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 for >1 0<<.An M-th order stable real-coefficient The unwrapped phase function is a continuous 4(z=1r=1 allpass transfer function satisfies: function of >1 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