Time Frequency Analysis and FPGA Implementation of Modified S- Transform for De-noising

Transcription

1 Vol. 4, No., June, 011 Time Frequency Analysis and FPGA Implementation of odified S- Transform for De-noising Birendra Biswal 1, Pradipta Kishore Dash, ilan Biswal 3 1 GR Institute of Technology, Rajam, Srikakulam 5317, A.P, India S O A University, DRC, Bhuwaneshwar , India 3 Silicon Institute of Technology, Bhuwaneshwar-75104, India birendra_biswal1@yahoo.co.in, {mpkdash.india, milan.biswal}@gmail.com, Abstract A new approach to modified S-transform based non-stationary power signal de-noising is presented in this paper. odified S-transform is employed in de-noising techniques to separate high frequency noise which is prevalent in practical signals like power disturbance signals from the low frequency undistorted signals. In this we have tried to bring out the advantage of the modified S-transform and its application in power signal analysis through various examples. S-transform with modified Gaussian window is found to provide excellent normalized frequency contours of the power signal disturbances suitable for accurate detection, localization and classification. The implementation of the de-noising scheme has been carried out in FPGA using non-stationary power signals. Key words Non-Stationary signals, power signal disturbances, modified S-transform (ST), modified Gaussian window, scaling factor, time-frequency resolution, de-noising, FPGA. 1. Introduction Various methods have been used to study non-stationary signals and both spectral as well as time localized information has been obtained. Amongst the various techniques available, short time Fourier transform (STFT) and wavelet transform (WT) are used most often. The short- time Fourier transform (STFT) views a signal in terms of a finite time series and uses a window of fixed width, and therefore, it is unable to provide effective frequency relative resolution. On the other hand the wavelet transform (WT) analyzes the signal in terms of the translations and dilatations of a basis wavelet function, thereby gives both time and scale information. Thus WT is effective in providing time-localized information but the spectral information is in terms of scales rather than the absolute frequency. Current advances in signal analysis have led to the development of a new method for non-stationary signal analysis called the S-transform. The S-transform as defined by Stock well et al [1] is an extension of the short- time Fourier transform along with phase corrected wavelet transform and it allows a signal to be analyzed in terms of both time and frequency simultaneously. It is being increasingly used in the analysis of power system disturbance signals and non-stationary signal analysis applications. Dash et al [] have used the S- transform to detect and classify the power disturbance signals. It has also been employed in the analysis of seismic signals by Schimmel and Gallart [3] and gear vibration signals by cfadden et al [4]. The proposed study uses a modification to the Gaussian window whose width can be adjusted to produce excellent time and frequency resolution. The localized 119

2 Vol. 4, No., June, 011 spectral information thus obtained is useful in designing filters which are capable of enhancing or attenuating specific signal components along time and frequency axes. The filtering methods have earlier been used in seismic signal analysis by Pinnegar and Eaton [5] and in brain activity detection applications by Goodyear et al [6]. A method of time-time analysis by Pinnegar et all [7] and the inverse S-transform in filter by Schimmeland et all [8] have also been used. It is shown in the paper that the forward and backward transforms of our proposed method exists and provide an excellent time frequency representation of the power signal. A de-noising scheme is precisely discussed and finally simulated with the help of ATLAB by comparing both wavelet transform and S-transform. Simulation result shows that the modified S-transform based de-noising scheme is more effective than the wavelet based denoising method. Recent advances in FPGA technology not only provides significant increase in the logic states, but also furnishes relatively significant amount of flexible internal RA modules. The internal RAs provide the advantage of on- chip storage for intermediate results and avoid the most time consuming external memory access operation. These advances are particularly useful for the application of interest in this work. In this a modified S-transform (ST) based power signal filtering using FPGA is implemented to exhibit the practicality of the transform.. The S transform The S-Transform of a continuous time series u(t) as given by Stockwell[1] is: i ft S(, f ) u( t) w( t, f ) e dt (1) Here the window function used is a Gaussian model given by f t k f w( t, f ) e, k 0 () k In which is the frequency, t and are the time variables, k is a scaling factor which controls the number of oscillations in the window. Taking advantage of this fact that the FT of a Gaussian is a Gaussian we can write the above equation as follows f i S(, f ) u( f ) e e d (3) has the same dimension as frequency. This way of writing the ST will be called the frequency ST..1 The Inverse S-Transform The Fourier transform U( f ) of a signal ut () and its inverse is given as: i ft U( f ) u( t) e dt (4) u( t) U( f ) e i ft df 10

3 Vol. 4, No., June, 011 Then the inverse S-transform is given by ift u( t) S(, f ) d e df (5). The S-Transform and It s Inverse in the Discrete Case Let u(n)= u(nt), n=0,,n-1 denote the discrete time series of length, corresponding to u(t) with time sampling interval of T. Let f s be the sampling frequency and f 0 be the frequency step, = f s / f 0 and m,... 1 is the index of frequency range. Then the discrete ST is given by 1 m( p n) k nm N 1 m i S( p, m) u( n) e e m (6) 0 k where p=0,,n-1 denotes the time index in the S-transform. The definition of the ST in the frequency domain (3) is used to compute the ST in discrete case. It is thus computed as For m=0 n np i 1 m n mk n0 S[ p, m] U e e S[ p,0] u( n) (8) The inverse transform is given as: 1 1 N 1 i mn u[ n] m / n0 S[ n, m] e (9) where all symbols stand for same parameters as (6). 3. The odified S-Transform We propose a slight modification to the Gaussian window being used and redefine its dilation parameter as: k, k 0 and c 0 c f (10) The S-transform can now be written as: c c ( t ) f f k i ft S(, f ) u( t) e e dt N1 n0 (11) k c f i U ( f ) e e d (1) The parameter C was initially introduced as a switch between STFT (C=0) and S- transform(c=1) but later we saw that could be better controlled and the window gave better resolution if C will be slightly more than 1.At the same time value of C < 0 can not be (7) 11

4 Vol. 4, No., June, 011 taken, since on doing so instead of being proportional to the time period of the signal becomes inversely proportional to it which takes it outside the basic definition of S- transform. We shall henceforth refer to (11) and (1) as modified S-transform (ST) to distinguish it from the one proposed by Stockwell. The contour plots of the amplitude spectrum of the modified S-transform (ST) have been shown in figures () (5), for different values of the scaling parameter C. The analyzed signal Fig.1 is a mixture of sinusoid of frequency order, having amplitude 1. The transients taken are of the order of 10 and 0 respectively, each having amplitude 1. The first transient occurs in between the sampling points 100 and 00 and the second lies in between the points 300 and 400. Thus the center or peak points for the transients are positioned at time points 150 and 350 respectively. Fig. shows the contour plots of the amplitude spectrum of the original ST. The contour plots of Fig.3 and Fig.5 has been obtained for C = 0.9 and C = 0.7 and it is observed that, when C<1, better frequency localization is obtained. For C=1.1, the window is squeezed around the center points, thus giving good time localization, while the contours are scattered over the frequency axis, which indicates poor spectral information as depicted in Fig.4. Thus it is observed that the scaling parameter C acts as a further fine tuning element for a particular resolution level. By tuning the parameter C highly effective ST based time and frequency analysis can be achieved. Fig1. Original time domain signal consisting of sinusoidal content of frequency order and two transients of frequency order10 and 0 centered at time points 150 and 350 respectively Fig.Contour plot of Original S-transform for C =1 Fig3. Contour plot of odified S-transform for C = 0.9 1

5 Vol. 4, No., June, 011 Fig4. Contour plot of odified S-transform for C = 1.1 Fig5. Contour plot of odified S-transform for C = Discrete Form of the odified S-Transform (DST) In discrete form the signal u(t) can be written as u(t)=u(nt) where n index of sample T time interval between two consecutive samples c c 1 m ( pn) nm i N 1 m k m0 S( p, m) u( n) e e (13) k For m=0 n np 1 c i m n mk n0 U e e N 1 n0 (14) S [ p, 0] u ( n ) (15) m, n, p,, N represent same parameters as in (6). 3. The Inverse ST As in case of S- transform, it can be shown that a time average of ST yields the Fourier transform of the original signal. Thus it can be shown that the proposed transformation is a representation of the local spectra that is fully invertible which is shown below: c c ( t ) f f k i ft S(, f ) u( t) e e dt k 13

6 Vol. 4, No., June, 011 So we get S, f ) u( t) For a Gaussian function c c ( t ) f f e k ift ( d e dt (16) k ( t ) f k k e d (17) f From the above we get c ( t ) f k k e d (18) c f Putting (18) in (16) we get i ft S(, f ) u( t) e dt (19) U( f) (0) The original equation can be obtained as: u( t) U( f ) e i ft df (1) i ft S(, f ) d e df () The inverse ST in discrete form (IDST)is thus given as: i mn 1 N 1 u[ n] n0 S[ n, m] e (3) 1 m 4. ST in De-noising Noise is an inherent property of all the practical systems. Sometimes these noise components create a hindrance in studying the finer details present in a stochastic signal. Thus removal of noise elements is vital in signal analysis. ST is highly suitable in separating the noise elements from the signal components as it acts as a power discriminator. Since the power content of the noise is usually lower than that of the signal, it is possible to remove them using ST. The procedure for de-noising the signal using ST is explained as follows: 1. The DST of a signal is performed.. A power threshold p is set. The ST coefficients greater than p are separated from the coefficients which are less than p, i.e. the coefficients are filtered according to their power contents. The higher power elements are the signal components, which are called S-approximation coefficients, and the low power 14

7 Vol. 4, No., June, 011 components are called the S-detail coefficients that consist of the non-sinusoidal noise elements. This can be written as T = T h +T l Where T h is the S-approximate coefficients and T l is the S-detail coefficients such that T h (f,) >= p and T l (f,) < p Here the threshold p is chosen suitably according to the ratio of the power of sinusoidal and non-sinusoidal elements. We have used p =0.1. A suitable noise threshold n is used and an approximate thresholding technique is performed. In our work we have used hard thresholding where the noise threshold is given as a function of the maximum power present in the transform matrix. Here, n d max{ T( f, )} (4) Where d lies between the ranges 0 d 1 3. Now all the S-approximate coefficients obtained in each step are added with the thresholded S-detailed coefficients and the IDST is performed to obtain the denoised signal. Here it is noted that the level of de-noising depends on the number of times the ST is performed. The presented algorithm can be established in a mathematical form by considering a noisy signal given as, u n ( n) u( n) ( n) (5) Where θ (n) is the noise element and u (n) is the pure signal. The odified S-transform of the noisy signal is given as ifn Tn ( f, ) { u( n) ( n)} w( n, f ) e dn (6) u( n) w( n, f ) e ifn dn ( n) w( n, f ) e T Ld TH ( f, ) TL ( f, ) T L f ( p ) (7) Where f ( p ) is an approximate thresholding function given as, 0, if TL ( f, ) p f ( p) 1, if TL ( f, ) p Now the de-noised threshold transform is T T T T T f ( p ) (8) d H Ld H L ifn and the time domain signal is obtained by taking the inverse S-transform given as d n) Tn ( f, ) u ( d (9) U ( f ) e d k The DST de-noising model is represented in Fig.6. c f e i dd dn 15

8 Vol. 4, No., June, 011 Fig.6 De-noising model By rearranging the order of integration and simplifying the equation we get ifn ud ( n) { Tn ( f, ) d} e df (30) Fig.7 Original noisy signal Fig.8 A noisy Doppler shift signal 16

9 Vol. 4, No., June, 011 Fig.9 Smooth signal obtained using S-approximation coefficients Fig.10 Smooth signal obtained using Symlet-4 wavelet function The original noisy signal is shown in Fig.7 and the noisy Doppler shift signal is shown in Fig.8.The de-noised signal obtained using both S-transform and wavelet transform is shown above in Fig.9 and in Fig.10 respectively. 4.1 De-noised Algorithm 1) Enter the data sequence for the signal to be filtered or provide the link if any external source of signal is to be added. ) Calculate the S-transform of the signal using equations (9) & (30) though both the equations can be used, and for simplicity of calculations we have used equation (30). 3) De-noising: the S-matrix obtained from step () can now be used to calculate the energy of different frequency components present. It must be remembered that the rows of the matrix represent different frequency voices. The energy can be calculated as: N E[ m] S( p, m). S *( p, m) (31) p0 1 Where m is the frequency index is the time index, E is the energy is the complex conjugate of S. Let Emax be the maximum energy present i.e. Emax max( E) (3) In order to de-noise the signal, a 90% threshold value is chosen. The frequencies whose energy are within a level of 90% from the maximum signal strength are taken to be a component of the original signal and retained and rest of the components are removed. The basic assumption underlying this process is that the original signal has an energy value much greater than that of the noise signal. So it is assumed that the frequency component having the maximum energy value is a component of the original signal. 4. Filtering Approach for De-noising Power Signal 1) Filter the de-noised signal obtained from above using the following filter function: 17

10 Vol. 4, No., June, 011 F( p, m) 1, m m c (33) 0, otherwise where m c is the frequency index of the desired frequency voice. The filtered function is thus given as, S ( p, m) S( p, m). F( p, m) (34) where f S gives the filtered matrix. f The filter is thus a notch filter that allows only the required signal component to pass through. The filter function can however be suitably modified for specific purpose say for example a low pass filter. Here the gain of the filter has been taken as 1 which can again be suitably modified. ) Calculate the inverse of the above signal using equation.(3) to obtain the time domain representation of the desired filtered signal. The simulation result of the non-stationary power signals used and the filtered signals obtained from the above algorithm have been included below. 4.3 Simulation Results The sampling rate for the collection of power signal disturbance data is chosen to be equal to 3.84 khz. Fig.11(a) oscillatory transient without noise (b) oscillatory transient with output SNR of 10dB(c) oscillatory transient with output SNR of -10dB (d)residual noise and disturbances after removal of the desired signal from 10dB SNR signal (e)residual noise and disturbances after removal of the desired signal from -10dB SNR signal(f)filter output. 18

11 Vol. 4, No., June, 011 Oscillatory transient is shown in Fig.11 (a) without noise and Fig.11 (b) shows Oscillatory transient after addition of white Gaussian noise with output SNR of 10dB. Fig.11(c) shows Oscillatory transient after addition of white Gaussian noise with output SNR of -10dB, and Fig.11 (d) shows residual noise and disturbances after removal of the desired signal from the signal with SNR equal to 10dB. Fig.11 (e) depicts residual noise and disturbances after removal of the desired signal with SNR equal to -10dB. Fig.11 (f) shows the filtered signal of desired frequency. 5. FPGA Implementation of odified S-transform A hardware implementation of the modified S-transform algorithm is carried out for simulated real-time applications. The Xilinx [9] SPARTAN XC3S4000 is selected as a target FPGA. It is chosen as it has a large number of input and output pins with ADC and DAC blocks, moderate size of RA blocks, and configurable logic blocks. For the hardware implementation, the same modified S-transform algorithm structure as described for atlab simulation was chosen and the codes were converted to very high speed integrated circuit hardware description language (VHDL). 1-bit wide bus signals are used in the network interconnections. The amount of resources spent by the modified S-transform computation logic after the synthesis is well below 10 % including the RA usage. Several clocks are used in order to convert integer values from/to standard logic vectors. In total, there are 11 clocks required in order to produce the final modified S-transform output. This number can in principle be further reduced for the real time application by removing the conversion between integer and standard logic vectors used in the current VHDL for easier simulation. 5.1 Design of S-Filter Implementing the Filter program in VHDL language requires the data samples to be converted into binary form & the exponential complex equations to be solved in simple trigonometric form. The discrete form of the modified S-transform equation is given by 1 n np n m n c i mk, NT n0 S mt U e e (35) Simplifying the exponential terms in the equation to its Trigonometric form we get n np C i m k e e sin (sinh b cosh b) cos (cosh bsinh b) Simplifying the exponential terms in the equation to its Trigonometric form we get c here a nm / N c b 0.5 ( m ( p n)) N (37) In VHDL language there are no predefined trigonometric functions such as sin, cos, sinh, cosh etc. The trigonometric functions can be designed by implementing their Power series expansion. The power-series expansion for the trigonometric functions that have been used are given below (36) x x x x sin( x) x..., x 3! 5! 7! 9! 19

12 Vol. 4, No., June, x x x x cos( x) 1..., x (38)! 4! 6! 8! x x x x cosh( x) 1..., x! 4! 6! 8! x x x x sinh( x) x..., x (39) 3! 5! 7! 9! In the program a buffer line is created and that line is passed through the whole circuit according to the circuit requirement. The buffer contains the modified S-transform based filter function and the decimal to binary conversion function. The circuit contains the description of architecture part of the program. 5. Data Flow Diagram Fig.1 shows the data flow diagram of the filter process. The input buffer contains the data points of the non stationary signal obtained from ATLAB by simulation. The input buffer is fed to the FPGA Chip where the whole algorithm and architecture of modified S- transform is defined. The signal is processed inside the FPGA chip according to the algorithm and output of the chip is in the binary form (1 bit). That 1 bit binary output is fed to the Digital to Analog Converter (DAC).The DAC output is fed to the CRO channel and the output is viewed in CRO. This process is kept within a continuous loop. Thus the input signal that is processed in the filter resembles in all respect an original time varying non stationary signal. Fig.1 Data flow diagram of the filtering process 130

13 Vol. 4, No., June, Buffer Description and Realization I. The real and imaginary values of the filter coefficients are stored in two separate matrices. II. The sampled values of the input signal are multiplied with the filter coefficients and are stored in two separate product matrices. III. The filtered signal is obtained by extracting the components of the required frequency voice from the modified S- matrix. IV. The sum of the filtered matrix over time is done and stored V. An Inverse Fourier transform is realized on the matrix obtained in IV to get back the original signal.the IFFT coefficients are stored in different matrices and the matrices are multiplied with the matrix from IV for getting the original signal in time domain. VI. The time domain signal we get is in the absolute form. The minimum value of the signal is calculated. At every alternate minimum, the signal is phase shifted by 180 and the result is stored in the signal reconstruction matrix. The resultant sample points are in the form of real data type. In order to be able to obtain the output in a CRO, the output has to be given to the DAC. For this we need to convert the real space output values to their equivalent binary form. In this a decimal to binary conversion function is considered and the output of which is fed to the DAC. The DAC output is then viewed on CRO. The buffer is so designed that it simultaneously takes 3 sample values from the input for processing. The block diagram of the buffer & the processes involved inside the buffer is given below in Fig 1.The RTL schematic diagram is shown in Fig.13. Fig.1 Block diagram of the buffer RTL schematic 131

14 Vol. 4, No., June, FPGA Implementation results Fig.13 RTL Schematic Diagram The FPGA implementation result is described below: Fig.14 (a) shows a signal having spike i.e. input to the FPGA Kit. The corresponding filter output is shown in Fig. 14(b). Fig. 15(a) shows a signal containing transient to which white Gaussian noise has been added with the resultant SNR of 10 db (signal power = 10 time noise power). The corresponding filtered output is shown in Fig.15 (b). Similarly 16(a) depicts a signal with transient to which a white Gaussian noise has been added with the resultant SNR of 10 db (signal power = noise power / 10). The filtered output is shown in Fig. 16(b). Fig.14 (a) Signal with Spike as input Fig.14 (b) Output signal 13

15 Vol. 4, No., June, 011 Fig.15 (a) Transient with 10dB output SNR as input Fig.15 (b) Output signal Fig.16 (a) Transient with -10dB output SNR as input 133

16 Vol. 4, No., June, Conclusion Fig.16 (b) Output signal This paper presents a new modification to the S-transform and it is shown that it retains all the necessary properties of the S-transform. Further it is implemented to design a frequency based de-noising filter to remove unwanted signals from the non-stationary power signals. Due to excellent time- frequency isolation property of the transform it is found to be highly effective even in noisy environment. The system is tested with white Gaussian noise with SNR of the signal varying from 10dB to -10dB. The FPGA implementation simulates the real-time environment by an infinite looping of the algorithm. The effectiveness of the filter is very high in noisy environment. In future the S-transform based filters may be used in highly sensitive signal processing areas like processing of biomedical signals, seismography, radar signal analysis, etc References [1] R.G. Stockwell, L.ansinha, and R.P. Lowe, Localization of the complex Spectrum: The S-Transform, IEEE Transaction on Signal Processing, vol.44, No.4, 1996, pp [] P, K Dash, B.K Panigrahi, and G.Panda Power Quality Analysis Using S-Transform IEEE Transaction on power delivery,vol 18,No,April 003. [3]. Schimmel and J.Gallart, Degree of polarization filter for frequency dependent signal enhancement through noise suppression Bull. Seism. Soc.Amer.,Vol.94,pp ,004. [4] cfadden P.D., Cook, J, G., and Forster, L.., 1999, Decomposition of gear vibration signals by the generalized S-transform: ech. Syst. Signal Process, 13, [5] C. R. Pinnegar and D.E Eaton Application of the S-transform to prestack noise attenuation filtering J. Geophysics Res. Vol. 108,no.B9,P.4,003. [6] B G. Goodyear, H. Zhu, R.A Brown, and R.itchell, Removal of phase artifacts from fri data using a Stock well transform filter improves brain activity detection agn. Resonance ed.,vol.51, pp [7] Pinnegar C.R., L. ansinha, A method of time-time analysis: The TT-transform,Elsevier, vol 13, April 003, pp [8]. Schimmeland J.Gallart The inverse S-transform in filters with time frequency localization IEEE Transaction on Signal processing.vol.53.no 11. Nov

17 Vol. 4, No., June, 011 Authors Birendra Biswal received the aster of Engg. Degree from University College of Engineering, Burla, India in 001 and PhD from Biju Patnaik University of Technology, India. Presently working as Professor and Chairman Post Graduate Studies in GR Institute of Technology, India. His primary research interest focuses on Digital Signal Processing, Adaptive Signal Processing, Statistical Signal Processing, Image Processing, Data ining, System Identification and achine Intelligence. P.K.Dash, Ph.D., D.Sc, SIEEE, FNAE, is currently working as Director (Research and Consultancy) in Sikha O Anusandhana University, Bhubaneswar, India. He is a visiting professor to USA, Canada, alaysia & Singapore. Under his Supervision more than 40 research scholars have completed their PhD successfully. His area of research includes Power System, Control Engineering, Data mining, Intelligent Signal Processing, Soft Computing, Adaptive Signal Processing & Advanced Signal Processing. ilan Biswal is pursuing his PhD in Sikha O Anusandhana University, India. Currently working as an Assistant Professor in Silicon Institute of technology, India. His primary research interest focuses on Digital Signal Processing, Adaptive Signal Processing, Statistical Signal Processing, and achine Intelligence. 135

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