a la physique de l'information - Lisa - Université d'Angers

Let us de£ne the improvement in performance of the
quantizer–estimator over that of the Wiener £lter alone as
estimation gain
Gest = Cors˜s
Corsˆs
. (25)
Substituting Eq. (6) into Eq. (23) and Eq. (22) into Eq. (24),
we have, after simpli£cation, a simple analytical expression
for the asymptotic estimation gain under the weak signal
assumption
G0 =
2f 2 η (γ)
, (26)
1 − Fη(γ)
where Gest tends to G0 when the input SNRin = σ 2 s tends
to zero. Therefore, when G0 > 1, an improvement of the
quantizer–estimator against the Wiener £lter estimator will
be expected if the weak signal assumption is valid.
Let us consider the input–output SNR gain achieved by
the 3–level quantizer de£ned as
GSNR = SNRout
SNRin
, (27)
with the ouput SNR de£ned as
T 2 T E[s y] / E[s s]
SNRout = N−1 i=0 var(yi)
. (28)
It can be shown [5, 4], under weak signal assumption, that
GSNR = 2f 2 η (γ)
. (29)
1 − Fη(γ)
As a result, maximizing the estimation gain G0 of Eq. (26)
is equivalent to maximizing the input–ouput signal to noise
ratio gain GSNR of the 3–level quantizer. A SNR gain greater
than unity is interesting in the detection context where a
higher SNR leads to a better detector performance. Here,
we see that SNR gain exceeding unity will also improve the
performance of an estimation task. The maximization, with
respect to the threshold γ, of the SNR gain GSNR of a 3–
level quantizer has been developed and applied to generalized
Gaussian and mixture of Gaussian noise pdf’s by [5, 4].
As an illustration, Fig. 2 shows the evolution of the optimal
estimation gain G0 for the generalized Gaussian noise
family and the mixture of Gaussian when the threshold γ is
chosen in order to maximize GSNR. It is shown by Fig. 2
that, if the noise is suf£ciently non-Gaussian and the quantizer
thresholds are optimally chosen, then G0 > 1, which
means that the quantizer–estimator will theoretically exhibit
better performance in the weak signal assumption than the
Wiener £lter estimator.
4. SIMULATION RESULTS
Now we propose to con£rm the theoretical results of the
previous section with numerical simulations. Monte Carlo
estimation gain G 0
1.6
1.4
1.2
1
0.8
0 0.2 0.4 0.6 0.8 1
mixing parameter α for mixture of Gaussian
Fig. 2a
estimation gain G 0
5
4
3
2
1
0 2 4 6 8 10
exponent p of generalized Gaussian
Fig. 2b
Fig. 2. Optimal estimation gain G0 under the weak signal
assumption for the generalized Gaussian (panel (b)) plotted
as a function of exponent p and the mixture of Gaussian
(panel (a)) as a function of the mixing parameter α while
the ratio of the standard deviations is £xed to β = 4. The
region above the dashed line shows an improvement in the
estimation of the quantizer–estimator over the Wiener £lter.
simulations have been done to compute the estimation gain
Gest of Eq. (25) of the quantizer–estimator over the Wiener
£lter in generalized Gaussian and mixture of Gaussian noise.
The vector length taken for the samples is N = 3000 and
the expectation operators in Eq. (25) are estimated with an
averaging taken over 300 trials.
In Fig. 3, we show the evolution of the estimation gain
Gest of Eq. (25) in given conditions for the noise and signal
and for different input signal to noise ratio SNRin (dB) =
10 log(σ 2 s ). In both Fig. 3a and Fig. 3b we £nd Gest > 1 for
suf£ciently small SNRin (approximately for SNRin (dB)
smaller than −3 dB in Fig. 3a and smaller than −5 dB in
Fig. 3b). Figure 3a illustrates that the analytical estimation
gain G0 calculated in Eq. (26) is indeed an asymptotic theoretical
result valid under the weak signal assumption. In
Fig. 3b, the estimation gain Gest is well below the asymptotic
value even for SNRin (dB) = −20 dB, although Gest
is clearly greater than unity. This can be explained theoretically
by including the third and higher order terms in the
approximate expressions for E [siyj] and E [yiyj] derived in
Sec. 3. In order to complete the illustration provided by
Fig. 3, we give in Tab. 1 and Tab. 2 the values of the correlations
Corsˆs and Cors˜s corresponding to some points of
Fig. 3; from these tables, one can appreciate how the estimation
performance of the Wiener £lter degrades when the
SNRin (dB) is decreasing and how the quantizer–estimator
arrests this degradation. We have tested a large number of
pdf among the generalized Gaussian and mixture of Gaussian
noise. We have also studied the in¤uence of the length
of the data set N and the signal correlation exponential decay
parameter ζ on the estimation improvement. All our
results con£rm the asymptotic analysis of the previous section.
For weak signals in suf£ciently non-Gaussian noise
the quantizer–estimator described in this paper outperforms
the Wiener £lter.
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