[PD] basic DSP stuff
Chuckk Hubbard
badmuthahubbard at gmail.com
Tue Nov 8 01:45:18 CET 2005
That's just it, it's convenient. Suddenly you have the rule for cosine of a
sum. But you get it by arbitrarily inserting a convention that, when you
multiply the x values for two complex numbers, they turn negative. I mean,
maybe it's not arbitrary, but I don't understand how it isn't. If a complex
number is just a way to express two numbers without relating them to each
other, then the decision that they should relate when multiplied seems kind
of weird, giving cos(a)cos(b) - sin(a)sin(b). Seems like cos(a)cos(b) +
i*sin(a)sin(b) would be more accurate.
I think sine and cosine are a little more real than i. Maybe you don't
actually have a triangle, but it's not like saying 2+2=5. Or i = 1/0.
As regards e, wouldn't it make as much sensel to use a and b as non-mixing
terms, but each equivalent to units? Instead of e^4+9i, have e^4a+9b?
I can accept that this i*i stuff works- I'm not saying I can't follow that-
but I'm not ready to stop asking about it.
-Chuckk
On 11/7/05, Martin Peach <martinrp at vax2.concordia.ca> wrote:
>
> Chuckk Hubbard wrote:
>
> > I understand complex numbers used to represent rotation, so as to
> > encode frequency in an easy to manipulate form, and I understand that
> > the imaginary part can be disposed of when converting back to real
> > signals and nothing is changed, but there's one bit that is hanging me
> > up:
> >
> > Using i is just a convention, a way to keep from mixing the two
> > numbers. The square root of -1 is not really involved in any of it,
> > because it doesn't exist. So why, when you multiply Z1 and Z2, do
> > i*sin(a) and i*sin(b) multiply to -sin(a)sin(b)?
> >
>
> The square root of minus one is a number that, when multiplied by
> itself, gives minus one:
> i*i = -1
> so i*sin(a)*i*sin(b) = -1*sin(a)sin(b)
> The terms containing just one i cancel out: that's very convenient. You
> never have to deal with 'naked' i so you don't have to worry if it
> exists or not.
> If you think of a wave that is momentarily passing through zero with
> momentarily no acceleration and then ask yourself 'where is the energy
> of the wave?' you may see why i is useful: the energy is all in the
> imaginary dimension for that one instant. Otherwise the law of
> conservation of energy would be violated.
>
> >
> > Also, the decision (this is coming from Miller's "Theory and
> > Techniques") to multiply the complex constant A by the unit-value
> > complex number Zn; this is convenient, but it seems rigged. Instead
> > of letting Z have whatever amplitude it really has, in which case
> > multiplying it (which already seems forced) by itself changes its
> > amplitude, you force it to be 1 and add on the amplitude later.
> > Nothing violated, but it seems artificial, like we fudge the numbers
> > to make it come out right. If it is artificial, and just a way of
> > simplifying sinusoid manipulation, then why even use
> > artificial mathematical operations to explain it?
>
> Well actually sin and cos are hacks as well, being simply the result of
> drawing giant circles and measuring the lengths of perpendiculars to the
> radius at many points. It was later discovered that the exponential
> function exp() is more natural and that sin and cos can be expressed in
> terms of e if the number i is introduced. This is a more sophisticated
> hack in that the numbers produced by powers of e can be discovered to
> any desired precision by doing a long series of multiplications without
> having to 'exit' mathematics to measure the length of a line.
>
> >
> > I already tried just moving ahead accepting these, but eventually
> > something always throws me.
>
> Wolfram in his "A New Kind of Science" claims that there are any number
> of mathematical systems possible and humans have just chosen the ones
> that work for them in this universe. Because really the universe is not
> understood by humans, they are just good at manipulating the symbols
> they use to model it, and these symbols arose from empirical interaction
> with the universe: they work. And I'm sure Nietszche would agree, even
> if we did understand it, who would we tell?
>
> >
> >
> > One more question... regarding how filters work, is there no intuitive
> > way to express it? No shortcut so that rpole~ and rzero~ will at
> > least make sense in theory before I push through all the math?
>
> Roughly that poles are resonant and zeros are damping at a particular
> frequency.
>
> >
> > -Chuckk
> >
> >
> > --
> > "It is not when truth is dirty, but when it is shallow, that the lover
> > of knowledge is reluctant to step into its waters."
> > -Friedrich Nietzsche, "Thus Spoke Zarathustra"
> >
> >------------------------------------------------------------------------
> >
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>
--
"It is not when truth is dirty, but when it is shallow, that the lover of
knowledge is reluctant to step into its waters."
-Friedrich Nietzsche, "Thus Spoke Zarathustra"
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