5.Adjustable Window Functions 5.Adjustable Window Functions 5.Adjustable Window Functions Dolph-Chebyshev window can be designed Gain response of a Dolph-Chebyshev window Properties of Dolph-Chebyshev window: with any specified relative sidelobe level of length 51 and relative sidelobe level of 50 .All sidelobes are of equal height while the main lobe width adjusted by dB is shown below choosing length appropriately Stopband approximation error of filters 2.056a-16.4 Filter order is estimated using N designed have essentially equiripple behavior where Aois the normalized transition 2.285(△0) For a given window length,it has the smallest bandwidth,e.g.for a lowpass filter A=,- main lobe width compared to other windows resulting in filters with the smallest transition band 5.Adjustable Window Functions 5.Adjustable Window Functions 5.Adjustable Window Functions Kaiser Window- Filter order is estimated using w)i ·In practice -M SnSM o W=g-8 1() A controls the minimum stopband 2.285(△四) where is an adjustable parameter and lo(u)is attenuation of the windowed filter response where A is the normalized transition the modified zeroth-order Bessel function of ·B is estimated using bandwidth the first kind: 6)=1+u2yTr 0.1102(a.-8.7八. for a,>50 」 B={0.5824a,-21)+0.07886(a,-21).for21sa,≤50 .Note lou)>0 for u being real 0. for a.<2131 5. Adjustable Window Functions Dolph-Chebyshev window can be designed with any specified relative sidelobe level while the main lobe width adjusted by choosing length appropriately Filter order is estimated using where is the normalized transition bandwidth, e.g, for a lowpass filter 2.056 16.4 2.285( ) s N )   ' ' '  s p 32 5. Adjustable Window Functions Gain response of a Dolph-Chebyshev window of length 51 and relative sidelobe level of 50 dB is shown below 0 0.2 0.4 0.6 0.8 1 -80 -70 -60 -50 -40 -30 -20 -10 0 Normalized Frequency Gain in dB Dolph-Chebyshev Window 33 5. Adjustable Window Functions Properties of Properties of Dolph-Chebyshev window: window: All sidelobes sidelobes are of equal height Stopband approximation error of filters designed have essentially equiripple equiripple behavior For a given window length, it has the smallest smallest main lobe main lobe width compared to other windows resulting in filters with the smallest transition smallest transition band 34 5. Adjustable Window Functions Kaiser Window Kaiser Window - where is an adjustable parameter and I0(u) is the modified zeroth-order Bessel function of the first kind: Note I0(u)>0 for u being real  2 0 0 1(/ ) () , ( ) I nM wn M n M I * *     * 2 0 1 ( / 2) () 1 ! r r u I u r      ! $ %  35 5. Adjustable Window Functions In practice £ controls the minimum stopband attenuation of the windowed filter response £ is estimated using 2 20 0 1 ( / 2) () 1 ! r r u I u  r     ! $ %  0.4 0.1102( 8.7), for 50 0.5824( 21) 0.07886( 21), for 21 50 0, for 21 s s s ss s ) ) *) ) ) )  ,         36 5. Adjustable Window Functions Filter order is estimated using where is the normalized transition bandwidth 8 2.285( ) s N )   ' '