S 7.4 IIR Digital Filter Design: Bilinear Transformation Method Digital filter design consists of 3 steps: (1)Develop the specifications of H(s)by applying the inverse bilinear transformation to specifications of G(z) (2)Design H(s) (3)Determine G(by applying bilinear transformation to H(s) As a result,the parameter T has no effect on G()and T=2 is chosen for convenience
§7.4 IIR Digital Filter Design: Bilinear Transformation Method • Digital filter design consists of 3 steps: (1) Develop the specifications of Ha(s) by applying the inverse bilinear transformation to specifications of G(z) (2) Design Ha(s) (3) Determine G(z) by applying bilinear transformation to Ha(s) • As a result, the parameter T has no effect on G(z) and T = 2 is chosen for convenience
S 7.4 IIR Digital Filter Design: Bilinear Transformation Method Mapping of s-plane into the z-plane Im 一1 Re< 0 0 1 s-plane 3-plane
§7.4 IIR Digital Filter Design: Bilinear Transformation Method • Mapping of s-plane into the z-plane
S 7.4 IIR Digital Filter Design: Bilinear Transformation Method For z=ej with T=2 we have -1-e-ho_eo/2(e/2-e-2) 1+e-joe-jol2(ejol2+e-jo2) j2sin(@/2) jtan(@/2) 2c0s(o/2) 0 or -tan(@/2) 0
§7.4 IIR Digital Filter Design: Bilinear Transformation Method • For z=ejω with T = 2 we have ( ) ( ) 1 1 / 2 / 2 / 2 / 2 / 2 / 2 − ω ω − ω − ω ω − ω − ω − ω + − = + − Ω = j j j j j j j j e e e e e e e e j tan( / 2) 2cos( / 2) 2sin( / 2) ω ω ω j j = = or Ω=tan(ω/2)
S 7.4 IIR Digital Filter Design: Bilinear Transformation Method Mapping is highly nonlinear Complete negative imaginary axis in the s- plane from =-oo to =0 is mapped into the lower half of the unit circle in the z-plane from z=-1 toz=1 Complete positive imaginary axis in the s- plane from =0 to =oo is mapped into the upper half of the unit circle in the z-plane from z=1 to Z=-1
§7.4 IIR Digital Filter Design: Bilinear Transformation Method • Mapping is highly nonlinear • Complete negative imaginary axis in the splane from Ω= - ∞ to Ω=0 is mapped into the lower half of the unit circle in the z-plane from z = -1 to z = 1 • Complete positive imaginary axis in the splane from Ω=0 to Ω= ∞ is mapped into the upper half of the unit circle in the z-plane from z = 1 to z = - 1
S 7.4 IIR Digital Filter Design: Bilinear Transformation Method Nonlinear 卫=ctan(w2) mapping introduces a 8 distortion in the frequency axis 0 called frequency Ra() warping H(eju) Effect of warping shown right 0102 0304
§7.4 IIR Digital Filter Design: Bilinear Transformation Method • Nonlinear mapping introduces a distortion in the frequency axis called frequency warping • Effect of warping shown right