Negative Freq: Conclude Relation Between x(t) and x*(w)

[Niles]

Hi

Say I have a real quantity given by

$$x(t) = \int_{ - \infty }^\infty {\tilde x(\omega )e^{ - i\omega t} d\omega }$$

Now I complex conjugate it (remember it is real)

$$x(t) = \int_{ - \infty }^\infty {\tilde x^* (\omega )e^{ + i\omega t} d\omega }$$

How is it that I from this can conclude that we must have the relation

$${\tilde x^* (\omega )} = {\tilde x(-\omega )}$$
?


Niles.

 

[Niles]

I figured it out. Just make a substitution, and it all becomes obvious.