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distortion in the reconstructed sequence, and distortion is much more likely to be noticed in a
musical piece than in a politician's speech.
In the best of all worlds, we would always use the end user of a particular source output
to assess quality and provide the feedback required for the design. In practice this is not often
possible, especiallywhen the end user is a human, because it is difficult to incorporate the human
response into mathematical design procedures. Also, there is difficulty in objectively reporting
the results. The people asked to assess one person's design may be more easygoing than the
people who were asked to assess another person's design. Even though the reconstructed
output using one person's design is rated “excellent” and the reconstructed output using the
other person's design is only rated “acceptable,” switching observers may change the ratings.
We could reduce this kind of bias by recruiting a large number of observers in the hope that
the various biases will cancel each other out. This is often the option used, especially in the
final stages of the design of compression systems. However, the rather cumbersome nature of
this process is limiting. We generally need a more practical method for looking at how close
the reconstructed signal is to the original.
A natural thing to do when looking at the fidelity of a reconstructed sequence is to look at
the differences between the original and reconstructed values—in other words, the distortion
introduced in the compression process. Two popular measures of distortion or difference
between the original and reconstructed sequences are the squared errormeasure and the absolute
differencemeasure. These are called
difference distortionmeasures
.If
{
x
n
}
is the source output
and
{
y
n
}
is the reconstructed sequence, then the squared error measure is given by
2
d
(
x
n
,
y
n
)
=
(
x
n
−
y
n
)
(1)
and the absolute difference measure is given by
(2)
In general, it is difficult to examine the difference on a term-by-term basis. Therefore, a
number of average measures are used to summarize the information in the difference sequence.
The most often used average measure is the average of the squared error measure. This is called
the
mean squared error
(mse). If we model the information sequence as a sequence of random
variables
d
(
x
n
,
y
n
)
= |
x
n
−
y
n
|
{
X
n
}
and the reconstruction sequence as the sequence of random variables
{
Y
n
}
,for
a sequence of length
N
we can define the mean squared error as
E
1
N
2
N
N
E
2
1
N
D
=
1
(
X
n
−
Y
n
)
=
(
X
n
−
Y
n
)
(3)
n
=
n
=
1
where the expectation is with respect to the joint distribution of
X
n
and
Y
n
. If we assume the
sequences are independent and identically distributed this expression simplifies to
E
2
D
=
(
X
−
Y
)
In practice we assume the information and reconstruction sequences are ergodic and replace
the ensemble averages in Equation (
3
) with time averages to obtain
N
1
N
2
d
2
σ
=
1
(
x
n
−
y
n
)
(4)
n
=
