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.