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I am currently working on analyzing a signal with the autocorrelation function given by:

$$R(\tau) = 100 \cos( 10000\pi\tau) (Λ(2000\tau))^2$$

I need assistance in plotting the power spectral density $G_f​(\omega)$ of this signal. Additionally, I am interested in calculating the total power of the signal and specifically determining its power within the frequency range of $10\texttt{kHz}\leq |\omega|≤12\texttt{kHz}$.

Could someone please provide guidance on how to approach this problem or share some code snippets in MATLAB/Python that could help me achieve these objectives?

Thank you in advance for your support!

I have attempted to implement the calculation of the power spectral density and total power using the given autocorrelation function in MATLAB/Python. Specifically, I've looked into functions like fft, pwelch, and related signal processing tools.

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  • $\begingroup$ Can you make $Λ$ clearer? what function is that? I'm assuming it's the triangular function ? $\endgroup$
    – Jdip
    Nov 29, 2023 at 23:08
  • $\begingroup$ @Jdip yeap, it is triangular function $\endgroup$ Nov 30, 2023 at 10:36

1 Answer 1

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By the Wiener–Khinchin theorem, the power spectral density (of a power signal) is equal to the Fourier transform of the autocorrelation function. Therefore, one way to proceed is to analytically determine the continuous-time Fourier transform of $R(\tau)$, and then use MATLAB or some other tool to plot the result. Doing that will require you to find the Fourier transform of $\Lambda(2000 \tau)$ in a table of transforms, and then apply the multiplication and modulation properties to find $G_f(\omega)$. This in turn will require that you perform frequency-domain convolution of a sinc function with itself, and that is very challenging.

Another option is to compute the Fourier transform numerically, using the fft function in MATLAB. To do that, you would need to define a vector containing samples of $R(\tau)$, apply the fft to find the discrete Fourier transform, and then scale the result by the time between samples so that it is calibrated to correspond to the continuous-time Fourier transform.

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