Digital Signal Processing Reference
In-Depth Information
is the geometric mean of the powers.
In effect, we know that as long as we have
M
positive real numbers, the arithmetic
mean is greater than or equal to the geometric mean and that this equivalence occurs
when the
M
numbers are equal as follows:
M−
1
1
/M
M
−
1
1
M
a
k
≥
a
k
k
=0
k
=0
If we put
a
k
=
σ
Y
k
2
−
2
b
k
, this relation becomes:
M−
1
1
/M
=
M−
1
1
/M
M−
1
1
M
2
−
2
M
−
1
σ
Y
k
2
−
2
b
k
≥
σ
Y
k
2
−
2
b
k
σ
Y
k
b
k
/M
k
=0
k
=0
k
=0
k
=0
M
−
1
1
M
σ
Y
k
2
−
2
b
k
≥
α
2
2
−
2
b
k
=0
The optimum value is achieved when all the intermediate terms in the sum
are equal. In this case, the inequality becomes an equality. Whatever the value of
k
is, we have:
σ
Y
k
2
−
2
b
k
=
α
2
2
−
2
b
[3.6]
such that:
2
b
k
=2
b
σ
Y
k
α
2
[3.7]
Hence, we find equation [3.5].
Equation [3.6] has an interesting interpretation. It means that the optimum scalar
quantization of each random variable
Y
k
(
m
) gives rise to the same MSE. We therefore
arrive at the following important conclusion: everything happened as though we were
seeking to make the reconstruction error as close as possible to a full-band white noise.
The quantization error power can be written as:
σ
Q
=
c
(1)
M−
1
1
/M
σ
Y
k
2
−
2
b
[3.8]
k
=0
