Gibb s Phenomenon Analysis on FIR Filter using Window Techniques

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1 86 Gibb s Phenomenon Analysis on FIR Filter using Window Techniques 1 Praveen Kumar Chakravarti, 2 Rajesh Mehra 1 M.E Scholar, ECE Department, NITTTR, Chandigarh 2 Associate Professor, ECE Department, NITTTR, Chandigarh ABSTRACT In This paper FIR filter has been designed to analysis the effect of Gibb s phenomenon. Three window techniques are used for design simulation Bohman window, Blackman-Harris window, Tukey window. The Gibb s phenomenon has been analyzed by changing filter order in all designed. Blackman-Harris provided minimum stop band attenuation in all three windows. Hence it provided minimum Gibb s phenomenon for FIR fitter Design. Gibb s phenomenon takes place in the form of undesirable oscillations near the band edge of the filter. It is observed that it is significant with large value of filter order. It is reduced with proper selection of window function. The particular window is selected depending upon the application. Key Words: Bohman window, Blackman-Harris window, DSP, FIR, Gibb s phenomenon, Order, Tukey window. I. INTRODUCTION Digital filters are the discrete time systems used mainly for filtering of arrays. The objective of filtering is to improve the quality of signals.they are preferred in many applications like speech processing, image processing. As compared to analog filters, Digital filters are having so many advantages: They are highly flexible and portable.they have negligible effect of environmental parameters. The frequency response of these can be attuned if they are implemented using a programmable processor.digital filters are classified as finite duration unit pulse response (FIR) filters and infinite duration unit pulse response (IIR).In the FIR system, impulse response has finite duration i.e.it has a finite number of non-zero terms.i.e.h(n) = 0 for n < 0 and n M. This h (n) exists only for the duration from 0 to M-1. Hence this is FIR. In IIR system, impulse response is of infinite duration i.e.h (n) = 0 for n < 0.This h (n) exists for the duration from 0 to.hence this is IIR system. [1-2] The difference equation of linear time invariant system is given by: y (n) =- + (1) They do not have feedback so there is no past output term y (n-i) in equation (1). Hence output for FIR is given by: y(n) = (2) FIR filters have several advantages over IIR filters. They have an exact linear phase. For this, the condition is h (n) =h (M-1-N). According to this condition, unit sample response is symmetric about its origin. They are always stable. They need higher value of order for similar magnitude response as compared to IIR filters. They can be implemented efficiently in hardware. IIR filters have feedback, so they are recursive. So difference equation for LTI system in equation (1) represents IIR filters. These filters are designed in analog domain and transforming the design into the digital domain.fir filters are used in filtering problems where linear phase within pass band is required. If it is not necessary, either an IIR or an FIR filter may be used.an IIR filter has less no. of side lobes in the stop band as compared to FIR with same no. of parameters. So IIR filter is preferable in application where phase distortion is tolerable. [1] -[3] Fig.1.Magnitude characteristics of filter Ideal filters are physically unrealizable for real time applications in digital signal processing. Fig.1 shows a

2 87 small ripple in pass band and also in stop band. Transition from pass band to stop band describes transition band. II. GIBB S PHENOMENON IN WINDOWS To design FIR filter there are many methods such as Fourier series method, frequency sampling method, window method and optimal design. Optimum design method can be carried by Remez exchange algorithm. A large computation is required so this method is unsuitable for real time applications. In Fourier series, the transfer function represents a non-causal digital filter of infinite duration. A finite duration causal filter can be determined by truncating the infinite duration impulse response, but the abrupt truncation of the Fourier series results in oscillations in the pass band and stop band.these undesirable oscillations can be minimized by multiplying the desired impulse response by suitable window function. [3] In this method, from the desired frequency response specification Hdr(w), corresponding unit sample response hdr(n) is determined using the following relation Where Hdr(w) = (4) Generally, unit sample response hdr(n) obtained from (3) is infinite in duration, so it must be reduced at some point say n= M-1 to yield an FIR filter of length M (i.e. 0 to M- 1). This truncation of hdr(n) to length M-1 is equal as multiplying hdr(n) by the window Thus the unit sample response of the FIR filter becomes (3) h(n) = hdr(n)*w(n) (5) Window function w(n), is used for filter design. There are several windows for FIR filter deign in digital signal processing. These are: i. Bohman window ii. Blackmanharris window iii. Tuckey window In this paper Bohman, Blackman-Harris, Tukey are used. These are following mathematical equations which govern above window functions. Bohman window: The equation for computing the coefficients of a Bohman window is given by wb(n )= cos + (6) Where 0 n N/2 and window length is L=N+1 Blackman-Harriswindow: The equation for calculating the coefficient of a minimum 4- term Blackman-Harris window is given by wbh(n)= (7) Here = = = = Tukeywindow: The equation for computing the coefficients of a tukey window is given by wt(n)= 1, 0 n α N/2 1/2,αN/2 n N/2 (8) The window length is L=N+1 Truncation of hdr(n) to length M-1 is same as multiplying hdr(n) by the rectangular window defined as wr(n) = 1, n M-1 0, otherwise (9) Equation (5) shows unit sample response of FIR filter. Frequency response of FIR filter is obtained with Fourier transform of equation (5).i.e H(ω) = F.T( hdr(n)*w(n) ) = Hdr(ω) * W(ω) (10) According to equation it shows that response of FIR filter is equal to convolution of desired frequency response with that of the window function. Here W(ω) is frequency domain representation of window function. It is calculated by W(ω) = (11) The side lobes of W(ω) create undesirable ringing effects in H(ω). These oscillations or ringing is produced because of side lobes in the frequency response of W(ω). These side lobes are generated because of steep discontinuity of window function. This oscillatory behavior near the band edge of the filter is called Gibb s phenomenon. In rectangular window side lobes are larger in size because discontinuity is abrupt.so ringing effect is maximum in rectangular window. Hence several windows are developed which contain taper and decays gradually

band attenuation. For low pass FIR filter design specifications are cutoff frequency of 0.

It is observed that pass band contains undesirable oscillation. Fig.

3 respectively. Here it is observed that Bohman has Side lobe attenuation of -43 db and it is -92.1 db for Blackman Harris.

6 FIR filter design with Bohman window Magnitude response of FIR filter designed with Bohman window is shown in fig.6 with N=31.

3 88 toward zero. [3]-[7] III. MATLAB BASED DESIGN Time domain and magnitude response of windows functions are defined with MATLAB for calculating minimum stop band attenuation. For low pass FIR filter design specifications are cutoff frequency of 0.4π rad/sample and order, N of the filter. Here FIR filter is designed using different values of N. Fig.2 Response of Bohman window Fig.5 FIR filter design with Bohman window Magnitude response of FIR filter designed with Bohman window is shown in fig.5. Here filter order is N=21. It is observed that pass band contains undesirable oscillation. Fig.3 Response of Blackman-Harrsis Time and frequency domain representation of Bohman and Blackman-Harris is shown in fig.2 and fig.3 respectively. Here it is observed that Bohman has Side lobe attenuation of -43 db and it is db for Blackman Harris. Here window length is 81 for both cases. Fig.6 FIR filter design with Bohman window Magnitude response of FIR filter designed with Bohman window is shown in fig.6 with N=31.It is observed that undesirable oscillations are increased with increasing the valve of N. Fig.4 Response of Tukey window Time and frequency domain representation of Tukey window is shown in figure.4.it has Side lobe attenuation of db as compared to above described window functions. Here window length is also 81. Fig.7 FIR filter design with Blackman-Harris window Magnitude response of FIR filter is shown in fig.7.here filter is designed with Blackman-Harris for N=21.Due to low side lobe attenuation as compared with Bohman

89 window, filter has low oscillations in pass band Fig.

fig.8 with N=31.It is observed that undesirable oscillations are increased with increasing the valve of N.

11 Bohman window response for N=21 and 31 The magnitude response of the window function is shown in fig.11 for N=21 and N=31.

9 FIR filter design with Tukey window Magnitude response of FIR filter is shown in fig.9.here filter is designed with Tukey window for N=21.

band. Low pass designed with different window functions is shown in fig.12 for filter order of 21.

10 FIR filter design with Tukey window Magnitude response of FIR filter designed with Tukey window is shown in fig.10 with N=31.

Here filter has large oscillation in pass band as compared with filters designed with Bohman and Blackman-Harris windows. IV.

4 89 window, filter has low oscillations in pass band Fig.8 FIR filter design with Blackman-Harris window Magnitude response of FIR filter designed with Blackman-Harris window is shown in fig.8 with N=31.It is observed that undesirable oscillations are increased with increasing the valve of N. but it is low as compared with fig.6. Fig.11 Bohman window response for N=21 and 31 The magnitude response of the window function is shown in fig.11 for N=21 and N=31.As N increases main lobe becomes narrower and side lobes remain unaffected. Fig.12 Magnitude response of FIR filter, N=21 Fig.9 FIR filter design with Tukey window Magnitude response of FIR filter is shown in fig.9.here filter is designed with Tukey window for N=21.Due to large side lobe attenuation as compared with Bohman window and blackman-harris window,filter has large oscillations in pass band. Low pass designed with different window functions is shown in fig.12 for filter order of 21.here pass band contains Gibb s effect due to different window functions. Fig.13 Magnitude response of FIR filter, N=31 Fig.10 FIR filter design with Tukey window Magnitude response of FIR filter designed with Tukey window is shown in fig.10 with N=31.It is observed that undesirable oscillations are increased with increasing the valve of N. Here filter has large oscillation in pass band as compared with filters designed with Bohman and Blackman-Harris windows. IV. RESULT ANALYSIS Analysis of Gibb s phenomenon for FIR filter Using three window functions (Bohman, Blackman-Harris, Tukey) has been done for different values of N. Low pass designed with different window functions is shown in fig.13 for filter order of 31.here pass band contain s large Gibb s effect due to large value of N. It is observed that transition width will be reduced. Fig.14 Gibb s phenomenon in pass band

5 90 From fig.11, it is observed that as N increased main lobe becomes narrower and side lobes remain unaffected with increases in N. But width of each side lobe increases with an increase in N. Therefore FIR filter designed with different window functions, shown in fig.13 has large Gibb s phenomenon as compared with FIR filter, shown in fig Saurabh Singh Rajput, Dr. S. S. Bhaduria Implementation of FIR Filter using Efficint Window Function and Its Application in Filtering a Speech Signal, International Journal of Electrical Electronics and Mechanical Controls, Volume 1, Issue 1, pp.1-12, November V. CONCLUSION It is observed that, as order increased frequency response of window function becomes narrower and smoothing reduced. It provided large side lobes in frequency response of window function with lower transition width. These large side lobes cause large oscillations in filter frequency response. This results Gibb s phenomenon. Blackman-Harris provided minimum stop band attenuation, hence it contains small side lobes as compared to Bohman and Blackman-Harris window functions. Hence it is observed filter has small effect of Gibb s phenomenon if it is designed with Blackman- Harris window. ACKNOWLEDGEMENT The authors would also like to thank Director, National Institute of Technical Teachers Training & Research, Chandigarh, India and Director, Meerut Institute of Engineering and Technology, Meerut for their constant inspirations and support throughout this research work. REFERENCES 1. S Salivahanan, Digital Signal Processing, Tata,Mc Graw Hill, Third Edition pp , J.S. Chitode, Digital Signal Processing, Technical Publication, pp. 8-27, Proakis, J.G. and Manolakis, D.G, Digital Signal Processing Principles, Algorithms and Applications, Pearson Education Ltd, Fourth Edition, pp , Sanjit K. Mitra, Digial Signal Processing, Tata,Mc Graw Hill, Third Edition, pp Tahseen flaih Hassan, Hazin salah Abiddulsatar, Design and Simulation of FIR Digital Filter using MATLAB and MAC filter, Journal of Engineering and Development Volume 17, Issue 5, pp , November Suhaib Ahmad, Design Analysis of High Pass FIR Filters using Hanning, Bartlett and Kaiser Windows, International Journal of Advanced Research in Computer Science and Software Engineering Volume 2, Issue 17, pp , November 2012.

FIR Filter Design using Different Window Techniques

FIR Filter Design using Different Window Techniques Kajal, Kanchan Gupta, Ashish Saini Dronacharya College of Engineering Abstract- Digital filter are widely used in the world of communication and computation.