Minimum Variance - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 175

The mean expression, whether the noise is white or not, shows that the MV
estimator is a biased estimator. The basic definition of the MV estimator summed up
by equation [7.3] brings us to an identical conclusion, starting from the frequency
domain:

+ f /2

= ∫ c

2

( )

() () ()

i

df

[7.23]

Bias P f A f S f

MV c f x

c

− f /2

c

with ()

i

x Sf

as defined in equation [7.2):

() for

= ⎨
⎩

Sf f <− f ε and f >+ f ε

i

()

x c c

Sf

x

0 else

The MV estimator is built on the minimization of this integral which tends to 0
but which is never null. The MV estimator by construction can be only a biased
estimator of the signal power. If we consider again the case of the noisy exponential
function (see equation [7.7]), at the frequency of the exponential, the impulse
response, the Fourier transform of the frequency response given by equation [7.15],
is equal to the vector

f E , hence the mean of the MV estimator is deduced from
equation [7.21], which is now written:

exp

2

=+

( )

()

2

EP f C for f = f [7.24]

MV c c exp

M

From this expression, the bias value is deduced:

2

σ

( ) ()

Bias P f = for f = f

MV c c exp

M

This expression is actually directly visible in equation [7.16]. This bias depends
on the noise variance and especially on the order M. This formula illustrates the fact
that the bias is not null, but it especially clearly shows the interest of a high order for
diminishing the bias influence. These results are illustrated in Figure 7.6, which
represents faraway cases. It is about the spectral analysis of a sinusoid embedded in
a low-level noise (signal to noise ratio of 20 dB) at orders varying from 3 to 70. The
representation is centered on the frequency of the sinusoid and the amplitudes are
linearly represented in order to better visualize the bias. The amplitude of the
spectrum minimum increases when M diminishes.

We will see in section 7.4 that, in order to have a non-biased estimator, it is
necessary to possess a priori information which is stronger than that of the
covariance matrix of the signal to be analyzed.