Part C:The Transfer Function Chapter 6C Part C ◆Definition ●●● ●●● Transfer Function Expression ●●● Frequency Response from Transfer Function z-Transform The Transfer Function Geometric Interpretation of Frequency Response Computation Stability Condition in Terms of Poles Locations 1.Definition 1.Definition 1.Definition Example 1 .The concept of transfer function is a Y(z)=- generalization of the frequency response H(e) - Consider the M-point moving average FIR The z-transform H(z)of the impulse 三国之刘必] filter with an impulse response response h(n)of the filter is called the 1/M,0<n<M-1 transfer function or the system function h(m)= 0.otherwise The transfer function is derived through the Its transfer function is then given by method similar to that of the frequency 1 1-2w z”-1 response FM0-2可Mz-(e-) =H(z)X(2) H()M
Chapter 6C z-Transform Part C The Transfer Function 3 Part C: The Transfer Function Definition Transfer Function Expression Frequency Response from Transfer Function Geometric Interpretation of Frequency Geometric Interpretation of Frequency Response Computation Stability Condition in Terms of Poles Locations Stability Condition in Terms of Poles Locations 4 1. Definition The concept of transfer function is a generalization of the frequency response The z-transform H(z) of the impulse response h(n) of the filter is called the transfer function or the system function The transfer function is derived through the method similar to that of the frequency response ( ) j H e 5 1. Definition ( ) () ()( ) () ( ) ( ) () ( ) () () () n n k n k n lnk l k k l l k k l Y z hkxn k z hk xn kz hk xlz hk xlz z HzXz 6 1. Definition Example 1 Consider the M-point moving average FIR filter with an impulse response Its transfer function is then given by 1/ 0 1 ( ) 0 otherwise M nM h n ˈˈ 1 1 1 0 11 1 ( ) (1 ) ( 1) M M M n M n z z Hz z M M z Mz z
1.Definition 1.Definition 1.Definition The transfer function has Mzeros on the unit Example 2 Note:Poles farthest from z=0 have a circle at z=e,K=0.1.2.....M-1 A causal LTI IIR digital filter is described magnitude 0.74 ROC:>0.74 .There are poles atz=0 and a single pole at by a constant coefficient difference equation. 2=1 given by The pole at z=1 y(m）-x(n-1)-12x(n-2)+x(n-3) exactly cancels the +1.3)(n-1)-1.04y(n-2)+0.222y(n-3) zero at z=1.The Its transfer function is therefore given by ROC is the entire z-plane except z z1-12z2+2 H(e)=Y(e) X(②)1-13z1+1.0422-0.22223 2.Frequency Response from Transfer 2.Frequency Response from Transfer 2.Frequency Response from Transfer Function Function Function .If the ROC of the transfer function H(z) It is convenient to visualize the contributions includes the unit circle,then the frequency .For a stable rational transfer function in the form of the zero factor (-and the pole response H(e)of the LTI digital filter can be obtained simply as follows: He)=凸nΠe- factor (z)from the factored form of do I(2-A) the frequency response.The magnitude H(e)=H(2) function is given by The factored form of the frequency response For a real coefficient transfer function H(z), is given by it can be shown that He)=凸emwΠ(- Πae- H(=H(e)H'() do I(em- 2aΠe-a =H(e)H(e)=H(2)H(z) 0 aΠaem-人
7 1. Definition The transfer function has M zeros on the unit circle at There are poles at z = 0 and a single pole at z = 1 2 / , 0,1, 2, , 1 j kM ze K M The pole at z = 1 exactly cancels the zero at z = 1. The ROC is the entire z-plane except z = 0 -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 7 Real Part Imaginary Part M=8 8 1. Definition Example 2 A causal LTI IIR digital filter is described by a constant coefficient difference equation, given by Its transfer function is therefore given by ( ) ( 1) 1.2 ( 2) ( 3) 1.3 ( 1) 1.04 ( 2) 0.222 ( 3) yn xn xn xn yn yn yn 1 23 12 3 ( ) 1.2 ( ) ( ) 1 1.3 1.04 0.222 Yz z z z H z Xz z z z 9 1. Definition Note: Poles farthest from z = 0 have a magnitude 0.74 ROC: z 0.74 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Real Part Imaginary Part 10 2. Frequency Response from Transfer Function If the ROC of the transfer function H(z) includes the unit circle, then the frequency response of the LTI digital filter can be obtained simply as follows: For a real coefficient transfer function H(z) , it can be shown that ( ) () j j z e He Hz ( ) j H e 2 * 1 () ()() ( ) ( ) () ( ) j j jj j j z e He He H e He He HzHz 11 2. Frequency Response from Transfer Function For a stable rational transfer function in the form The factored form of the frequency response is given by 0 ( ) 1 0 1 ( ) M N M k k N k k p z Hz z d z ( ) 1 0 0 1 ( ) M j k k j j NM N j k k p e He e d e 12 2. Frequency Response from Transfer Function It is convenient to visualize the contributions of the zero factor zero factor and the pole factor from the factored form of the frequency response. The magnitude function is given by z k ( ) 1 0 0 1 1 0 0 1 ( ) M j k k j j NM N j k k M j k k N j k k p e He e d e p e d e z k
2.Frequency Response from Transfer 3.Geometric Interpretation of 3.Geometric Interpretation of Function Frequency Response Computation Frequency Response Computation The product omgnude of M voctors .The phase response for a rational transfer .Thus,an approximate plot of the magnitude H(e The product of d magnitudes of N vectors and phase responses of the transfer function function is of the form of an LTI digital filter can be developed by arg H(e)=arg(po/do)+(N-M) examining the pole and zero locations 之arg(em-5)-之arg(em-A) term zero or pole vector (e-pe) Now,a zero (pole)vector has the smallest magnitude when w=中 represents a vector starting at the point ==pe and ending on the unit circle at=e 13 15 3.Geometric Interpretation of 4.Stability Condition in Terms of 4.Stability Condition in Terms of Frequency Response Computation Poles Locations Poles Locations .To highly attenuate signal components in a .A causal LTI digital filter is BIBO stable if .The ROC of the z-transform (z)of the specified frequency range,we need to place and only if its impulse response h(n)is impulse response sequence h(n)is defined by zeros very close to or on the unit circle in absolutely summable,i.e., values of =r for which h()is absolutely this range. summable Likewise,to highly emphasize signal s=Σnlk .Thus,if the ROC includes the unit circle1, components in a specified frequency range, We now develop a stability condition in terms then the digital filter is stable,and vice versa we need to place poles very close to or on of the pole locations of the transfer function For LTI system causality we require that the unit circle in this range. H() h(n)=0,for n<0.This implies that the ROC of H(z)must be outside some circle of radius R. 多 18
13 2. Frequency Response from Transfer Function The phase response for a rational transfer function is of the form 0 0 1 1 arg ( ) arg( / ) ( ) arg( ) arg( ) j M N j j k k k k He p d N M e e 14 3. Geometric Interpretation of Frequency Response Computation 1 0 0 1 ( ) M j k k j N j k k p e H e d e constant term zero or pole vector ( ) j j e e represents a vector starting at the point and ending on the unit circle at j z e j z e -1 1 Re z jIm z j -j The product of magnitudes of M vectors The product of magnitudes of N vectors j e 15 3. Geometric Interpretation of Frequency Response Computation Thus, an approximate plot of the magnitude and phase responses of the transfer function of an LTI digital filter can be developed by examining the pole and zero locations Now, a zero (pole) vector has the smallest magnitude when ¹ =¶ 16 3. Geometric Interpretation of Frequency Response Computation To highly attenuate signal components in a specified frequency range, we need to place zeros very close to or on the unit circle in this range. Likewise, to highly emphasize signal components in a specified frequency range, we need to place poles very close to or on the unit circle in this range. 17 4. Stability Condition in Terms of Poles Locations A causal LTI digital filter is BIBO stable if and only if its impulse response h(n) is absolutely summable, i.e., We now develop a stability condition in terms of the pole locations of the transfer function H(z) ( ) n h n S 18 4. Stability Condition in Terms of Poles Locations The ROC of the z-transform H(z) of the impulse response sequence h(n) is defined by values of |z|=r for which h(n)r-n is absolutely summable Thus, if the ROC includes the unit circle |z|= 1, then the digital filter is stable, and vice versa For LTI system causality we require that h(n)=0,for n <0. This implies that the ROC of H(z) must be outside some circle of radius Rx-
4.Stability Condition in Terms of 4.Stability Condition in Terms of 4.Stability Condition in Terms of Poles Locations Poles Locations Poles Locations Theorer里 Proof:(reduction to absurdity) Exampl业e A causality LTI system is stable if and only if the Under what conditions,the following system is system function //z)has all its poles inside the unit stable or causal circle.It is a easy way to judge the causality and -3这 H(z)= stability of an LTI system.So,the ROC will include ◆Solution: 2-5z1+2z☒ the unit circle and entire z-plane including the point =0 Step 1---Determine the zeros and poles of H(z) An FIR digital filter with bounded impulse -31 -3z UCirw response is always stable H2)=2-5+222z-102-2可 19 2zeo-0,Za=0.5n2 21 4.Stability Condition in Terms of 4.Stability Condition in Terms of Poles Locations Poles Locations Step 2--plot the zeros and poles on the z-plane Step4----Discuss the system's stability and causality Step 3----Determine Case 1:ROC1=(lz2)Because the unit does not lie in this area and the ROC is outside of the circle with R0C3-0中2☑ -Green Area 22 radius 2,the system is causal and unstable
19 4. Stability Condition in Terms of Poles Locations Theorem A causality LTI system is stable if and only if the A causality LTI system is stable if and only if the system function system function H(z) has all its poles inside the unit has all its poles inside the unit circle. It is a easy way to judge the causality and circle. It is a easy way to judge the causality and stability of an LTI system. So, the ROC will include stability of an LTI system. So, the ROC will include the unit circle and entire z-plane including the point plane including the point An FIR digital filter with bounded impulse response is always stable z 20 4. Stability Condition in Terms of Poles Locations Proof: (reduction to absurdity) 1 Re z jIm z h h h Unit Circle 21 4. Stability Condition in Terms of Poles Locations Example Under what conditions, the following system is stable or causal ? Solution: Step 1----Determine the zeros and poles of H(z) 1 1 2 3 ( ) 25 2 z H z z z 1 1 2 3 3 ( ) 25 2 2 1 2 z z H z zz zz zzero=0, zpole=0.5, 2 22 4. Stability Condition in Terms of Poles Locations Step 2----plot the zeros and poles on the z-plane ROC1={|z|2} —Green Area 1 Re z jIm z 1/2 2 h h zero poles Step 3----Determine all possible ROCs according to the distribution of the zeros and poles 23 4. Stability Condition in Terms of Poles Locations Step4----Discuss the system’s stability and causality Case 1: ROC1={|z|2} Because the unit does not lie in this area and the ROC is outside of the circle with radius 2, the system is causal and unstable.