Databases Reference
In-Depth Information
Output
Output
Output
Input
Input
Input
Compressor
Uniform quantizer
Expander
F I GU R E 9 . 22
Block diagram for companded quantization.
of this chapter). A somewhat different approach is to use a nonlinear mapping to flatten the
performance curve shown in Figure
9.21
. In order to study this approach, we need to view the
nonuniform quantizer in a slightly different manner.
9.6.2 Companded Quantization
Instead of making the step size small, we could make the interval in which the input lies with
high probability large—that is, expand the region in which the input lands in proportion to the
probability with which it lands there. This is the idea behind companded quantization. This
quantization approach can be represented by the block diagram shown in Figure
9.22
. The input
is first mapped through a
compressor
function. This function “stretches” the high-probability
regions close to the origin and correspondingly “compresses” the low-probability regions away
from the origin. Thus, regions close to the origin in the input to the compressor occupy a greater
fraction of the total region covered by the compressor. If the output of the compressor function
is quantized using a uniform quantizer and the quantized value is transformed via an
expander
function, the overall effect is the same as using a nonuniform quantizer. To see this, we devise
a simple compander and see how the process functions.
Example9.6.1:
Suppose we have a source that can be modeled as a random variable taking values in the
interval
with more probability mass near the origin than away from it. We want to
quantize this using the quantizer of Figure
9.3
. Let us try to flatten out this distribution using
the following compander and then compare the companded quantization with straightforward
uniform quantization. The compressor characteristic we will use is given by the following
equation:
[−
4
,
4
]
⎧
⎨
−
2
x
if
1
x
1
2
x
3
4
3
c
(
x
)
=
+
x
>
1
(35)
⎩
2
x
3
4
3
−
x
<
−
1
