# my friend the complex number

douglas irving repetto glmrboy
Mon Dec 8 01:03:15 EST 1997

```hi!

let's say i do an fft of some real signal, and one of the complex pairs i
get out (in the form x + yi) is this:

-2.62 + -8.04i

to get the magnitude of the band i do this:

mag = sqrt(x^2 + y^2)
mag = R = sqrt(-2.62^2 + -8.04^2) = 8.4561220426387

great.  to get the phase i do:

phase = atan(y/x)
phase = atan(-8.04/-2.62) = 1.2557771591334

so now i have a magnitude/phase pair.  now if i want to do an ifft on
the pair i have to go back to complex like this:

x = Rcos(phase)
x = 8.4561220426387 * cos(1.2557771591334) = 2.62

whoops, wrong sign!

y = Rsin(phase)
y = 8.4561220426387 * sin(1.2557771591334) = 8.04

whoops again!

i end up with 2.62 + 8.04i - both x and y have changed their sign!
what's going on here?  i'm really really sure that i haven't discovered an error
in FFT theory, so i must have discovered (another) error in my head.

it seems that when i compute the magnitude i will always lose the
signs, since i'm squaring both numbers.  and when i compute the phase, i will
lose the signs if both numbers are negative.  ???  should i not be
getting negative numbers in my complex pairs to begin with?  clearly i'm lost.

thank you!

douglas

```