1.Preliminary Considerations Chapter 9 Part A >Digital Filter Specifications ●●● ●●● Selection of the Filter Type Basic Approaches to Digital Filter Design IIR Digital Filter Design Preliminary Considerations Estimation of the Filter Order Scaling the Digital Filter 1.1 Digital Filter Specifications 1.1 Digital Filter Specifications 1.1 Digital Filter Specifications Objective: Usually,either the magnitude and/or the Digital Filter Design Steps Determination of a realizable transfer function phase (delay)response is specified for the G(z)approximating a given frequency design of digital filter for most applications. A choice between IIR and FIR digital filter response specification is an important step in has to be made In most practical applications,the problem of the development of a digital filter interest is the development of a realizable Derivation of a realizable transfer function If an IIR filter is desired,G(z)should be a approximation to a given magnitude response G(2) stable rational function specification. Realization of G(z)using a suitable filter Digital filter design is the process of deriving structure the transfer function G(z) 4
1 Chapter 9 IIR Digital Filter Design 2 Part A Preliminary Considerations 3 1. Preliminary Considerations Digital Filter Specifications Digital Filter Specifications Selection of the Filter Type Selection of the Filter Type Basic Approaches to Digital Filter Design Estimation of the Filter Order Estimation of the Filter Order Scaling the Digital Filter Scaling the Digital Filter 4 1.1 Digital Filter Specifications Objective : Objective : Determination of a realizable 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 rational function Digital filter design is the process of deriving the transfer function G(z) 5 1.1 Digital Filter Specifications Usually, either the magnitude magnitude and/or the phase (delay) response phase (delay) response is specified for the design of digital filter for most applications. In most practical applications, the problem of interest is the development of a realizable approximation to a given magnitude response specification. 6 1.1 Digital Filter Specifications Digital Filter Design Steps Digital Filter Design Steps A choice between IIR and FIR digital filter has to be made ķ Derivation of a realizable transfer function G(z) ĸ Realization of G(z) using a suitable filter structure
1.1 Digital Filter Specifications 1.1 Digital Filter Specifications 1.1 Digital Filter Specifications Digital Filter Specifications Normalized Specifications .Frequency specifications are normalized using ◆Passband: the sampling rate: Maximum value of the magnitude 1+6 1-6,1+6,, (gain)is assumed to be unity or the minimum value of the 0。= F 2x5=2πFT 。Stopband: sA,鸟sase V+ loss function is 0 dB _2tF.=2xF.T ●Peak Passband fipple a=20logn dB 0= F a,--20log(1-)dB am三-20logo1-26,) .=x corresponds to half the sampling rate, .Minimum Stopband atlenuation 226 dB F2 --20loga(8,)dB Q:What is the condition for non-overlapping? 1.3 Basic Approaches to Digital Filter 1.2 Selection of the Filter Type 1.2 Selection of the Filter Type Design .IIR filters: IIR Filter Design .FIR filters: Better attenuation properties An analog filter transfer function H(s)is Linear phase response Closed form approximation formulas transformed into the desired digital filter Stability with quantized coefficients transfer function G(z) Nonlinear phase response Analog approximation techniques are highly Higher order required than using IIR filters xInstability with finite wordlength computation advanced ☑Lower order .They usually yield closed-form solutions N is typically of the oder of tens (or more) 名 12
7 1.1 Digital Filter Specifications Digital Filter Specifications Digital Filter Specifications Passband: Stopband: Peak Passband ripple: Minimum Stopband attenuation 1 ( )1 , j p G e p p () , j G e s s 10 p p 20log (1 ) dB 10 s s 20log ( ) dB ( ) j G e 0 p s 1 p 1 p s c pass band stop band Transition band 8 1.1 Digital Filter Specifications Normalized Specifications Normalized Specifications ( ) j G e 0 p s 1 2 1 1 1 A c pass band stop band Transition band Maximum value of the magnitude (gain) is assumed to be unity or the minimum value of the loss function is 0 dB 2 max 10 20log 1 dB max 10 20log (1 2 ) 2 dB p p 9 1.1 Digital Filter Specifications Frequency specifications are normalized using the sampling rate: ¹=± corresponds to half the sampling rate, FT/2 2 2 p p p p T T F F T F F 2 2 s s s s T T F F T F F Q: What is the condition for non-overlapping? 10 FIR filters: + Linear phase response + Stability with quantized coefficients ˉHigher order required than using IIR filters 1.2 Selection of the Filter Type Ĝ h Ĝ 11 1.2 Selection of the Filter Type IIR filters: + Better attenuation properties + Closed form approximation formulas - Nonlinear phase response - Instability with finite wordlength computation Lower order NFIR/NIIR is typically of the order of tens (or more) Ĝ h Ĝ h Ĝ 12 1.3 Basic Approaches to Digital Filter Design IIR Filter Design An analog filter transfer function Ha(s) is transformed into the desired digital filter transfer function G(z) Analog approximation techniques are highly advanced They usually yield closed-form solutions
1.3 Basic Approaches to Digital Filter 1.3 Basic Approaches to Digital Filter 1.3 Basic Approaches to Digital Filter Design Design Design IIR Filter Design IIR Filter Design IIR Filter Design Extensive tables are available for analog filter The basic idea behind the conversion of an Requirements for the transform are: design or the methods are easy to program analog prototype transfer function H(s)to a digital filter transfer function G()is to apply >The imaginary axis (j)of the s-plane is .Digital filters often replace (or simulate) mapped onto the unit circle in the z-plane analog filters a mapping from the s-domain to the z-domain Stable H(s)must be transformed into a stable H.(s)= B()G(e)=P( so that the essential properties of the analog frequency response are preserved. G) D.(d) D(z) 13 15 1.3 Basic Approaches to Digital Filter 1.3 Basic Approaches to Digital Filter Design Design 1.4 Estimation of the Filter Order FIR Filter Design FIR Filter Design .IIR Design-Filter order is solved from the .No analog prototype filters are available FIR transfer function: approximation formulas FIR filter design is based on a direct approximation of the specified magnitude m) .FIR Design-Several formulas proposed for estimating the minimum length of the impulse response The corresponding frequency response: response H(em)=∑Mnea Kaiser:N= -20logo（V6,d)-13 .A linear phase response is usually required 14.6(@,-0,)/2x Linear phase requirement: h(n=±h(N-1-n) 多 18
13 1.3 Basic Approaches to Digital Filter Design IIR Filter Design Extensive tables are available for analog filter design or the methods are easy to program Digital filters often replace (or simulate simulate) analog filters ( ) ( ) ( ) a a a P s H s D s ( ) ( ) ( ) P z G z D z 14 1.3 Basic Approaches to Digital Filter Design IIR Filter Design The basic idea behind the conversion of an analog prototype transfer function Ha(s) to a digital filter transfer function G(z) is to apply a mapping from the s-domain to the z-domain so that the essential properties of the analog frequency response are preserved. 15 1.3 Basic Approaches to Digital Filter Design IIR Filter Design Requirements for the transform are: The imaginary axis imaginary axis (j) of the s-plane is mapped onto the unit circle unit circle in the z-plane Stable Ha(s) must be transformed into a stable G(z) 16 1.3 Basic Approaches to Digital Filter Design FIR Filter Design No analog prototype filters are available FIR filter design is based on a direct approximation of the specified magnitude response A linear phase response is usually required 17 1.3 Basic Approaches to Digital Filter Design FIR Filter Design FIR transfer function: The corresponding frequency response: Linear phase requirement: 1 0 () () N n n H z hnz 1 0 ( ) () N j jn n He hne hn hN n () ( 1 ) 18 1.4 Estimation of the Filter Order IIR Design IIR Design -- Filter order is solved from the approximation formulas FIR Design FIR Design -- Several formulas proposed for estimating the minimum length of the impulse response Kaiser: Kaiser: 20log 13 10 14.6( ) / 2 p s s p N
1.4 Estimation of the Filter Order 1.5 Scaling the Digital Filter 1.5 Scaling the Digital Filter Nis inversely proportional to the normalized G(z)has to be scaled in magnitude so that the Lowpass filter:Unity gain at zero frequency transition width and does not depend on the maximum gain in the passband is unity 0=0(or2=1) location of the transition band Notice that the scaling coefficient K does not Ndepends also on the product of and affect the shape of the magnitude response, KG(eL。=KG(eL→K=1/G0 i.e.,it does not affect the locations of poles ·Highpass filter:Unity gain at=π(or=- and zeros in the z-plane 1) KG(e)=KG()K=1/G(-1) 19 20 21
19 1.4 Estimation of the Filter Order N is inversely proportional to the normalized transition width and does not depend on the location of the transition band N depends also on the product of and p s 20 1.5 Scaling the Digital Filter G(z) has to be scaled in magnitude so that the maximum gain in the passband is unity Notice that the scaling coefficient K does not affect the shape of the magnitude response, i.e., it does not affect the locations of poles and zeros in the z-plane 21 1.5 Scaling the Digital Filter Lowpass Lowpass filter: filter: Unity gain at zero frequency ¹=0 (or z=1) Highpass Highpass filter: filter: Unity gain at ¹= (or z=ˉ 1) 0 0 1 ( ) () j z KG e KG z K G 1/ (1) 1 ( ) () j z KG e KG z K G 1/ ( 1)