Detection of coherent radar returns of unknown Doppler shift

Abstract
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.

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