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Data Compression for Types of Nopaper Recorders Based on Some Wavelets.

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Dev. Chem. Eng. Mineral Process., 9 ( l R ) , pp.109-114, 2001.
Data Compression for Types of Nopaper
Recorders Based on Some Wavelets
Jianxiang Jin and Jianxin Zhu#*
Dept. of Control Eng., Zhejiang Univ., Hangzhou, P. R. China
#Dept. of Math., Zhejiang Univ., Hangzhoii 310027, P. R. China
In this paper, an efficient compression algorithm based on some wavelets for
types of nopaper recorders is provided. Numerical results demonstrate that
the algorithm is better applied to the compression of historical data, for example, temperature, liquid level, flow rate and pressure. Therefore, the algorithm
reduces the need for storage of these data and improves the functions of the
recorders to a maximum.
Introduction
In recent years, by the development of computer technology, a new recorder,
namely, a nopaper recorder has been produced, it seems to solve all problems
existing in traditional recorders. However, if a lot of historical data are stored
directly, a great number of storers are needed. Thus, it is important to inverstigate how to reduce the need for storage of these data when given a recreated
precision and a compression ratio.
Theory
According to signal properties with oscillation, the signals are expanded approximately by wavelet bases [1,2]. Considering the simplicity of the processing time
and program, we adopt Daubechies' compactly supported orthonormal wavelet
bases { ( p n ( 2 Q z - k ) } [3]. As q is a large enough number:
+m
y(Z)
1
Ck(pn(2'Z
- k ) , 0 5 Z 5 m.
- . .(1)
k=--oo
Note that cpn(u)= 0 as u 5 0 or u 2 2n
29m-1
y(z) x
1
C k ( p n ( 2 ' Z - k),
- 1. Then
o < z 5 m.
2-2n
By the equation of dauble scaling function, that is,
2n-1
yn(Z)
= fi
1h , j ~ n ( 2 z
-j),
j=O
* Author for correspondence (email: zjx@math.zju.edu.cn).
109
J. Jin and J. Wu
where { hj}:zi' represent filtering coefficients of the Daubechies' compactly supported wavelet p n ( x ) , we construct an iterative formula:
2n-1
cpik'(z)= Jz C hjcpLk-1)(2~- j),
(k = 1,2,
,jo),
x E R.
. (2)
j=O
Let
where k is an iterative order, 0 i 5 (2n - 1)2k, ( 0 5):z 5 2n - 1 ).
According to (2), we can obtain {cp$")(xiGO))}to
(2n-l)Po
, and
where x E [xi
9,
( T = 0, 1 , 2 , . - - ,(2n - 1)2Jo - 1 ).
It is proved that cpik'(x) + q n ( x ) as k * +oo
Considering { ( j , y j ) } s o
'
[3].
{ ({, y j ) } g 0 , then from (1) we obtain
L
2-271
1
where 1- = p.
For simplicity, we suppose
is an integer. If na 2 - 2n > 0 and i > *,
then
- 1- (2 - 2n) 1 c rn
1. So, we can realize the compression.
+
Denote n o = 2
T
- 2n, nl = - 1 and
= &min
€ R OSjSrn
max ~ y j
s.t.
1 10,
+
+
n2
= nl - no
C
t k r p n ( S1 - k ) l ,
k=no
where n3 represents the number
points after compression (n3 5
110
+ 1, then let
. . . (3)
of necessary
722).
Data Compressionfor Types of Nopaper Recorders
The problem (3) can be solved by the Abdelmalek algorithm [4]. By analysis,
we obtain the following facts:
Lemma 1. Suppose A = ( u , j ) z X zis a matrix, and B = ( b i j ) z x z is a nonsingular submatrix of A , then
-n
where E i > ii.
Lemma 2. [5] Let A E RrX' is an invertible matrix. Suppose that
( u + A u ) E Rrxl satisfy the following equations:
{
Au=b
(A AA)(u
+
u
and
+ Au) = b
where b # 0 and b E R r X 1 , then
According to Lemma 1 and Lemma 2, we can obtain the following results:
Lemma 3. Let 3 5 i 5 m , 10 5 m 5 160 and f be an integer. The equation
obtained by the Abdelmalek algorithm to solve (3) is as follows:
**T
( ;)
... (4)
= fB*.
Here the notations B*,x* and f p in the Abdelmalek algorithm are the same
as the ones in [4], B*T is a matrix of order k + 1, and k = n3 5 n2.
Then
I
n = 2,
n = 3,
n = 4,
n = 5,
n = 6,
n = 7,
+12
k +12
k
k
k
k
k
+1 2
+12
+12
+12
4, Cond(B*T), 2 2.65 x 10';
6, Cond(B*T),
1.52 x lo3;
8, Cond(BfT), 2 1.11 x 10';
10, Cond(B*T), 2 4.60 x lo7;
12, Cond(B*T), 2 4.75 x 10';
14, Cond(B**), 2 1.32 x 10l2.
N
N
Proposition. Let n = 5 , i E { 'i I 11 515 m , 1 ( mod(2) ) is odd },
be an integer. Then the equation (4)is abnormal.
32 5 m 5 160 and
Thus, we should avoid taking any odd i (mod(2)) for good stability.
Due to noise, the signals often contain a lot of tooth-like parts. It is very
difficult to compress these signals. To solve this problem, we propose two approximate methods (I) and (11) as follows.
(I). First the signals are filtered from { y i } z o to { y i } z 0 , second the signals
{ y i } z 0 are compressed by (3), here:
N
N
J. Jin and J. Zhu
N
{ Y,'
gj+l=
a ~i +(1
- a)yi+lt
(i = 0, I , . - . , m- 1) and (0
< o < 1).
YO,
(11). Finding an approximate envelop of the signals by the following steps.
Step 1: for {(j,y j ) , 0 5 j 5 m } , finding out their maximal points and minimal
points with the criterion
Yi+l
- 2 ~ +i Yi-1
{ >c 0,
0, ( i , y i ) is a minimal point;
( i , y i ) is a maximal point.
Step 2: if the numbers of signals among two neighboring maximal points (minimal points) are greater than lo, then these signals are combined into maximal
points (minimal points).
Step 3: for the minimal points and maximal points, we construct two cubic spline
interpolation polynomials, which are represented as s1 (x) and s2(z) respectively.
So, the signal function can be approximately represented as follows:
Y(X)
= W ( X ) + (1 - w ) s 2 ( 5 )
where 0 5 w 5 1 and 0 5 x 5 m.
For a standard curve y = e-*,
sinx, let
xj
=
-j,
( j = 0 , 1 , 2 , . . . ,m ) and
5 0.00375, that is, the limit of relative error is
0.005. Through computation, we may choose n = 5. The computation results
are as follows:
' m = 32,
1 = 16, n 3 = 10, E = 0.009199;
m = 64,
1 = 32,
n g = 11, E = 0.009613;
m = 64,
1 = 16,
n3 = 13, E = 0.005800;
m = 96,
1 = 32, ng = 12, E = 0.011698;
1 = 48,
m = 96,
n3 = 11, E = 0.009609;
m = 128, 1 = 64,
ng = 10, E = 0.009639;
m = 128, 1 = 128, ng = 10, E = 0.029729;
m = 160, 1 = 40,
n3 = 13, E = 0.005709;
m = 160, 1 = 80,
ng = 11, E = 0.009648;
m = 160, 1 = 20,
n3 = 17, E = 0.002191 (relatively adaptive).
j o = 4. It is required that E
But if
{
zj =
1 0 . ~f , (j = 0,1,2,. . . ,m ) here zj E [0,5 x 2 ~ 1 then
,
m = 160, 1 = 20, n3 = 17, E = 0.093104;
m = 160, 1 = 16, n3 = 19, E = 0.035359.
These results show that the effect of compression is related to the density between
given points .
Application
We have dealt with some real data, for example, temperature, liquid level, flow
rate and pressure. For the temperature data, choosing m = 100, n = 5,
j o = 5 and .f = 50, we can obtain n3 = n2 = 10, E = 1.105755, the relative
112
Data Compressionfor Types of Nopaper Recorders
error=0.00027 and the compression ratio=l0.1 where the time step=lO (second).
Corresponding to Figure 1, which describes the original data of liquid level,
Figure 2 shows the compressed result, where m = 100, n = 5, jo = 5 , 1 = 50, the
time step=lO (second), n3 = n2 = 10, E = 3.848262, the relative error=0.00094
and the compression ratio= 10.1.
L
1995
1970
20
40
60
80
100
t
Figure 1 . Origind data of liquid level.
t
Figure 2. Compressed data of liquid level.
The original data of flow rate are shown in Figure 3. Figure 4 demonstrates
the compressed result, where m. = 100, n = 5, jo = 5 , 1 = 50, the time step=lO
(second), 723 = n2 = 10, E = 14.191399, the relative error=0.00346 and the
compression ratio=lO.l.
2450
F
2400
2050
1
0
20
40
60
80
1
100
Figure 3. Original data of flow rate.
2400
I
F
2400
2350
0
20
40
60
80
I
100
t
Figure 4 . Compressed data of flow rate.
113
J. Jin and J. Zhu
P
1900
r
1650
-
'.*.-..'
a.
1400
t
m
Figure 5. Origind data of pressure.
P-
1900
1660
a
00
y
%+JO
1100
20
60
40
80
100
t
Figure 6. Compressed data of pressure.
Figure 6 shows the compressed result for the original data of pressure described by Figure 5, where rn = 100, n = 8, j o = 4, 1 = 30, the time step=lO
(second), n3 = n 2 = 17, E = 18.399235, the relative error=0.00449 and the
compression ratio=5.94.
Above results demonstrate that our method can be better applied to the
compression of the historical data for nopaper recorders.
Conclusions
For some historical data of types of nopaper recorders, such as temperature,
liquid level, flow rate and pressure, they can be successfully compressed by means
of our developed method, where their relative errors are less than 0.005 and most
of their compression ratios are greater than 10. Thus, this technique of data
compression can improve the nopaper recorders and reduce the need of storage
to a maximum.
Acknowledgement
This work was supported by the Natural Science Foundation of Zhejiang Province
(No. 198016).
References
1. Yan Zhuang, and John SBaras. 1994. Optimal wavelet basis selection for signal representation. SPIE (Wavelet Applications),2242,200-211.
2. Murtaza Ali, and Ahmed H. Twefik. 1994. Real time implementation of second generation of audio multilevel information coding. SPIE (Wavelet Applications),2242,212-223.
3. DaubechiesJngrid. 1992. Ten Lectures on Wavelets. Capital City Pres ,USA.
4. Shen Xiechang. 1984. Implement for Best Appro& with Polynomial. Science and Technolgy Press,Shanghsi.
5 . P. G. Ciarlet. 1990. Numerical Analysis of Matrix and Optimization. High Education
Press,Ekijing.
Received: 12 June 1999; Accepted after revision: 12 May 2000.
114
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