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)? 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? I already tried just moving ahead accepting these, but eventually something always throws me. 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? -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"
Chuckk Hubbard wrote:
i*sin(a) and i*sin(b) multiply to -sin(a)sin(b)?
let's forget the sin*sin and look at the i*i part only: since i = sqrt(-1) by definition, i*i is -1 (slowly: i*i= sqrt(-1)*sqrt(-1) = sqrt(-1 * -1) = sqrt((-1)^2) = -1)
Thus, i*sin(a) * i*sin(b) = -1 * sin(a)*sin(b)
br, piotr
Hallo, Piotr Majdak hat gesagt: // Piotr Majdak wrote:
Chuckk Hubbard wrote:
i*sin(a) and i*sin(b) multiply to -sin(a)sin(b)?
let's forget the sin*sin and look at the i*i part only: since i = sqrt(-1) by definition, i*i is -1 (slowly: i*i= sqrt(-1)*sqrt(-1) = sqrt(-1 * -1) = sqrt((-1)^2) = -1)
Your equations are a bit, uhm, misleading, because (-1)^2 is 1, so sqrt(-1 * -1) = sqrt(1) = 1
Anyway, i is the unit, which fulfills the equation: i*i = -1
Ciao
Frank Barknecht _ ______footils.org_ __goto10.org__
Chuckk Hubbard wrote:
One more question... regarding how filters work, is there no intuitive way to express it?
For me, the most intuitive way to explain a filter is looking at convolution. Every linear filter can be described as a convolution of a signal with the impulse response of a filter. Although, any pole in a filter leads to a impulse response with infinite length and for the explanation it's easier to look at a finite impulse response.
A great link to explain filtering by convolution is the "joy of convolution": http://www.jhu.edu/~signals/convolve/ There is a java applet where you can play with convolution a little bit...
br,
Piotr
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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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@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"
PD-list@iem.at mailing list UNSUBSCRIBE and account-management ->
-- "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"
Hallo, Chuckk Hubbard hat gesagt: // Chuckk Hubbard wrote:
That's just it, it's convenient.
Convenience is very important in math. For example, in some geometric problems it's more convenient to work with polar coordinates instead of cartesian. A large part of math (and all other sciences, too) consists of finding simpler methods and explanations than you had before (while still staying "correct").
The motivation to introduce complex numbers and that funky imaginary unit i (or j in DSP books) is, well, complex. For a detailed introduction you might want to review somehting like that: http://en.wikibooks.org/wiki/Algebra:Complex_numbers Especially the part with the geometric interpretation and Euler's formula is useful to understand some of the motivation.
Another nice and short introduction I found is this: http://www.physics.umd.edu/courses/Phys374/fall04/files/ComplexNumbers.pdf
It's written for physicists, so it's a bit dumbed down to be more hands-on. ;) (I'm allowed to say this as I studied physics myself.)
Ciao
Frank Barknecht _ ______footils.org_ __goto10.org__
On Tue, 8 Nov 2005, Frank Barknecht wrote:
unit i (or j in DSP books) is, well, complex. For a detailed introduction you might want to review somehting like that: http://en.wikibooks.org/wiki/Algebra:Complex_numbers http://www.physics.umd.edu/courses/Phys374/fall04/files/ComplexNumbers.pdf
There is also an introduction on Julius Smith homepage:
http://ccrma.stanford.edu/~jos/mdft/Complex_Numbers.html
really good stuff there in general.
Guenter
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Chuckk Hubbard -
Frank Barknecht -
günter geiger -
Martin Peach -
Piotr Majdak