from "Signals and Systems Demystified", 2006, page 142, Example 6-3: (http://www.gatestudymaterial.com/study-material/signals%20and%20systems/text%20books/SIGNALS%20AND%20SYSTEMS%20BY%20DAVID%20MCMOHAN.pdf)
Given signal:
$$x(t)=e^{-at}u(t)$$
has autocorrelation function:
$$R_{xx}(\tau)=e^{-a\tau}/(2a)$$
Find the energy content of the signal:
So I take the Fourier transform of $R_{xx}(\tau)$ to obtain energy spectral density $S_{xx}(\omega)$:
$$F\{R_{xx}(\tau)\}=(1/(2a))\text{ } F\{e^{-a\tau}\}$$ $$S_{xx}(\omega) = (1/(2a))\text{ }(2a/(a+j\omega))$$ $$S_{xx}(\omega) = 1 / (a+j\omega)$$
Now according to the book, equation 6.9, "if x(t) is real valued, then the energy spectral density (S) is:
$$ S_{xx}(\omega) = | X(\omega)|^{2}$$
However, when I look at my Fourier transform I find a complex number:
$$S_{xx}(\omega) = 1 / (a+j\omega)$$
my question is: what is going on? why is it complex when the book says it should be a real magnitude?
(I suppose I could look inside a real "signals and system" textbook and obtain the answer. But, I've already went through half the "Demystified" book to turn around. I'm just curious, what they are doing here.)