Physics/Astronomy power spectrum from waveforms

[fasterthanjoao]

phi(t)=A_o*e^(2*pi*i*mu_0*t) (label:1)

Calculate the power spectra of the following waveforms (using 1):

1) for all t
2) a pulse duration of two tau for |t|<tau, and for phi(t)=0

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3)an exponentially decaying sinusoid:
phi(t)=A_0*(e^(-t/(2*tau)))*e^(2*pi*i*mu_0*t) for t>0
and phi(t)=0 for t<0.



note: A_0, mu_0 and tau are constants.phi(t) should be the signal in time domain. the question also specifies to sketch the real part of the waveform in the time domain and the power spectrum in the frequency domain.

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For the first part of the question, I think I'm right in saying that the power in this wave is proportional to the amplitude squared, so |A_0|^2, and is concentrated at mu_0, and I'm working this out basically from what I know of sinusoids...

For the second part I'm not really sure what its asking, but i think i need to integrate, since the actual power spectrum is over all freqencies (so from minus to plus infinity), the signal in time domain (phi(t)) but bleh, I'm not really sure where I'm going after that.

(for that integral i get to (after simplifying)

A(mu)= int\phi(t)*e^(-2*pi*i*mu*t) dt (between minus/plus infinity)

)

sorry for the lack of latex, I'm a little rusty and it'd take me a while to work it out.. i'll try and get it but i'd like the post up asap.

thanks.

 

[Claude Bile]

The question looks like it's asking you to perform Fourier transforms on some function of time f(t), to obtain a spectral function which is amplitude as a function of frequency F(w). So to do this question, you need to know what the Fourier transform of each of these waveforms are.

The power spectrum is just the modulus squared of F(w).

Also, mu_0 is a standard constant, namely the permeability of free space. I think you mean nu_0.

Claude.

 

[quicknote]

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Hello! Thank you for your question. I would like to provide a response to your query regarding the power spectrum of a waveform in physics and astronomy.

Firstly, the power spectrum of a waveform is a representation of the power or energy contained in the waveform at different frequencies. It is calculated by taking the Fourier transform of the waveform in the time domain.

Now, for the given waveform phi(t)=A_o*e^(2*pi*i*mu_0*t), the power spectrum would be concentrated at the frequency mu_0, with a power proportional to the square of the amplitude A_o. This is because the waveform is a sinusoid with a frequency of mu_0 and the power in a sinusoidal waveform is proportional to the square of its amplitude.

For the second part of the question, where the waveform is a pulse with a duration of 2 tau, the power spectrum would still be concentrated at the frequency mu_0, but the power would be spread out over a wider range of frequencies due to the shorter duration of the pulse. The power spectrum would also have a peak at mu_0 since that is the dominant frequency in the waveform.

Finally, for the third part of the question, where the waveform is an exponentially decaying sinusoid, the power spectrum would have a peak at the frequency mu_0 and would also show a gradual decrease in power as the frequency increases. This is because the exponential decay in the waveform causes the amplitude to decrease at higher frequencies.

To sketch the real part of the waveform in the time domain, you would plot the amplitude A_o as a function of time t. To sketch the power spectrum in the frequency domain, you would plot the power as a function of frequency mu. The power spectrum would have a peak at mu_0, and the shape of the graph would depend on the specific waveform.

I hope this response has been helpful in understanding the power spectrum of waveforms in physics and astronomy. Please let me know if you have any further questions or clarifications. Thank you.