Calculating the power spectrum

[ACLerok]

Homework Statement

Calculate the power spectrum in dBm of a zero offset, 10MHz square wave with amplitude A from DC to 50MHz.

Homework Equations

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The Attempt at a Solution

I was given this problem but am not sure how to go about solving it. Is the power spectrum the same as the power spectral density? I don't need the solution rather some tips and formulas that can be used to solve it. Thanks

 

[antonantal]

When you have a periodic signal you use the Fourier series to approximate the signal as a sum of sinusoidal signals at frequencies of 0 (the DC component), F (the frequency of the periodic signal), and positive integer multiples of F. The coefficients of the Fourier series are the complex amplitudes of this sinusoidals. Knowing the complex amplitudes at each frequency you can calculate the power at each frequency (considering a load of 1 ohm), and this would be the power spectrum. Ofcourse you can't calculate the whole spectrum because it has an infinity of components. But in this problem you are asked to calculate it from DC (0 Hz) to 50MHz.

If you had a non-periodic signal you would have used the Fourier transform to calculate it's spectral density of complex amplitude from which you would have calculated the power spectral density.

So, when you have a periodic signal you calculate it's power spectrum and when you have a non-periodic signal you calculate it's power spectral density.

This is because the spectrum of a periodic signal is discrete as opposed to that of a non-periodic signal which is continuous (and in the case of a continuous spectrum it's not handy to tell the power at each frequency).

 

[ACLerok]

so in calculating the power spectrum, the amplitude A is not relevant? Once i have the Fourier series representation of the square wave, I can just take the 3rd term (for the 3rd harmonic) and use the Power equation to calculate the power generated at that harmonic?

 

[antonantal]

so in calculating the power spectrum, the amplitude A is not relevant? Once i have the Fourier series representation of the square wave, I can just take the 3rd term (for the 3rd harmonic) and use the Power equation to calculate the power generated at that harmonic?

Each term of the Fourier series depends on the amplitude A.

 

[moderate]

I thought I would add to this thread instead of making another one.

My question is this: why do we calculate power spectrums at all?

This makes sense in the context of voltage (power = V²/R), but I've seen this used in numerous other contexts with other units for the signal.

For example, some might calculate the power spectrum for the signal intensity of an MRI image of the brain.

Why?

Thanks in advance for any insight!