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"