As we have seen for both the deterministic and the random cases, a signal composed of a sum of sines shows “peaks” in its spectrum. The object of this chapter is to study methods for estimating their frequencies and their amplitudes when the signal is corrupted by noise.
In this chapter, random processes will be denoted by lowercase letters so as to reserve capital letters for Fourier transforms.
Consider an observation x(n) = s(n;θ) + b(n) where s(n;θ) = α1e2jπf1n is a complex harmonic signal, where b(n) is a white, centered, WSS, complex random signal, and where θ refers to the parameters (α1, f1). We are going to try to estimate the complex parameter α1 and the parameter f1, which belongs to (0, 1), based on a sequence of N noised observations.
In practice, the noise b(n) is used to take into account the measurement errors, but also the possibility that we are not quite sure of the model used for the signal s(n;θ). This occurs when we have a priori information at our disposal on the wanted signal, for example with an active radar, where s(n) represents the signal emitted then sent back by the target. It is also the case with speech when some of the noises originating from the vocal cords are described as a sum of sines.
The least squares method, the general presentation of which is given in Chapter 11, consists of calculating the values ...