Simple Digital Filters Chapter 7B Part B Simple FIR Digital Filters ●●● ●●● ●●● Simple IIR Digital Filters LTI Discrete-Time Systems Simple Digital Filters in the Transform-Domain 88 ◆Comb Filters 1.Simple FIR Digital Filters 1.Simple FIR Digital Filters 1.1 Lowpass FIR Digital Filters The simplest lowpass FIR digital filter is the Later in the course we shall review various FIR digital filters considered here have 2-point moving-average filter given by methods of designing frequency-selective integer-valued impulse response coefficients filters satisfying prescribed specifications (quantified) a)=0+- 22 We now describe several low-order FIR and These filters are employed in a number of The above transfer function has a zero at z=- IIR digital filters with reasonable selective practical applications,primarily because of 1 and a pole at z=0 frequency responses that often are satisfactory their simplicity,which makes them amenable Note that here the pole vector has a unity in a number of applications to inexpensive hardware implementations magnitude for all values of @thus H(e)=0.5+1 4
Chapter 7B LTI Discrete-Time Systems in the Transform-Domain Part B Simple Digital Filters 3 Simple Digital Filters Simple FIR Digital Filters Simple FIR Digital Filters Simple IIR Digital Filters Simple IIR Digital Filters Comb Filters Comb Filters 4 1. Simple FIR Digital Filters Later in the course we shall review various methods of designing frequency-selective filters satisfying prescribed specifications We now describe several low-order FIR and IIR digital filters with reasonable selective frequency responses that often are satisfactory in a number of applications 5 1. Simple FIR Digital Filters FIR digital filters considered here have integer-valued impulse response coefficients (quantified) These filters are employed in a number of practical applications, primarily because of their simplicity, which makes them amenable to inexpensive inexpensive hardware hardware implementations 6 1.1 Lowpass FIR Digital Filters The simplest lowpass FIR digital filter is the 2-point moving-average average filter given by The above transfer function has a zero at z=ˉ 1 and a pole at z = 0 Note that here the pole vector has a unity magnitude for all values of , thus 1 0 1 1 ( ) (1 ) 2 2z Hz z z 0 ( ) 0.5 1 j j He e
1.1 Lowpass FIR Digital Filters 1.1 Lowpass FIR Digital Filters 1.1 Lowpass FIR Digital Filters A cascade of 3 sections-an improved scheme ●As increases from0 Frst-orde FIR Loupass F to,the magnitude of /5、 Notice: the zero vector decreases from a value 00 The cascade of first. of 2,the diameter of the 05 order sections yields a sharper magnitude unit circle,to 0 03 passband_ -stopband response but t at the expense of a ●We can work out the decrease in the width frequency response of the passband H(e)=e cos(/2) 0601020的04094070m991 normalized digital angutar frequency @20log(/=-3 3-dB cutoft frequency 02@04 00 1.1 Lowpass FIR Digital Filters 1.2 Highpass FIR Digital Filters 1.2 Highpass FIR Digital Filters M-order FIR Lowpass (M-order moving-average)Filter Mordet FIR Loupass Faee The simplest highpass FIR filter is obtained Improved highpass magnitude response can from the simplest lowpass FIR filter by again be obtained by cascading several replacing z with- sections of the first-order highpass filter .This results in H()=(1-2) Alterately,a higher-order highpass filter of the form h(n)= 年8间 Corresponding frequency response is given by H(e)=je-isin(@/2) aaa2r- is obtained by replacing with-in the transfer function of a moving average filter 12
7 1.1 Lowpass FIR Digital Filters -1 1 Re z jIm z j -j As increases from 0 to , the magnitude of the zero vector decreases from a value of 2, the diameter of the unit circle, to 0 We can work out the frequency response / 2 0 ( ) cos( / 2) j j He e 8 1.1 Lowpass FIR Digital Filters c 3-dB cutoff frequency passband stopband 1/ 2 20log 1/ 2 3 dB normalized digital angular frequency c 9 1.1 Lowpass FIR Digital Filters A cascade of 3 sections—an improved scheme 1/ 2 Notice: The cascade of firstorder sections yields a sharper magnitude response but at the expense of a decrease in the width of the passband c 10 1.1 Lowpass FIR Digital Filters M-order FIR order FIR Lowpass (M-order moving-average) Filter 1/ 2 0 0 1 ( ) 1 M m m H z z M 0 1 1 () () 1 M hn R n M 11 1.2 Highpass FIR Digital Filters The simplest highpass FIR filter is obtained from the simplest lowpass FIR filter by replacing z with ˉz This results in Corresponding frequency response is given by 1 1 1 ( ) (1 ) 2 Hz z / 2 1( ) sin( / 2) j j H e je 12 1.2 Highpass FIR Digital Filters Improved highpass magnitude response can again be obtained by cascading several sections of the first-order highpass filter Alternately, a higher-order highpass filter of the form is obtained by replacing z with ˉz in the transfer function of a moving average filter 1 0 1 () 1 1 M n n n Hz z M
1.2 Highpass FIR Digital Filters 2.Simple IIR Digital Filters 2.1 Lowpass lIR Digital Filters M-order FIR Hgpess Faer Lowpass IIR Digital Filters A first-order causal lowpass IIR digital filter has a transfer function given by Highpass IIR Digital Filters Hie()= 1-a1+z Bandpass IIR Digital Filters ,21-az where a<1 for stability Bandstop IIR Digital Filters The above transfer function has a zero at 2=- I i.e.,at which is in the stopband Higher-order IIR Digital Filters .H(z)has a real pole atz= 2.1 Lowpass lIR Digital Filters 2.1 Lowpass lIR Digital Filters 2.2 Highpass lIR Digital Filters 。As)increases from0toπ，the magnitude of the zero vector decreases from a value of 2 .A first-order causal highpass IIR digital filter has a transfer function given by to 0,whereas,for a positive value of a,the Hm(2）= 1+01-2 magnitude of the pole vector increases from value of 1-a to 1+a 21-a2 where a<1 for stability The maximum value of the magnitude function is I at =0,and the minimum value The above transfer function has a zero at z=I is0at0=π i.e.,at =0 which is in the stopband 1
13 1.2 Highpass FIR Digital Filters 1/ 2 14 2. Simple IIR Digital Filters Lowpass Lowpass IIR Digital Filters IIR Digital Filters Highpass Highpass IIR Digital Filters IIR Digital Filters Bandpass Bandpass IIR Digital Filters IIR Digital Filters Bandstop IIR Digital Filters IIR Digital Filters Higher-order IIR Digital Filters order IIR Digital Filters 15 2.1 Lowpass IIR Digital Filters A first-order causal lowpass IIR digital filter has a transfer function given by where |a| < 1 for stability The above transfer function has a zero at z=ˉ 1 i.e., at which is in the stopband has a real pole at 1 1 1 1 ( ) 2 1 LP z H z z ( ) H z LP z 1 16 2.1 Lowpass IIR Digital Filters As w increases from 0 to p, the magnitude of the zero vector decreases from a value of 2 to 0, whereas, for a positive value of a , the magnitude of the pole vector increases from a value of to The maximum value of the magnitude function is 1 at w = 0, and the minimum value is 0 at w = p 0 1 1 17 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 First-order IIR Lowpass Filter / Magnitude =0.8 =0.7 =0.5 2.1 Lowpass IIR Digital Filters 10 -2 10 -1 10 0 -20 -15 -10 -5 0 First-order IIR Lowpass Filter / Gain, dB =0.5 =0.7 =0.8 18 2.2 Highpass IIR Digital Filters A first-order causal highpass IIR digital filter has a transfer function given by where |a| < 1 for stability The above transfer function has a zero at z=1 i.e., at w = 0 which is in the stopband 1 1 1 1 ( ) 2 1 HP z H z z 1 0
2.2 Highpass lIR Digital Filters 2.3 Bandpass lIR Digital Filters 2.3 Bandpass lIR Digital Filters Magnitude and gain responses of,(z)are ·Hr(e)goes to zero at=0ando=π A 2nd-order bandpass digital transfer function shown below is given by It assumes a maximum value of I at=, 1-a 1-22 called the center frequency of the bandpass Ha(2)=21-B1+a)2+a2 filter,where Its squared magnitude function is The frequencies and @where the squared Har(e)= magnitude becomes 1/2 are called the 3-dB cutoff frequencies (1-a)21-cos2) The difference between the two cutoff 2[1+B(1+a)+a2-2B(1+a)cos@+2acos20] frequencies,is called the 3-dB bandwidth 20 2.3 Bandpass IIR Digital Filters 2.4 Bandstop lIR Digital Filters 2.4 Bandstop lIR Digital Filters The transfer function is a BR function if<l .A 2nd-order bandstop digital filter has a Here,the magnitude function takes the and <1 transfer function given by maximum value of I at =0 and= Has(2)= I+a 1-2B2-+2- .It goes to 0 at=,where called the 21-B1+a)z+a2 notch frequency,is given by cosB .The transfer function is a BR function if a< The digital transfer function is more and <1 commonly called a notch filter Its magnitude response is plotted in the next The difference between the two cutoff slide frequencies is called the 3-dB notch bandwidth
19 2.2 Highpass IIR Digital Filters Magnitude and gain responses of are shown below ( ) H z LP 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 First-order IIR Highpass Filter / Magnitude =0.8 =0.7 =0.5 10 -2 10 -1 10 0 -20 -15 -10 -5 0 First-order IIR Highpass Filter / Gain,dB =0.8 =0.7 =0.5 20 2.3 Bandpass IIR Digital Filters A 2nd-order bandpass digital transfer function is given by Its squared magnitude function is 2 1 2 1 1 ( ) 2 1 (1 ) BP z H z z z 2 2 2 22 2 ( ) (1 ) (1 cos 2 ) 2[1 (1 ) 2 (1 ) cos 2 cos 2 ] j H e BP 21 2.3 Bandpass IIR Digital Filters goes to zero at and It assumes a maximum value of 1 at , called the center frequency of the bandpass filter, where The frequencies and where the squared magnitude becomes 1/2 are called the 3-dB cutoff frequencies cutoff frequencies The difference between the two cutoff frequencies, is called the 3-dB bandwidth 2 ( ) j H e BP 0 0 c1 c2 22 2.3 Bandpass IIR Digital Filters The transfer function is a BR function if and 1 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Second-order IIR Bandpass Filter ( =0.34) / Magnitude =0.2 =0.5 =0.8 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Second-order IIR Bandpass Filter ( =0.6) / Magnitude =0.1 =0.5 =0.8 23 2.4 Bandstop IIR Digital Filters A 2nd-order bandstop digital filter has a transfer function given by The transfer function is a BR function if and Its magnitude response is plotted in the next slide 1 2 1 2 1 12 ( ) 2 1 (1 ) BS z z H z z z 1 1 24 2.4 Bandstop IIR Digital Filters Here, the magnitude function takes the maximum value of 1 at w = 0 and w = p It goes to 0 at , where , called the notch frequency, is given by The digital transfer function is more commonly called a notch filter notch filter The difference between the two cutoff frequencies is called the 3-dB notch bandwidth 0 0 0 1 0 cos
2.4 Bandstop lIR Digital Filters 2.5 Higher-Order IIR Digital Filters 2.5 Higher-Order IIR Digital Filters By cascading the simple digital filters The corresponding squared-magnitude discussed so far,we can implement digital function is given by filters with sharper magnitude responses Gr( (1-a)21+coso) Consider a cascade of K first-order lowpass 21+2-2ac0s】 sections characterized by the transfer function .To determine the relation between its 3-dB Gw(2)= 1-a1+z1 cutoff frequency o and the parameter a ,we set 21-az 1-a)'(1+cos0,) 7 2(1+a-2ac0so) 2.5 Higher-Order lIR Digital Filters 3.Comb Filters 3.Comb Filters which when solved for a,yields for a .The simple filters discussed so far are .In its most general form,a comb filter has a stable G(z): characterized either by a single passband frequency response that is a periodic function _1+(1-C)cos0.-sin0.V2C-C2) of o with a period2/L.where L is a and/or a single stopband Q=- positive integer 1-C+c0s0 There are applications where filters with where C=2(-x multiple passbands and stopbands are required If H(z)is a filter with a single passband and/or a single stopband,a comb filter can be easily It should be noted that the expression for The comb filter is an example of such filters generated from it by replacing each delay in its given earlier reduces to realization with L delays resulting in a a=I-sino for K=1 structure with a transfer function given by G(2H) cos 10
25 2.4 Bandstop IIR Digital Filters 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Second-order IIR Bandstop Filter ( =0.5) / Magnitude =0.2 =0.5 =0.8 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Second-order IIR Bandstop Filter ( =0.5) / Magnitude =0.2 =0.5 =0.8 26 2.5 Higher-Order IIR Digital Filters By cascading the simple digital filters discussed so far, we can implement digital filters with sharper magnitude responses Consider a cascade of K first-order lowpass sections characterized by the transfer function 1 1 1 1 ( ) 2 1 K LP z G z z 27 2.5 Higher-Order IIR Digital Filters The corresponding squared-magnitude function is given by To determine the relation between its 3-dB cutoff frequency and the parameter ,we set 2 2 2 (1 ) (1 cos ) ( ) 2(1 2 cos ) K j G e LP c 2 2 (1 ) (1 cos ) 1 2(1 2 cos ) 2 K c c 28 2.5 Higher-Order IIR Digital Filters which when solved for a, yields for a stable : where It should be noted that the expression for a given earlier reduces to for K=1 2 1 (1 )cos sin 2 ) 1 cos c c c C CC C ( 1)/ 2 K K C ( ) G z LP 1 sin cos c c 29 3. Comb Filters The simple filters discussed so far are characterized either by a single passband and/or a single stopband There are applications where filters with multiple multiple passbands and stopbands are required The comb filter comb filter is an example of such filters 30 3. Comb Filters In its most general form, a comb filter has a frequency response that is a periodic function of w with a period 2p/L , where L is a positive integer If H(z) is a filter with a single passband and/or a single stopband, a comb filter can be easily generated from it by replacing each delay in its realization with L delays resulting in a structure with a transfer function given by G(z)=H(zL) 2 / L
3.Comb Filters 3.Comb Filters 3.Comb Filters 。IfH()exhibits a peak at。,thenG(e For example,the comb filter generated from For example,the comb filter generated from will exhibit L peaks at.k/L,0≤k≤L-1in the prototype lowpass FIR filterH(=)=(1/2)(1+) the prototype highpass FIR filter H,(=)=(1/2X1-=) the frequency range0≤o≤2π has a transfer function G()=(1/2)(1+) has a transfer function G()=(1/2X1-z) 。是 .Likewise,if H()has a notch at@,thenH) .G(e)has L notches at ·lG,()has L peaks at will have L notches at a,k/L,0≤k≤L-lin =(+1/L and L @=(2k+1)/L and L the frequency range0sos2 peaks at w=2kπ/L notches at w=2k/L 0s L-1 in the 0skL-1 in the A comb filter can be generated from either an frequency range frequency range FIR or an IIR prototype filter 0≤ms2x 0≤0≤2x 3.Comb Filters 3.Comb Filters 3.Comb Filters Depending on applications,comb filters with This filter has a peak magnitude at =0,and 0wm由rnm绿iat moviag avarage Prulalype other types of periodic magnitude responses M-1 notches at=2πl/M,1≤I≤M-1 15M=3 can be easily generated by appropriately choosing the prototype filter The corresponding comb filter has a transfer function For example,the M-point moving average G(z)= 1-2- filter 1-2w Ma1-2-) H2)=M0-2 whose magnitude has L peaks at @=2k/L, has been used as a prototype 0≤k≤L-1andL(M-1)notches at)=2kπ/LM 1≤k≤L(M-1)
31 3. Comb Filters If exhibits a peak at , then will exhibit L peaks at , in the frequency range Likewise, if has a notch at ,then will have L notches at , in the frequency range A comb filter can be generated from either an FIR or an IIR prototype filter ( ) j H e p ( ) j G e / p k L 0 1 k L 0 2 ( ) j H e 0 ( ) j H e 0 k L/ 0 1 k L 0 2 32 has L notches at and L peaks at , , in the frequency range 3. Comb Filters For example, the comb filter generated from the prototype lowpass FIR filter has a transfer function 1 0 H ( ) (1/ 2)(1 ) z z 0 ( ) j G e (2 1) / k L 0 1 k L 0 2 2 / k L 0 ( ) (1/ 2)(1 ) L Gz z 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Comb Filter from Lowpass Prototype (L=5) / Magnitude 33 has L peaks at and L notches at , in the frequency range 3. Comb Filters For example, the comb filter generated from the prototype highpass FIR filter has a transfer function 1 1 H ( ) (1/ 2)(1 ) z z 1( ) j G e (2 1) / k L 0 1 k L 0 2 2 / k L 1( ) (1/ 2)(1 ) L Gz z 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Comb Filter from Highpass Prototype (L=5) / Magnitude 34 3. Comb Filters Depending on applications, comb filters with other types of periodic magnitude responses can be easily generated by appropriately choosing the prototype filter For example, the M-point moving average filter has been used as a prototype 1 1 ( ) (1 ) M z H z M z 35 3. Comb Filters This filter has a peak magnitude at w = 0, and Mˉ1 notches at , The corresponding comb filter has a transfer function whose magnitude has L peaks at , and L(Mˉ1 ) notches at 1 ( ) (1 ) LML z G z M z 0 2 / l M 1 1 l M 0 1 k L 2 / k L 2 / k LM 1 ( 1) k LM 36 3. Comb Filters 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Comb Filter from M-point moving avarage Prototype / Magnitude L=5 M=3