On the Filter Problem of the Power-Spectrum Analyzer

The function of a power-spectrum analyzer is to predict from a single finite length of wave record, which has the statistical properties of filtered random noise (e.g. ocean wave, turbulence etc.), the average power per unit bandwidth of an ensemble of such records at various frequencies. The signal derived from repeating the record is heterodyned with another signal whose frequency is scanned at a uniform rate across the entire spectrum and the resultant wave is passed through a narrow filter and then detected by a square-law detector. Two problems arise: 1. Due to its finite bandwidth, the filter performs a necessary weighted average on the power spectrum. What is the best filter response to minimize the intrinsic error associated with the prediction of an average characteristic of an ensemble from a single record? What practical filter is closest to the ideally best? 2. How fast can the frequency be scanned without appreciably deviating the filter response? Definite solutions are given to the above problems. Eqs. (26) and (27) together with Table I give the lowest probable error for filters with various shapes of response curve. Eq. (32) defines the ideal fliter which minimizes this error. The ideal filter can be very closely approximated by cascading a single resonant circuit to a pair of critically coupled resonant circuits with a Q-value √2 times that of the former.