电子科技大学：《数字信号处理 Digital Signal Processing》课程教学资源（英文讲义）Chapter 07 Digital Filter Design

Chapter 7 Digital Filter design

Chapter 7 Digital Filter Design

Objective- Determination of a realizable transfer function G(z) approximating a given irequency 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)

87.1 Digital Filter Specifications Usually, either the magnitude and/or the phase(delay response is specified for the design of digital filter for most applications In some situations, the unit sample response or the step response may be specified In most practical applications, the problem of interest is the development of a realizable approximation to a given magnitude response specification

§7.1 Digital Filter Specifications • Usually, either the magnitude and/or the phase (delay) response is specified for the design of digital filter for most applications • In some situations, the unit sample response or the step response may be specified • In most practical applications, the problem of interest is the development of a realizable approximation to a given magnitude response specification

87.1 Digital Filter Specifications .. We discuss in this course only the magnitude approximation problem There are four basic types of ideal filters with magnitude responses as shown below HLp(e /o) HHP(e/) 0 兀 兀 HBp(e) π-0c2-0cl Ocl (c2 O 兀-0c2-0cl0cl0c2π

§7.1 Digital Filter Specifications • We discuss in this course only the magnitude approximation problem • There are four basic types of ideal filters with magnitude responses as shown below π 1 ω 0 ωc –ωc HLP(e jω) − π π ω 0 ωc –ωc 1 HHP(e jω) − π − π π ω –1 –ωc1 ωc1 –ωc2 ωc2 HBP (e jω) − π π ω 1 –ωc1 ωc1 –ωc2 ωc2 HBS(e jω)

87.1 Digital Filter Specifications As the impulse response corresponding to each of these ideal filters is noncausal and of infinite length, these filters are not realizable In practice, the magnitude response specifications of a digital filter in the passband and in the stopband are given with some acceptable tolerances In addition a transition band is specified between the passband and ° stopband

§7.1 Digital Filter Specifications • As the impulse response corresponding to each of these ideal filters is noncausal and of infinite length, these filters are not realizable • In practice, the magnitude response specifications of a digital filter in the passband and in the stopband are given with some acceptable tolerances • In addition, a transition band is specified between the passband and stopband

87.1 Digital Filter Specifications For example, the magnitude response G(elo) of a digital lowpass filter may be given as indicated below (e Passband. Stopband-*

§7.1 Digital Filter Specifications • For example, the magnitude response |G(ejω)| of a digital lowpass filter may be given as indicated below

87.1 Digital Filter Specifications A S indicated in the iigure, In the passband, defined by0≤0≤0pwe require that G(ej@)=1 with an error +8 1.e 1-8≤Ge)s1+S,|o≤0 In the stopband, defined by≤0≤π,we require that G(ejo)=o with an error 8 i.e., G(ej@)s 8n, @ s slosh

§7.1 Digital Filter Specifications • As indicated in the figure, in the passband, defined by 0≤ω≤ωp , we require that |G(ejω)|≅1 with an error ±δp , i.e., 1- δp ≤ |G(ejω)| ≤ 1+ δp , | ω| ≤ ωp • In the stopband, defined by ωs ≤ω≤π, we require that |G(ejω)|≅0 with an error δs i.e., |G(ejω)| ≤ δp , ωs ≤|ω|≤π

87.1 Digital Filter Specifications passband edge frequency O- stopband edge frequency p-peak ripple value in the passband 8- peak ripple value in the stopband Since g(elo) is a periodic function of a, and G(eo)l of a real-coefficient digital filter is an even function of o As a result, filter specifications are given only for the frequency range 0 sasn

§7.1 Digital Filter Specifications • ωp - passband edge frequency • ωs - stopband edge frequency • δp - peak ripple value in the passband • δs - peak ripple value in the stopband • Since G(ejω) is a periodic function of ω, and |G(ejω)| of a real-coefficient digital filter is an even function of ω • As a result, filter specifications are given only for the frequency range 0 ≤|ω|≤π

87.1 Digital Filter Specifications Specifications are often given in terms of loss function G(o)=-20logoIG(ejo)lin d B Peak passband ripple O1=2090(1-8n)dB Minimum stopband attenuation 03=-20log0(、)dB

§7.1 Digital Filter Specifications • Specifications are often given in terms of loss function G(ω)=-20log10 |G(ejω)| in dB • Peak passband ripple αp= -20log10 (1- δp ) dB • Minimum stopband attenuation αs= -20log10 (δs ) dB

87.1 Digital Filter Specifications magnitude specifications may alternately be given in a normalized form as indicated below Alejo +ε Stopband- Transiton

§7.1 Digital Filter Specifications • Magnitude specifications may alternately be given in a normalized form as indicated below