4.Fixed Window Functions 4.Fixed Window Functions 4.Fixed Window Functions Various window functions:(rised cosine) Plots of magnitudes of the DTFTs of these windows for M=25 are shown below: Magnitude spectrum of each window Hann: characterized by a main lobe centered at =0 (n)= 1+c08 MsnsM followed by a series of sidelobes with 2 2M+1 decreasing amplitudes Hamming: 4n)=0.54+0.46c0s 2πn -AM≤n≤M Parameters predicting the performance of a 2M+1 Blackman: window in filter design are: 2元 1)Main lobe width n=0.42+0.5cos +0.08c0s 2M+1 2M+1 2)Relative sidelobe level -MsnsM 4.Fixed Window Functions 4.Fixed Window Functions 4.Fixed Window Functions 。Main lobe width-△given by the distance Lowpass Filter Design by Windowing ·Observe H,(ea+a+H,(ea-aa三l between zero crossings on both sides of main ·Thus H,(em)≥0.5 lobe 1+6 Passband and stopband ripples are the same Relative sidelobe level-A given by the difference in dB between amplitudes of largest Distance between the locations of the sidelobe and main lobe maximum passband deviation and minimum stopband value≈△w 。Width of transition band△w=w,一op<△n 419 4. Fixed Window Functions Various window functions: (rised cosine) Hann: Hamming: Hamming: Blackman: Blackman: 1 2 ( ) 1 cos , 2 21 n wn M n M M          !   $ % " #  2 ( ) 0.54 0.46cos , 2 1 n wn M n M M          " #  2 4 ( ) 0.42 0.5cos 0.08cos 21 21 n n w n M M M nM         "# "#     20 4. Fixed Window Functions Plots of magnitudes of the DTFTs of these windows for M = 25 are shown below: 0 0.5 1 1.5 2 2.5 3 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Rectangular Window 0 0.5 1 1.5 2 2.5 3 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Hann Window 0 0.5 1 1.5 2 2.5 3 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Blackman Window 0 0.5 1 1.5 2 2.5 3 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Hamming Window 21 4. Fixed Window Functions Magnitude spectrum of each window characterized by a main lobe centered at¹=0 followed by a series of sidelobes with decreasing amplitudes Parameters predicting the performance of a window in filter design are: 1) Main lobe width 2) Relative 2) Relative sidelobe sidelobe level 22 4. Fixed Window Functions Main lobe width Main lobe width - ML given by the distance between zero crossings on both sides of main lobe Relative Relative sidelobe sidelobe level - Asl given by the difference in dB between amplitudes of largest sidelobe and main lobe 23 4. Fixed Window Functions Lowpass Lowpass Filter Design by Windowing ( ) j H e t  ( ) j H e d  s p  1& 1& & c & ' 'ML  c ( ) ( ) c j e    24 4. Fixed Window Functions Observe Thus Passband and stopband ripples are the same Distance between the locations of the maximum passband deviation and minimum stopband value ĬML Width of transition band ¹= ¹sˉ¹p<ML     ( ) ( )1 c c j j He He t t ' '  ( ( ) 0.5 c j H e t (