Conjugation in Fourier Transform

Question

I have a very simple question.

In Oppenheim book, it says that:

If CT Fourier transform of $x(t)$ is $X(j\omega)$ then, CT Fourier transform of $x^*(t)$ is $X^*(-j\omega)$.

What I can't understand is what is $X^*(-j\omega)$? Or what is $X^*(j\omega)$? Can you tell me what I should understand when I see $X^*(-j\omega)$ or $X^*(j\omega)$?

For example $x(t) = (a + bj)t$ and $X(j\omega) = (c + dj)\omega$ what should be $X^*(-j\omega)$?

Thanks

Answer 1

$X^*(j\omega)$ is the complex conjugate of $X(j\omega)$. So if

$$X(j\omega)=X_R(\omega)+jX_I(\omega)$$

then

$$X^*(j\omega)=X_R(\omega)-jX_I(\omega)$$

and

$$X^*(-j\omega)=X_R(-\omega)-jX_I(-\omega)$$

where $X_R(\omega)$ and $X_I(\omega)$ refer to the real and imaginary parts of $X(j\omega)$, respectively. [Note that they are not the respective Fourier transforms of the real and imaginary parts of the time domain signal.]

Also, the notation $X^*(j\omega)$ is to be understood as $\left[X(j\omega)\right]^*$, i.e., the whole expression for $X(j\omega)$ is conjugated. Similarly for $X^*(-j\omega)$, i.e., $X^*(-j\omega)=\left[X(-j\omega)\right]^*$.

To gain some more understanding, consider the definition of the Fourier transform:

$$X(j\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt$$

So we have

$$X(-j\omega)=\int_{-\infty}^{\infty}x(t)e^{j\omega t}dt$$

and

$$X^*(-j\omega)=\int_{-\infty}^{\infty}x^*(t)e^{-j\omega t}dt=\mathcal{F}\{x^*(t)\}$$

as claimed in the quote of your question.