- #1

CivilSigma

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## 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