Spectral signal analysis is the field concerned with the estimation of the
distribution of a signal (more exactly its power) in the frequency domain.
The Power Spectrum Density (PSD) S_X of X_n is defined as the squared discrete
time Fourier transform of X_n
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Under suitable technical conditions, this is equal to the discrete time Fourier
transform of the autocorrelation function for stationary signals in the wide
sense:
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This is called the power spectrum density because integrating it over the
frequency domain gives you the average power of X and because it can be
proved that S_X is always positive for any f.
Useful complements
A random signal
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is said to be (strictly) stationary if its
distribution does not depend on the time. For time-discrete signals
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of distribution
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, this is written:
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Other stationary concepts can be defined, when only some moments of the
signal do not depend on time. A widely used concept is weakly stationary
signals. A signal is said to be weakly stationary (or stationary in the wide
sense) if its means and covariance do not depend on time, and its covariance
function
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depends only on the time difference n-k:
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