The detection reliability of a single-bit sampling cross- correlator for detecting random Gaussian reflections
Howard, John Edward
In this thesis the detection reliability of a single-bit, digital, sampling cross-correlator used for detecting either single-hit or analog bandlimited Gaussian signals is investigated. This is done by deriving the exact output probability mass function of the cross-correlator, this directly yields the Detection and False Alarm probabilities. The cross-correlator output mass function is derived for the following cases: (a) a single-bit bandlimited Gaussian signal cross-correlated with an attenuated reflection corrupted by wideband Gaussian noise, and (b) and in many applications the interfering signal can be periodic in nature; thus the output mass function is also considered. (c) for a single-bit bandlimited Gaussian signal cross-correlated with no attenuated reflection corrupted by a random phase sinusoid. In all cases except (d), the cross-correlated function is derived first, and then the probability mass functions are derived for both burst and continuous transmitted signal operation. In (d) the cross-correlation function cannot he derived in a closed form, and a series approximation is given. However, the zero delay (i.e. peak) cross-correlation function is derived exactly, and this yields information on the detection probabilities to be expected. The cross-correlator output probability mass functions are discussed qualitatively in this case. It is found that in general the detection reliability obtained using single-bit bandlimited Gaussian signals is higher than that achievable with analog signals, and that a random phase sine wave has a more adverse effect on the cross-correlator's detection performance than wideband Gaussian noire has. The theoretical derivations of (a), (b) and (c) are verified by extremely close agreement with experimental results taken on a specially built single-bit, sampling cross-correlator. The cross-correlator's performance under multiple reflection condit¬ions is considered, and the cross-correlation function of a single-bit or an analog bandlimited Gaussian signal with two attenuated reflections corrupted by wideband Gaussian noise is derived. An extension of the theory to more than two reflections is discussed in both cases. The derivation of the cross-correlator output mass functions is considered for both burst and continuous signal operation. It is shown that under conditions where there are two overlapping single-bit reflection, in a low extraneous noise environment, there is a high probability of missing the smaller of the two reflections completely, even though it may be only slighter smaller than the larger one. This defect does not occur with analog Gaussian signals, and, although the peaks in their case are not so sharp or well-defined, under these conditions analog signals offer distinct advantage over single-bit signals. The practical application of the detection scheme to acoustics is briefly discussed, and it is found that the Gaussian signal centre freq¬uency and the cross-correlator sampling frequency must he matched. A sampling frequency of between one and ten times the signal centre frequency yields satisfactory results. There are several constraints on the signal bandwidth, and octave bandwidth, are found to offer a good compromise.
A Thesis Submitted to th e Faculty of Engineering , University of th e Witwatersrand, Johannesburg , for the Degree of Doctor of Philosophy Johannesburg 1974