Chapter 7 Filter Design Techniques ◆7.0 Introduction 7.1 Filter Specifications 7.2 Design of Discrete-Time IIR Filters From Continuous-Time Filters 7.3 Discrete-Time Butterworth,Chebyshev and Elliptic Filters 7.4 Frequency Transformations of Lowpass IIR Filters 7.5 Design of FIR Filters by Windowing 7.6 Examples of FIR Filters Design by the Kaiser Window Method 2
2 Chapter 7 Filter Design Techniques ◆7.0 Introduction ◆7.1 Filter Specifications ◆7.2 Design of Discrete-Time IIR Filters From Continuous-Time Filters ◆7.3 Discrete-Time Butterworth, Chebyshev and Elliptic Filters ◆7.4 Frequency Transformations of Lowpass IIR Filters ◆7.5 Design of FIR Filters by Windowing ◆7.6 Examples of FIR Filters Design by the Kaiser Window Method
7.0 Introduction Frequency-selective filters pass only certain frequencies Any discrete-time system that modifies certain frequencies is called a filter. We concentrate on design of causal Frequency-selective filters (IIR,FIR). IH(e) 1+81 1-81 He） Passband Transition Stopband -2π-π-0.0 0 p 0e元 2n 4
4 7.0 Introduction ◆Frequency-selective filters pass only certain frequencies ◆Any discrete-time system that modifies certain frequencies is called a filter. ◆We concentrate on design of causal Frequency-selective filters (IIR, FIR). ( ) j H e − 2 − −c 0 c 2 1
5.1.1 Ideal Frequency-Selective Filters idea owpas fiter: ◆Noncausal,not in computationally realizable ◆no phase distortion H( -2π-7π-00 2π 5
5.1.1 Ideal Frequency-Selective Filters ◆Ideal lowpass filter: 5 ◆Noncausal, not computationally realizable ◆no phase distortion sin , c lp n h n n = − n ( ) , | | , 0, | | j 1 c lp c H e = ( ) j H e − 2 − −c 0 c 2 1
5.1.1 Ideal Frequency-Selective Filters Ideal highpass filter l-eH(e-e3e ◆Noncausal,not hmln]=8fn] sinan -o0<n<o0 computationally 元n realizable ◆no phase distortion H(e) -2元 0 2元 6
5.1.1 Ideal Frequency-Selective Filters ◆Ideal highpass filter sin , c h n n hp n n n = − − − 2 − −c 0 c 2 ( ) H ej 1 6 ◆Noncausal, not computationally realizable ◆no phase distortion ( ) ( ) 0, | | 1 1, | | j c hp c j Hlp e H e − = =
5.1.1 Ideal Frequency-Selective Filters Ideal bandpass filter hae）h.e）.ey6: others hon-sinon s nan -oo<n<oo πn πn .Noncausal,not computationally realizable ◆no phase distortion 1H(e） -π-02 02π 7
5.1.1 Ideal Frequency-Selective Filters ◆Ideal bandpass filter 0 c1 c1 − − ( ) j H e 1 −c2 c2 7 ◆Noncausal, not computationally realizable ◆no phase distortion 2 1 sin , sin c c bp n n h n n n n = − − ( ) 1 ( ) ( ) 1 2 2 , | | 0, j 1 lp j p c c l bp H ej oth s H H e er e − = =
5.1.1 Ideal Frequency-Selective Filters Ideal bandstop filter h,e）+,e(e-8 01@Kw2 others hos n]=n] sinan sinaan ，-0<n<0 πn πn .Noncausal,not computationally realizable ◆no phase distortion 1H(el) -π-02 一0%0 0c2 8
◆Ideal bandstop filter − 0 ( ) H ej 1 −c2 −c1 c1 c2 5.1.1 Ideal Frequency-Selective Filters 8 ◆Noncausal, not computationally realizable ◆no phase distortion 2 1 sin [ ] sin , bs c n c n n n n n n h = + − − ( ) ( ) ( ) 0, | | 1 2 1, j c c b j hp j lp H es oth r H H e e s e + = =
Stages of Filter Design 1.Determine the specification of the desired properties of the system(filter). 2.The approximation of the specifications using(designing)a causal discrete-time system. 3.The realization of the system. ◆Our focus is on second step.(实验侧重3) Specifications are typically given in the frequency domain. 9
9 Stages of Filter Design ◆1. Determine the specification of the desired properties of the system(filter). ◆2.The approximation of the specifications using(designing) a causal discrete-time system. ◆3. The realization of the system. ◆Our focus is on second step.(实验侧重3) ◆Specifications are typically given in the frequency domain
7.1 Filter Specifications THe figure depicts the typical representation of tolerance limits associated with approximating a discrete-time ideal lowpass filter. H(ej)l 容差极限 In passband, tolerance scheme容限图 1+δp H(may vary symmetrically around1,δp1=δp2 1 1-823H(elo)3Hp,laop 1- or1-ozH(e/lls0δ1=0 Passband Transition Stopband In stopband,H(eo)l≤δs,wzo cutoff frequency edge frequency 0 T 10通带边界（截止）频率 阻带边界（截止）频率
10 7.1 Filter Specifications tolerance scheme 容限图 ◆The figure depicts the typical representation of tolerance limits associated with approximating a discrete-time ideal lowpass filter. edge frequency |H(e j)| may vary symmetrically around 1, δp1=δp2 or ,δp1=0 cutoff frequency 1 c 容差极限 通带边界(截止)频率 阻带边界(截止)频率 P S 1- 1 p2 ( ) H j e ， p 1- 1+ p p 2 1 ( ) H j e ， p In passband, In stopband, |H(e j)|≤δs , ||≥s
7.1 Filter Specifications nd constraints on phase response other than implicit requirements of stability and causality for IIR filter. The constraint for FIR filters:linear phase. H(ej)l 容差极限 In passband, tolerance scheme容限图 1+δp H()may vary symmetrically around 1,1=p2 1 1- 1-82H(el@)31+p,laop P2 or1-o2H(eoKl,osδ1=0 Passband Transition Stopband In stopband,lH(eo)l≤δs,ozos cutoff frequency edge frequency 0 W T 1通带边界（截止）频率 阻带边界（截止）频率
11 7.1 Filter Specifications tolerance scheme 容限图 edge frequency |H(e j)| may vary symmetrically around 1, δp1=δp2 or ,δp1=0 cutoff frequency 1 c 容差极限 通带边界(截止)频率 阻带边界(截止)频率 P S 1- 1 p2 ( ) H j e ， p 1- 1+ p p 2 1 ( ) H j e ， p In passband, In stopband, |H(e j)|≤δs , ||≥s ◆no constraints on phase response other than implicit requirements of stability and causality for IIR filter. ◆The constraint for FIR filters: linear phase
Ex.7.1 Determine Specifications for a Discrete- Time Filter H(ei). given: Specifications of the continuous-time filter: ◆1.passband1-0.01<He(/2kl+0.01,for02≤2π(2000) ◆2.stopband m(2<0.001,for2π(3000)≤2 f®-47 CID H(ej) DIC xa(t) x(n] y[n] T=104s 2=27=2x00000) T For bandlimited input,aliasing avoided when sampling frequency is high enough 12
12 ◆Specifications of the continuous-time filter: ◆1. passband ◆2. stopband 1 0.01 1 0. − + H j for eff( ) 01, 0 2 20 ( 00) H j for eff( ) 0.001, 2 3000 ( ) 4 T s 10− = ( ) 2 S 2 10000 T = = ( ) ( ) , , 0, j T eff H e H T j T = For bandlimited input, aliasing avoided when sampling frequency is high enough Ex. 7.1 Determine Specifications for a DiscreteTime Filter H(e j) . given: If 可作为