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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