Digital Signal Processing Reference
In-Depth Information
Chapter 1
Scalar Quantization
1.1. Introduction
A,
+
A
].
Defining a scalar quantization with a resolution of
b
bits per sample requires three
operations:
- partitioning the range [
Let us consider a discrete-time signal
x
(
n
) with values in the range [
−
A,
+
A
] into
L
=2
b
−
non-overlapping intervals
Θ
1
···
Θ
L
Δ
1
···
Δ
L
{
,
- numbering the partitioned intervals
}
of length
{
}
i
1
···
i
L
,
- selecting the reproduction value for each interval, the set of these reproduction
values forms a dictionary (codebook)
1
C
=
{
}
x
1
···
x
L
{
}
.
Encoding (in the transmitter) consists of deciding which interval
x
(
n
) belongs
to and then associating it with the corresponding number
i
(
n
)
∈{
···
L
=2
b
}
.
It is the number of the chosen interval, the symbol, which is transmitted or stored.
The decoding procedure (at the receiver) involves associating the corresponding
reproduction value
x
(
n
)=
x
i
(
n
)
from the set of reproduction values {
x
1
1
x
L
}
with the number
i
(
n
). More formally, we observe that quantization is a non-bijective
mapping to [
···
A,
+
A
] in a finite set
C
with an assignment rule:
x
(
n
)=
x
i
(
n
)
∈{
−
Θ
i
The process is irreversible and involves loss of information, a quantization error
which is defined as
q
(
n
)=
x
(
n
)
x
1
···
x
L
}
iff
x
(
n
)
∈
−
x
(
n
). The definition of a distortion measure
1. In scalar quantization, we usually speak about quantization levels, quantization steps, and
decision thresholds. This language is also adopted for vector quantization.
