finding energy content of signal from "energy spectral density" function

Question

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.)

Answer 1

The so called auto-correlation function that's used in the example (which was found in a previous example) is not correct.

First note that it's the deterministic auto-correlation computation, compared to its theoretical counterpart which requires that the signals be random for their ACF to be defined at all.

Anyway, so the following integral signifies one definition of deterministic ACF of a given deterministic signal $x(t)$ as :

$$R_{xx}(\tau) = \int_{-\infty}^{\infty} x(t)x(t-\tau) dt$$

applying this definition to the signal $x(t) = e^{-a t} u(t)$ with $a >0$ we have:

$$ \begin{align} R_{xx}(\tau) &= \int_{-\infty}^{\infty} x(t)x(t-\tau) dt \\ \\ &= \int_{-\infty}^{\infty} e^{-a t} u(t) e^{-a (t-\tau)} u(t-\tau) dt \\ \\ &= e^{a \tau} \int_{-\infty}^{\infty} e^{-2 a t} u(t) u(t-\tau) dt \\ \\ &= e^{a \tau} \int_{\max\{0,\tau\}}^{\infty} e^{-2 a t} dt \\ \\ &= \begin{cases}{ e^{a\tau} \int_{\tau}^{\infty} e^{-2at} dt ~~~,~~~\tau \geq 0 \\ \\ \\ e^{a\tau} \int_{0}^{\infty} e^{-2at} dt ~~~,~~~\tau < 0 }\end{cases}\\ \\ &= \begin{cases}{ e^{a\tau} \frac{e^{-2a\tau}}{2a} ~~~,~~~\tau \geq 0 \\ \\ \\ e^{a\tau} \frac{1}{2a} dt ~~~,~~~\tau < 0 }\end{cases}\\ \\ &= \begin{cases}{ \frac{e^{-a\tau}}{2a} ~~~,~~~\tau \geq 0 \\ \\ \\ \frac{e^{a\tau}}{2a} dt ~~~,~~~\tau < 0 }\end{cases}\\ \\ R_{xx}(\tau) &= \frac{e^{-a |\tau|}}{2a} \\ \\ \end{align} $$