Spectrometry and Autocorrelation

The spectral power density, at a frequency f, of an electrical signal is simply the average power per unit bandwidth when the signal is passed through a lossless narrow bandpass filter centered at f. In the limit that the bandwidth goes to zero, this can be taken as the definition of the spectral power density. In radio astronomy, however, it is common to use a digital processing method wherein uniformly-spaced samples of the signal are used to estimate an average of the signal's autocorrelation function. This function is then Fourier transformed to produce an estimate of the spectral power density. In the limit that the averaging time goes to infinity, this is an equivalent definition of the spectral power density. The equivalence is reviewed below.