  Up: Spectral analysis of stellar networks

Subsections

# 6 MUSIC and the Cramer-Rao lower bound

In this section we show the relation between the MUSIC estimator and the Cramer-Rao bound following the notation and the conditions proposed by Stoica and Nehorai in their paper (Stoica & Nehorai 1990).

## 6.1 The model

The problem under consideration is to determine the parameters of the following model: (12)

where are the vectors of the observed data, are the unknown vectors and is the added noise; the matrix and the vector are given by (13)

where varies with the applications. Our aim is to estimate the unknown parameters of . The dimension n of is supposed to be known a priori and the estimate of the parameters of is easy once is known.

Now, we reformulate MUSIC to follow the above notation. The MUSIC estimate is given by the position of the n smallest values of the following function: (14)
From Eq. (14) we can define the estimation error of a given parameter. has (for big N) an asintotic Gaussian distribution, with mean and with the following covariance matrix: (15)

where is the real part of x, where (16)

and where U is implicitly defined by: (17)

where P is the covariance matrix of . The elements of the diagonal of the matrix are the variances of the estimation error. On the other hand, the Cramer-Rao lower bound of the covariance matrix of every estimator of , for large N, is given by: (18)

Therefore the statistical efficiency can be defined with the condition that P is diagonal as: (19)

where the equality is reached when m increases if and only if (20)

For P non-diagonal, .

To adapt the model used in the spectral analysis (21)

where M is the total number of samples, to Eq. (14) we make the following transformations, after fixing an integer m>p: (22)

In this way our model satisfies the conditions of (Stoica & Nehorai 1990). Moreover, Eqs. (22) depend on the choice of m which influences the minimum error variance.

## 6.2 Comparison between PCA-MUSIC and the Cramer-Rao lower bound

In this subsection we compare the n.e. method with the Cramer-Rao lower bound, by varying the frequencies distance, the parameters M and m and the noise variance.

From the experiments it derives that, fixed M and m, by varying the noise (white Gaussian) variance, the n.e. estimate is more accurate for small values of the noise variance as shown in Figs. 1-3. For small, the noise variance is far from the bound. By increasing m the estimate improves, but there is a sensitivity to the noise (Figs. 4-6). By varying M, there is a sensitivity of the estimator to the number of points and to m (Figs. 7-8). In fact, if we have a quite large number of points we reach the bound as illustrated in Figs. 9-10. Figure 1: CRB and standard deviation of n.e. estimates; abscissa is the distance between the frequencies and . CRB (down); standard deviation of n.e. (up) with m=5, and M=50 Figure 2: CRB and standard deviation of n.e. estimates; abscissa is the distance between the frequencies and . CRB (down); standard deviation of n.e. (up) with m=5, and M=50 Figure 3: CRB and standard deviation of n.e. estimates; abscissa is the distance between the frequencies and . CRB (down); standard deviation of n.e. (up) with m=5, and M=50 Figure 4: CRB and standard deviation of n.e. estimates; abscissa is the distance between the frequencies and . CRB (down); standard deviation of n.e. (up) with m=20, and M=50 Figure 5: CRB and standard deviation of n.e. estimates; abscissa is the distance between the frequencies and . CRB (down); standard deviation of n.e. (up) with m=20, and M=50 Figure 6: CRB and standard deviation of n.e. estimates; abscissa is the distance between the frequencies and . CRB (down); standard deviation of n.e. (up) with m=20, and M=50 Figure 7: CRB and standard deviation of n.e. estimates; abscissa is the distance between the frequencies and . CRB (down); standard deviation of n.e. (up) with m=5, and M=20 Figure 8: CRB and standard deviation of n.e. estimates; abscissa is the distance between the frequencies and . CRB (down); standard deviation of n.e. (up) with m=10, and M=20 Figure 9: CRB and standard deviation of n.e. estimates; abscissa is the distance between the frequencies and . CRB (down); standard deviation of n.e. (up) with m=20, and M=100 Figure 10: CRB and standard deviation of n.e. estimates; abscissa is the distance between the frequencies and . CRB (down); standard deviation of n.e. (up) with m=50, and M=100

Therefore, the n.e. estimate depends on both the increase of m and the number of points in the input sequence. Increasing the number of points, we improve the estimate and the error approximates the Cramer-Rao bound. On the other hand, for noise variances very small, the estimate reaches a very good performance. Finally, we see that in all the experiments shown in the figures we reach the bound with a good approximation, and we can conclude that the n.e. method is statistically efficient.  Up: Spectral analysis of stellar networks

Copyright The European Southern Observatory (ESO)