Please provide any amount of assistance on this question: R_{XX}(t, t+tau) = e^{-a|tau|}, a>0, a constant

[PA3040D]

Dear Experts

I do not have a good idea on how to tackle this question. My theoretical knowledge in this area is also very low. I would highly appreciate any amount of assistance

 

[mmm4444bot]

I do not have a good idea on how to tackle this question. My theoretical knowledge in this area is also very low. I would highly appreciate any amount of assistance.

Have you already learned the meanings of the words and phrases? Were I to look into it, I would need to do that first. The meanings of α and C must be relevant.

Please clarify the definition for R_xx(t, t+τ). The exponent in the image is not clear.

 

[mario99]

Dear Experts

I do not have a good idea on how to tackle this question. My theoretical knowledge in this area is also very low. I would highly appreciate any amount of assistance



View attachment 38055

The Power Spectral Density function, [imath]S_{X}(f)[/imath] is the Fourier transform of the Autocorrelation function, [imath]R_{X}(\tau)[/imath].

 

[PA3040D]

Have you already learned the meanings of the words and phrases? Were I to look into it, I would need to do that first. The meanings of α and C must be relevant.

Please clarify the definition for R_xx(t, t+τ). The exponent in the image is not clear.

I would highly appreciate your response. Actually, I haven't looked into this area yet. It would be greatly appreciated if you could provide suitable YouTube video links for reference. Could you please share the keywords to search?
Thanks in advance.

 

[mario99]

It does not matter you will call it [imath]R_{X}(\tau) \ \ \ [/imath] or [imath]\ \ \ R_{XX}(t,t + \tau)[/imath].

You have:

[imath]\displaystyle R_{X}(\tau) = e^{-a|\tau|}[/imath]

[imath]\displaystyle S_{X}(f) = \mathcal{F}\{R_{X}(\tau)\} = \int_{-\infty}^{\infty} R_{X}(\tau) \ e^{-i2\pi f \tau} \ d\tau[/imath]

The solution is straightforward.

 

[PA3040D]

It does not matter you will call it [imath]R_{X}(\tau) \ \ \ [/imath] or [imath]\ \ \ R_{XX}(t,t + \tau)[/imath].

You have:

[imath]\displaystyle R_{X}(\tau) = e^{-a|\tau|}[/imath]

[imath]\displaystyle S_{X}(f) = \mathcal{F}\{R_{X}(\tau)\} = \int_{-\infty}^{\infty} R_{X}(\tau) \ e^{-i2\pi f \tau} \ d\tau[/imath]

The solution is straightforward.

This will be the good help to get he idea
Great Thanks

 

[PA3040D]

Have you already learned the meanings of the words and phrases? Were I to look into it, I would need to do that first. The meanings of α and C must be relevant.

Please clarify the definition for R_xx(t, t+τ). The exponent in the image is not clear.

Sorry I misunderstood your question

 

[PA3040D]

It does not matter you will call it [imath]R_{X}(\tau) \ \ \ [/imath] or [imath]\ \ \ R_{XX}(t,t + \tau)[/imath].

You have:

[imath]\displaystyle R_{X}(\tau) = e^{-a|\tau|}[/imath]

[imath]\displaystyle S_{X}(f) = \mathcal{F}\{R_{X}(\tau)\} = \int_{-\infty}^{\infty} R_{X}(\tau) \ e^{-i2\pi f \tau} \ d\tau[/imath]

The solution is straightforward.

Please advise from this onwards

 

[PA3040D]

Hoped this will be correct
e^0 = 1 here

 

[mario99]

\(\displaystyle \int_{-\infty}^{\infty} e^{-a|\tau|} = \int_{-\infty}^{0} e^{a\tau} + \int_{0}^{\infty} e^{-a\tau}\)

Can you see your mistake?

 

[PA3040D]

Thanks you for the reply and I got the mistake

But how −τ become τ

is it when τ < 0 right

-a−τ = at
due to limits ....Am I correct ?

Hoped your advise

 

[mario99]

[imath]|\tau| = \begin{cases}\ \ \ \tau, \ & \tau \geq 0\\-\tau, \ & \tau < 0\end{cases} [/imath]

 

[PA3040D]

Dear experts, could you please assist in completing the remaining tasks to find the autocorrelation function

 

[mario99]

First, the autocorrelation function was already given. Second, the problem had only one task and it has been solved.