2.2.2. ESPRIT

This parametric algorithm (estimation of signal parameter via rotational invariance technique), introduced in [22–24], exploits the rotational invariance property, which is valid for the signal eigenvectors (**x**) of the sample sequence covariance matrix. Similar to the MUSIC algorithm, ESPRIT needs an estimation of the signal covariance matrix. Thanks to the knowledge of the number of components, the eigenvectors corresponding to signal components can be separated from the noise eigenvectors. Each signal eigenvector can be written as:

$$\begin{aligned} \mathbf{x}\_k &= [\mathbf{x}(0), \mathbf{x}(1), \dots, \mathbf{x}(N-2), \mathbf{x}(N-1)] \\ &= \mathbf{A}\_\mathbf{k} \times [1, e^{j\omega\_\mathbf{k}}, e^{j2\omega\_\mathbf{k}}, \dots, e^{j(N-1)\omega\_\mathbf{k}}] \\ &= [s\_1, \mathbf{x}(N-1)] = [\mathbf{x}(0), s\_2], \end{aligned} \tag{8}$$

where **Ak** is the coefficients vector. The *s*<sup>2</sup> = *s*1*ejω*<sup>1</sup> rotational invariance property is valid, so all the signal eigenvectors and the signal components can be collected into the matrix, *U*, as well as into the signal components in the following matrices:

$$\Gamma\_1 \quad = \begin{bmatrix} I\_{M-1} \vert 0\_{M-1} \end{bmatrix} \vert\_{\left(M\_1\right)\times M} \tag{9a}$$

$$
\Gamma\_2 \quad = \begin{bmatrix} 0\_{M-1} \left| I\_{M-1} \right. \end{bmatrix} (M\_1) \times M \tag{9b}
$$

Considering the rotational invariance property for each signal eigenvector, the selection matrices, Γ<sup>1</sup> (9a) and Γ<sup>2</sup> (9b), can be used to obtain the following system:

$$[\Gamma\_1 \mathcal{U}] \Phi = \Gamma\_2 \mathcal{U},\tag{10}$$

where <sup>Φ</sup> <sup>=</sup> diag{*ejω*<sup>1</sup> ,*ejω*<sup>2</sup> , ...,*ejωNs* }. It is possible to obtain the frequencies of the components belonging to the signal solving this system with a least square technique.

#### 2.2.3. IWPA

This method, proposed in [25], is based on the iteration of the weighted phase average algorithm (WPA). Considering the case of a signal with only one spectral component, *x*(*t*) = *A*0*cos*(2*π f*0*t* + *φ*0), a coarse estimation, ˆ *f*0, of the frequency, *f*0, can be obtained in the first place, as the maximum of the amplitude of the DFT sequence, *X*(*k*). The

signal is then divided into *M* non-overlapping segments of length *P*: *xs*(*n*) = *x*(*n* + *s* · *P*), 0 ≤ *n* ≤ *P* − 1.

In the simple but effective case of two segments and *P* = *N*/2, the spectra of the two segments, *x*<sup>1</sup> and *x*2, are evaluated at frequency ˆ *f*0, and it can be shown that the fractional bin deviation, *δ*, can be estimated as:

$$\delta = \frac{N}{2\pi \cdot P} \left( \underline{\int X\_1(\hat{f}\_0)} - \underline{\int X\_2(\hat{f}\_0)} \right). \tag{11}$$

The IWPA algorithm, at each iteration, applies the WPA to obtain the frequency estimation of the strongest component, while amplitude and phase of this component are obtained through a least square technique. In the next step, the estimated component is subtracted from the samples of the previous iteration in the time domain. The number of iterations has to be equal to the number of components, so that a new component can be estimated at each iteration. The IWPA algorithm can be easily converted into a non-parametric algorithm by iterating its processing steps until the level of the residual decreases below a threshold.
