IIR Digital Filter Design Chapter 9B Part B Impulse Invariance Method ●●● ●●● >Bilinear Transform Method >Spectral Transformations of IIR Filters IIR Digital Filter Design IIR Digital Filter Design >Lowpass-to-Lowpass Transformation 1.Impulse Invariance Method 1.Impulse Invariance Method 1.Impulse Invariance Method Definition- The relation between ZT and ST The relation between ZT and ST The impulse response of the digital filter is H.()=[h0e"h identical to the impulse response of an analog H(e)=h(n)B.()=h(nT)e prototype filter at sampling instants i,0=∑hnT)du-nn h(m)=h(nT)21=0,l,2 .Analog transfer function:H(s) h,(t)=ST-H.(s)月 B.()-me H(=H(e")=(s) .The impulse response of the digital filter is: z=e,s=Inz h(n）=h(nT),t=0,1,2,.. -差nf-ea 4 -h(uTer T
1 Chapter 9B IIR Digital Filter Design 2 Part B IIR Digital Filter Design 3 IIR Digital Filter Design Impulse Invariance Method Bilinear Transform Method Spectral Transformations of IIR Filters Spectral Transformations of IIR Filters Lowpass-to-Lowpass Transformation 4 1. Impulse Invariance Method Definition – The impulse response of the digital filter is identical to the impulse response of an analog prototype filter at sampling instants Analog transfer function: Ha(s) The impulse response of the digital filter is: ( ) ( ), 0,1,2, a h n h nT t 1 ( ) ST { ( )} a a ht H s 5 1. Impulse Invariance Method The relation between ZT and ST ( ) () st H s h t e dt a a ˆ ˆ ( ) () ( ) ( ) () ( ) ( ) st st aa a n st a n nsT a n H s h t e dt h nT t nT e dt h nT t nT e dt h nT e ˆ () ( ) ( ) a a n h t h nT t nT 6 1. Impulse Invariance Method The relation between ZT and ST ˆ () ( ) nsT a a n H s h nT e () () n n H z hnz ( ) ( ), 0,1,2, a h n h nT t ˆ () ( ) () sT sT Hz He H s z e a 1 ln sT z es z T ˈ
1.Impulse Invariance Method 1.Impulse Invariance Method 1.Impulse Invariance Method The relation between ZT and ST The digital filter transfer function A(z)is: According to the sampling theorem H(")is 月.U0=A,(slm H(2)=ZT(h(n))=ZT(h(nT)) a periodic version of H(j) 月.Um=22H.U0-n,) 2(-浮 .Transformation from s-plane to z-plane:2=e ·Fors=0otji0:z=rem=eea,=r=e网 42- The frequency responses are obtained by ·Mapping relations substituting z=e"and s=j: I r= 0=T+2kπ 号2(-劉 e-2(n-) Ⅱem=e◆ =T。+rj 1.Impulse Invariance Method 1.Impulse Invariance Method 1.Impulse Invariance Method ·Mapping上r-emeans Thus,the impulse invariance mapping has the Mapping亚a=pT+2kr=TQ+T A point on the frequency axis in the s-plane (=0) desired properties: is mapped to a point on the unit circle in the r-plane Frequency axis i corresponds to unit circle .A point on the left-halfs-plane with is mapped .Stability is preserved to z-plane with 1,i.e.,the left-half s-plane is mapped inside the unit circle .Similarly,A point on the right-half s-planc with is mapped to z-planc with .i.c.,the right- half s-plane is mapped outside the unit circle s-plane z-plane 12
7 1. Impulse Invariance Method The relation between ZT and ST 1 2 ˆ () ( ) a a ss k H j H j kj T T ˆ ˆ ( ) () a a s j H j Hs 1 2 ˆ ( ) a a k H s H s kj T T 1 2 ( ) z esT a k H z H s kj T T 8 1. Impulse Invariance Method The digital filter transfer function H(z) is: The frequency responses are obtained by substituting z=ej¹and s=j : 1 ln ( ) ZT( ( )) ZT( ( )) 1 2 a a k s z T H z h n h nT k Hsj T T 1 2 ( ) j a k k He H j j T T 9 1. Impulse Invariance Method According to the sampling theorem H(ej¹) is a periodic version of Ha( j) Transformation from s-plane to z-plane: For s=³0+j0 : Mapping relations I II sT z e 00 0 , j T j T T z re e e z r e 0T r e 0 j j T e e 0 0 2 2 T k k T T 10 1. Impulse Invariance Method Mapping I: means A point on the frequency axis in the s-plane (³0=0) is mapped to a point on the unit circle in the unit circle in the z-plane A point on the left-half s-plane with ³00 is mapped to z-plane with |z|>1, i.e., the righthalf s-plane is mapped outside the unit circle 0T r e 11 1. Impulse Invariance Method Thus, the impulse invariance mapping has the desired properties: Frequency axis j corresponds to unit circle Stability is preserved 12 1. Impulse Invariance Method j 0 /T /T 3 / T 3 / T j Imz 1 Re z s-plane z-plane mapping 2 2 k TkT T Mapping II:
1.Impulse Invariance Method 1.Impulse Invariance Method 1.Impulse Invariance Method Due to sampling the mapping is many-to-one Assume that H(s)has the form of .H(z)converges ife0,indicating The strips of length 2 /T are all mapped onto H(s)=4 that H (s)is stable the unit circle The corresponding signal in time-domain is Generalizing to higher order(N)analog .Only if h(r)is a band-limited signal,no alias transfer functions will occur h(1)=ST-(H(s)=Ae-u(r) H.=克4 Hence,this method is not suitable for ·By sampling h(i) s+ar highpass and bandstop filters design h(n)=h(nT)=Ae-"u(nT) h,0-∑Aen'ut0 H)=∑n=em 1-e2 →He)=月 1-er2可 1.Impulse Invariance Method 1.Impulse Invariance Method 2.Bilinear Transform Method Example 。Magnitude Response Definition- First Order Butterworth Filter Designed Using the To avoid aliasing,the mapping from s-plane Impulse Invariant Method (T=1) to z-plane should be one-to-one,i.e.,a single H.(s)= 8+1 +h.(t)=et)+H(z）= point in the s-plane should be mapped to a l-e2可 unique point in the z-plane and vice versa zero at z=0 1)The entire jn-axis should be mapped onto pole at z=1/e the unit circle 2)The entire left-half s-plane should be mapped inside the unit circle
13 1. Impulse Invariance Method Due to sampling the mapping is many-to-one The strips of length 2±/T are all mapped onto the unit circle Only if ha(t) is a band-limited signal, no alias will occur Hence, this method is not suitable for highpass highpass and bandstop filters design 14 1. Impulse Invariance Method Assume that Ha(s) has the form of The corresponding signal in time-domain is By sampling ha(t) ( ) a A H s s 1 ( ) ST { ( )} ( ) t a a h t H s Ae u t () ( ) ( ) nT a h n h nT Ae u nT 1 0 () () 1 n nT n T n n A H z hnz A e z e z 15 1. Impulse Invariance Method H(z) converges if or ¢>0, indicating that Ha(s) is stable Generalizing to higher order (N) analog transfer functions 1 T e 0 1 ( ) N k a k k A H s s 1 () () k N t a k k h t Ae ut 1 1 ( ) 1 N k T k A H z e z 16 1. Impulse Invariance Method Example First Order Butterworth Filter Designed Using the Impulse Invariant Method (T=1) zero at z=0 pole at z=1/e 1 ( ) 1 H s a s () () t a h t e ut 1 1 1 ( ) 1 H z e z 17 1. Impulse Invariance Method Magnitude Response 18 2. Bilinear Transform Method Definition – To avoid aliasing, the mapping from s-plane to z-plane should be one-to-one, i.e., a single point in the s-plane should be mapped to a unique point in the z-plane and vice versa 1) The entire j-axis should be mapped onto the unit circle 2) The entire left-half s-plane should be mapped inside the unit circle
2.Bilinear Transform Method 2.Bilinear Transform Method 2.Bilinear Transform Method 2)Employ impulse invariance method to s'-plane with One-to-one mapping from s to s' mapping 2=47 n'=m 1 s-plane z-plane mapping 77 Derivation of the bilinear transform: 1)One-to-one mapping froms tos'which compresses the entire s-plane into the strip -T<Im(s)<T s-plane s'plane -17 9 2.Bilinear Transform Method 2.Bilinear Transform Method 2.Bilinear Transform Method The normalized frequency now The desired transformation from s to z(via s) The bilinear transform:s= 21-21 corresponds toΩ"T T1+2 @=2 tan- @=2tan- 2T) 2 子m The s-plane transfer function H(s)gives a .Thus,the entire j-axis is compressed to the ●As we know e-e 1-e2 plane transfer function interval (-for in a one-to-one /tanx= ete-Ite2 G()=H(s儿2- manner ·Solving z gives: T14 ·Hence The mapping is highly nonlinear 2 0）_21-em .However,for small ='T it is =j片ta2)厂T1+e网 21-2 /-到 approximately linear 。Lets=jQ andz=e“,we can arrive ats= The bilinear transform T1+2-1 23 24
19 2. Bilinear Transform Method Derivation of the bilinear transform: Derivation of the bilinear transform: 1) One-to-one mapping from s to s’ which compresses the entire s-plane into the strip ˉ±/T < Im (s’) < ±/T j 0 j z Im 1 Re z s-plane z-plane mapping 20 2. Bilinear Transform Method 2) Employ impulse invariance method to s’-plane with z=es’T j 0 s-plane s’-plane j 0 /T /T mapping 21 2. Bilinear Transform Method 2 1 ' tan 2 T T One-to-one mapping from s to s’ 0 /T /T ' 22 2. Bilinear Transform Method The normalized frequency ¹ now corresponds to ’T Thus, the entire j-axis is compressed to the interval (ˉ±,±) for¹ in a one-to-one manner The mapping is highly nonlinear However, for small ¹=’T it is approximately linear 1 2 tan 2 T 23 2. Bilinear Transform Method The desired transformation from s to z (via s’) As we know Hence Let s=j and z=ej¹ ,we can arrive at 1 2 tan 2 T 2 tan T 2 2 2 1 tan 1 jx jx jx jx jx jx ee e j x ee e 2 21 tan 2 1 j j e j j T Te 1 1 2 1 1 z s T z The bilinear transform 24 2. Bilinear Transform Method The bilinear transform: The bilinear transform: The s-plane transfer function Ha(s) gives a zplane transfer function Solving z gives: 1 1 2 1 1 () () z a s T z Gz H s 1 1 2 2 T T z s s 1 1 2 1 1 z s T z
2.Bilinear Transform Method 2.Bilinear Transform Method 2.Bilinear Transform Method ①jn-axis,Res=0:this gives中l Frequency Warping To design a digital filter meeting the desired (digital) The frequency axis from s-plane is mapped onto the specifications we have tor unit circle Distortion due to Left-half s-plane,Re(s)0;1+(T2)s>-(72)s prewarped critical frequencies or 1 3 Transform H(s)to G(=)using the bilinear Right-half s-plane is mapped outside the unit circle transformation 3.Spectral Transformations of 2.Bilinear Transform Method 2.Bilinear Transform Method IIR Filters Example ·Magnitude Response Transformation of a given digital IIR lowpass First Order Butterworth Filter Designed by the transfer function G(z)to another digital Bilinear Transformation transfer function Gp(z) 1 H(s)= →H(z)= .Prototype lowpass G,(z);variable 2-1 3+1 +12- 21-2 Transformed filter G();variable2 1+2 →Halm=3- T1++1 Transformation from z-domain to z'-domain: F) zero atz=-1 .The entire frequency axis from the s-plane is mapped onto the Now,G(z)is transformed to Gp(z)through unit circle in the z-plane one-to-one No AL/ASING! pole at z=1/3 G2G(F))
25 2. Bilinear Transform Method ķ j-axis, Re(s)=0; this gives |z|=1 The frequency axis from s The frequency axis from s-plane is mapped onto the plane is mapped onto the unit circle ĸ Left-half s-plane, Re(s)0; |1+(T/2)s| > |1ˉ(T/2)s| or |z|>1 Right-half s-plane is mapped outside the unit circle 26 2. Bilinear Transform Method Frequency Warping tan( / 2) ( ) H j a 1 2 3 4 ( ) j H e 12 3 4 Distortion due to nonlinearity of the mapping 2 tan T 2 27 2. Bilinear Transform Method To design a digital filter meeting the desired (digital) To design a digital filter meeting the desired (digital) specifications we have to˖ ķ Prewarp the critical bandedge frequencies (¹p and ¹s) to analog frequencies (p and s) ĸ Design an analog prototype filter Ha(s) using the prewarped critical frequencies Ĺ Transform Ha(s) to G(z) using the bilinear transformation 28 2. Bilinear Transform Method Example First Order Butterworth Filter Designed by the Bilinear Transformation zero at z=ˉ1 pole at z=1/3 1 ( ) 1 H s a s 1 1 1 1 ( ) 3 T z H z z 1 1 1 2 1 1 1 1 1 ( ) 1 2 1 1 1 z s T z H z s z T z 29 2. Bilinear Transform Method Magnitude Response The entire frequency axis from the s-plane is mapped onto the unit circle in the z-plane one-to-one NO ALIASING ! NO ALIASING ! 30 3. Spectral Transformations of IIR Filters Transformation of a given digital IIR lowpass transfer function GL(z) to another digital transfer function GD(z) Prototype lowpass GL(z) ; variable zˉ1 Transformed filter GD(z’); variable z’ˉ1 Transformation from z-domain to z’-domain: z=F(z’) Now, GL(z) is transformed to GD(z’) through GD(z’)= GL(F(z’))
3.Spectral Transformations of 3.Spectral Transformations of 4.Lowpass-to-Lowpass IIR Filters IIR Filters Transformation To transform a rational G,(z)into a rational ●The requirements >1if>1 .G(z)with cutoff frequency is transformed Gp(z),F(z)must be a rational function in z F(z==1if=1 to another lowpass filter Gp()with The inside of the z-plane should be mapped into the inside of z'-plane <L if 2 <1 2=F(=1-a2 2'-a .F(z)must be a stable allpass function with a real In order to map a lowpass magnitude response e-ju e-ie -a to one of the four basic types of magnitude The most general form of F(z)with real 1-ae-jo responses,points on the unit circle in z-plane coefficients is given by should be mapped onto the unit circle in z' plane 1-a2 4.Lowpass-to-Lowpass 5.Computer-Aided Design of IIR 5.Computer-Aided Design of lIR Transformation Digital Filters Digital Filters The IIR and FIR filter design techniques Basic idea behind the computer-based is discussed so far can be easily implemented on iterative technique a computer Let H(e)denote the frequency response of 2 In addition,there are a number of filter design the digital filter (z)to be designed + algorithms that rely on some type of approximating the desired frequency 2 optimization techniques that are used to response D(e),given as a piecewise linear minimize the error between the desired function of o,in some sense frequency response and that of the computer generated filter
31 3. Spectral Transformations of IIR Filters To transform a rational GL(z) into a rational GD(z’), F(z’) must be a rational function in z’ The inside of the z-plane should be mapped into the inside of z’-plane In order to map a lowpass magnitude response to one of the four basic types of magnitude responses, points on the unit circle in z-plane should be mapped onto the unit circle in z’- plane 32 3. Spectral Transformations of IIR Filters The requirements Fˉ1(z’) must be a stable allpass allpass function The most general form of Fˉ1(z’) with real coefficients is given by 1, if 1 ( ') 1, if 1 1, if 1 z Fz z z * 1 1 1 ' ( ') ' L l l l z F z z 33 4. Lowpass-to-Lowpass Transformation GL(z) with cutoff frequency ¹c is transformed to another lowpass filter GD(z’) with ¹’c with real 1 1 1 ' ( ') ' z z Fz z ' ' 1 j j j e e e 1 ' tan tan 21 2 34 4. Lowpass-to-Lowpass Transformation ' sin 2 ' sin 2 c c c c 35 5. Computer-Aided Design of IIR Digital Filters The IIR and FIR filter design techniques discussed so far can be easily implemented on a computer In addition, there are a number of filter design algorithms that rely on some type of optimization techniques that are used to minimize the error between the desired frequency response and that of the computer generated filter 36 5. Computer-Aided Design of IIR Digital Filters Basic idea behind the computer-based is iterative technique Let denote the frequency response of the digital filter H(z) to be designed approximating the desired frequency response , given as a piecewise linear function of , in some sense ( ) j H e ( ) j D e
5.Computer-Aided Design of lIR 5.Computer-Aided Design of IIR 5.Computer-Aided Design of IIR Digital Filters Digital Filters Digital Filters Objective--Determine iteratively the Chebyshev or minimax criterion Least-p Criterion coefficients of (z)so that the difference between D(ei)and H(e)over closed Minimizes the peak absolute value of the ·Minimizes weighted error: subintervals of 0sos is minimized s=max E(@) W(e)[H(e)-D(e)Tdo This difference usually specified as a where R is the set of disjoint frequency bands over the specified frequency range R withp a weighted error function in the range 0ss,on which D(e)is positive integer E(@)=W(e)[H(ei)-D(e) defined p=2 yields the least-squares criterion where W(e)is some user-specified For example,for a lowpass filter design,R is ·Asp→oo,the least p-th solution approaches weighting function the disjoint union of (0,@)and ( the minimax solution 5.Computer-Aided Design of lIR Digital Filters In practice,the p-th power error measure is approximated as e)-D where ,1sisK,is a suitably chosen dense grid of digital angular frequencies For linear-phase FIR filter design,H(ei)and D(e)are zero-phase frequency responses .For IIR filter design,H(e)and D(e)are magnitude functions
37 5. Computer-Aided Design of IIR Digital Filters Objective Objective -- Determine iteratively the coefficients of H(z) so that the difference between and over closed subintervals of is minimized This difference usually specified as a weighted error function where is some user-specified weighting function ( ) j H e ( ) j D e 0 () ( ) ( ) ( ) jj j E We He De ! " # ( ) j W e 38 5. Computer-Aided Design of IIR Digital Filters Chebyshev or minimax criterion Minimizes the peak absolute value of the weighted error: where R is the set of disjoint frequency bands in the range , on which is defined For example, for a lowpass filter design, R is the disjoint union of and max ( ) E $ % R 0 ( ) j D e (0, ) p ( ,) s 39 5. Computer-Aided Design of IIR Digital Filters Least-p Criterion Minimizes over the specified frequency range R with p a positive integer p=2 yields the least-squares criterion As pėĞ, the least p-th solution approaches the minimax solution () () () P jj j We He De d $ % ! R " # 40 5. Computer-Aided Design of IIR Digital Filters In practice, the p-th power error measure is approximated as where , , is a suitably chosen dense grid of digital angular frequencies For linear-phase FIR filter design, and are zero-phase frequency responses For IIR filter design, and are magnitude functions & ' 1 ()() () ii i K P jj j i We He De $ ! " # i 1 i K ( ) j H e ( ) j D e ( ) j H e ( ) j D e