A comparative study on main lobe and side lobe of frequency response curve for FIR Filter using Window Techniques

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1 Proc. of Int. Conf. on Computing, Communication & Manufacturing 4 A comparative study on main lobe and side lobe of frequency response curve for FIR Filter using Window Techniques Sudipto Bhaumik, Sourav Chandra, Samir Bhowmik, Subhojit Malik, Writi Mitra Hooghly Engineering and Technology College, Electronics & Communication Engineering Department, Hooghly,India sudipto.bhaumik3@gmail.com, s.chandra.tiluri@gmail.com, samir59@gmail.com Hooghly Engineering and Technology College, Electronics & Communication Engineering Department, Hooghly,India subhojitmalik@gmail.com, writi mitra@gmail.com Abstract Finite Impulse Response (FIR) filters are Digital filters which act as frequency selective systems. The design of FIR filter is a non-recursive structure because there is no feedback connection. The response of FIR filter depends on the present and past input samples. In this paper FIR filters are designed by methods. The desired time domain response with infinite number of sequence is truncated at some point by multiplying by a sequence. The length of the resultant sequence will be fixed and finite. w the use of function is reasonably straight forward to get filter impulse response with minimal computational effort. There are many sequences like Rectangular Window, Hanning Window, Hamming Window, Blackman Window, Kaiser Window, Bohman Window, Taylor Window and Tukey Window etc. These s are helping to approximate the desired characteristics. Basically the function is a weighting function for an -point sequence. The spectrum of any can be obtained by taking Fourier Transform and the obtained frequency response curve can be low pass, high pass, band pass and band stop type. w the width of the main lobe of the response curve is inversely proportional to the length of the sequence. In this paper, the width of the main lobe is being varied by changing the value of the length () of any function. The characteristic features for different types of filters are studied and the generated frequency responses are compared with respect to the length of the sequence. Index Terms Digital Filter, Finite Impulse Response, Window function, Frequency Response Curve, Central Lobe, Side Lobe I. ITRODUCTIO In advanced communication system, digital filter plays an important role. Depending upon the impulse response, the digital filter can be two types: Infinite Impulse Response (IIR) and Finite Impulse Response (FIR) filters. In impulse response, the attenuation level is very low or ideally zero for desired signal components and the attenuation level is high for unwanted signal components. In this paper, the design of FIR filters is discussed using method. The smoothness in pass band and stop band will be obtained when sharp transition in edges will occur. By taking proper value of number of taps the following s are taken: Rectangular, Blackman, Hamming, Bartlett, Modified Bartlett- Hanning, Bohman, Chebyshev, Flat Top, Gaussian, uttall defined minimum 4-term Blackman-Harris, Taylor and Tukey [-3]. All the normalized magnitude response curves for low pass, high pass, band pass and band reject filters are compared for 8

2 different number of taps. From magnitude response curves, the maximally flat value in main lobe and minimal side lobe value are observed and compared. II. FIR FILTER DESIG The design of FIR filter can be done by using functions in complex domain [4-6]. The impulse response h(n) of FIR filter for -samples can be obtained by multiplying desired impulse response h d (n) with the function w(n) and it is given in equation () where the desired impulse response h d (n) is obtained by taking Inverse Fourier transform of desired frequency response H d (e jω ),shown in equation () hn = hd( n) wn () + π jω jωn hd( n) = Hd ( e ) e dω () π π w the selection of function is important. The following functions are used for testing and discussed briefly.. Rectangular Window: This function is defined by ω( n) = ω n, n ( ) represents the width i.e. the number of samples in discrete-time. 3. Hamming Window: The is given by the function nπ ω( n) = α β cos with, α =.54, β = α = Bartlett-Hanning : The function is n nπ ω ( n) = a a acos a =.6; a =.48; a = Chebyshev Window: The is described in frequency domain by the expression k cheb( L, β * cos ( π * )) L ω k = ; with, β = cosh * a cosh cheb( L, β ) L and Cheb(m,x) denoting the m-th order Chebyshev polynomial calculated at the point x. 9. Gaussian Window: The function of Gaussian given by ω ( n) = exp at n ( )/ σ ( )/. Blackman Window: The following equation defines the Blackman of length n ω n =.4.5cos nπ / +.8cos 4 nπ /, ( ) ( ) n M 4. Taylor Window: This is given by [7] ω n+ n ( n) = ( ) exp n r ( ) ( ) where, n 6. Bohman Window: The equation for computing coefficient of Bohman function is ω( x) = ( x) cos( π x) + sin ( π x), x Where x is a length L vector of π linearly spaced values generated using linspace the first and last elements of bohman are forced to be identically zero. 8. Flattop Window: The flattop function is expressed by the equation, nπ 4nπ 6nπ 8nπ ( n) a a a a3 a4 ω = cos + cos cos + ; a =, a =.93, a =.9, a3 =.388, a =.8 4. uttal Window: The uttal function is given by nπ 4nπ 6nπ ω n = a a cos + a cos a cos 3 a = , a = , a = , a =

3 . Bartlett : The function is given by For even value of n, n ; n ω ( n) = n ; n For odd value of n, n ; n ω ( n) = n ; + n.. Tukey Window: This function is given by, n α( ) + cos π, n α ( ) α( ) α ω( n) =, n ( ) n α + cos π +,( ) n ( ) α ( ) α III. IMPLEMETATIO OF THE DESIG The normalized magnitude response curves are obtained and observed for low pass filter (LPF), high pass filter (HPF), band pass filter (BPF) and band reject filter (BRF) using the above mentioned functions and by varying the number of samples for designing the FIR filters. To implement and design of FIR filter MATLAB 7 has been used. The number of samples are taken as =5 and =. The maximum magnitude in the main lobe and the maximum magnitude in the side lobe are observed for every case and are shown in Table IA and Table IB respectively. Different Window Functions TABLE IA. TABLE FOR COMPARATIVE STUDY OF MAXIMUM MAGITUDE OF MAI LOBE Maximum of Main Lobes LPF HPF BPF BRF Value of Value of Value of Value of Rectangular Blackman Hamming Bartlett Modified Bartlett-Hanning Bohman Chebyshev Flat Top Gaussian Blackman-Harris Taylor Tukey

4 Different Window Functions TABLE IB. TABLE FOR COMPARATIVE STUDY OF MAXIMUM MAGITUDE OF SIDE LOBE Maximum of side lobes LPF HPF BPF BRF Value of Value of Value of Value of Rectangular Blackman Hamming Bartlett Modified Bartlett-Hanning Bohman A A A A A A A A Chebyshev A A A A A A A A Flat Top A A A A A A A A Gaussian Blackman-Harris A A A A A A A A Taylor Tukey The graphs show the nature of normalized magnitude curves for different s with different value of. Characteristics of filter For =5 For =.6 Frequency response of LPF having =5.6 Frequency response of LPF having = LPF data data data data data data HPF data data data Frequency response of HPF having = data data data Frequency response of HPF having =

5 Characteristics of filter For =5 For = BPF Frequency response of BPF having =5 data data data Frequency response of BPF having = data data data Frequency response of BRF having =5.6 Frequency response of BRF having = data data.4.4 BRF..8.6 data data..8.6 data data Figure : rmalized Response Curves of FIR Filters using different s COCLUSIOS The computational effort is minimal if we use method to obtain the filter impulse response. From the comparison tables we observe that the Flat-top and Blackman are giving better response because the maximum desired value of main lobe is near to the normalized value which should be one and the maximum magnitude for side lobe is nearly zero. The response curve corresponds to Bohman and Chebyshev are also reaching towards the desired value as well. So the response of Fir filter is best when the Flat-top and Blackman are used. ACKOWLEDGMET The authors would like to thank the Department of Electronics and Communication Engineering and the authority of Hooghly Engineering and Technology College, Hooghly, India for providing the facilities and encouraging to carry out research work. REFERECES [] Roark R.M., Escabi M., Design of FIR filters with exceptional pass band and stop band smoothness using a new transitional, Circuits and Systems ISCAS, Geneva, IEEE International Symposium Volume [] Roy T.K., Morshed M., Performance Analysis of low pass FIR filters Design using Kaiser, Gaussian and Tukey function methods, International Conference on Advances in Electrical Engineering (ICAEE), 3 [3] Ben-Dau Tseng, Specifications of ideal prototype filters for designing linear phase FIR filters, 3th Asilomer Conference on Signals, Systems and Computers, 996 [4] Kurth R., Design of FIR filters to complex frequency response specifications, IEEE International Conference ICAASSP 8 on Acoustic, Speech, and Processing [5] Sahesteh M.G., Mottaghi-Kashtiban M., An efficient function for design of FIR filters using IIR FILTERS IEEE Conference EUROCO 9 [6] Chen X, Parks T.W., Design of FIR filters in the complex domain, IEEE Transactions on Acoustic, Speech, and Signal Processing, Volume:35, Issue: [7] Xiao F., Tang X.H., Zangh X.J., Comparison of Taylor finite difference and finite difference and their application in FDTD,Journal of Computational and Applied Mathematics, September,4, pp:

F I R Filter (Finite Impulse Response)

F I R Filter (Finite Impulse Response) Ir. Dadang Gunawan, Ph.D Electrical Engineering University of Indonesia The Outline 7.1 State-of-the-art 7.2 Type of Linear Phase Filter 7.3 Summary of 4 Types FIR