How to Compute Auto-correlation and Spectral Density of a Damped Sine Wave?

[CivilSigma]

Homework Statement

I am computing the auto correlation and spectral density functions of the following signal:

$$f(t)=Ae^{-ct}sin(\omega t)$$

$$AutoCorrelation = R_x(\tau) = \int_{-\infty}^{\infty} f(x)f(x+\tau) \cdot \frac{1}{T} dx$$
$$SpectralDensity = S_x(\omega) = \frac{1}{2\pi} \int_{\infty}^{\infty} R_x(\tau)\cdot e^{-i2\pi \omega \tau} d\tau$$

where T is the period of the function, and omega is the natural circular frequency.

My lecture notes suggest that the solution follows the following form:
(Gxx is the spectral function in the picture)

https://imgur.com/a/G11HPRw

The Attempt at a Solution

I have no problem expanding out the integral and simplifying to get the integrand (and verifying with Wolfram Alpha). However, I am having a hard time when it comes to evaluating the integrand at the limits as I am diverging to infinity and moreover, my solution looks no where as close to the suggested one.

Am I missing a critical concept/step in my evaluation?

https://imgur.com/a/G11HPRw

 

[Ray Vickson]

Homework Statement

I am computing the auto correlation and spectral density functions of the following signal:

$$f(t)=Ae^{-ct}sin(\omega t)$$

$$AutoCorrelation = R_x(\tau) = \int_{-\infty}^{\infty} f(x)f(x+\tau) \cdot \frac{1}{T} dx$$
$$SpectralDensity = S_x(\omega) = \frac{1}{2\pi} \int_{\infty}^{\infty} R_x(\tau)\cdot e^{-i2\pi \omega \tau} d\tau$$

where T is the period of the function, and omega is the natural circular frequency.

My lecture notes suggest that the solution follows the following form:
(Gxx is the spectral function in the picture)

https://imgur.com/a/G11HPRw

The Attempt at a Solution

I have no problem expanding out the integral and simplifying to get the integrand (and verifying with Wolfram Alpha). However, I am having a hard time when it comes to evaluating the integrand at the limits as I am diverging to infinity and moreover, my solution looks no where as close to the suggested one.

Am I missing a critical concept/step in my evaluation?

https://imgur.com/a/G11HPRw

While the linked image is almost unreadable, it looks like it is using $f(t) = A e^{-c|t|} \sin(\omega t)$; that is, it uses $e^{-c|t|},$ not your $e^{-ct}.$