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).

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) |

(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.

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.

Copyright The European Southern Observatory (ESO)