Chapter 7 Digital Filter Design
Chapter 7 Digital Filter Design
Objective-Determination of a realizable transfer function G(z)approximating a given frequency response specification is an important step in the development of a digital filter If an IIR filter is desired,G(z)should be a stable real rational function Digital filter design is the process of deriving the transfer function G(z)
• Objective - Determination of a realizable transfer function G(z) approximating a given frequency response specification is an important step in the development of a digital filter • If an IIR filter is desired, G(z) should be a stable real rational function • Digital filter design is the process of deriving the transfer function G(z)
S 7.2 Selection of Filter Type The transfer function H(z)meeting the frequency response specifications should be a causal transfer function For IIR digital filter design,the IIR transfer function is a real rational function of z1: H(z)= +p2+p222+…+pM2M d4+dz+d2z2+…+dwzw H(z)must be a stable transfer function and must be of lowest order N for reduced computational complexity
§7.2 Selection of Filter Type • The transfer function H(z) meeting the frequency response specifications should be a causal transfer function • For IIR digital filter design, the IIR transfer function is a real rational function of z-1: N N M M d d z d z d z p p z p z p z H z − − − − − − + + + + + + + + = 2 2 1 0 1 2 2 1 0 1 ( ) • H(z) must be a stable transfer function and must be of lowest order N for reduced computational complexity
S 7.2 Selection of Filter Type For FIR digital filter design,the FIR transfer function is a polynomial in z1 with real coefficients: H(z)=∑hin]z-" n=0 For reduced computational complexity,degree N of H(z)must be as small as possible If a linear phase is desired,the filter coefficients must satisfy the constraint: h[n=±hN-n
§7.2 Selection of Filter Type For reduced computational complexity, degree N of H(z) must be as small as possible • If a linear phase is desired, the filter coefficients must satisfy the constraint: h[n] = ± h[N-n] = ∑ = − N n n H z h n z 0 ( ) [ ] • For FIR digital filter design, the FIR transfer function is a polynomial in z-1 with real coefficients:
7.2 Selection of Filter Type Advantages in using an FIR filter- (1)Can be designed with exact linear phase, (2)Filter structure always stable with quantized coefficients Disadvantages in using an FIR filter-Order of an FIR filter,in most cases,is considerably higher than the order of an equivalent IIR filter meeting the same specifications,and FIR filter has thus higher computational complexity
§7.2 Selection of Filter Type • Advantages in using an FIR filter - (1) Can be designed with exact linear phase, (2) Filter structure always stable with quantized coefficients • Disadvantages in using an FIR filter - Order of an FIR filter, in most cases, is considerably higher than the order of an equivalent IIR filter meeting the same specifications, and FIR filter has thus higher computational complexity
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
