What 'exactly' is power spectral density for discrete signal? I was always under the assumption that taking the Fourier transform of the signal, and then the ratio of desired freq range magnitude over entire freq range gives the power ratio for that freq range which is the same as power spectral density. Is that wrong? Reading a student paper got me confused as it says to computes PSD and then 'absolute and relative spectral powers in desired bands' as well. Are they different? If yes, how does one compute it?
Answer
I have no idea what your calculation of power spectral density gives since I cannot understand it.
If a signal $x(t)$ has Fourier transform $X(f)$, its power spectral density is $|X(f)|^2 = S_X(f)$. The absolute spectral power in the band of frequencies from $f_0$ Hz to $f_1$ Hz is the total power in that band of frequencies, that is, the total power delivered at the output of an ideal (unit gain) bandpass filter that passes all frequencies from $f_0$ Hz to $f_1$ Hz and stops everything else. Thus, $$\text{Absolute Spectral Power in Band} = \int_{-f_1}^{-f_0} S_X(f)\,\mathrm df + \int_{f_0}^{f_1} S_X(f)\,\mathrm df.$$ The relative spectral power measures the ratio of the total power in the band (i.e., absolute spectral power) to the total power in the signal. Thus, $$\text{Relative Spectral Power in Band} = \frac{\displaystyle\int_{-f_1}^{-f_0} S_X(f)\,\mathrm df + \int_{f_0}^{f_1} S_X(f)\,\mathrm df}{\displaystyle\int_{-\infty}^{\infty} S_X(f)\,\mathrm df}.$$
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