A comparison of average-likelihood and maximum-likelihood ratio tests for detecting radar targets of unknown Doppler frequency

In a coherent search rader, the pulse-to-pulse Doppler shift of a signal is generally not known a priori. Given the distribution of this parameter, the best test variable for detection is the average of the likelihood ratio with respect to target Doppler frequency. Most coherent search radars employ a maximum-likelihood ratio detector, that is, a bank of independent Doppler filters, for detection. The average-likelihood and maximum-likelihood tests are compared here for a target with the Rayleigh amplitude distribution. It is shown that, over a wide range of detection and false-alarm probabilities, the performances of the two tests do not differ significantly. For this target model, the likelihood ratio has the Pareto distribution, which arises in some statistical problems in economics. The new results obtained here for the distribution of the sum of two or more Pareto-distributed variables are of considerable general interest.