Minimum Variance - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 189

a)

. .

. . .

b)

. . . . .

Figure 7.11. Spectral analysis by the NMV estimator of the same signal as that used for
Figure 7.5. a) Superposition of the true spectrum (in thin line) and of the NMV estimation at
order 12. b) Normalization factor in Hz. Vertical axes: spectra in dB.
Horizontal axes: frequency in Hz

This equation is an eigenvalue equation. At each frequency f c , the impulse
response calculated by the Capon estimator is an eigenvector of the signal
covariance matrix. The associated eigenvalue is the value of the spectrum estimated
by the normalized NMV estimator at this frequency f c . This link with the eigenvalue
theory [LAG 86] indicates that the NMV method is essentially a pure frequency
estimation method because its calculation comes back to a calculation of
eigenvalues. Figure 7.11 illustrates this point of view.

This equation has a second point of interest. It makes it possible to study the
convergence of the NMV estimator. Let us consider the following theorem
[GRE 58].

Let {λ 0 , λ 1 , …, λ M- 1 } be the set of eigenvalues and {v 0 , v 1 , …, v M- 1 } the set of
eigenvectors of the autocorrelation matrix R x , this matrix being Toeplitz matrix, if
→∞

M , then, for I = 0, …, M - 1, we have the following convergence property:

( )

λ → SiMT /

1 x s