Databases Reference
In-Depth Information
20
18
16
14
12
10
SNR (dB)
8
6
4
2
0
−2
0
0.5 1.0 1.5 2.0
Ratio of input variance to design variance
2.5
3.0
3.5
4.0
F I GU R E 9 . 11
Effect of variance mismatch on the performance of a 4-bit uniform
quantizer.
T A B L E 9 . 4
Demonstration of the effect of mismatch using eight-level quantizers
(dB). Each input distribution has unit variance.
Input Distribution Uniform Quantizer Gaussian Quantizer Laplacian Quantizer Gamma Quantizer
Uniform
18
.
06
15
.
56
13
.
29
12
.
41
Gaussian
12
.
40
14
.
27
13
.
37
12
.
73
Laplacian
8
.
80
10
.
79
11
.
39
11
.
28
Gamma
6
.
98
8
.
06
8
.
64
8
.
76
x
X
2
that for a distribution with zero mean, the variance is given by
, which is also a
measure of the power in the signal
X
. As we can see from the figure, the signal-to-noise ratio
is maximum when the input signal variance matches the variance assumed when designing
the quantizer. From the plot we also see that there is an asymmetry; the SNR is considerably
worse when the input variance is lower than the assumed variance. This is because the SNR is a
ratio of the input variance and the mean squared quantization error. When the input variance is
smaller than the assumed variance, the mean squared quantization error actually drops because
there is less overload noise. However, because the input variance is low, the ratio is small.
When the input variance is higher than the assumed variance, the
msqe
increases substantially,
but because the input power is also increasing, the ratio does not decrease as dramatically.
To see this more clearly, we have plotted the mean squared error versus the signal variance
separately in Figure
9.12
. We can see from these figures that the decrease in signal-to-noise
ratio does not always correlate directly with an increase in
msqe
.
The second kind of mismatch is where the input distribution does not match the distribution
assumed when designing the quantizer. In Table
9.4
we have listed the SNR when inputs
with different distributions are quantized using several different eight-level quantizers. The
quantizers were designed assuming a particular input distribution.
σ
=
E
[
]
