Ensemble-averaged randomly scaled radar or sonar pulses are considered and the statistical theory for the corresponding phase and amplitude modulations are developed. Random scaling expresses varying radar cross-sections for scattering objects or varying antenna gain of a sweeping emitter. The noise variance of the modulations depends on the distribution function of the scaling, and how to minimise the variance by rejecting pulses below a certain amplitude threshold is shown. The theory is asymptotic in the sense that it is more accurate for increasing signal-to-noise ratios (SNRs). In a test case with uniformly distributed scaling, sufficient accuracy is reached for an average SNR larger than ~5 for the phase average and ~15 for the amplitude average
[Show abstract][Hide abstract] ABSTRACT: As the test case in Sect. 4.2 shows, it was justified to apply the asymptotic estimates in Sect. 3 for both phase and amplitude averaging for sufficient levels of SNR ranging from roughly 10. Although this SNR is reasonable for many practical purposes, the instantaneous signal to noise ratio varies throughout the radar/sonar pulse with the instantaneous amplitude. Parts of the rising and falling flanks of the pulses will then correspond to short time intervals in which the theory should not be applied. We adopted a smooth scaling distribution p(a) in our analysis. In a practical situation, only the scaling histogram is available. The normalised histogram approximates p(a;a0) and the optimal scaling threshold can be obtained by the discrete analog to eq. (23). On the other hand, the optimum scaling threshold can of course be computed by brute force, i.e. by straightforward estimation of the variance based on available pulse signals and rejecting those pulses that contribute to a degraded ensemble average. One interesting possible future investigation is to evaluate the brute force and theoretically driven approaches in practical situations and compare them in terms of efficiency and reliability. In Sect. 2.3 we defined and obtained a mathematical expression for the mean square error (MSE) of the ensemble average of a quantized, noisy signal. The MSE is a signalindependent measure of the average signal variance. When the signals over which we average are randomly scaled, there is no obvious way of defining the MSE. One way of circumventing this problem is to, as we did in Sect. 4.2 above, calculate variances of a large number of randomly selected points and then taking the average in order to achieve variances that are roughly signal-independent (Fig. 8). In the future, more sophisticated definitions of average variance that account for random scaling as well as quantization and stochastic noise should be developed. Direct averaging with subsequent amplitude and phase calculation (Method I) provides the same results as Method II in the large SNR limit. Method I is potentially a more efficient averaging method, since amplitude and phase need not be computed for each pulse. However, signal degradation is more sensitive to alignment errors of the pulses before averaging; the sensitivity to precise alignment increases for increased carrier frequency due to larger phase errors for the same time lag error. This problem is much reduced when one performs averaging on amplitude and phase modulations directly (Method II).
Advances in Sonar Technology, 02/2009; , ISBN: 978-3-902613-48-6
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