# [PD] basic DSP stuff

Frank Barknecht fbar at footils.org
Sat Nov 12 10:03:06 CET 2005

```Hallo,
Chuckk Hubbard hat gesagt: // Chuckk Hubbard wrote:

>  I don't much care, I don't want to define my own math, I just want to know
> why we mix the real and imaginary parts of a product in one specific case.
>  Z1*Z2=r1r2(cos a + isin a)(cosb + isin b)
> = r1r2 (cosa*cosb)-(sina*sinb) + i(sina*cosb + cosa*sinb)
> = r1r2 (cos(a + b) + isin(a + b))

While doing computations, you can consider i to behave just like a
"normal" real number, except if you multiply it with itself, then the
result is i*i=-1. That's all there is to know about i.

Because these are the rules for multiplying complex numbers, with
z=(a,b)=a+ib you have:

z1*z2 = (a1,b1) * (a2,b2)
:= (a1 + i*b1) * (a2 + i*b2)
= a1a2 + a1*i*b2 + i*b1*a2 + i*i*b1*b2
= a1a2 + i(a1b2 + a2b1)     - b1b2
= (a1a2 - b1b2) + i(a1b2 + a2b1)
= a3 + i*b3
= z3

so z3 is the complex number pair (a3,b3) or (a1a2 - b1b2, a1b2 +
a2b1).

Ciao
--
Frank Barknecht                 _ ______footils.org_ __goto10.org__

```