Introduction

In this chapter, we apply the theory of Chapter 2 to sinusoidal models with fixed frequencies. In Section 3.2, the likelihood function under Gaussian noise assumptions is derived, for both white and coloured noise cases, and the relationships between the resulting maximum likelihood estimators and local maximisers of the periodogram is explored. The problem of estimating the fundamental frequency of a periodic signal in additive noise is also discussed. The asymptotic properties of these estimators are derived in Section 3.3. The results of a number of simulations are then used to judge the accuracy of the asymptotic theory in ‘small samples’.

The exact CRB for the single sinusoid case is computed in Section 3.4 and this is used in Section 3.5 to obtain accurate asymptotic theory for two special cases. In the first case, we assume that there are two sinusoids, with frequencies very close together. In fact, we assume that they are so close together that we expect sidelobe interference, and that the periodogram will not resolve the frequencies accurately. Although the difference between the frequencies is taken to be of the form, where T is the sample size, we show that the maximum likelihood estimators of the two frequencies still have the usual orders of accuracy.