Databases Reference
In-Depth Information
1.0
Original
Difference
0.8
0.6
0.4
0.2
0
−0.2
0
0.6
0.8
0
5
6
0
1
2
3
4
F I GU R E 11 . 1
Sinusoid and sample-to-sample differences.
does not change a great deal
from one sample to the next. This means that both the dynamic range and the variance of
the sequence of differences
In many sources of interest, the sampled source output
{
x
n
}
are significantly smaller than that of the source
output sequence. Furthermore, for correlated sources the distribution of
d
n
is highly peaked at
zero. We made use of this skew, and resulting loss in entropy, for the lossless compression of
images in Chapter 7. Given the relationship between the variance of the quantizer input and
the incurred quantization error, it is also useful, in terms of lossy compression, to look at ways
to encode the difference from one sample to the next rather than encoding the actual sample
value. Techniques that transmit information by encoding differences are called
differential
encoding techniques
.
{
d
n
=
x
n
−
x
n
−
1
}
Example11.2.1:
Consider the half cycle of a sinusoid shown in Figure
11.1
that has been sampled at the rate
of 30 samples per cycle. The value of the sinusoid ranges between 1 and
1. If we wanted
to quantize the sinusoid using a uniform four-level quantizer, we would use a step size of 0.5,
which would result in quantization errors in the range
−
. Ifwetakethesample-
to-sample differences (excluding the first sample), the differences lie in the range
[−
0
.
25
,
0
.
25
]
.
To quantize this range of values with a four-level quantizer requires a step size of 0.1, which
results in quantization noise in the range
[−
0
.
2
,
0
.
2
]
[−
0
.
05
,
0
.
05
]
.
The sinusoidal signal in the previous example is somewhat contrived. However, if we look
at some of the real-world sources that wewant to encode, we see that the dynamic range that con-
tainsmost of the differences is significantly smaller than the dynamic range of the source output.
Example11.2.2:
Figure
11.2
is the histogram of the Sinan image. Notice that the pixel values vary over almost
the entire range of 0 to 255. To represent these values exactly, we need 8 bits per pixel. To
