Hey /sci/, got a n00b math/signals question:

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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?

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>>7638652

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)

= i

= 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))

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>>7638665

thx. what's the reasoning for multiplying by b (or by f(x) in the last example)?

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>>7638685

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.

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>>7638697

yes. because i used sin(x) and that maps to a unit circle. but if an arbitrary function does not map to a unit circle, should the magnitude be sqrt(cos(a)^2+sin(a)^2) ?

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>>7638713

That always yields 1, so probably not.

When you take the magnitude to be f(x), you gain a bijection between ]0,2*pi[XR and the complex numbers.

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>>7638863

> ]0,2*pi[XR

can you please explain what this means

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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.

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>>7638898

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.

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>>7638910

i see

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