Minimum Variance - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 171

averaged. The frequency response is a Dirichlet's kernel (see equation [7.15]). The
output power given by equation [7.16] is dependent on the order M [LAC 71, SHE
91]. This dependence in 1/ M is responsible for a bias, which diminishes when the
order increases (see section 7.2.2).

For the other cases, it is necessary to make approximations. If the signal to noise
ratio is weak (C 2 /σ 2 << 1), Q tends to 0 and equations [7.12] and [7.14] are written:

() ( )

A f ≈ Df − f [7.17]

f c

c

() 2

σ

[7.18]

Pf =

MV c

M

This drastic approximation also leads to a simple periodogram, non-weighted
and non-averaged with, as for a frequency response, a Dirichlet's kernel centered on
the MV filter frequency.

The other case is more interesting. It is about studying the case of the signal on
noise bigger than 1(C 2 /σ 2 >> 1). In this case, the product QM tends to 1 and the
frequency response is no longer a Dirichlet's kernel. Equation [7.12] highlights the
basic principle of the minimum variance estimator. While the denominator is
constant, the numerator is the difference between two Dirichlet's kernels. The first is
centered on the filter frequency f c and the second is centered on the exponential
frequency f exp . This second kernel is also smoothed by a kernel which depends on
the difference ( f c - f exp ) between these two frequencies. Let us examine the behavior
of this impulse response according to the relative position of these two frequencies.

The filter frequency f c is far from the frequency of the exponential f exp

The smoothing factor D ( f c - f exp ) tends to 0. Then, equation [7.12] reduces to:

( )

⎧ ⎪
≈ ⎨

Df − f if f ≠ f

c exp

()

Af [7.19]

f

c

0 if f → f

⎪ ⎩ exp

In this case, the response is a Dirichlet's kernel except at the frequency of the
exponential. This latter point is the fundamental difference with a Fourier estimator.