ARM BASED WAVELET TRANSFORM IMPLEMENTATION FOR EMBEDDED SYSTEM APPLİCATİONS 1 FEDORA LIA DIAS, 2 JAGADANAND G 1,2 Department of Electrical Engineering, National Institute of Technology, Calicut, India Abstract- Implementation of wavelet transform into the field of embedded system gives a wider scope of application due to the limitations of Fourier transform. In this paper, we present the use of an embedded system; arm processor LPC2148 based development board for implementing wavelet transform. Reduced system, improved performance and low power consumption has made ARM architecture the leading one in many market segments but very less research has been done on using ARM processor in the field of signal processing. Lifting algorithm for Daubechies- 4 wavelet has been presented for finding wavelet coefficients. The algorithm was tested for current drawn by a three phase induction motor. Also wavelet transform was studied in MATLAB wavelet toolbox. Keywords- Wavelet Transform, Lifting Algorithm, ARM processor, embedded systems. I. INTRODUCTION Technology in the recent century has provided portable and customizable devices, the system architectures of which are a powerful tool to develop friendly visual environments for industrial applications such as control and monitoring tasks. Processing speed and the size of the system are the main aspects while developing industrial tools. Therefore for real time application it is preferred to have direct access to the instruction set of the processor to ensure fast data processing. The most dominant architecture in embedded market is based on ARM which provides reduced system, improved performance and low power consumption. We present in this paper, an example of the use of an embedded system, LPC2148 pro development board for implementing wavelet transform. It is a powerful development platform with LPC2148 ARM7TDMI microcontroller with 512K on chip memory. It is ideal for developing embedded applications involving high speed wireless communication (Zig bee / Bluetooth / Wi-Fi), USB based data logging, real time data monitoring and control, interactive control panels etc. SD can be used for saving data. Wavelet transforms offer a method for solving difficult problems in mathematics, physics, and engineering, with modern applications as diverse as wave propagation, data compression, signal processing, image processing, pattern recognition, computer graphics, the detection of aircraft and submarines and other medical image technology. Wavelets allow complex information such as music, speech, images and patterns to be decomposed into elementary forms at different positions and scales and subsequently reconstructed with high precision. Wavelets have gained advantage over traditional Fourier method in analyzing physical situation where function is piecewise smooth with isolated discontinuities and sharp spikes. It gives good performance in detecting faults in industrial machines. Research on implementing wavelet transform has been conducted using DSP and FPGA for images. The wavelet multi-resolution analysis of a signal performs a filter bank decomposition of the signal using low pass and high pass filters. This permits to use the basic operations of the ARM processor by implementing a series of addition and multiplication operations. The lifting algorithm being the most simplest and efficient to find wavelet coefficients is implemented in this paper. Amongst the various types of mother wavelets, the performance of Daubechies 4 wavelet has been presented in this paper. The paper is organized as follows. Theory of wavelet transform is presented in section II. Study of wavelets in MATLAB is also provided in this section. Lifting algorithm is described in section III. Real time implementation of lifting algorithm is discussed in section IV. Conclusion is contained in Section V. II. WAVELET TRANSFORM Theory of wavelet transform The Wavelet analysis is capable of revealing aspects of data that other signal analysis techniques miss aspects like trends, breakdown points, discontinuities in higher derivatives and self-similarity. Further, because it affords a different view of data than those presented by traditional techniques, wavelet analysis can often compress or denoise a signal without appreciable degradation.a wavelet is a waveform of effectively limited duration that has an average value of zero.comparing wavelets with sine waves, which are the basis of Fourier analysis, sinusoids do not have limited duration they extend from minus to plus infinity. And where sinusoids are smooth and predictable, wavelets tend to be irregular and asymmetric.in [1], the advantages of wavelet transform over Fourier transform are presented. Fourier analysis consists of breaking up a signal into 122
sine waves of various frequencies. Similarly, wavelet analysis is the breaking up of a signal into shifted and scaled versions of the original (or mother) wavelet as shown in Fig (1). There are two main approaches to present wavelet theory: The integral transform approach (continuous time) and the multi-resolution analysis (MRA)/filter bank approach (discrete time). An efficient way to implement discrete wavelet transform (DWT) using filters was developed in 1988 by Mallat. The Mallat algorithm is in fact a classical scheme known in the signal processing community as a 2- channel sub band coder. This original work focused on orthonormal systems where one set of basic functions was used for both analysis and synthesis [2]. In 1992, Daubechies and Cohen developed the idea of biorthogonal wavelets where the analysis basis and synthesis basis are different. This increases the flexibility in wavelet design [3]. There are number of basic function that can be used as the mother wavelet for wavelet Transformation. Since the mother wavelet produces all wavelet functions used in the transformation through translation and scaling, it determines the characteristics of the resulting Wavelet Transform. Fig2. Filtering process Therefore, the details of the particular application should be chosen in order to use the wavelet transform effectively.daubechies wavelet is shown in Fig 3. Fig1. Shifted and scaled versions of wavelet The continuous wavelet transform (CWT) is defined as the sum over all time of the signal multiplied by scaled, shifted versions of the wavelet function C(scale, position) = f(t) Ψ(scale, position, t)dt The results of the CWT are many wavelet coefficients C, which are a function of scale and position. Multiplying each coefficient by the appropriately scaled and shifted wavelet yields the constituent wavelets of the original signal. Calculating wavelet coefficients at every possible scale is a fair amount of work, and it generates an awful lot of data. In discrete wavelet transform (DWT) only a subset of scales and positions are chosen for the calculations.they are chosen in the powers of two, so called dyadic scales and positions. The original signal, S passes through two complementary filters and emerges as two signals, approximations and details.the approximations are the high-scale, low-frequency components of the signal. The details are the low-scale, high-frequency components. The filtering process, at its most basic level is shown in fig 2. The decomposition can be implemented using filtering and downsampling, and can be iterated with successive approximation. Fig.3. Daubechies wavelet Study of wavelet in MATLAB Wavelet coefficients on sine wave were obtained using wavelet toolbox in MATLAB. A noise was introduced in this sine wave and change in wavelet coefficients was observed as shown in Fig 4. III. LIFTING ALGORITHM The lifting scheme is a way of generating a new set of biorthogonal wavelets, it transpires that all biorthogonal filter can be generated using the lifting scheme. It has some additional advantages in comparison with the classical wavelets. All constructions are derived in spatial domain in contrast with classical method which relies on frequency domain. Spatial domain does not require Fourier analysis as prerequisite.the Lifting scheme does not require complex mathematical calculations that are required in traditional methods. In classical wavelet transform technique filter coefficients are convolved with input signal. Each step of wavelet transform applies the scaling filter coefficients and wavelet coefficients to data input. If input data has length N then it will result into N/2 smooth values & N/2 detail 123
values. N/2 smooth values are further processed. This method demands more amount of computation. As filter length increases, computational complexity increases. Constructing wavelets using lifting scheme consists of three phases: Split, Predict and Update.Consider the sequence of samples. Split phase splits the entire set of data into two frames;one frame with even index samples and the other odd. Fig.4. Wavelet coefficients of a normal and a distorted sine wave a new (k) = a old (k) + 3 2 d new (k+1) 4 3 d new (k) + 4 d new (k) = a new (k-1) + d new (k) a new (k) = 2 3 a new (k) d new (k) = 2 3 d new (k) (1) Where x (k) represents the input samples, a (k) s and d (k) s represent approximate and detailed coefficients respectively. IV. REAL TIME IMPLEMENTATION OF LIFTING ALGORITHM IN ARM PROCESSOR The wavelet transform is defined as a multi resolution analysis of a finite energy function. The scaling and wavelet filters and lifting filtering steps can be implemented by integer arithmetic using only register shifts and summation. Block diagram of implementation is shown in Fig 6. Splitting is called lazy wavelet because no mathematical operations are done. In the predict phase, odd set is predicted form even set. This is called dual lifting. The update phase called primal lifting will update even set using wavelet coefficient to calculate scaling function. Fig (1) shows the basic steps of lifting algorithm. X[n] is the original signal, d[n] and a[n] represent detail and approximate coefficients respectively, where P and U are predicting and Updating factor respectively [4]. Fig.6. Block diagram ARM Processor was programmed by first configuring the ADC using ADCR register. There are two in built ADC modules, one with 8 channels and the other one with 6 channels.lpc2148 has two 10 bit analog to digital converters. Basic clocking for these converters is provided by the VPB clock. A programmable divider is included in each converter, to scale this clock to the 4.5 MHz (max). A 10 bit conversion requires 11 of these clocks. It can take maximum voltage of 3.3v. Conversion is obtained by successive approximation. Fig.5. Split, Predict and Update stages in lifting algorithm Every finite impulse response filter wavelet can be factored into lifting steps. Daubechies [5] wavelet transform can be implemented with lifting scheme as shown below. The Daubechies4 lifting algorithm can be implemented by the following steps. a old (k) = x (2k) d old (k) = x (2k+1) d new (k) = d old (k) - 3 (a old ) The level shifted input is given to ADC0 Channel 1 and the DC shift voltage (1.5V) is given to ADC0 Channel 2. The output of both the channels were subtracted and scaled to get the real value of the current drawn by the motor. Using PINSEL register, a particular channel was selected. 10 bit accuracy was selected using CLK. The conversion results were stored in ADDR register. Current drawn by a 3Hp three phase induction motor was sensed using Hall Effect sensors LA 55 -P. It has a conversion ratio of 1:1000. The circuit diagram of current sensors is shown in Fig7. 124
For further verification the wavelet coefficients generated were transmitted to PC through UART communication and plotted using Makerplot software and the values were displayed using serial terminal. Graphical representation of approximate and detail coefficient are shown in fig 11. Fig.7. Current Sensor Circuit Since the ADC in LPC2148 is unipolar, the current sensed was level shifted by 1.5v. The design of Level shifter is shown in Fig 8. The signal was sampled at the rate of 5 khz i.e. with a delay of 0.199msec. So we get 100 samples per cycle Fig.8. Level Shifter The design power supply unit is shown in Fig 9. It gives +15v and -15v to level shifter circuit and +5v through USB for LPC2148 development board. Fig.11.Graphical representation of wavelet coefficients The wavelet coefficients were observed in serial terminal software as shown in Fig12. It shows the approximate and detail coefficients of first level decomposition for each cycle. The experimental setup is shown in fig 13. Fig.9. power supply unit The results were compared with standard wavelet toolbox in MATLAB as shown in fig 10. Fig.12.wavelet coefficients Fig.10. Comparison of Wavelet coefficients using c code for lifting algorithm and matlab toolbox Fig.13. Experimental Setup 125
CONCLUSION Lifting algorithm was tested and verified with MATLAB toolbox. Further it was implemented in ARM processor and wavelet coefficients were plotted. This can be used for various applications such as fault detection and image processing. REFERENCES [1] M. Sifuzzaman, M.R. Islam and M.Z. Ali, Application of Wavelet Transform and its Advantages Compared to Fourier Transform Journal of Physical Sciences, Vol. 13, 2009, 121-134 ISSN: 0972-8791 [2] Mallat, S. G. A theory for multiresolution signal decomposition; the wavelet representation, IEEE Transaction on pattern analysis and machine intelligence, Vol. 11, No 7, Jul. 1989, pp. 674-693 [3] Daubechies, I. Ten lectures on wavelets, Capital city Press 1992 [4] W. Sweldens, The lifting scheme: A construction of second generation wavelets, SIAM J. Math. Anal., vol. 29, pp. 511 546, 1988. [5] I. Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math., vol. 41, pp. 909 996, 1988R. Nicole, Title of paper with only first word capitalized, J. Name Stand. Abbrev., in press. 126