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

In summary, the conversation focuses on computing the auto correlation and spectral density functions of a given signal. The equations for the auto correlation and spectral density are provided, along with the suggested solution for the spectral function. The person is having trouble evaluating the integrand at the limits and is wondering if they are missing a critical concept or step in their evaluation. It is suggested that the linked image may be using a different form of the function than the one provided.
  • #1
CivilSigma
227
58

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

 
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  • #2
CivilSigma said:

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}.##
 

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

1. What is auto-correlation integral?

Auto-correlation integral is a mathematical tool used to measure the similarity between a signal and a delayed version of itself. It is often used in signal processing and time series analysis to identify patterns and relationships within a dataset.

2. How is auto-correlation integral calculated?

The auto-correlation integral is calculated by taking the integral of the product of a signal and a delayed version of itself over a specified time interval. This is typically done using numerical integration techniques such as the trapezoidal rule or Simpson's rule.

3. What is the significance of auto-correlation integral?

The auto-correlation integral can provide insights into the underlying dynamics of a system. It can reveal periodicities, trends, and other patterns that may not be apparent from a simple visual inspection of the data.

4. How is auto-correlation integral used in scientific research?

Auto-correlation integral is commonly used in fields such as physics, engineering, and economics to analyze time series data. It can be used to study the behavior of complex systems, identify underlying patterns, and make predictions about future behavior.

5. Are there any limitations to using auto-correlation integral?

Like any statistical tool, auto-correlation integral has its limitations. It assumes that the data is stationary and that there is a linear relationship between the signal and its delayed version. It is also sensitive to noise and outliers in the data, which can affect the accuracy of the results.

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