CHAPTER 7 filter design techniques 7.0 introduction 7.Idesign of discrete-time IIR filters from continuous-time filters filter design by impuls e invariance 7.1.2 filter design by bilinear transform 7.1.3 not low pass filter design and other method 7.2 design of Fir filters by windowing 7.3 summary
7.0 introduction 7.1design of discrete-time IIR filters from continuous-time filters 7.1.1 filter design by impulse invariance 7.1.2 filter design by bilinear transform 7.1.3 not low pass filter design and other method 7.2 design of FIR filters by windowing. 7.3 summary CHAPTER 7 filter design techniques
7.0 introduction ideal frequency selective filter 2其-c2Ⅱ c 2 (a)理想低通 (b)理想高通 H(eJ° C 2其 2 (c)理想带通 (①理想带隕 band-stop filter: notch filter; quality factor of band-pass filter=pass-band width/ center frequency
7.0 introduction ideal frequency selective filter band-stop filter：notch filter；quality factor of band-pass filter=pass-band width/center frequency
CD H(e D/C r, (t yIn ya(t) Figure 7.1 impulse response of ideal low-pass filter, noncausal, unrealizable sin(on) ha(n) Oc h dle Jo dOn O 丌J- Specifications for filter design given in frequency domain pl ase: inear magnitude: given by a tolerance scheme analog or digital, absolute or relative
Figure 7.1 = = = − − , ...... sin( ) ( ) 2 1 ( ) n n n h n H e e d j j n c d d c c impulse response of ideal low-pass filter，noncausal，unrealizable Specifications for filter design ：given in frequency domain phase：linear? magnitude：given by a tolerance scheme analog or digital，absolute or relative
Hefr(in2) absolute specification 1+δ magnitude response of equivalent analog system Passband i Transition Stopband δ He/) 1+δ 3dB cutoff Passband i Transition Stopband stope Figure 7. toleranc monotonous\ descent T toft -Feetenev
Figure 7.2 magnitude response of equivalent analog system monotonous descent 1/ 2 c passband tolerance c stopband tolerance passband cutoff frequency stopband cutoff frequency 3dB cutoff frequency absolute specification
relative specifications: maximum magnitude in passband is normalized to l, viz. odB 20*log10(1 0 maximum attenuation in passband 20*log106>0 minimum attenuation in stopband BdB cutoff frequency: H(e/)=l/√2 20log1o H(ec)=3dB magnitude response of equivalent analog system H(eJo)l Heff(jQ2) T } T digital specification, finally: @p=Qp 1, as=Q25T
relative specifications：maximum magnitude in passband is normalized to 1, viz. 0dB 20*log 0 20*log 1 0 10 10 = − = − − s s p p （ ） maximum attenuation in passband minimum attenuation in stopband 3dB cutoff frequency： H e dB H e c c j j 20log | ( ) | 3 | ( ) | 1/ 2 − 10 = = p = p T,s = s T = = T T H e H j T j eff 0 | | ( ) | | | ( ) magnitude response of equivalent analog system: digital specification, finally:
Specifications for bandpass and bandstop filters up and down passband cutoff frequency up and down stopband cutoff frequency Design steps: (1) decide specifications according to application 2) decide type according to specification: generally, if the phase is required, choose Fir (3) approach specifications using causal and stable discrete time system VIZ. design H(zO)or hn, nonuniform (4) choose a software or hardware realization structure. take effects of limited word length into consideration H()or hn
（1）decide specifications according to application （2）decide type according to specification：generally , if the phase is required , choose FIR. （3）approach specifications using causal and stable discretetime system： viz. design H(z0) or h[n]，nonuniform （4）choose a software or hardware realization structure, take effects of limited word length into consideration H(z) or h[n] Design steps： Specifications for bandpass and bandstop filters： up and down passband cutoff frequency， up and down stopband cutoff frequency
7.1 design of discrete -time ir filters from continuous-time filters ala log y low pass>digital low pass frequency transform ala log y high pc A/D ass igital high pass attention: original analog filter and equivalent analog filter is different their frequency response is not al ways the same 7.1.0 introduction of analogy filter 7.1. filter design by impulse invariance 7.1.2 filter design by bilinear transform
7.1 design of discrete-time IIR filters from continuous-time filters ala y high pass digital high pass frequency transform ala y low pass digital low pass A D A D ⎯ ⎯→ ⎯ ⎯→ / / log log 7.1.0 introduction of analogy filter 7.1.1filter design by impulse invariance 7.1.2 filter design by bilinear transform attention：original analog filter and equivalent analog filter is different, their frequency response is not always the same
7.1.0 introduction of analog filter Ha〔 Ha〔2川 comparison 1. wave 2. the same order increase performance 3. increase design 0(a)巴特沃炳波器90 0(b椭圆型能波器9 comple Ha〔川 Ha〔)川 c)切比雪夫型滤波器 (d切比雪夫型滤波器 (A)(C) small aliasing in the impulse invariance design technique
7.1.0 introduction of analog filter (A)（C）small aliasing in the impulse invariance design technique comparison: 1.wave 2.the same order, increase performance 3.increase design complexity
BW design formula: specification>system function magnitude frequency function: H (Q2)I 1+(2/2)2sH(S)H2(-s)= take the specifications into system 10 function and get the results from N= los /9 equation group ORΩ 10 confirm the poles of system function k=0……N-1 (in the left half plane H(S)=I get system function
− − = 2 1 10 10 10 1 10 1 log s p p s N = − = + = N c c s j c c H j H (s)H ( s)| 1 ( / ) 1 | ( )| 2 2 N p c p 2 1 10 10 1 − = N s c s 2 1 10 10 1 − = , 0 1 2 1 2 2 1 = = − + + s e k N N k j k c − = − = − = − = 1 0 1 0 1 ( ) 1 ( ) N k k N c N k k c s s s s H s BW design formula: specification→system function magnitude frequency function: take the specifications into system function and get the results from equation group: OR confirm the poles of system function: （in the left half plane） get system function：
EXAMPLE design a low pass analogy filter Q.=2T.100Orad/s@=2T. 2000rad /s.a=ldB.a=15dB N,Wc]= buttord(2000p124000*pi,1,15,s) BS, As]=butter(N, WC,S) H,W-freqs(Bs, As plot(w/2/pi, 20*(log10(abs(dD)) axis(1000,2000,-160]) grid on OUTPUT Wc=8.1932e+003 BS=10e+015* 00004.5063 As=1.0e+015*0.00000.00000.00000.00144.5063 6+,stbs+bs+b, H( a +++ tas
[N,Wc]=buttord(2000*pi,4000*pi,1,15, ' s ') [Bs,As]=butter(N,Wc, ' s ') [H,W]=freqs(Bs,As); plot(W/2/pi,20*(log10(abs(H)))) axis([1000,2000,-16,0]) grid on design a low pass analogy filter: p = 2 1000rad /s,s = 2 2000rad /s, p =1dB,s =15dB EXAMPLE OUTPUT: N = 4 Wc = 8.1932e+003 Bs = 1.0e+015 * 0 0 0 0 4.5063 As = 1.0e+015 * 0.0000 0.0000 0.0000 0.0014 4.5063 4 4 3 3 2 0 1 2 4 4 3 3 2 0 1 2 ( ) a a s a s a s a s b b s b s b s b s H s c + + + + + + + + =