Hey /sci/, got a n00b math/signals question:
If I have a simple signal sin(x), I can convert a sample taken at some point (x,y) on that signal to a complex number by using Euler's formula. e.g. the point (pi/2, 1) on the sine curve becomes complex number 0 + 1i on the unit circle in the complex plane.
What if the signal is not a simple sin(x), but is some arbitrary non-periodic signal? How do you convert an (x,y) sample into a complex number?
Pretty sure about this, but not completely. With your notation, a point (a, b) would become
b * e ^ a*i
= b (cos(a) + i * sin(a))
so in your example,
(pi/2, 1) => 1 * e ^ (pi * i / 2)
= cos(pi/2) + i * sin(pi/2)
= 0 + 1i
for a general function f, its representation at a point x is given from applying the above formula to (x, f(x)).
(x, f(x))->f(x) * e ^ x * i
= f(x)(cos(x) + i * sin(x))
That's the part that was making me not completely sure. It's a system consistent with the example you gave, but I'm not sure it's correct.
I'm thinking that f(x) will be the magnitude of the complex vector and x will be the vector's angle.
everything i read in signal processing books is talking about sin(2*pi*f*t + phi(t) ) idealized signals. i am trying to understand how samples of real signals (e.g. radio) are converted into complex numbers.
The Cartesian product of the open interval with lower bound zero and upper bound 2*pi and the real numbers, or the set of all pairs with a first element in ]0,2*pi[ and second element in the real numbers, R.
I should actually replace R with R+, the set of all strictly positive real numbers. Sorry.
Basically, for every x and y such that 0<x<2*pi and y>0, there is a unique value associated to (x,y) in the complex numbers. Namely, y*(cos(x) + i * sin(x)).
This is becoming more related to analysis or set theory though, so it may not be so useful.