7.3 Digital Filter Design: Basic approaches Most common approach to IIR filter design (1)Convert the digital filter specifications into an analog prototype lowpass filter specifications (2)Determine the analog lowpass filter transfer function H(s) (3)Transform H.(s)into the desired digital transfer function G(z)
§7.3 Digital Filter Design: Basic Approaches • Most common approach to IIR filter design – • (1) Convert the digital filter specifications into an analog prototype lowpass filter specifications • (2) Determine the analog lowpass filter transfer function Ha(s) • (3) Transform Ha(s) into the desired digital transfer function G(z)
7.3 Digital Filter Design: Basic Approaches An analog transfer function to be denoted as Ha(s)=Pa(s)/Da(s) where the subscript“a”specifically indicates the analog domain A digital transfer function derived from H(s)shall be denoted as G(Z=P(Z/D(Z
§7.3 Digital Filter Design: Basic Approaches • An analog transfer function to be denoted as Ha(s)= Pa(s) / Da(s) where the subscript “a” specifically indicates the analog domain • A digital transfer function derived from Ha(s) shall be denoted as G(z)=P(z)/D(z)
7.3 Digital Filter Design: Basic Approaches Basic idea behind the conversion of H(s)into G(Z)is to apply a mapping from the s-domain to the z-domain so that essential properties of the analog frequency response are preserved Thus mapping function should be such that Imaginary (j)axis in the s-plane be mapped onto the unit circle of the z-plane -A stable analog transfer function be mapped into a stable digital transfer function
§7.3 Digital Filter Design: Basic Approaches • Basic idea behind the conversion of Ha(s) into G(z) is to apply a mapping from the s-domain to the z-domain so that essential properties of the analog frequency response are preserved • Thus mapping function should be such that – Imaginary (jΩ ) axis in the s-plane be mapped onto the unit circle of the z-plane – A stable analog transfer function be mapped into a stable digital transfer function
7.3 Digital Filter Design: Basic Approaches FIR filter design is based on a direct approximation of the specified magnitude response,with the often added requirement that the phase be linear The design of an FIR filter of order N may be accomplished by finding either the length-(N+1)impulse response samples (h[n]}or the (N+1)samples of its frequency response H(ei)
§7.3 Digital Filter Design: Basic Approaches • FIR filter design is based on a direct approximation of the specified magnitude response, with the often added requirement that the phase be linear • The design of an FIR filter of order N may be accomplished by finding either the length-(N+1) impulse response samples {h[n]} or the (N+1) samples of its frequency response H(ejω)
S 7.4 IIR Digital Filter Design: Bilinear Transformation Method Bilinear transformation :到 1+S Z三 1-S Above transformation maps a single point in the s-plane to a unique point in the z- plane and vice-versa Relation between G(z)and H.(s)is then given by ce)--非
§7.4 IIR Digital Filter Design: Bilinear Transformation Method Above transformation maps a single point in the s-plane to a unique point in the zplane and vice-versa • Relation between G(z) and Ha(s) is then given by − + − − = = 1 1 1 ( ) ( ) 2 1 z z T s a G z H s + − = − − 1 1 1 2 1 z z T s s s z − + = 1 1 • Bilinear transformation