Contents Part A Types of Transfer Functions 1.Based on Magnitude Characteristics Chapter 7A 1.Transfer Function Classification Based on ●●● Magnitude Characteristics 1.1 Ideal Filters 1.2 Bounded Real Transfer Functions 2.Transfer Function Classification Based on 1.3 Allpass Transfer Functions LTI Discrete-Time Systems Phase Characteristics 2.Based on Phase Characteristics in the Transform Domain 3.Types of Linear-Phase Transfer Functions 1.1 Zero-Phase Transfer Funetions 4.Sample Digital Filters 1.2 Linear-Phase Transfer Functions 1.3 Minimum-Phase and Maximum-Phase Transfer Functions Types of Transfer Functions Types of Transfer Functions 1.1 Ideal Filters In the case of digital transfer functions with Based on the shape of the magnitude The time-domain classification of an LTI function,four types of ideal filters are digital transfer function sequence is based on frequency-selective frequency responses,there usually defined:lowpass.highpass, the length of its impulse response: are two types of classifications bandpass and bandstop --Finite impulse response(FIR)transfer (1)Classification based on the shape of the A digital filter designed to pass signal function magnitude function ( components of certain frequencies without --Infinite impulse response (IIR)transfer (2)Classification based on the form of the distortion should have a frequency response function phase function ( equal to one at these frequencies,and should have a frequency response equal to zero at all other frequencies 5
Chapter 7A LTI Discrete-Time Systems in the Transform Domain 2 Contents 1. Transfer Function Classification Based on 1. Transfer Function Classification Based on Magnitude Characteristics Magnitude Characteristics 2. Transfer Function Classification Based on 2. Transfer Function Classification Based on Phase Characteristics Phase Characteristics 3. Types of Linear 3. Types of Linear-Phase Transfer Functions Phase Transfer Functions 4. Sample Digital Filters 4. Sample Digital Filters 3 Part A Types of Transfer Functions 1. Based on Magnitude Characteristics 1. Based on Magnitude Characteristics 1.1 Ideal Filters 1.1 Ideal Filters 1.2 Bounded Real Transfer Functions 1.2 Bounded Real Transfer Functions 1.3 Allpass Allpass Transfer Functions Transfer Functions 2. Based on Phase Characteristics 2. Based on Phase Characteristics 1.1 Zero-Phase Transfer Functions Phase Transfer Functions 1.2 Linear-Phase Transfer Functions Phase Transfer Functions 1.3 Minimum-Phase and Maximum-Phase Transfer Functions Transfer Functions 4 Types of Transfer Functions The time-domain classification of an LTI digital transfer function sequence is based on the length of its impulse response: -- Finite impulse response (FIR) transfer function -- Infinite impulse response (IIR) transfer function 5 Types of Transfer Functions In the case of digital transfer functions with frequency-selective frequency responses, there are two types of classifications (1) Classification based on the shape of the magnitude function |H(ej¹)| (2) Classification based on the form of the phase function ©(¹) 6 1.1 Ideal Filters Based on the shape of the magnitude functionˈfour types of ideal filters are usually defined˖lowpass lowpass , highpass highpass, bandpass bandpass and bandstop A digital filter designed to pass signal components of certain frequencies without distortion should have a frequency response equal to one at these frequencies, and should have a frequency response equal to zero at all other frequencies
1.1 ldeal Filters 1.1 Ideal Filters 1.1 Ideal Filters The range of frequencies where the frequency H Earlier in the course we derived the inverse response takes the value of one is called the DTFT of the frequency response of the ideal passband lowpass filter: The range of frequencies where the frequency h(n)=sinon -0≤n≤0 response takes the value of zero is called the Lowpass Highpas stopband He) We have also shown that the above impulse Frequency responses of the four popular types of ideal digital filters with real impulse response coefficients are shown in the next . response is not absolutely summable,and hence,the corresponding transfer function is not BIBO stable slide 7 1.1 Ideal Filters 1.1 Ideal Filters 1.1 Ideal Filters Also,is not causal and is of doubly infinite .To develop stable and realizable transfer Moreover.the length functions,the ideal frequency response magnitude response is The remaining three ideal filters are also specifications are relaxed by including a allowed to vary by a small amount both in 146, characterized by doubly infinite,noncausal transition band between the passband and the the passband and the 1-6 impulse responses and are not absolutely stopband. stopband. summable This permits the magnitude response to decay ·Typical magnitude Thus,the ideal filters with the ideal "brick slowly from its maximum value in the response specifications wall"frequency responses cannot be realized passband to the zero value in the stopband. of a lowpass filter are with finite dimensional LTI filters shown in the figure
7 1.1 Ideal Filters The range of frequencies where the frequency response takes the value of one is called the passband The range of frequencies where the frequency response takes the value of zero is called the stopband Frequency responses of the four popular types of ideal digital filters with real impulse response coefficients are shown in the next slide 8 1.1 Ideal Filters ( ) j H e BS 0 1 Bandstop ( ) j H e LP c 0 c 1 Lowpass ( ) j H e HP c 0 c 1 Highpass ( ) j H e BP 0 c1 c2 1 Bandpass c2 c1 c2 c1 c1 c2 9 1.1 Ideal Filters Earlier in the course we derived the inverse DTFT of the frequency response of the ideal lowpass filter: We have also shown that the above impulse response is not absolutely not absolutely summable summable, and hence, the corresponding transfer function is not BIBO stable not BIBO stable sin () , c LP n hn n n 10 1.1 Ideal Filters Also, is not causal not causal and is of doubly infinite length The remaining three ideal filters are also characterized by doubly infinite, doubly infinite, noncausal noncausal impulse responses and are not absolutely not absolutely summable summable Thus, the ideal filters with the ideal “brick wall” frequency responses cannot be realized with finite dimensional LTI filters 11 1.1 Ideal Filters To develop stable and realizable transfer functions, the ideal frequency response specifications are relaxed by including a transition band between the passband and the stopband. This permits the magnitude response to decay slowly from its maximum value in the passband to the zero value in the stopband. 12 1.1 Ideal Filters Moreover, the magnitude response is allowed to vary by a small amount both in the passband and the stopband. Typical magnitude response specifications of a lowpass filter are shown in the figure. ( ) j HLP e 0 p s 1 p 1 p s c pass band stop band Transition band
1.2 Bounded Real Transfer Functions 1.2 Bounded Real Transfer Functions 1.2 Bounded Real Transfer Functions A causal stable real-coefficient transfer function A(z)is defined as a bounded real(BR) .Then the condition H)s1implies that Thus,for all finite-energy inputs,the output energy is less than or equal to the input energy transfer function if Y(e)sX(e) implying that a digital filter characterized by a H(e)s1 for all values of BR transfer function can be viewed as a ●Integrating the above from-rtoπ，and passive structure Let x(n)and y(n)denote,respectively.the applying Parseval's relation we get input and output of a digital filter characterized If H(e)=1,then the output energy is equal by a BR transfer function H(z)with X(e) 2bnfs2uaf to the input energy,and such a digital filter is and Y(ei)denoting their DTFTs therefore a lossless system 1.2 Bounded Real Transfer Functions 1.3 Allpass Transfer Function 1.3 Allpass Transfer Function Definition A causal stable real-coefficient transfer Hence,Az)can be written as function H(z)with H(e)=1 is thus called a An IIR transfer function A(z)with unity magnitude response for all frequencies,i.e., u()=Du) lossless bounded real(LBR)transfer function D.(z) The BR and LBR transfer functions are the (e)=1.for all o .Note from the above that if=zis a pole of a real coefficient allpass transfer function,then keys to the realization of digital filters with is called an allpass transfer function it has a zero at 2=1/ low coefficient sensitivity .An M-th order causal real-coefficient allpass The numerator of a real-coefficient allpass transfer function is of the form transfer function is said to be the mirror- 4v(回）=t+d++dEa+2"Dwe image polynomial of the denominator,and dtdE++dw2a+d-Dw闹 vice versa 1
13 1.2 Bounded Real Transfer Functions A causal stable real-coefficient transfer function H(z) is defined as a bounded real (BR) bounded real (BR) transfer function if Let x(n) and y(n) denote, respectively, the input and output of a digital filter characterized by a BR transfer function H(z) with and denoting their DTFTs ( ) 1 for all values of j H e ( ) j X e ( ) j Y e 14 1.2 Bounded Real Transfer Functions Then the condition implies that Integrating the above from to , and applying Parseval’s relation we get ( )1 j H e 2 2 () () j j Ye Xe 2 2 () () n n y n xn 15 1.2 Bounded Real Transfer Functions Thus, for all finite-energy inputs, the output energy is less than or equal to the input energy implying that a digital filter characterized by a BR transfer function can be viewed as a passive structure passive structure If , then the output energy is equal to the input energy, and such a digital filter is therefore a lossless system ( )1 j H e 16 1.2 Bounded Real Transfer Functions A causal stable real-coefficient transfer function H(z) with is thus called a lossless bounded real (LBR) transfer function The BR and LBR transfer functions are the keys to the realization of digital filters with low coefficient sensitivity ( )1 j H e 17 1.3 Allpass Transfer Function Definition An IIR transfer function A(z) with unity magnitude response for all frequencies, i.e., is called an allpass allpass transfer function An M-th order causal real-coefficient allpass transfer function is of the form 2 ( ) 1, for all j A e 1 1 1 1 1 1 1 1 ( ) 1 M M M M M M M M M d d z dz z A z dz d z d z ( ) DM z 1 ( ) M M z D z 18 1.3 Allpass Transfer Function Hence, AM(z) can be written as Note from the above that if z=z0 is a pole of a real coefficient allpass transfer function, then it has a zero at z=1/z0 The numerator of a real-coefficient allpass transfer function is said to be the mirrorimage polynomial image polynomial of the denominator, and vice versa 1 ( ) ( ) ( ) M M M M z D z A z D z
1.3 Allpass Transfer Function 1.3 Allpass Transfer Function 1.3 Allpass Transfer Function ·The expression 4e=±De .To show that)=1,we observe that Now,the poles of a causal stable transfer D(=) implies that the poles and zeros of a real 4(e)=t"De倒 function must lie inside the unit circle in the z-plane D(2-) coefficient allpass function exhibit mirror- 。Therefore, Hence,all zeros of a causal stable allpass 4u()()-Du(D( transfer function must lie outside the unit image symmetry in the z-plane circle in a mirror-image symmetry with its An example Du(z)Du(2) poles situated inside the unit circle 。Hence,, -0.2+0.182+0.422+23 4(e)=4u(=)4u()=1 Figure in the next slide shows the principal A()= 1+0.4z+0.1823-0.22 value of the phase of the former example 1.3 Allpass Transfer Function 1.3 Allpass Transfer Function 1.3 Allpass Transfer Function Properties Let r(@)denote the group delay function of A causal stable real-coefficient allpass an allpass filter A(),i.e., transfer function is a lossless bounded real (LBR)function or,equivalently,a causal (o)=-4[o.(] stable allpass filter is a lossless structure do The unwrapped phase function of a stable allpass The magnitude function of a stable allpass function is a monotonically decreasing function of ·Note the discontinuity by the amount of2πin function A(z)satisfies: so that r(@)is everywhere positive in the range the phase (@ 1 01 for <1 23 oylo-Ms
19 -1 -0.5 0 0.5 1 1.5 2 2.5 -1.5 -1 -0.5 0 0.5 1 1.5 Real Part Imaginary Part 1.3 Allpass Transfer Function The expression implies that the poles and zeros of a real coefficient allpass function exhibit mirrorimage symmetry in the z-plane An example An example 1 ( ) ( ) ( ) M M M M z D z A z D z 1 23 3 1 23 0.2 0.18 0.4 ( ) 1 0.4 0.18 0.2 z z z A z z z z 20 1.3 Allpass Transfer Function To show that , we observe that Therefore, Hence, 1 1 ( ) ( ) ( ) M M M M z D z A z D z 2 () 1 j A e 1 1 1 ( ) () ( ) () 1 () ( ) M M M M M M M M z D z zD z Az Az Dz Dz 2 1 ( ) ( ) () 1 j j M M z e Ae A z A z 21 1.3 Allpass Transfer Function Now, the poles of a causal stable transfer function must lie inside the unit circle in the z-plane Hence, all zeros of a causal stable allpass transfer function must lie outside the unit circle in a mirror-image symmetry with its poles situated inside the unit circle Figure in the next slide shows the principal value of the phase of the former example 22 1.3 Allpass Transfer Function Note the discontinuity by the amount of in the phase The unwrapped phase function is a continuous function of 2 ( ) 0 0.2 0.4 0.6 0.8 1 -4 -3 -2 -1 0 1 2 3 4 / Phase Reponse (in rads/s) Principle Value of Phase 0 0.2 0.4 0.6 0.8 1 -10 -8 -6 -4 -2 0 / Phase Reponse (in rads/s) Unwrapped Phase 23 1.3 Allpass Transfer Function Properties Properties A causal stable real-coefficient allpass transfer function is a lossless bounded real (LBR) function or, equivalently, a causal stable allpass filter is a lossless structure The magnitude function of a stable allpass function A(z) satisfies: 1 1 () 1 1 1 1 for z A z for z for z 24 1.3 Allpass Transfer Function Let t(w) denote the group delay function of an allpass filter A(z) , i.e., The unwrapped phase function of a stable allpass function is a monotonically decreasing function of w so that t(w) is everywhere positive in the range . An M-th order stable real-coefficient allpass transfer function satisfies: ( ) () ( ) j c d e d 0 ( ) 0 ( )d M
1.3 Allpass Transfer Function 1.3 Allpass Transfer Function Part A Types of Transfer Functions A Simple Application 1.Based on Magnitude Charncteristics .A simple but often used application of an G 1.1 Ideal Filters allpass filter is as a delay equalizer 12 Bounded Real Transfer Functions .Let G(z)be the transfer function of a digital Sincee)=1,we have 1.3 Allpass Transfer Functions filter designed to meet a prescribed magnitude response G(e)(e=G(ei) 2.Based on Phase Characteristics The nonlinear phase response of G(z)can be .Overall group delay is the given by the sum of 2.1 Zero-Phase Transfer Fanctioms 22 Linear-Phase Tramsfer Functions corrected by cascading it with an allpass filter the group delays of G(z)and A(z) 之3 linicum-Phase and》aximui-Phas A(z)so that the overall cascade has a constant Transfer Functions group delay in the band of interest 2 2.1 Zero-Phase Transfer Functions 2.1 Zero-Phase Transfer Functions 2.1 Zero-Phase Transfer Functions A second classification of a transfer function One zero-phase filtering scheme is sketched is with respect to its phase characteristics However,it is not possible to design a causal below In many applications,it is necessary that the digital filter with a zero phase(pp.287-288) digital filter designed does not distort the For non-real-time processing of real-valued x) V(e)U)) e-】 phase of the input signal components with input signals of finite length,zero-phase frequencies in the passband From the figure,we can arrive at filtering can be very simply implemented by .One way to avoid any phase distortion is to relaxing the causality requirement Y(ei)=W'(e)=H(e(e)=H(ei(e) make the frequency response of the filter real =H'(e)H(e)X(e)H(eX(e) and nonnegative,i.e.,to design the filter with a zero phase characteristic 29 Real and Zero-Phase
25 1.3 Allpass Transfer Function A Simple Application A simple but often used application of an allpass filter is as a delay equalizer delay equalizer Let G(z) be the transfer function of a digital filter designed to meet a prescribed magnitude response The nonlinear phase response of G(z) can be corrected by cascading it with an allpass filter A(z) so that the overall cascade has a constant group delay in the band of interest 26 1.3 Allpass Transfer Function Since , we have Overall group delay is the given by the sum of the group delays of G(z) and A(z) G(z) A(z) 2 () 1 j A e ( )( ) ( ) jj j Ge Ae Ge 27 Part A Types of Transfer Functions 1. Based on Magnitude Characteristics 1. Based on Magnitude Characteristics 1.1 Ideal Filters 1.1 Ideal Filters 1.2 Bounded Real Transfer Functions 1.2 Bounded Real Transfer Functions 1.3 Allpass Allpass Transfer Functions Transfer Functions 2. Based on Phase Characteristics 2. Based on Phase Characteristics 2.1 Zero-Phase Transfer Functions Phase Transfer Functions 2.2 Linear-Phase Transfer Functions Phase Transfer Functions 2.3 Minimum-Phase and Maximum-Phase Transfer Functions Transfer Functions 28 2.1 Zero-Phase Transfer Functions A second classification of a transfer function is with respect to its phase characteristics In many applications, it is necessary that the digital filter designed does not distort the phase of the input signal components with frequencies in the passband One way to avoid any phase distortion is to make the frequency response of the filter real and nonnegative, i.e., to design the filter with a zero phase zero phase characteristic 29 2.1 Zero-Phase Transfer Functions However, it is not possible to design a causal digital filter with a zero phase (pp. 287-288) For non-real-time processing of real-valued input signals of finite length, zero-phase filtering can be very simply implemented by relaxing the causality requirement 30 2.1 Zero-Phase Transfer Functions One zero-phase filtering scheme is sketched below From the figure, we can arrive at x(n) H(z) Folding H(z) Folding v(n) u(n) w(n) y(n) ( ) j X e ( ) j V e ( ) j U e ( ) j W e ( ) j Y e * ** * 2 * ( ) ( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( ) j j j j jj jjj j j Ye W e H e U e H e Ve H e He Xe He Xe Real and Zero-Phase
2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions .Linear-Phase H(ei)=e-joo If D is an integer,then y(n)is identical to x(n), If it is desired to pass input signal components H(ei)=1 r(@)=D but delayed by D samples in a certain frequency range undistorted in .The output y(n)of this filter to an input If D is not an integer,y(n),being delayed by a both magnitude and phase,then the transfer x(n)=Aeicm is then given by function should exhibit a unity magnitude fractional part,is not identical to x(n) y(n)=Ae-jobei=Aejo-D) response and a linear-phase response in the .In the latter case,the waveform of the band of interest .Ifx (r)and y (1)represent the continuous time underlying continuous-time output is identical signals whose sampled versions,sampled at t= to the waveform of the underlying continuous- Figure in the next slide shows the frequency nT.are d(n)and vn)given above,then the delay time input and delayed D units of time response if a lowpass filter with a linear-phase between x (r)and y (r)is precisely the group characteristic in the passband delay of amount D 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions oT It is nearly impossible to design a linear-phase The above transfer function has a linear-phase,if IIR transfer function its impulse response h(n)is either symmetric,i.e., It is always possible to design an FIR transfer n)=hN-1-a,0sn≤N-1 function with an exact linear-phase response or is antisymmetric,i.e., Consider a causal FIR transfer function (z) h(n)=-h(N-1-n),0snsN-1 Since the signal components in the stopband of length N,i.e.,of order N-1 There are two types linear phase are blocked,the phase response in the -r@, 0=0 stopband can be of any shape 0)=0-t@= →do@l.-r 2 -wA=- do 2
31 2.2 Linear-Phase Transfer Functions Linear-Phase The output y(n) of this filter to an input is then given by If xa(t) and ya(t) represent the continuous time signals whose sampled versions, sampled at t = nT, are x(n) and y(n) given above, then the delay between xa(t) and ya(t) is precisely the group delay of amount D ( ) j jD He e ( )1 j H e ( ) D ( ) j n x n Ae ( ) ( ) jD jn j nD y n Ae e Ae 32 2.2 Linear-Phase Transfer Functions If D is an integer, then y(n) is identical to x(n), but delayed by D samples If D is not an integer, y(n) , being delayed by a fractional part, is not identical to x(n) In the latter case, the waveform of the underlying continuous-time output is identical to the waveform of the underlying continuoustime input and delayed D units of time 33 2.2 Linear-Phase Transfer Functions If it is desired to pass input signal components in a certain frequency range undistorted in undistorted in both magnitude and phase both magnitude and phase, then the transfer function should exhibit a unity magnitude unity magnitude response and a linear-phase response in the band of interest Figure in the next slide shows the frequency response if a lowpass filter with a linear-phase characteristic in the passband 34 2.2 Linear-Phase Transfer Functions ( ) j H e LP c 0 c 1 arg ( ) j H e LP c 0 c Since the signal components in the stopband are blocked, the phase response in the stopband can be of any shape 35 2.2 Linear-Phase Transfer Functions It is nearly impossible to design a linear-phase IIR transfer function It is always possible to design an FIR transfer function with an exact linear-phase response Consider a causal FIR transfer function H(z) of length N, i.e., of order Nˉ1 1 0 () () N n n H z hnz 36 2.2 Linear-Phase Transfer Functions The above transfer function has a linear-phase, if its impulse response h(n) is either symmetric, i.e., or is antisymmetric, i.e., There are two types linear phase hn hN n n N ( ) ( 1 ), 0 1 hn hN n n N ( ) ( 1 ), 0 1 0 0 0 , 0 ( ) , 2 2 d ( ) d
2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions Proof If this transfer function has a linear phase,such >h(n)sinc as w)=-t0 He=岁me-分n)cos om-j·分n)sin on tan ro=Sin tw cos ro We obtain the following relationship 风n)cose期 )sin on i.e., If /(m)is a real sequence,we have 分o h(月)sin cn cos ro=0 ()arg tan 》n)sin h(n)coson or (@)arg tan (n)cos Taking tan(.)on both sides of the above equation 岁siml0c-no=0★ 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions .sin[(r-m)]is odd-symmetry on r=m Since the length of the impulse response can be either even or odd,we can define four ·Letr=(N-l)/2,thus equation★holds ifh(m is even-symmetry on n=(N-1)/2 types of linear-phase FIR transfer functions For an antisymmetric FIR filter of odd length, ◆In other words,.hm)=hW-l-n,0sn≤N-l i.e.,Nodd h(N-1)/2))=0 Similarly,if 0(@)=-/2-ro ,we can arrive at We examine next the each of the 4 cases h(n)=-h(N-1-n),0snsN-1 0 r
37 2.2 Linear-Phase Transfer Functions Proof If h(n) is a real sequence, we have 11 1 00 0 ( ) ( ) ( )cos ( )sin NN N j jn nn n H e hne hn n j hn n 1 0 1 0 ( )sin ( ) arg tan ( )cos N n N n hn n hn n 38 2.2 Linear-Phase Transfer Functions If this transfer function has a linear phase, such as We obtain the following relationship Taking tan(.) on both sides of the above equation ( ) 1 0 1 0 ( )sin ( ) arg tan ( )cos N n N n hn n hn n 39 2.2 Linear-Phase Transfer Functions i.e., or 1 0 1 0 ( )sin sin tan cos ( )cos N n N n hn n hn n 1 1 0 0 ( )cos sin ( )sin cos 0 N N n n hn n hn n 1 0 ( )sin ( ) 0 N n hn n 40 2.2 Linear-Phase Transfer Functions is odd-symmetry on Let , thus equation holds if h(n) is even-symmetry on In other words, Similarly, if ,we can arrive at sin ( ) n n ( 1) / 2 N n N ( 1) / 2 hn hN n n N ( ) ( 1 ), 0 1 ( ) /2 hn hN n n N ( ) ( 1 ), 0 1 41 2.2 Linear-Phase Transfer Functions Since the length of the impulse response can be either even or odd, we can define four types of linear-phase FIR transfer functions For an antisymmetric FIR filter of odd length, i.e., N odd : h{(Nˉ1)/2)}= 0 We examine next the each of the 4 cases 42 2.2 Linear-Phase Transfer Functions h n( ) n 0 1 2 4 5 7 6 3 8 Center of even-symmetry h n( ) n 0 1 2 7 4 5 6 8 3 Center of odd-symmetry h n( ) n 0 1 2 4 7 5 3 6 Center of even-symmetry h n( ) n 0 1 2 4 7 5 6 3 Center of odd-symmetry Type 1 N=9 Type 4 N=8 Type 3 N=9 Type 2 N=8
2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions Consider first an FIR filter with a symmetric .A real-coefficient polynomial H(z)satisfying 。It follows the relation H(z)=±z-w-Hz' impulse response:h(n)=h(N-1-n) the above condition is called a mirror-image that if z=z is a zero of H(2),so is=1/z Its transfer function can be written as polynomial (MIP) V-I N-I Moreover,for an FIR filter with a real impulse H(e)=∑m)z=∑h(N-1-n)z In the case of anti-symmetric impulse response,the zeros of H(z)occur in complex =0 m-0 response,the corresponding expression is conjugate pairs By making a change of variable m=N-1- n,we can write H(2)=-2-N-DH(2-) Hence,a zero at z=z is associated with a zero He=艺Mm2-=2me which is called an antimirror-image at z=z* polynomial(AIP) =3-WH() 44 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions Am ●Since a zero at=士1 is its own reciprocal,,it can appear only singly Likewise,for a Type 3 or 4 filter, H1)=-H(1) Now a Type 2 FIR filter satisfies H(z)=z-w-H(2- implying H(z)must have a zero at z=1 Ret with degree N-1 odd .On the other hand,only the Type 3 FIR filter is restricted to have a zero at z=-1 since here ·Hence,H(-1)=(-l)w-H(-1)=-H(-1) the degree N-I is even and hence, implying (-1)=0,i.e.,H(z)must have a H(-1)=-(-1)w-H(-1)=-H(-1) Zero at z=-1 4
43 2.2 Linear-Phase Transfer Functions Consider first an FIR filter with a symmetric impulse response: Its transfer function can be written as By making a change of variable m=Nˉ1ˉ n ,we can write hn hN n () ( 1 ) 1 1 0 0 () () ( 1 ) N N n n n n H z hnz hN nz 1 1 ( 1 ) ( 1) 0 0 ( 1) 1 () ( ) ( ) ( ) N N Nm N m m m N H z hmz z hmz z Hz 44 2.2 Linear-Phase Transfer Functions A real-coefficient polynomial H(z) satisfying the above condition is called a mirror-image polynomial (MIP) polynomial In the case of anti-symmetric impulse response, the corresponding expression is which is called an antimirror antimirror-image polynomial (AIP) polynomial ( 1) 1 () ( ) N Hz z Hz 45 2.2 Linear-Phase Transfer Functions It follows the relation that if z=zi is a zero of H(z), so is z=1/zi Moreover, for an FIR filter with a real impulse response, the zeros of H(z) occur in complex conjugate pairs Hence, a zero at z=zi is associated with a zero at z=zi* ( 1) 1 () ( ) N Hz z Hz 46 2.2 Linear-Phase Transfer Functions Re z jIm z 1z * 1z 1 1 z * 1 z 2 z 2 1 z 3 z * 3 z 4 z 47 Since a zero at z=f1 is its own reciprocal, it can appear only singly Now a Type 2 FIR filter satisfies with degree Nˉ1 odd Hence, implying H(ˉ1)=0 ,i.e., H(z) must have a zero at z=ˉ1 2.2 Linear-Phase Transfer Functions ( 1) 1 () ( ) N Hz z Hz ( 1) ( 1) ( 1) ( 1) ( 1) N H HH 48 Likewise, for a Type 3 or 4 filter, implying H(z) must have a zero at z=1 On the other hand, only the Type 3 FIR filter is restricted to have a zero at z=ˉ1 since here the degree Nˉ1 is even and hence, 2.2 Linear-Phase Transfer Functions H H (1) (1) ( 1) ( 1) ( 1) ( 1) ( 1) N H HH
2.3 Minimum-Phase and Maximum- 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions Phase Transfer Functions Typical zero locations shown below Summary A causal stable transfer function with all .A Type 2 FIR filter cannot be used to design a zeros outside the unit circle has an excess highpass filter since it always has a zero1 phase compared to a causal transfer function .A Type 3 FIR filter has zeros at both==I and==-1, with identical magnitude but having all zeros and hence cannot be used to design either a lowpass inside the unit circle or a highpass or a bandstop filter zer or ploe vector A Type 4 FIR filter is not appropriate to design a △arg =2 inside the unit circle lowpass filter due to the presence of a zero at z=I .Type I FIR filter has no such restrictions and can be used to design almost any type of filter △arg rero or ploe vector outside the unit circle 2.3 Minimum-Phase and Maximum- 2.3 Minimum-Phase and Maximum- 2.3 Minimum-Phase and Maximum- Phase Transfer Functions Phase Transfer Functions Phase Transfer Functions It assumed that a causal stable transfer A causal stable transfer function with all 。Questions function has Mzeros (with m inside the UC zeros inside the unit circle (m=0)is called a An LTI system is said to be minimum-phase if the and mo outside the UC)and Npoles(with n minimum-phase transfer function system and its inverse are causal and stable. inside the UC and no outside the UC) A causal stable transfer function with all (A(z)B(z=1) AargHe =2πm-2πm,+2a(N-M) zeros outside the unit circle(m=M)is called .Is a causal stable allpass filter minimum or a maximum-phase transfer function maximum phase? =2π(6-mg） A transfer function with zeros inside and .What is case for a linear phase FIR filter? Since N=n and i.e.,no=0,we have outside the unit circle is called a mixed-phase Aarg[m(m transfer function
49 Typical zero locations shown below 2.2 Linear-Phase Transfer Functions Re z jIm z Re z jIm z Re z jIm z Re z jIm z 1 1 1 1 j e j e j0 e j0 e Type 1 N odd Type 4 N even Type 3 N odd Type 2 N even 50 Summary A Type 2 FIR filter cannot be used to design a highpass highpass filter since it always has a zero z=ˉ1 A Type 3 FIR filter has zeros at both z = 1 and z=ˉ1, and hence cannot be used to design either a lowpass lowpass or a highpass highpass or a bandstop filter A Type 4 FIR filter is not appropriate to design a lowpass lowpass filter due to the presence of a zero at z = 1 Type 1 FIR filter has no such restrictions and can be used to design almost any type of filter 2.2 Linear-Phase Transfer Functions 51 2.3 Minimum-Phase and MaximumPhase Transfer Functions A causal stable transfer function with all zeros outside outside the unit circle has an excess phase compared to a causal transfer function with identical magnitude but having all zeros inside the unit circle jIm z Re z 2 0 zero or ploe vector inside the unit circle arg 2 ! 2 0 zero or ploe vector outside the unit circle arg 0 ! unit circle 52 2.3 Minimum-Phase and MaximumPhase Transfer Functions It assumed that a causal stable transfer function has M zeros (with mi inside the UC and m0 outside the UC) and N poles (with ni inside the UC and n0 outside the UC) Since N= ni and i.e., n0=0 , we have " # 2 0 0 0 arg ( ) 2 2 2 ( ) 2 j He m n N M i i n m ! 2 0 0 arg ( ) 2 j He m ! 53 2.3 Minimum-Phase and MaximumPhase Transfer Functions A causal stable transfer function with all zeros inside the unit circle (mo=0) is called a minimum-phase transfer function A causal stable transfer function with all zeros outside the unit circle (mo=M) is called a maximum-phase transfer function A transfer function with zeros inside and outside the unit circle is called a mixed-phase transfer transfer function 54 2.3 Minimum-Phase and MaximumPhase Transfer Functions Questions An LTI system is said to be minimum-phase if the system and its inverse are causal and stable. (A(z)B(z)=1) Is a causal stable allpass filter minimum or maximum phase? What is case for a linear phase FIR filter?