Selectivity Concentration and Integration Efficiency

The relationship between the maximum achievable peakedness of the selectivity characteristic and the signal detection performance of a spectrum analyzer whose input is a finite duration sinusoid immersed in white noise is considered. The spectrum analyzer is assumed to be the equivalent of a bank of filters whose outputs are measured at a single point in time. Such systems have application to Doppler radars and speech analysis. In the Doppler radar application, the peakedness of the selectivity characteristic is a measure of frequency estimation performance capabilities of the spectrum analyzer system against single targets. The signal detection performance of a system may be characterized by its integration efficiency, which is a relative measure of the system´s signal-to-noise ratio using the matched filter as a standard. An expression for integration efficiency in terms of filter impulsive response is developed. The selectivity characteristic is defined as the spectrum analyzer normalized output signal power. An expression for the selectivity characteristic is also developed in terms of the filter impulsive response. The mathematical problem considered is the minimization of the even moments of signal concentration for fixed integration efficiencies. The result obtained by variational techniques is the specification of an impulsive response that minimizes the selectivity concentration for a fixed integration efficiency. The utility of this criterion is that it permits a trade off of accuracy of signal frequency estimation performance vs signal detectability performance. The second moment case is treated completely, and a special case of higher order moments is also examined. Graphical results of the selectivity characteristics for some special cases are also presented and compared with the results of the spectral shaping windows of Blackman and Tukey.