Analysis of compressor functions for Laplacian source’s scalar compandоr construction
А simple and complete analysis of nonuniform scalar quantizers based on the companding technique, so-called compandors, is presented. The performance of the scalar compandors for different definition of compressor functions are considered and compared with the performance of optimal Lloyd-Max’s scal...
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Інститут проблем реєстрації інформації НАН України
2006
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| Назва видання: | Реєстрація, зберігання і обробка даних |
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| Цитувати: | Analysis of compressor functions for Laplacian source’s scalar compandоr construction / Z. Peric, J. Nikolic // Реєстрація, зберігання і оброб. даних. — 2006. — Т. 8, № 2. — С. 15-24. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-508372013-11-05T03:06:56Z Analysis of compressor functions for Laplacian source’s scalar compandоr construction Peric, Z. Nikolic, J. Математичні методи обробки даних А simple and complete analysis of nonuniform scalar quantizers based on the companding technique, so-called compandors, is presented. The performance of the scalar compandors for different definition of compressor functions are considered and compared with the performance of optimal Lloyd-Max’s scalar quantizers. There are several definitions of compressor functions. Two of them, that are functions of the support region of the compandor, are presented in this paper. The support region of the compandor is defined with the maximum amplitude of the input signal that enables optimal compandor’s load. In order to design the scalar compandor having the best performances i.e. with the smallest distortion, it is necessary to determine optimally its support region. Представлено простий і повний аналіз скалярних квантувателів зі змінним кроком, що базується на засобах компандування, так званих компандерах. Розглянуто характеристики скалярних компандерів при різних визначеннях функцій стиснення й наведено порівняння з характеристиками оптимальних скалярних квантувателів Ллойда-Мекса. Існує кілька визначень функції стиснення. Представлено два з них, які є функціями області підтримки компандера. Область підтримки компандера визначено максимальною амплітудою вхідного сигналу, що дає можливість оптимального завантаження компандера. Для розробки скалярного компандера з оптимальними характеристиками, тобто з найменшим викривленням, необхідно оптимально визначити його область підтримки. Представлен простой и полный анализ скалярных квантователей с переменным шагом, базирующийся на средствах компандирования, так называемых компандерах. Рассмотрены характеристики скалярных компандеров при различных определениях функций сжатия и приведены сравнения с характеристиками оптимальных скалярных квантователей Ллойда–Мэкса. Существует несколько определений функции сжатия. Представлены два из них, которые являются функциями области поддержки компандера. Область поддержки компандера определена максимальной амплитудой входного сигнала, что дает возможность оптимальной загрузки компандера. Для разработки скалярного компандера с оптимальными характеристиками, то есть с наименьшим искажением, необходимо оптимально определить его область поддержки. 2006 Article Analysis of compressor functions for Laplacian source’s scalar compandоr construction / Z. Peric, J. Nikolic // Реєстрація, зберігання і оброб. даних. — 2006. — Т. 8, № 2. — С. 15-24. — Бібліогр.: 7 назв. — англ. 1560-9189 http://dspace.nbuv.gov.ua/handle/123456789/50837 004.93 en Реєстрація, зберігання і обробка даних Інститут проблем реєстрації інформації НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Математичні методи обробки даних Математичні методи обробки даних |
| spellingShingle |
Математичні методи обробки даних Математичні методи обробки даних Peric, Z. Nikolic, J. Analysis of compressor functions for Laplacian source’s scalar compandоr construction Реєстрація, зберігання і обробка даних |
| description |
А simple and complete analysis of nonuniform scalar quantizers based on the companding technique, so-called compandors, is presented. The performance of the scalar compandors for different definition of compressor functions are considered and compared with the performance of optimal Lloyd-Max’s scalar quantizers. There are several definitions of compressor functions. Two of them, that are functions of the support region of the compandor, are presented in this paper. The support region of the compandor is defined with the maximum amplitude of the input signal that enables optimal compandor’s load. In order to design the scalar compandor having the best performances i.e. with the smallest distortion, it is necessary to determine optimally its support region. |
| format |
Article |
| author |
Peric, Z. Nikolic, J. |
| author_facet |
Peric, Z. Nikolic, J. |
| author_sort |
Peric, Z. |
| title |
Analysis of compressor functions for Laplacian source’s scalar compandоr construction |
| title_short |
Analysis of compressor functions for Laplacian source’s scalar compandоr construction |
| title_full |
Analysis of compressor functions for Laplacian source’s scalar compandоr construction |
| title_fullStr |
Analysis of compressor functions for Laplacian source’s scalar compandоr construction |
| title_full_unstemmed |
Analysis of compressor functions for Laplacian source’s scalar compandоr construction |
| title_sort |
analysis of compressor functions for laplacian source’s scalar compandоr construction |
| publisher |
Інститут проблем реєстрації інформації НАН України |
| publishDate |
2006 |
| topic_facet |
Математичні методи обробки даних |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/50837 |
| citation_txt |
Analysis of compressor functions for Laplacian source’s scalar compandоr construction / Z. Peric, J. Nikolic // Реєстрація, зберігання і оброб. даних. — 2006. — Т. 8, № 2. — С. 15-24. — Бібліогр.: 7 назв. — англ. |
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Реєстрація, зберігання і обробка даних |
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AT pericz analysisofcompressorfunctionsforlaplaciansourcesscalarcompandorconstruction AT nikolicj analysisofcompressorfunctionsforlaplaciansourcesscalarcompandorconstruction |
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2025-07-04T12:41:24Z |
| last_indexed |
2025-07-04T12:41:24Z |
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1836720205666975744 |
| fulltext |
Математичні методи обробки даних
ISSN 1560-9189 Реєстрація, зберігання і обробка даних, 2006, Т. 8, № 2 15
UDC 004.93
Zoran Peric, Jelena Nikolic
Faculty of Electronic Engineering
Aleksandra Medvedeva, 14, 18000 Nis, Serbia and Montenegro
E-mail: peric@elfak.ni.ac.yu; njelena@elfak.ni.ac.yu
Analysis of compressor functions for Laplacian source’s
scalar compandor construction
А simple and complete analysis of nonuniform scalar quantizers based on
the companding technique, so-called compandors, is presented. The perfor-
mance of the scalar compandors for different definition of compressor func-
tions are considered and compared with the performance of optimal Lloyd-
Max’s scalar quantizers. There are several definitions of compressor func-
tions. Two of them, that are functions of the support region of the compan-
dor, are presented in this paper. The support region of the compandor is de-
fined with the maximum amplitude of the input signal that enables optimal
compandor’s load. In order to design the scalar compandor having the best
performances i.e. with the smallest distortion, it is necessary to determine
optimally its support region.
Key words: сompressor functions, scalar compandor, support region.
Introduction
A vast amount of research has been made in the area of quantization. The optimal
quantization problem, even for the simplest case-uniform scalar quantization, is very
actual in contemporary signal processing [1, 2]. Namely, optimal designing of scalar
quantizers requires optimal determining of the support region. Therefore, optimal de-
termining of the support region has been considered by a lot of researchers [1, 2]. A full
understanding of optimal quantization is not possible without a clear understanding of
quantizer’s support because there are different definitions of the support region [1, 2]. In
this paper we consider nonuniform scalar quantizers based on the companding tech-
nique. Namely, nonuniform quantizer consisting of a compressor, a uniform quantizer,
and expandor in cascade is called compandor Fig. 1 [3]. The companding technique is
easily realized, therefore it has wide application. Because of its usefulness in analyzing
and optimizing nonuniform quantizers having a large number of levels the companding
technique has been considered in [3, 4]. Moreover, in this paper we determine the sup-
port region of the scalar compandor, [–tmax, tmax], defined with the maximum amplitude
© Zoran Peric, Jelena Nikolic
Zoran Peric, Jelena Nikolic
16
of the input signal tmax. Optimal determining of the support region of the compandor
depends on the compressor function used for the compandor designing. Currently, de-
termination of the support region for Lloyd-Max’s quantizers is primarily of theoretical
importance. Lloyd-Max’s style algorithms for designing optimal scalar quantizers [4, 5]
begin with an estimate of the support region — the better the estimate, the more rapid
the algorithm convergence. In this paper the compandors’ distortions for different com-
pressor functions are calculated and compared with the Lloyd-Max’s quantizers distor-
tions. In such a way it is possible to chose the compressor function which provides de-
signing of scalar compandor having the best performances.
Nonuniform scalar quantization
Let us consider an N-level nonuniform scalar quantizer Q for the Laplacian input
signals. Scalar quantizer Q is defined with Q: R → C, as a functional mapping of the set
of real numbers R onto the set of the output representation. The set of the output repre-
sentation constitutes the code book:
{ } RyyyyC N ̺ ,...,,, 321 (1)
that has the size |C| = N. The output values, yj, are called the representation levels. The
nonuniform scalar quantizer Q is defined with the set of output values and with the par-
tition of the input range of values onto N cells i.e. intervals αj, j = 1,2,..., N. Cells αj are
defined with the decision thresholds {t0, t1,…, tN}, such that αj = ( tj–1, tj], j = 1,2,..., N.
Cells α2,..., αN–1 are referred to as the inner cells, while α1 and αN are referred to as the
outer cells. The negative thresholds and the representation levels are symmetric to their
nonnegative counterparts. A quantized signal has value yj when the original signal be-
longs to the quantization cell αj. Hence, N-level scalar quantizer is defined as a func-
tional mapping of an input value x onto an output representation, such as:
jyxQ =)( , jx aÎ . (2)
When the inner cells are equally sized, the quantizer is called uniform quantizer. Other-
wise, the quantizer is nonuniform. A general model for any nonuniform quantizer with a
finite number of levels can be structured as illustrated in Fig. 1 [1], where c(x) and
c–1(x) are compressor and expandor functions respectively. Namely, nonuniform quanti-
zation can be achieved by compressing the signal x using nonuniform compressor cha-
racteristic c(·) (also called companding law), by quantizing the compressed signal c(x)
employing a uniform quantizer, and by expanding the quantized version of the com-
pressed signal using a nonuniform transfer characteristic c–1(·) inverse to that of the
compressor. The overall structure of a nonuniform quantizer consisting of a compressor,
a uniform quantizer, and expandor in cascade is called compandor.
Analysis of compressor functions for Laplacian source’s scalar compandоr construction
ISSN 1560-9189 Реєстрація, зберігання і обробка даних, 2006, Т. 8, № 2 17
Fig. 1. Block diagram of the companding technique
The quantizer distortion
An optimal N-level nonuniform scalar quantizer, for a source characterized as a
continuous random variable with probability density p(x) is a quantizer that minimizes
total distortion. The quantizer distortion is defined as the expected mean square error
between original and quantized signal. Total distortion consists of two components, in-
ner and outer distortion. Symbolically,
oi DDD += , (3)
where the inner and outer distortions are defined as:
( )
3
3/1
2
1
1
)(
212
1
÷
ø
öç
è
æ
-
= ò
-
--
N
N
t
ti dxxp
N
D , (4)
( ) ( )ò
¥
-
-=
1
22
Nt
No dxxpyxD . (5)
The support region of the scalar compandor, denoted here [–tmax,+ tmax], is the interval
where quantization errors are small, or at least bounded. In this paper we consider the
Laplacian input signals with unrestricted amplitude range. Determination of the support
region enables quantizers to be adapted to the amplitudes of input signals. Laplacian
probability density function of the original random variable x with unit variance can be
expressed by:
( ) xexp 2
2
1 -= . (6)
By substituting (6) in (4) and (5), the expressions for determining inner and outer distor-
tions are derived as follows:
( )
3
1
2 3
2exp1
22
9
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
--
-
= -N
i
t
N
D , (7)
y Q(c(x)) c(x)
c(x) Q с–1 (x)
x
Zoran Peric, Jelena Nikolic
18
( ) ( )( )2
11
2
11 22122exp NNNNNNo ytytttD ++-++-= ---- , (8)
and the total distortion of the compandor may be rewritten such as:
( )
3
1
2 3
2
exp1
22
9
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
--
-
= -Nt
N
D +
(9)
+ ( ) ( )( )2
11
2
11 2212 2exp NNNNNN ytyttt ++-++- ---- .
A quantizer is optimal in the sense that no other N-point scalar quantizer can obtain
lower distortion.
Definition and determining of the compandor’s support region
Let us consider different definitions of compressor functions. First, denoted c0(x),
is similar to the one in [3]:
( )
( )
( )dxxp
dxxp
tc
jt
j
21
3
1
3
1
0
ò
ò
¥+
¥-
¥-+-= , (10)
( )
( )
( )dxxp
dxxp
yc
jy
j
21
3
1
3
1
0
ò
ò
¥+
¥-
¥-+-= . (11)
For the values of input signal x within (–∞, ∞) range, the values of c0(tj) range [–1,1].
Thresholds tj j = 1,2,..., N – 1 and quantization points yj j = 1,2,..., N can be determined
from:
( )
N
jtc j
210 +-= , (12)
( ) ( )
N
j
yc j
2
12
10
-
+-= . (13)
Therefore tN–1, denoted tN–1
(0), can be determined by equating (10) and (12):
Analysis of compressor functions for Laplacian source’s scalar compandоr construction
ISSN 1560-9189 Реєстрація, зберігання і обробка даних, 2006, Т. 8, № 2 19
( )
÷
ø
ö
ç
è
æ=- 2
ln
2
30
1
NtN . (14)
Also, yN, denoted yN
(0), can be determined by equating (11) and (13):
( ) ( )NyN ln
2
30 = . (15)
By using this definition of compressor function it is not possible to optimize the quan-
tizer’s load, because it is not possible to determine tmax. By generalizing compressor
function c0(x), it is easy to achieve another one, denoted here c(x):
( )
( )
( )dxxp
dxxp
tc t
t
t
t
j
j
ò
ò
+
-
-+-=
max
max
max
3
1
3
1
21 . (16)
In such a case, when the values of input signal x are within [–tmax, +tmax] range, the val-
ues of c(tj) range [–1,1]. It is very easy to show that the same quantizers will be de-
signed by using generalized compressor function c(x) and compressor function c1(x) [3]:
( )
( )
( )dxxp
dxxp
tttc t
t
t
t
j
j
ò
ò
+
-
-+-=
max
max
max
3
1
3
1
maxmax1 2 , (17)
( )
( )
( )dxxp
dxxp
ttyc t
t
y
t
j
j
ò
ò
+
-
-+-=
max
max
max
3
1
3
1
maxmax1 2 . (18)
Now, it is obvious that when the values of input signal x are within the [–tmax, +tmax]
range, the values of c1(tj) are copied into the [–tmax, +tmax] range by using thus defined
compressor function. Thresholds tj j = 1,2,..., N – 1 and quantization points yj j = 1,2,...,
N can now be determined from:
( ) maxmax1
2 t
N
jttc j +-= , (19)
Zoran Peric, Jelena Nikolic
20
( ) ( )
maxmax1
2
12
t
N
j
tyc j
-
+-= . (20)
Therefore tN–1, denoted tN–1
(1), can be determined by equating (17) and (19):
( )
( ) ÷
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
ç
è
æ
÷÷
ø
ö
çç
è
æ
--+
=-
max
1
1
3
2exp22
ln
2
3
tN
NtN , (21)
and yN, denoted yN
(1), by equating (18) and (20):
( )
( ) ÷
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
ç
è
æ
÷÷
ø
ö
çç
è
æ
--+
=
max
1
3
2exp11
ln
2
3
tN
NyN . (22)
Good approximation of Lloyd-Max quantizers inner distortion can be achieved by using
Bennett’s integral [6, 7] ranging [–tN–1, +tN–1] which is the expression (4) for the inner
compandor distortion. Therefore, a good heuristic hypothesis will be based on equating
the thresholds tN–1
(1) = tN–1
dif for the compandor and Lloyd-Max quantizer. Let us mark
the Lloyd-Max quantizer threshold tN–1 with tN–1
dif. The dependence of this threshold on
the number of quantizaion levels was determined in [2] by minimizing the total distor-
tion such as:
=-
dif
Nt 1 3
ln
2
3 N . (23)
Therefore, considering the expression (21) and relation tN–1
(1) = tN–1
dif, the values of tmax
for different N, can be calculated from:
( ) 2
3
2exp
2ln
2
3
1
1
max
-÷÷
ø
ö
çç
è
æ
-
-
=
-NtN
Nt . (24)
In such a way, the estimate of the quantizer’s optimal load, i.e. the realization of optimal
compandor can be achieved.
Let us consider another method of finding the support region of the compandor. By
putting the first derivative of compressor function c0(x), it is easy to calculate the slope
of this compressor characteristic. Particularly, for x = yN
(0) we get:
Analysis of compressor functions for Laplacian source’s scalar compandоr construction
ISSN 1560-9189 Реєстрація, зберігання і обробка даних, 2006, Т. 8, № 2 21
( )( )
N
yc N 3
20
0 =¢ . (25)
By using the following relation that is valid for large number of quantization cells N [4]:
( )( ) ( )0
0
0
2
N
N N
yc
D
Ȣ , (26)
the width of the N-th cell αN, denoted here ΔN
(0), can be determined such as ΔN
(0) ≈ 3 2 .
Therefore, the initial value for tmax, denoted tmax
(0), can be estimated from:
( ) ( ) ( )00
1
0
max NNtt D+= - . (27)
Now, let us calculate the slope of compressor characteristic c1(x), in x = yN
(1,i):
( )( )
( )
( )
( )
÷÷
ø
ö
çç
è
æ
-
÷
÷
ø
ö
ç
ç
è
æ
÷÷
ø
ö
çç
è
æ
--
=¢ i
N
i
i
i
N y
t
tyc ,1
max
max,1
1 3
2exp
3
2exp1
3
2 , (28)
where yN
(1,i) is defined as:
( )
( ) ( )
÷
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
ç
è
æ
÷÷
ø
ö
çç
è
æ
--+
=
i
i
N
tN
Ny
max
,1
3
2exp11
ln
2
3 . (29)
The iterative process starts for i = 0, and we take the initial value for tmax
(0) from (27).
Similar to (26) the following relation is valid:
( )( )
( )
( )i
N
i
i
N N
tyc ,1
max,1
1
2
D
Ȣ . (30)
Combining (28), (29) and (30) we get:
( )
( )
( ) ( )
÷÷
ø
ö
çç
è
æ
--+
÷÷
ø
ö
çç
è
æ
--
»D
i
i
i
N
tN
t
max
max
,1
3
2exp11
3
2exp1
23 . (31)
Zoran Peric, Jelena Nikolic
22
Namely, tmax
(i+1) can now be calculated as a sum of tN–1
(1,i) and ΔN
(1,i):
( )
( ) ( )
+
÷
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
ç
è
æ
÷÷
ø
ö
çç
è
æ
--+
=+
i
i
tN
Nt
max
1
max
3
2exp22
ln
2
3
(32)
+
( )
( ) ( )
÷÷
ø
ö
çç
è
æ
--+
÷÷
ø
ö
çç
è
æ
--
i
i
tN
t
max
max
3
2exp11
3
2exp1
23 .
Let the expression (9) has the values for tN–1 and yN determined in (14) and (15) re-
spectively; then the distortion marked as Dc0 can be calculated. When the values for tN–1
and yN in (9) are determined by using (21), (23) and (24), the distortion is marked as
Dc1. If the total distortion value is calculated by using the values of tN–1 and yN, which
are obtained iteratively, the total distortion will be marked as Dc1
(1,7). Namely, the itera-
tions are interrupted after the seventh iteration because for i = 7 we get Dc1
(1,i+1) ≥
≥ Dc1
(1,i). Consequently, the next iteration does not allow further reduction of distortion.
Finally, the distortion marked as DLM stands for the Lloyd-Max quantizer distortion. For
the calculation of the last distortion we observe the values for tN–1 that are given in [1]
and take into the calculation the fact that the distance between tN–1 and yN is 1/ 2 , also
shown in [1].
Numerical results
Table 1 compares the values of distortions Dc0, Dc1, Dc1
(1,7) and DLM. From the Ta-
ble 1 it is obvious that the values of Lloyd-Max distortion for N = 128, N = 256 and
N = 512 are the nearest to the corresponding values of the compandor distortion when
compressor function c1(x) is used and the support region is obtained iteratively. Also,
the distortion values are nearly equal to the Lloyd-Max distortion values when we ap-
plied distortion minimizing method and compressor function c1(x). Hence, it is obvious
that c1(x) enables designing of the scalar compandor having the best performance. Addi-
tionaly, the values of maximum input signals amplitudes, i.e. the values of the support
regions, are listed in Table 2 and Table 3. Finally, the maximal amplitudes dependence
on the number of bits per sample R (R = log2N) is given in Fig. 2.
Table 1. Distortions Dc0, Dc1, Dc1
(1,7) and DLM
N Dc0 Dc1 Dc1
(1,7) DLM
128 2,7450 10–4 2,7071 10–4 2,7062 10–4 2,7042 10–4
256 6,8644 10–5 6,8168 10–5 6,8157 10–5 6,8132 10–5
512 1,7164 10–5 1,7104 10–5 1,7103 10–5 1,7099 10–5
Analysis of compressor functions for Laplacian source’s scalar compandоr construction
ISSN 1560-9189 Реєстрація, зберігання і обробка даних, 2006, Т. 8, № 2 23
Table 2. Distortion minimized Compandоr Parameters for Compressor Function c1(x)
N tN–1
(1) = tN–1
dif tmax yN
(1) Dc1
128 7,9622 10,2593 8,8139 2,7069 10–4
256 9,4326 11,7465 10,2886 6,8167 10–5
512 10,9030 13,2253 11,7611 1,7104 10–5
Table 3. Iteratively obtained Compandоr Parameters for Compressor Function c1(x)
N yN
(1,7) ΔN
(1,7) tN–1
(1,7) tmax
(8) Dc1
(1,7)
128 8,6268 1,9164 7,8361 9,7525 2,7062 10–4
256 10,0895 1,9185 9,2981 11,2161 6,8157 10–5
512 11,5561 1,9196 10,7643 12,6839 1,7103 10–5
Fig. 2. The maximal amplitudes dependence on the number of bits per sample
Conclusion
In this paper the systematic analysis of different definitions of compressor func-
tions for scalar compandor construction are carried out. Simply expressions in closed
forms for the support region of the scalar compandоr for Laplacian source are obtained.
Also, the expression for the determining of the total distortion is derived. Hence, numer-
ical values of the total distortion are calculated when the scalar compandor is realised by
using different definitions of compressor functions. The results demonstrate that by us-
M
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Zoran Peric, Jelena Nikolic
24
ing the compressor function c1(x) the calculated distortion the least differs from the op-
timal Lloyd-Max’s quantizers distortion. Therefore, choosing the compressor function
c1(x) it is possible to design the scalar compandor having the best performance.
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Received 19.04.2006
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