.
If you understand the Fourier transform as a linear combination of projections, then you
understand the FT. If you can see the exponential basis as a set (or, better yet, a group) of
periodic functions, and you know modulo arithmetic, then you are capable of "truly understanding"
the FFT. Probably it's not a matter of years... though, it does depend on your taste/distaste
for math.
you wrote:
>> However, I'm not usually into totally abstract math that
>> doesn't have any practical value in real life or on the computer...
I think you would be surprised to find that seemingly useless abstract math can be extremely
useful for digital audio. Some anecdotal evidence: after reading somewhat abstract (yet applied)
books like [1] and [2], I finally understand what it is about the exponential functions that
allows us to make the FT into the FFT. The exponentials are what abstract algebraists call
"characters" -- homomorphisms with absolute value 1. Knowing this, you see that *any* character
basis decomposition -- not just the Fourier transform -- can be turned into a fast transform (a la
Cooley-Tukey). To me, such deeper insights resulting, from abstract formulations, are well worth
the extra effort. But then, I suppose the amount of extra effort required is a function of your
taste/distaste for math.
For whatever it's worth, you have my vote: do more math! :-)
-chipd
[1] Tolimieri and An, "Time-frequency Representations," Birkhauser.
[2] An and Tolimieri, "Group Filters and Image Processing," Psypher Press.
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