4.Fixed Window Functions 4.Fixed Window Functions 4.Fixed Window Functions To ensure a fast transition from passband to Table 10.2:Properties of fixed window functions In the case of rectangular,Hann,Hamming, stopband,window should have a very small and Blackman windows,the value of ripple ·Rectangular window-△a=4π/(2M+l) main-lobe width does not depend on filter length or cutoff 4t=13.3dB.a,=20.9dB.△o=0.92x/M To reduce the passband and stopband ripple 6, frequency ,and is essentially constant ·Hann window·△i位=8x/2M+) the area under the sidelobes should be very 。In addition,△w≈clM 4=31.5dB,a,=43.9dB.△0=311π/M small where c is a constant for most practical ·Hamming window-△z=8r/(2M+l) Unfortunately,these two requirements are purposes contradictory 4r=42.7dB.a,=54.5dB.△m=3.32x/M ·Blackman window-△=l2r/《2M+) 4=58.1dB.a,=753dB,△0=5.56π/M 27 4.Fixed Window Functions 4.Fixed Window Functions 5.Adjustable Window Functions Filter Design Steps- ·Lowpass filter of length51ando=π/2 Dolph-Chebyshev Window- ①Set0=(0。+0,)/2 减三 2M+1y台 2 Choose window based on specified a, amplitude of sidelobe -MSnsM main lobe amplitude ③Estimate Musing A@≈clM where B=cosh2 cosh-'》 1 An increase in the main lobe width is associated with an increase in the width of the transition band cos(/cos"r),for s1 A decrease in the sidelobe amplitude results in an increase in and T(r)= the stopband attenuation cosh(/cos-x),for>125 4. Fixed Window Functions To ensure a fast transition from passband to stopband, window should have a very small main-lobe width To reduce the passband and stopband ripple¥, the area under the the area under the sidelobes sidelobes should be very small Unfortunately, these two requirements are contradictory 26 4. Fixed Window Functions In the case of rectangular, Hann, Hamming, and Blackman windows, the value of ripple value of ripple does not depend on filter length or cutoff frequency¹c, and is essentially constant is essentially constant In addition, ¹Ĭc/M where c is a constant for most practical purposes 27 4. Fixed Window Functions Table 10.2: Properties of fixed window functions 28 4. Fixed Window Functions Filter Design Steps Filter Design Steps - ķ Set ĸ Choose window based on specified Ĺ Estimate M using ¹Ĭc/M ( )/2 c ps   ) s 29 4. Fixed Window Functions Lowpass filter of length 51 and / 2  c  An increase in the main lobe width is associated with an An increase in the main lobe width is associated with an increase in the width of the transition band A decrease in the A decrease in the sidelobe amplitude results in an increase in amplitude results in an increase in the stopband attenuation 0 0.2 0.4 0.6 0.8 1 -120 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Hann Window 0 0.2 0.4 0.6 0.8 1 -120 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Hamming Window 0 0.2 0.4 0.6 0.8 1 -120 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Blackman Window 30 5. Adjustable Window Functions Dolph-Chebyshev Window – where and 1 11 2 ( ) 2 cos cos 21 21 21 M k k k nk wn T M MM M nM   * +         !      $ % " #" #    amplitude of sidelobe main lobe amplitude +  1 11 cosh cosh 2M * +       " # 1 1 cos( cos ), for 1 ( ) cosh( cos ), for 1 l lx x T x lx x      ,