with Improved Symmetry In Gain Using Optimal Pole Reposition Technique

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1 International Journal of Electrical, Electronics and Mechanical Fundamentals, Vol. 03, Issue 01, Sept 2012 ISSN: IIR Multiple Notch filter f with Improved Symmetry In Gain Using Optimal Pole Reposition Technique Prince Bansal 1, Ramnish Kumar 2 1 M. Tech Student, E.C.E Deptt., Guru Jambheswar University of Science and Technology, Hisar, India kaunprince@yahoo.com 2 Assistant Professor, E.C.E Deptt., Guru Jambheswar University of Science and Technology, Hisar, India ramnish_ramnish@yahoo.com Abstract This paper presents the optimal pole reposition technique for designing a stable IIR multiple notch filter with improved symmetry in gain using optimal pole reposition technique. We have to create the tow notch filter for it step to find suitable pole position with a sharp notch at particular frequency. After that connect two IIR notch filter to get a IIR multiple notch filter to notch two frequencies. Similarly two frequency notch filter are connected to form a IIR multiple notch filter to notch four frequencies. Zeros are placed at the unit circle and get a good response we must have to change the value of pole radius. The value of pole radius is chosen in such a way that we get a symmetrical response at notch frequency and can control the pass band gain at dc, π and ω c frequencies. Keywords: Multiple notch filter, pole reposition technique, IIR, FIR. INTRODUCTION: Notch filter have been an effective means for eliminating a particular frequency or very narrow band in a lot of application like cutting noise in TV broadcast, ECG and rejecting the interference in GSM. Notch filter can be IIR or FIR. But the main limitation of FIR is that it does not have a narrow band response. There are various method for the construction of notch filter such as transforming analog notch filter, implementing from all pass filter and optimal pole reposition technique. But optimal pole reposition technique is the most efficient technique and easy. But there is main limitation in optimal pole reposition technique is that it has constraint on asymmetric and uncontrollable gain. That can be eradicated by placing pole at the appropriate position. Single notch filter are employed to notch a single frequency. Multiple notch filter are used to notch multiple frequencies interferences. Another classification of notch filter is given as fixed notch filter and adaptive notch filter. The fixed notch filter can be applied where unwanted frequencies are known otherwise adaptive ones will be employed. For the construction of a multiple IIR notch filter a number of single notch filter are to be cascaded. Therefore the following article is introduced to design a multiple notch IIR filter which will have controllable pass band gain between the notch frequencies. THEORIES: Previous design: The frequency response specification of ideal single notch filter is given by: H (e Jω ) = 0, = 1 h (1) Where ω 0 is the cutoff frequency of notch filter or pole zero angle on the z-plane The transfer function of single notch filter will be given as follows: H (z) =!" # # (2) b 0 represents the constant coefficient of an IIR notch filter and r is distance between pole and the origin whose value can vary from 0 to 1. H(z) is Transfer function of the given function. 201

2 International Journal of Electrical, Electronics and Mechanical Fundamentals, Vol. 03, Issue 01, Sept 2012 ISSN: Fig.1 shows that before applying pole reposition technique the notch filter has uncontrollable passband gain and asymmetrical about notch frequency. Fig.2: Magnitude response of IIR multiple notch filter derived from cascading single notch filters before applying pole reposition technique. Fig.1: Magnitude response of conventional IIR filter before applying pole reposition technique Multiple notch filters are used for removing frequency interferences, to construct the IIR notch filter cascading of IIR single notch employed,where the frequency response multiple notch filter is given by : H (ejω) =$ 0, 1, = ℎ,.., Fig(3) shows that the filter has controllable pass band gain at frequencies at frequency dc,ωc1 and π but it does not have controllable pass band gain ωc2. It also shows the asymmetry at the notch frequencies. multiple multiple filter is of IIR (3) Transfer function for a multiple notch filter is given by: H (z) = ) '* ' (!" ( # (# (4) Fig.3: Cascading of three single notch filter before finding a proper pole position Fig (2) shows the magnitude response results of previous design for two frequencies. These results also shows that the filter has uncontrollable pass band gain at dc, ωc and π frequency. Fig. also shows that the filter also shows no symmetricity at the notch frequencies 202

3 International Journal of Electrical, Electronics and Mechanical Fundamentals, Vol. 03, Issue 01, Sept 2012 ISSN: Propose design: In the previous design we can see the magnitude response which do not have a controllable passband gain and design is also not symmetrical with respect to the notch frequency due to lacking of exact pole position. Zero position will not be changed because these are fixed on the unit circle Transfer function for multiple IIR filter using pole reposition technique is given by: H(z)= ) ( '* ' (5) (!"+ ( # ( # ω n= improved pole angle. ω ci= central frequency = (ω i +ω i+1 )/2 (6) Here we will solve the transfer function for four frequencies k1,k2,k3,k4 are the gain of four filters r1,r2,r3,r4 are the pole postion and ω1,ω2,ω3,ω4 are the four frequencies to be notched, a1 and c1 are bandwidth factor. b1,b2,b3 and b4 are the filter coefficient. ω c1,ω c2 and ω c3 are the three central frequencies. ω c1 is between ω1 and ω2, ω c2 is between ω2 and ω3, ω c3 is between ω3 and ω4 ω c1 = (ω1+ω2)/2; ω c2= (ω2+ω3)/2; ω c3 =(ω3+ω4)/2; When the transfer function is solved for various values of n than we have to find out value of b1, b2, b3,b4 and cos+, cos+, cos+,, cos+ -. First find out the value of b1,b2,b3,b4 b3=k3*(1-2*r3*w4+r3^2)/(2-2*x3); (7) b4=k4*(1-2*r4*w5+r4^2)/(2-2*x4); (8) Similarly we can find out the value of b1,b2. Than find out the value of cos+,cos+,cos+,,cos+ - Cosω 3=. / 0(!" /!" / ) / (!" /!" / ) Cosω 4=. 3 0(!" 3!" 3 ) 3 (!" 3!" 3 ) Similarly we can also find the value of cos+,cos+ Let m3= (!" /!" / ) (!" /!" / ) Let m4= (!" 3!" 3 ) (!" 3!" 3 ) Cosω 3=. / 0 / 43 (13) Cosω 4= (14) m3 and m4 are used to simplify the equations. (9) (10) (11) (12) RESULTS: IIR multiple notch filter to notch four frequency is defined in the equation below: H(e Jω ) = 0, =.28,.38,.68,.78 1, h (15) Value of pole radius r to be chosen is.99, and k1 chosen is 1. The design example have specified the notch frequencies at.2π.,4π,.6π,.7π and their individual transfer function obtained in matlab programming is as shown in the equations given below : 203

03, Issue 01, Sept 2012 ISSN: 2278-3989 0.99-1.

99 z -2 H1(z)= ---------------------------------

164 z -1 + 0.9801 z -2 0.99 + 0.6119 z -1 + 0.

1 + 0.6119 z -1 + 0.9801 z -2 0.9901+ 1.

4 International Journal of Electrical, Electronics and Mechanical Fundamentals, Vol. 03, Issue 01, Sept 2012 ISSN: z z -2 H1(z)= (16) z z z z -2 H2(z)= (17) z z z z -2 H3(z)= (18) z z z z -2 H4(z)= (19) z z -2 fig(5).mag. response of notch filter2, ω=.3π Fig(6).Mag.response of notch filter 3, ω=.6π Fig(4) Mag. response of notch filter 1 ω=.2π Fig(7).mag. response of notch filter ω=.7π 204

programming in matlab is as shown in the equation(20) and the magnitude response is shown in Fig(8) while pole zero plot is shown in figure in Fig(9) Pole zero plot of

9608 z -1 +1.555 z -2-1.555 z -3 +2.515 z -4-1.555 z -5 + 1.555 z -6 0.9608 z -7 + 0.9608 z -8 H(z)= ----------------------------------------- (20) 1-0.9901 z -1 + 1.

9227 z -8 Magnitude plot in figure(8) shows that the notch filter have a controllable passband gain at dc, π and ω ci frequencies. Fig(9).

in symmetric and controllable pass band gains according to specification. The aim is to get a proper pole position at the pole zero plot.

magnitude is zero. In the graph we get better result in comparision of previous results without any complicated design procedure.

5 International Journal of Electrical, Electronics and Mechanical Fundamentals, Vol. 03, Issue 01, Sept 2012 ISSN: After cascading all IIR notch filter to notch four frequencies,the transfer function of multiple IIR notch filter obtained by programming in matlab is as shown in the equation(20) and the magnitude response is shown in Fig(8) while pole zero plot is shown in figure in Fig(9) Pole zero plot of multiple IIR notch filter is shown is as shown in figure(9) shows that the optimal position of pole has been obtaibned which is very close to unit circle z z z z z z z z -8 H(z)= (20) z z z z z z z z -8 Magnitude plot in figure(8) shows that the notch filter have a controllable passband gain at dc, π and ω ci frequencies. Fig(9).Pole Zero plot of given transfer function CONCLUSION: Graph result have shown that the new design of IIR notch filter by applying modified pole reposition technique resulted in symmetric and controllable pass band gains according to specification. The aim is to get a proper pole position at the pole zero plot. The proposed IIR filter design can guarantee that the magnitude response at DC,π and ω ci can be controlled to have uniformly flat pass band gains and at notch frequency gain magnitude is zero. In the graph we get better result in comparision of previous results without any complicated design procedure. However the disadvantage of this approach is its increasing complexity with increase in the number of frequencies. For the future research direction we can also consider the other parameter like gain and bandwidth factors. Fig(8).Magnitude response of given transfer function of multiple notch filter for r=.99 REFERENCES: [1] J. G. Proakis and D. G. Manolakis, Digital Signal Processing Principle, Algorithms, and Applications", Prentice, [2] C. Charoenlarpnopparut, P. Charoen, A. Thamrongmas, S. Samurpark, and P. Boonyanant, 205

6 International Journal of Electrical, Electronics and Mechanical Fundamentals, Vol. 03, Issue 01, Sept 2012 ISSN: High-qualtiy factor, double notch, IIR digital Filter design using optimal pole re-position technique with controllable passband gains", ECTI-2009, 6th, International Conference, Vol. 2, 6-9 May, pp , [3] C. C. Tseng and S. C. Pei, Stable IIR Notch Filter Design with Optimal Pole Placement", IEEE Trans. Signal Processing, Vol. 49, No. 11, pp , November, 2001 [4] Surapun Yimman and Kobchai Dejhan, IIR Multiple Notch Filter Design with Optimum Pole Position", ISCIT 06, International Symposium, [5] Yashwant V. Joshi and S. C. Dutta Roy, Design of IIR Multiple Notch Filters Based on All-Pass Filters", IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, Vol. 46, No. 2, February, [6] Surapun Yimman,Watcharapong Hinjit,Weerasak Ussawongaraya,Payao Thoopluangand Kobchai Dejhan Design and implementation of IIR Multiple Notch Filter with modified pole zero placement algorithm", The Journal of KMITNB, vol.15. No.2, Apr-Jun.2005 [7] S. K. Mitra, Digital Signal Processing Computer-Base Approach", McGraw- Hill, 2001 [8] S. C. Pei and C. C. Tseng, IIR multiple notch filter design based on all-pass filter, IEEE Trans. Circuits Syst. II, vol. 44, pp , Feb

Optimal FIR filters Analysis using Matlab

Optimal FIR filters Analysis using Matlab International Journal of Computer Engineering and Information Technology VOL. 4, NO. 1, SEPTEMBER 2015, 82 86 Available online at: www.ijceit.org E-ISSN 2412-8856 (Online) Optimal FIR filters Analysis