2009 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2009), October 4-6, 2009, Kuala Lumpur, Malaysia
Piecewise-polynomial Bases and Specialized Processor for Digital Signal Processing Hsein-Ping Kew Department of Ubiquitous Engineering, Dongseo University, Busan, South Korea. E-mail:
[email protected]
Hakinjon Zaynodonov Division of Design & IT, Dongseo University, Busan, South Korea. E-mail:
[email protected]
Abstract—In the work, the results of research for the systems of basic function of Haar and Harmut are resulted. The results of modeling and the comparative analysis of possibilities of these bases are resulted also. The structure of the specialized processor of signals processing in piecewisepolynomial bases is developed and tested herein. By modeling in МАТLAB environment with application standard Simulink, components possibilities of the offered algorithms and structure of the specialized processor are shown. Keywords — Effect of Compression; Fast Transformation; Piecewise-polynomial Bases; Signals Processing
I. INTRODUCTION For construction of patterns for the signals received from actual objects, traditional harmonious functions are widely applied. It explains that many signals received from actual objects can be easily presented by a set of sine wave and cosine fluctuations for which the device of Fourier analysis is used. The result is the transition from time to frequency functions. However, representation of time function for sine and cosine functions is only one of the many representations. Any full system of orthogonal functions can be applied for the decomposition in numbers which correspond to Fourier series. In technical appendices, wide distributions have received orthogonal systems of basic functions which there are fast transformations algorithms. Transition to piecewise-quadratic basic functions brings a lot of advantages. It allows to improve the accuracy of approximation, to reduce the quantity of factors for approximation and lastly save a memory size. I. THE BASIC THEORETICAL DATA Elementary functions which are decisions of the simple differential equations, find a very wide application in practical engineering problems. Usually in the engineering literature under the terminology elementary, it is understood in general as simple the functions of one or two variables having limited quantity of extrema, without points of break, with the limited steepness in the set limits of change of argument [1]. They serve for construction of mathematical models of the signals received from real objects.
978-1-4244-4683-4/09/$25.00 ©2009 IEEE
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Do-Un Jeong Division of Computer & Information Engineering, Dongseo University, Busan, South Korea E-mail:
[email protected]
In the first case, frequency defined by the periods discharged a signal, which particularly in gravimetric prospecting lies within the range from a shares of second to about several seconds [2], [3]. In the second case, the duration of the signal is from 10 ms to 400 ms with frequency range from 10 kHz to 1700 kHz. Usually the frequency range of precursor-signal divides at low frequency from 10 up to 50 kHz and high frequency from 50 up to 1700 kHz. Orthogonal system of basic functions which are set on a valid axis, for which there also exist algorithm of fast transformation have received wide circulation in technical appendices. They can be broken into two classes which is global basic functions and allocative basic function. Global basic functions are such whose values are not equal to zero on one subinterval. Examples of global basic functions are Walsh Functions [4], [5], numerical [6], [7], [8], sawtooth and etc. Allocative basic functions are nonzero values which are set on the enclosed segment. Examples of allocative basic function are Haar functions [4], [5], [7] and Harmut’s [5], [7], [9]. Splitting of the valid axis is usually done by doubled the rational. Later considering the interval [0, 1] or [1, 0] and use the concept of a binary piece, which is as a result of division of a given interval 2р equal parts (p = 1, 2...): h
h
.
(1)
where j 0,1, … , 2 , k j 2 For example, binary pieces can serve as intervals [0; 1], [ ; , ; . The length of a binary piece h equals to
h
2
.
(2)
, are their lefts and right half respectively where and also representing binary pieces of
;
,
;
.
(3)
2009 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2009), October 4-6, 2009, Kuala Lumpur, Malaysia
The system of un-rationed Haar functions to the form is defined:
1 1
(4)
0
.
1. It is necessary to note, that har x The number P is the order of Haar’s function it is known that Haar’s series.
∑
.
.
(5)
This can provide as uniform including the uniform best and mean-square approximation. All depends on the method of calculating the factors. Haar and Harmut’s bases draw the attention of experts for two main reasons which are firstly the reduction of the number of factors necessary for approximation with a given accuracy in relation to the general number of binary pieces. Secondly is absence of "long" operations in expression. Operations of addition, subtraction and shift are used only. Researches of the methods is done for reducing the volume of tables of coefficients, improvement of parameters of "smoothness" by obvious means leads to systems of piecewise-polynomial basic functions of higher degree. Piecewise-linear basic functions which are Shauder’s function are most easily got as a result of integration with a variable top limit of Haar’s orthogonal piecewise-constant functions [8], [10], [11]:
shd x
2
har r dr .
(6)
It is necessary to consider also, that shd x 1 and shd x x. Often in practical appendices series of piecewise-linear functions with the purpose of getting the amplitudes of all basic functions that are equal to unit, are convenient to operate with “normalized" systems:
shd x
2
har r dr .
algorithms, but practical use of spectra in adamars ordering is inconvenient ordering, since such spectrum does not provide uniform convergence of the partial sums for continuous functions which are usually considered as signals. The approximation of continuous signals with partial sums of a spectrum provides uniform convergence with classical ordering functions of Haar [13]. As dyadic ordering differs from classical only by the arrangement of functions inside groups, it also has this property. Dyadic the arrangement of Haar functions provides greater uniformity of approximation as rejection of members of some occurs not on one hand, and in regular intervals on all range of definition. har 1 1 0 -1
0,5
0 -1
hain 2 1 0 1 -1
har 3 1 0 -1
hain3 1 0 1 -1
har 2 1
0,5
0,5
har 4 1 0 -1 har 5 1 0 -1 har 6 1 0 -1
0,5
0,5
0,5
har 7 1
(7)
where 0,1, … ; and 0,1,2, … The har x functions can be placed in an orderly manner (Walsh’s functions can analogically be arranged) in three ways. Firstly, is to arrange binary factions’ k in the natural order of increment Adamarov’s ordering. Secondly, arrange k in ascending order of inverted code dyadic ordering. Thirdly which is lastly, grouping k in group with number of categories after a comma 1, 2, 3, etc., and inside of each group to arrange fractions in ascending order - classical ordering. Fig. 1(а) shows piecewise-constant bases of Haar; Fig. 1(b) shows the piecewise-linear bases of Haar and Fig. 1(c) show the piecewise-parabolic bases of Haar. Adamars ordering of Haar functions leads to the reception of the optimal fast transformation of Haar (FTH)
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hain 1 1 0 1 -1
0 -1
0,5
hain 4 1 0 1 -1 hain 5 1 0 1 -1 hain6 1 0 1 -1 hain 7 1 1 0 -1
0,5
1
haid 2 1 0 1 -1
0,5
0,5
0,5
0,5
0,5
0,5
haid 1 1 0 -1
1
1
1
haid3 1 0 -1 haid 4 1 0 -1 haid 5 1 0 -1 haid6 1 0 1 -1 haid 7 1 0 1 -1
0,5
1
0,5
1
1
0,5
1
0,25
0,25
1
0,5
0,5
0,75
1
0,75
1
Figure 1. (a) Piecewise-constant; (b) Piecewise-linear; (c) Piecewisequadratic bases of Haar
Harmut [12], [13] examined a system piecewiseconstant orthogonal basic functional occupying the position between Walsh and Haar systems. The most simple are functions of system harm x obtained from Haar functions by double accession precisely same under the form, but on an adjacent binary piece on the right, and one of them joins with a + sign, and another with a - sign. As a result, a system of evens and odds relative to the middle of the binary interval functions, whose integration leads to a system of piecewise-linear functions and also even and odds.
2009 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2009), October 4-6, 2009, Kuala Lumpur, Malaysia
hın x
2
hrm r dr .
(8)
Charts of functions of system hrm x are shown on Fig. 2(a). Fig. 2(b) show the charts of system hin x are shown. Fig. 2(c) show the system of hid x . It is possible to see that they are related to localized category. In many practical appendices connected with recovery of functions between readouts, the possibilities of continuous piecewise-linear bases are not enough. This can be explained by two basic reasons which are firstly because of low speed of convergence of the approximations caused by comparatively big error of piecewise-linear interpolation which often leads to significant loss on factors. Secondly, because of non smoothnesses of approximation where the first derivative of disruptive functions of basis and there is the absence of the concept of curvature which causes essential restrictions. For example, it is possible to skip the maxima and minima of functions.
signal processing which is the problem in achieving the maximal speed. The structure of the specialized processor in parallelconveyor-based, carrying out fast transformation of Haar on algorithm Cooley-Tukey is shown in Fig. 3. The structure consists of a block of buffer registers for storage N/2 of readout data (where N-quality of entrance readout), four processor blocks and an ultrafast operative memory (RAM) for storage both intermediate and final factors. The degree of universality of the proposed computing structure is caused by the application of the piecewisequadratic basic functions, allowing approximation with required accuracy of signals which is still under research. High speed and relative simplicity of hardware realization are reached in many aspects. With the use of the parallelization principles, conveyorization calculation and overlapping of operation of input with processing is performed.
Figure 3. Structure of the specialized processor for digital signal processing
Figure 2. (a) Piecewise-constant; (b) Piecewise-linear; (c) Piecewisequadratic bases of Haar
II. MODELING STRUCTURE OF THE SPECIALIZED PROCESSOR FOR SIGNAL PROCESSING IN PIECEWISEPOLYNOMIAL BASES Researched graph of fast spectral transformation in localized bases show that their various modifications define sets of versions of special-purpose computing structure. A number of works were devoted to issues the development of structures and principles of their construction, thus the opportunities of work in real time is achieve. However, there is a major problem in digital
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The development of the computing structures which are carrying out transformations on piecewise-quadratic functions and transition to piecewise-quadratic functions Haar and Harmut allow to improve the accuracy of approximation, to reduce quantity of the factors necessary for approximation and lastly save the memory size. Application of parallelization and conveyorization principle and overlapping of input operation with processing able to increase the speed of specialized computing structure. High speed parallel-conveyor structure of specialized processor was developed for the performance of fast transformations in piecewise-polynomial. This is different by wide introduction of principle of parallelization,
2009 IEEE Symposium on Industrial Eleectronics and Applications (ISIEA 2009), October 4-6, 20099, Kuala Lumpur, Malaysia
conveyorization and overlapping of inpput operation with processing. III. STRUCTURE OF THE SPECIALIIZED PROCESSOR In Fig. 4, the Simulink-model of o the specialized processor is shown. For modeling of the proposed structure in the environment of MATLAB with application of resources Simulink, folllowing blocks are used. It show that two blocks of Display and Constant for feigning of an entry signal and Zero-Ordder Hold block for simulation of digitization and quanntization of input analogue signal. Display block is used for fo numeric display for input values; constant block is for f constant value parameters. A demux block is used to split s vector signals into scalars or smaller vectors. In thee process, there is several add and subtract inputs blocks. Later, L a mux block is used for multiplex scalar signals. Finally F the output signal is display for the control of signalls at various stages of processing and for link with MATLA AB environment.
the following. If in FTW algoriithm for spectral processing an array from N=2n values of an initial signal, n.2n addition and subtraction operrations in FTH algorithms required number of operations is i reduced up to 2(n-1). For fast Harmut transformation, thee parameter value is 3N-4. Fig. 5 and Fig. 6 have shown s the results of the comparative analysis on effectt of signal compression in the form of the diagram. Piecew wise-constant bases (PCB), piecewise-linear bases (PLB)) and piecewise-quadratic bases (PQB) were taken foor research purpose. For processing of these bases, fuunctions consisting of the combination of elementary functions f (CEF), a signal received as a result of geophyysical researches (CR) and also a file received as a ressult of bench test of car components (RBCC) are taken.
Figure 5. Comparative anaalysis for Haar’s bases
Figure 4. Simulink-model of specializzed processor
IV.
RESULTS OF THE COMPARATIIVE ANALYSIS OF PIECEWISE-POLYNOMIAL BASES B
In this work, results of the compaarative analysis of possibilities of piecewise-constant, pieecewise-linear and piecewise-quadratic local bases are shoown. As criterions of comparison the factor were takenn are quantity of necessary operations, effect of signal compression and approximation errors. The requirements for fast spectrral transformation algorithm are minimum quantity of opeerations, simplicity of each operation and a minimum voluume demanded of operative memory. By comparing the fast transformationn of Walsh (FTW) and FTH algorithms using the requuired quantity of addition and subtraction operations, it is possible to say
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Figure 6. Comparative anallysis for Harmut’s bases
2009 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2009), October 4-6, 2009, Kuala Lumpur, Malaysia
The comparative analysis of all Haar and Harmut’s piecewise bases shows that with transition from the lowest degree to higher degrees, it improves the effect of compression. For Haar bases, the effect of compression for PCB makes from 1.8% up to 2.7%, for PLB it makes from 3.2% up to 7.1% and for PQB it makes from 5.5% up to 10.9%. It show that the effect of compression is constantly grows. For Harmut bases, the effect of compression for PCB makes from 4.8% up to 12.3%, for PLB it makes from 7.3% up to 29.3% and for PQB it makes from 17.2% up to 56.2%. The effect of compression for Harmut bases case is constantly grows too. The comparative analysis of Haar and harmut’s bases shows that from the point of view for effect of signal compression, Harmut’s bases gives better results. V. CONCLUSION Thus, as a result of research of systems of piecewiseconstant, piecewise-linear and piecewise-quadratic bases their limitations are shown. The research result of piecewise-constant, piecewise-linear and piecewisequadratic bases and their limitation is shown. For eliminating the limitation of piecewise-polynomial bases and improving the approximation properties of investigated bases, the fast transformation algorithm in piecewise-quadratic bases based on the application of parabolic basic spline is proposed. Transition to Haar and Harmut’s piecewise-polynomial bases allows the research done on the effect of signal compression. Harmut’s bases give better effect on signal compression compare to Haar’s bases. The effects of signal compression increase the accuracy of approximation and also increase the achievement in highefficiency computing structures on their basis [7], [9]. Thus, it able to save the memory size for the spectral factors storage especially during the processing of real experimental files from 5% to 24% and during the processing of elementary functions as well as functions consisting of their combinations from 11% to 70%. The proposal structure of the specialized processor realizes an investigated class of signals with demanded accuracy, possesses the big factors of compression at the expense of application of piecewise-quadratic basis. Application for results of researches in various systems of modeling, the controlling and management has shown that developed piecewise-polynomial methods of approximation of signals can be successfully realized
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algorithmically in the form of a system of program components, and also in hardware in the form of subsystems of digital signals processors. ACKNOWLEDGMENT This research was financially supported by the program in the Regional Innovation Center is conducted by the Ministry of Knowledge Economy of the Korean Government. REFERENCES [1] [2] [3]
[4] [5] [6] [7]
[8]
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