This paper derives a solution to the equation\rho (u, s) = N_{0}h(s, u) + \int_{0}^{T} \rho (t, s)h(t, u) dt, for the case in which the kernel is the covariance function of a stationary random process whose spectral density is a rational function of frequency.

This paper considers the radar detection of short-duration, pulsed signals shifted in frequency by a target moving at a constant but unknown velocity. The filter operation required is derived by using the likelihood ratio, averaged over the unknown frequency shift. For large signal-to-noise ratios (SNRs) and white Gaussian noise, the performance of the filter is essentially identical to that of maximum likelihood detectors (e.g., periodograms [1]-[3]), and the choice of filter is principally based on convenience. However, for small SNRs, or when cumulative probability of detection is of interest, the average likelihood receiver is preferable. The detection characteristic, given and discussed in Section III, is seen to consist of the sum of pairwise products of the in-phase and out-of-phase components of the individual pulses, added to the sum of the pulse envelopes. For very small integrated SNRs, little improvement over the phase-incoherent (square law) detector is possible, since the velocity estimates have very high variances. For small SNR per pulse but many pulses, the advantage over incoherent detection may be considerable; this remains to be investigated. For large SNR, the discussion of maximum likelihood receivers is germane. The filter has two properties that are convenient for the circuit designer. First, only various simple iterated sums of correlations of each received pulse with a sine wave at the carrier frequency, and with a cosine wave at the carrier frequency, are required. Thus, if m pulses are received,4mnumbers must be formed and stored. Second, the computation of the likelihood ratio may be carried out by a differential procedure. If the likelihood ratio formpulses,L(m), is available,L(m+1)is formed by adding2(m+1)numbers toL(m). Thus, only2m2-step numerical operations, not2^{2m}are needed after each pulse. This property suggests that the test might be carried out sequentially. If sequential analysis is performed, digital rather than analog procedures might prove more convenient, since the required amounts of delay (i.e., of memory duration) cannot be specified in advance.

In this paper we consider the estimation of the delay of a signal of unknown phase obscured by white noise and by another signal of unknown phase and known delay. We also consider the situation where both delays are unknown. Use is made of Shannon's sampling analysis^{1}as developed by Woodward.^{2}We conclude that the relative carrier phase of the two signals must be first estimated in order to maximize the {\em a posteriori} probability of the estimate, but that knowledge of the absolute phase is only of secondary value.

Abstract : Consideration is given to a simple sequential test for the presence of a signal known except for one parameter i. This parameter may assume only one of k discrete values, k < infinity. The distribution of the true parameter value is assumed to be uniform over the parameter space. It is shown that the test is almost a maximum likelihood test. Next it is shown how to set the test parameters in order to achieve desired conditional error probabilities, and an approximate expression is obtained for the distribution of the test termination time. (Author)