Asymptotic Distributions of the Correlator and Maximum Likelihood Estimators of Nonlinear Signal Parameters

In time delay estimation the correlator or, equivalently, matched filter estimator is widely used. Examples of its usage can be found in the global positioning system (GPS), radars and code division multiple access (CDMA) communication systems. Although widely used its performance is not studied in general case until recently. Partially this study is done in this thesis. If interfering signals like multipath or multiple access signals exist in addition to additive white Gaussian noise, as in GPS and CDMA, the correlator is not a maximum likelihood (ML) estimator. However, it is known that the correlator produces consistent estimates in the existence of multipath interference if the delay separation is larger than the correlation time of the signal (in direct sequence spread spectrum applications such as GPS and CDMA, the correlation time approximately equals the chip duration of the spreading code). It also performs well in the existence of multiple access interference (MAI), if the powers of the MAI signals are equal to the power of the desired signal. In this thesis the asymptotic distribution of the correlator estimator is derived in multisignal environments. Using the result, it can be analytically shown, that in these benign interference cases the exact ML estimator and the correlator estimators perform equally well in the sense that their asymptotic covariance matrices are equal. The thesis also verifies the well known result that if the signals are orthogonal, then the correlator and ML estimators perform equally. In addition, the correlator’s asymptotic performance is investigated also in the inconsistent case by slightly extending the earlier results found in the literature. Also the resolution of the correlator estimator is investigated. It is numerically shown that the correlator estimator can produce consistent estimators even if the delay separation is less that the chip duration, which is commonly believed to be the resolution limit of the correlator. This can happen in fading channels where the multipath amplitudes are uncorrelated or just slightly correlated. This result seems to be fairly unknown. In addition to the classical ML estimator, where all the unknowns are assumed to be deterministic, also an improved ML estimator is investigated. This other ML estimator is obtained by assuming that the amplitudes are Gaussian distributed. It is an improved estimator in the sense that its asymptotic covariance, say CML, is less positive definite than that of the classical ML estimator CCML, i.e., CCML−CML is positive semidefinite. More importantly, this result is valid independent of the fact are the amplitudes really deterministic or Gaussian. This well known result is shown in this thesis to be valid also if the signals contain more than one unknown parameter, which occurs, for example, in direction-of-arrival estimation when two angles per arrival are to be estimated.