For a real signal, the power spectrum will have Hermitian symmetry. If I consider the one-sided spectrum (as negative frequencies, or anything above the Nyquist frequency are redundant) must I then multiply my power spectrum by a factor of 2? I have seen this done but I can't understand why the "redundant", symmetrical portion of the spectrum should factor into the power density? Maybe this is a matter of convention? Please correct my assumptions if necessary.
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$\begingroup$ The power spectrum of a real-valued signal must be a real-valued nonnegative even function of $f$. Yes it has Hermitian symmetry but in a trivial sense since there are no imaginary components to worry about and the real components are even functions of $f$. $\endgroup$– Dilip SarwateCommented Sep 18, 2013 at 3:20
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2 Answers
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This stems from Parseval's theorem. For any signal, $\int_{-\infty}^{+\infty}|x(t)|^{2}\mathrm dt = \int_{-\infty}^{+\infty}|X(f)|^{2}\mathrm df$ and hence for a real signal, the one sided PSD must be multiplied by a factor of 2 because of Symmetry. Intuitively, the Fourier Analysis of any signal provides us with a spectrum which has Negative frequencies too because of the fact that Sines and Cosines can be constructed using Euler's theorem of using Complex Exponentials of positive and negative frequencies; and it so happens for any real signal that all you need are complex exponentials which are conjugates of each other to construct them.
The negative portion is not exactly "redundant". The signal still owns those "negative" frequencies and hence anything present there accounts to its energy.
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I've found this paper on Spectrum and spectral density estimation to be a great resource in understanding some of these subtle points when dealing with the spectrum, windowing, and overlap. Refer to section 9 for a discussion on the scaling.
Long story short, efficient FFT routines don't compute the redundant frequencies, so in order to properly represent the spectrum for the first half you must multiply by a factor of 2 to make up for that energy. Also, depending on if you want the power spectrum or power spectral density there are other scaling factors required.
