Biomedical Signal processing Chapter 5 Transform Analysis of

Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems Zhongguo Liu, Biomedical Engineering School of Control Science and Engineering, Shandong University 山东省精品课程《生物医学信号处理(双语)》 http: //course. sdu. edu. cn/bdsp. html 1

Chapter 5 Transform Analysis of Linear Time-Invariant Systems u 5. 0 Introduction u 5. 1 Frequency Response of LTI Systems u 5. 2 System Functions For Systems Characterized by Linear Constant-coefficient Difference equation u 5. 3 Frequency Response for Rational System Functions u 5. 4 Relationship Between Magnitude and Phase u 5. 5 All-Pass System u 5. 6 Minimum-Phase Systems u 5. 7 Linear Systems with Generalized Linear Phase 2

5. 0 Introduction u. An LTI system can be characterized in time domain by impulse response . u. Output of the LTI system: u. With Z-transform and Fourier Transform, an LTI system can be characterized Ø in Z-domain by system function Ø in frequency-domain by Frequency response 6

5. 1 Frequency Response of LTI Systems system 5. 1. 1 Frequency Response Phase and Group Delay input signal u. Frequency response modified in a useful way, Y(ejw) desirable + u. Magnitude response modified in a (gain) deleterious manner distortions u. Phase response (phase shift) 7

Principal Value（主值） u. The phase angle of any complex number is not uniquely defined, since 2πr can be added without affecting the complex number. u denote Principal Value of the phase of ambiguous phase 不确定的相位 12 arbitrary

解卷绕的 continuous (unwrapped) phase curve is denoted as arg [H (e jw)] Principal Value（主值） we refer to ARG [H(ejw)] as the "wrapped" phase, 卷绕的 13

Group Delay(群延迟，grd ) u. Another particularly useful representation of phase is through the group delay: correct !all value ambiguous phase appropriate r(w) continuous unique r(w) group delay can be obtained from the principal value, except at the discontinuities, or, from ambiguous phase: ignoring impulses caused by discontinuities. 14

5. 1. 1 Frequency Response Phase and Group Delay u. To understand the effect of the phase and the group delay of a linear system, first consider the ideal delay system: u. The impulse response u. The frequency response 15

effect of phase and group delay of a linear system u. For ideal delay system The group delay represents a convenient measure of the linearity of the phase. 16

5. 1. 1 Frequency Response Phase and Group Delay u delay distortion is a rather mild form of phase distortion, its effect is to shift the sequence in time. we accept linear phase response rather than zero phase response in designing approximations to ideal filters. u. Ideal lowpass filter with linear phase delay is better u. The impulse response (delayed by time nd ) windowed h LP[n] can be 17

5. 1. 1 Frequency Response Phase and Group Delay u. Ideal lowpass filter with linear phase delay u. The impulse response (delayed by time nd ) nd =5 noncausal windowed h. LP [n] can be causal, 18 approximations to ideal filters.

effect of phase and group delay of a linear system u. Given a narrowband input x[n]=s[n]cos(w 0 n) for a system with frequency response H(ejw), assume X(e jw) is nonzero only around w =w 0 Group Delay it can be shown (see Problem 5. 63(二版5. 57)) that the response y[n] to x[n] is the time delay of the envelope s[n] is . 19

effect of phase and group delay of a linear system narrowband input x[n]=s[n]cos(w 0 n) ufor a system H(e jw), Group Delay Output the time delay of the envelope s[n] is . 20

5. 1. 2 Illustration of Effects of Group Delay and Attenuation u. Consider the system having system function: 18阶, 8个 2重极点, 2个单极点 × H 2(z) allpass system 21

5. 1. 2 Illustration of Effects of Group Delay and Attenuation uzeros: , 8个 2重: u. Poles: , 8个 2重: H 2(z) represents an allpass system H 2(z) introduces a large amount of group delay over a narrow band of frequencies. 22 Figure 5. 2 Pole-zero plot for the filter

5. 1. 2 Illustration of Effects of Group Delay and Attenuation Frequency Response of the system Figure 5. 3 (a) Principle Value of Phase Response: 是全通系统作用 zeros: poles: (b) Unwrapped Phase Response -0. 23π 23 0. 17π

5. 1. 2 Illustration of Effects of Group Delay and Attenuation for Frequency Response of the system Figure 5. 3 -0. 23π - 0. 17π jw) (a) Group delay of H(z) , H(e 0. 4π 0. 8π 0. 2πn 0. 4πn 0. 8πn (b) Magnitude of Frequency Response 24 0. 17π 0. 23π 0. 4π 0. 8π 2 1. 5 1

5. 1. 2 Illustration of Effects of Group Delay and Attenuation u. An input signal x[n] consisting of three narrowband pulses separated in time, the pulses are given by 20 40 60 uwhere each sinusoid is shaped into a finite-duration pulse by the 61 -point envelope sequence: 25 Hamming Window

5. 1. 2 Illustration of Effects of Group Delay and Attenuation 0. 2πn 0. 4πn 0. 8πn Figure 5. 5 (a) Waveform of signal x[n] 两信号乘 � 的�� 是两信号各自�� 的卷� 26 (b) Magnitude of DTFT of x[n]

Group delay 5. 1. 2 Illustration of Effects of Group Delay and Attenuation Magnitude Frequency Response Figure 5. 5 (a) Waveform of signal x[n] delay 10 samples 27 Figure 5. 6 Output signal of the system delay 150 samplesfor input x[n]

5. 2 System Functions For LTI Systems Characterized by Linear Constant. Coefficient Difference Equation u. Linear Constant-coefficient Difference equation For LTI Systems 28 Linear Time Invariat, if not

5. 2 System Functions For Systems Characterized by Linear Constantcoefficient Difference equation u. For an LTI system: uits poles and zeros: 30

Ex. 5. 1 find difference equation for second-order System function Solution: 31

5. 2. 1 Stability and Causality u. The difference equation of a LTI system does not uniquely specify the impulse response. LTI u. Each ROC of the system function will lead to a different impulse response; ubut they will all correspond to the same difference equation. Ø 第 3章 3. 2节有介绍系统时域特性与H(z)收敛域关系的内 32容, 3. 5再联系差分方程也简单介绍了一下, 这里强化.

Causality u. For a causal system the impulse response must be right-sided sequence. u. The region of convergence (ROC) of must be outside the outermost pole. 33

Stability u. For a stable system The impulse response must be absolutely summable, i. e. , z-plane unit circle u. ROC of includes the unit circle 34

Ex. 5. 2 Determine the ROC, Stability and causality for LTI system: Solution: upoles: 1/2, 2; zeros(two)： 0 35

Example 5. 2 Determining the ROC 1) 2) 3) 36 poles: 1/2, 2; zeros(two): 0

Causal and Stable system u. Causal: ROC must be outside the outermost pole. u. Stable: ROC includes the unit circle. u. Causal and stable: all the poles of the system function are inside the unit circle; ROC is outside the outermost pole, and 37

5. 2. 2 Inverse Systems u. For LTI system , define the inverse system to be a system cascaded with , and satisfies: x[n] y[n]=x[n] or u. Time domain: u. Frequency response u. Not all systems have an inverse. Ideal LPF hasn’t. 38 x[n]

5. 2. 2 Inverse Systems systems with rational system functions: u. ROC of and ROC of must overlap, for convolution theorem to hold: 39

Ex. 5. 3 find Inverse System for First-Order System Solution: 40

Ex. 5. 4 find Inverse for System with a Zero in the ROC Solution: 1) 2) 41

Minimum-phase Systems u. An LTI system is stable and causal and also has a stable and causal inverse if and only if both the poles and the zeros of are inside the unit circle, u. Such systems are referred as minimumphase systems 42

5. 2. 3 Impulse Response for Rational System Functions u. For a LTI system with only 1 st-order poles: If causal, ucan be infinite impulse response (IIR) uor finite impulse response (FIR) 43

FIR System convolution u. Difference equation: 44

Ex. 5. 5 A Simple FIR System Determine System function, zero-pole plot, stability, difference equation For h[n]: Solution: (7重), 46

Example 5. 5 A Simple FIR System u. Difference equation 47

5. 3 Frequency Response for Rational System Functions u. If a stable LTI system has a rational system function u. Its frequency response is: 48

5. 3 Frequency Response for Rational System Functions magnitude-squared frequency response (function): 49

Frequency Response: Log magnitude 50

Phase Response for a rational system function continuous phase group delay : (5. 51 ) 1 st-Order Systems: Single Pole or Ze 52

Phase Response, group delay 53

5. 3. 1 Frequency Response of a Single Zero or Pole: 1 st-Order System 1. formular method 2. Geometrical method magnitude-squared frequency response : +: zero factor; - : pole 55

5. 3. 1 Frequency Response of a Single Zero or Pole Phase Response : 对总体相位的贡献: (群延迟) 零点因式取正+; 极点因式取负- group delay: 56 +: for zero factor; -: for pole factor

Figure 5 -9 Log Magnitude response for a single zero with r=0. 9 57

Fig. 5 -9 for a single zero with r=0. 9 Phase response Group Delay 0. 45 0. 46 0. 47 58 58

2. Geometrical method frequency response is associated with vector diagrams in the complex plane and pole-zero plots 59 z=ejw极点附近取值时, 幅频响应极大 z在零点附近取值时, 幅频响应极小

2. Geometrical method frequency response is associated with vector diagrams in the complex plane and pole-zero plots z=ejw极点附近取值时, 幅频响应极大 z在零点附近取值时, 幅频响应极小 60

Magnitude response for a single zero with Log Magnitude 61 z=ejw极点附近取值时, 幅频响应极大

Phase, Group Delay for a single zero with 64

5. 3. 2 Examples with Multiple Poles and Zeros（self study） jw在极点附近取值时, 幅频响应当z=e 70

5. 3. 2 Examples with Multiple Poles and Zeros（self study） z=ejw极点附近取值时, 幅频响应极大 71

5. 4 Relationship Between Magnitude and Phase u. In filter designing, Frequency response H(ejw) of LTI system is usually not given: Given ? u. In general, knowledge about the magnitude provides no information about the phase, and vice versa. u. If the magnitude of the frequency response and the number of poles and zeros are known, uthen there are only a finite number of choices for the associated phase. 72

5. 4 Relationship Between Magnitude and Phase u. If the number of poles and zeros and the phase are known, then, to within a scale factor, there are only a finite number of choices for the magnitude. uniquely number of poles-zeros within a scale factor u. Under a constraint referred to as minimum phase, the frequency-response magnitude and the specifies the phase uniquely, frequency-response phase specifies the magnitude to within a scale factor. 73

5. 4 Relationship Between Magnitude and Phase z与(1/z*)关于单位圆镜像对称 magnitude-squared system function 零点和极点各自都是共轭倒数对 74

5. 4 Relationship Between Magnitude and Phase u. The poles and zeros of occur in conjugate reciprocal pairs, with one element of each pair associated with and the other associated with If is causal and stable, then all its poles are inside the unit circle， the poles of H (z) can be identified from the poles of C(z). ubut its zeros are not uniquely identified by C(z) 75

Ex. 5. 9 Different Systems with the Same C(z) u. Prove that two systems have same magnitude-squared system function: Solution: 76

Ex. 5. 9 Different Systems with the Same C(z) 77

Ex. 5. 10 given Determine zeros and poles of stable, causal system H(z), if coefficients are real, and number of poles, zeros is known(3) exclude 78 H(z): real poles(zeros), or complex conjugate pairs

Ex. 5. 10 given Determine zeros and poles of stable, causal system H(z), if coefficients are real , and number of poles, zeros is known(3) Solution: H(z) poles: p 1, p 2, p 3; H(z) Zeros: (z 1, z 2, z 3), or (z 1, z 2, z 6), or (z 4, z 5, z 3), or (z 4, z 5, z 6), 79 H(z): real pole(zero), or complex conjugate pairs

5. 5 All-Pass(全通) System u. A stable system function of the form u. Zero: , Pole: unit circle frequency-response magnitude is unity 80

General Form of All-Pass System uall-pass system: A system for which the frequency-response magnitude is a constant. u. It passes all frequency components of its input with a constant gain A (is not restricted to be unity in this text). u An all-pass system is always stable, since when frequency response characteristics (such as allpass) are discussed, it is naturally assumed that the Fourier transform exists, thus stability is implied. 81

General Form of All-Pass System with real-valued h[n] 82

5. 5 All-Pass System uphase response 83 83

5. 5 All-Pass System u group delay of a causal all-pass system is positive 84

5. 5 All-Pass System u group delay of a causal all-pass system is positive for causal, stable system, every term of the sum is positive, 85

nonpositivity of the unwrapped phase of causal all-pass systems with real-valued h[n] Proof: (5 -82) positive real const 86

Example 5. 11 analyse frequency-response of 1 st-Order All-Pass System Log magnitude 87

Example 5. 11 First-Order All. Pass System: "wrapped" phase 89

Example 5. 11 First-Order All. Pass System: “unwrapped" phase 90

Ex. 5. 11 First-Order All-Pass System: group delay “unwrapped" phase group delay 91

Second-Order All-Pass System with poles at and . 93

Ex. 5. 11 Second-Order All-Pass System : Magnitude "wrapped" Phase Group delay 94

Fig. 5. 18 fourth order all-pass system 95

Frequency response of Fig. 5. 18 Magnitude "wrapped" Phase Group delay 96

Application of All-Pass Systems u. Used as compensators for phase or group delay (Chapter 7) u. Be useful in transforming frequencyselective lowpass filters into other frequency-selective forms and in obtaining variable-cutoff frequencyselective filters (chapter 7) u. Be useful in theory of minimumphase systems (Section 5. 6) 97

5. 6 Minimum-Phase Systems u. For an LTI system: uits poles and zeros: H (z) Stable, Causal no restriction u. Its inverse: 98 Hinv(z) Stable, Causal

5. 6 Minimum-Phase Systems u. For a stable and causal LTI system, all the poles must be inside the unit circle. u. If its inverse system is also stable and causal, all the zeros must be inside the unit circle. Unit Circle 100 Unit Circle

5. 6 Minimum-Phase Systems u. Minimum-phase system: all the poles and zeros of an LTI system are inside unit circle, so the system and its inverse is stable and causal. Unit Circle 101

5. 6. 1 Minimum-Phase and All-Pass Decomposition u. Any (stable, causal) rational system function H(z) can be expressed as: all the poles are inside the unit circle (no zeros on unit circle ) can be generalized to include more zeros Proof: u. Suppose H(z) has one zero outside the unit and the remaining circle at , poles and zeros are inside the unit circle. conjugate reciprocal: c 103

Ex. 5. 12 find Minimum-Phase/All-Pass Decomposition of stable, causal system (1) Solution: reflect the zero to conjugate reciprocal locations inside the unit circle: 105

Ex. 5. 12 find Minimum-Phase/ All-Pass Decomposition of stable, causal system (2) Solution: reflect the zero to conjugate reciprocal locations inside the unit circle: 106

Ex. 5. 12 Minimum-Phase/All-Pass Decomposition of stable, causal system zeros conjugate reciprocal 107

5. 6. 2 Frequency-Response Compensation u. When a signal has been distorted by an LTI system with undesirable frequency response, perfect compensation: minimum-phase system If poles and zeros of Hd(z) are inside the unit circle : 108 inverse

5. 6. 2 Frequency-Response Compensation u. If isn’t minimum-phase, its inverse can’t be stable and causal, then we decompose so is stable and causal. magnitude is compensated phase distortion while phase response is modified by 109

Ex. 5. 13 Compensation of an FIR System Solution : zeros: outside the unit circle, decomposition is needed: 110

Frequency response of Magnitude "wrapped" Phase Group delay 111

Ex. 5. 13 Compensation of an FIR System 112

Frequency response of Magnitude "wrapped" Phase Group delay 113 Minimum Phase-Lag

Frequency response of Magnitude "wrapped" Maxmum Phase-Lag Phase Group delay 114

5. 6. 3 Properties of Minimum-Phase Systems u 1. Minimum Phase-Lag Property phase-lag function 115 For all systems that have a given magnitude response , minimum-phase system has the Minimum Phase-Lag.

u 1. Minimum Phase-Lag Property uto make the interpretation of Minimum it is Phase-Lag systems more precise, necessary to impose the additional constraint that be positive at to remove the ambiguity usince u. Its system function with same poles and zeros, is also a minimum-phase system, according to its defination, but the phase is altered by π. 116

5. 6. 3 Properties of Minimum-Phase Systems u 2. Minimum Group-Delay Property For all systems that have a given magnitude response , minimum-phase system has the Minimum Group Delay. 117

5. 6. 3 Properties of Minimum -Phase Systems u 3. Minimum Energy-Delay Property u. For any causal, stable, LTI systems If then (Prob 5. 65(2 nd)) (Parseval’s theorem) 118

Minimum Energy-Delay Property u. For any causal LTI systems, define the partial energy of the impulse response (Prob 5. 66(2 nd)) (Prob 5. 72(3 rd)) For all systems that have a given magnitude response , minimum-phase system has the Minimum Energy-Delay. 119

Proof of Minimum Energy-Delay Property Proof: 120

Fig. 5. 27 minimum -phase maximum -phase Four systems, all having the same frequency-response magnitude. Zeros are at all combinations of the complex conjugate zero pairs and their reciprocals. 121

Minimum-Phase System and Maximum-Phase System u A maximum-phase system is the opposite of a minimum-phase system. A causal and stable LTI system is a maximum-phase system if its inverse is causal and unstable. (From Wikipedia) u Maximum-Phase System: poles are all in the unit circle, zeros are all outside the unit circle. It’s causal and stable. u (noncausal)Maximum-Phase System: anticausal, stable System whose System function has all its poles and zeros outside the unit circle. (problem 5. 63 (2 nd), 5. 69 (3 rd)). 122

Maximum energy-delay systems are also often called maximum-phase systems. minimum-phase sequence ha[n] maximum-phase sequence hb[n]. Sequences corresponding to the pole-zero plots of Fig. 5. 27 123

the maximum energy delay occurs for the system that has all its zeros outside the unit circle. Maximum energy-delay systems are also often called maximum-phase systems. Fig. 5. 29 Partial energies for the four sequences of Fig. 5. 27. (Note that Ea[n] is for the minimum-phase sequence ha[n] and Eb[n] is for the maximum-phase sequence hb[n]. 124

5. 7 Linear Systems with Generalized Linear Phase u. In designing filters, it’s desired to have nearly constant magnitude response and zero phase in passband. u. For causal systems, zero phase is not attainable, and some phase distortion must be allowed. 125

5. 7 Linear Systems with Generalized Linear Phase u. The effect of linear phase (constant group delay) with integer slope is a simple time shift. u. A nonlinear phase, on the other hand, can have a major effect on the shape of a signal, even when the frequencyresponse magnitude is constant 126

5. 7. 1 System with Linear Phase 127

Interpretation of comes from sampling a continuous-time signal, if (4 -25) Figure 4. 15 128

Linear-phase ideal lowpass filter time domain The corresponding impulse response is 131

Ex. 5. 14 symmetry of impulse response of Ideal Lowpass with Linear Phase in three cases: α is integer; 2α is not integer Solution: (1) 132

Ex. 5. 14 symmetry of impulse response of Ideal Lowpass with Linear Phase (2) is an integer, 134

Ex. 5. 14 symmetry of impulse response of Ideal Lowpass with Linear Phase (3) is not an integer 135

Ex. 5. 14 symmetry of impulse response of Ideal Lowpass with Linear Phase u 136

5. 7. 2 Generalized Linear phase For moving average system (Ex. 2. 16, 37页, 原版45页) if negtive, it’s not (strictly speaking) a linear-phase system, since π is added to the phase. It’s the form: it is referred to Generalized Linear phase system 137

if a system with h[n] has generalized linear phase real 138

if a system with h[n] has generalized linear phase u. This equation is a necessary condition on h[n], for the system to have constant group delay. u. It is not a sufficient condition, and, owing to its implicit nature, it does not tell us how to find a linear-phase system. 139

satisfy One set of condition: even symmetry M even 0 M/2=α M M odd Shown in Type I, II FIR 140 0 M/2=α M=5

satisfy Another set of condition : odd symmetry M even M 0 Shown in. Type III, IV FIR M/2 M=3 0 M/2 M odd 141

5. 7. 3 Causal Generalized Linear-Phase Systems 142

5. 7. 3 Causal Generalized Linear-Phase Systems u. Causal FIR systems have generalized linear phase if they have impulse response length and satisfy Frequency response: 143

u. If 5. 7. 3 Causal Generalized Linear-Phase Systems symmetric uthen may be negative It’s sufficient condition, not necessary condition 144

u. If 5. 7. 3 Causal Generalized Linear-Phase Systems Antisymmetric uthen may be negative It’s sufficient condition, not necessary condition 145

5. 7. 3 Causal FIR Linear-Phase Systems u satisfies: symmetric or Antisymmetric impulse response h[M n] = ±h[n] for n = 0, 1, …, M M is even Type I symmetric M is odd Type II h[M n] = h[n] M/2 0 147 0 M/2 M=5 Type IV Type III Antisymmetric h[M n] = h[n] M=6 M=3 M=6 0 M/2

Type I FIR Linear-Phase Systems u. Symmetric impulse response u. M: even integer, u. M/2 : integer. h[n ] 148 0 M/2 2 1 K=0 1 2 M=10 M/2

Type I FIR Linear-Phase Systems 149

Type I FIR Linear-Phase Systems u. Symmetric impulse response 0 M/2 150 u. M: even integer, u. M/2 : integer. M/2 2 1 K=0 1 2 M=10 M/2

Type II FIR Linear-Phase Systems u. Symmetric impulse response 0 M/2 M=5 u. M: odd integer. integer plus one-half. 153

Type III FIR Linear-Phase Systems u. Antisymmetric impulse response u. M: even integer. u integer. 154 M=4 0 M/2

Type IV FIR Linear-Phase Systems u. Antisymmetric impulse response u. M: odd integer. u integer plus one-half. 155 M=3 0 M/2

Ex. 5. 15 determine H(ejw) of Type I FIR Linear-Phase Systems Solution: 156

Frequency response Type I Magnitude "wrapped" Phase Group delay 157

Ex. 5. 16 determine H(ejw) of Type II FIR Linear-Phase Systems Solution: 158

Frequency response Type II Magnitude "wrapped" Phase Group delay 159

Frequency response Type I Magnitude "wrapped" Phase Group delay 160

Ex. 5. 17 determine H(ejw) of Type III FIR Linear-Phase Systems Solution: 161

Ex. 5. 17 Frequency response Type III Magnitude "wrapped" Phase Group delay 162

Ex. 5. 18 determine H(ejw) of Type IV FIR Linear-Phase Systems Solution: 163

Ex. 5. 18 Frequency response Type IV Magnitude "wrapped" Phase Group delay 164

Locations of Zeros for causal FIR Linear-Phase Systems has a pole of order M: z=0 u. For Type I and II, u. For Type III and IV, 168

Type I and II u. If is a zero of , factor uthen for causal FIR Linear-Phase Systems, z 0≠ 0, since 0 is pole. has factor u. This implies that if is a zero of , then is also a zero of u. The same result for Type III and IV 169

Type I and II real coefficient equation has conjugate complex roots pair: u. When is real and is a zero of , will also be a zero of , . so will . So there are four possible complex zeros: usame result for Type III and IV 170

Type I and II , Type III and IV u. When is real, each complex zero not on the unit circle will be part of a set of four conjugate reciprocal zeros of the form ucomplex zeros on the unit circle Type I 171 Type II

Type I and II , Type III and IV uif a zero of is real, and not on the unit circle, the reciprocal is also a zero, and so have the factors of the form Type I 172 Type II usame result for Type III and IV

Type I and II u. The case of a zero at is important in designing filter of some types of frequency responses (such as high-pass, low-pass filter). u. If M is even Type I can be HP filter usually u. If M is odd, Type II cannot be HP filter uso z=-1 must be zero of Type II generalized linear-phase systems. uz=1 may be not zero for both, Both can be LP filter 173

Type I and II Type I can be HP, LP, BS filter cannot be HP, BS filter can be LP, BP filter HP(high pass filter), LP(low pass filter), BP(band pass filter), BS(band stop filter) 175

Type III and IV u. The case of u. For both M is even and odd, u must be zero of Type III and IV generalized linear-phase systems. Type III 177 cannot be LP filter Type IV

Type III and IV u. The case of u. If M is even, u must be zero of Type III generalized linear-phase systems. z=-1 may be u. If M is odd (Type IV), not zero. Type III 178 cannot be LP, Type IV HP filter can be BP filter

(a) Type I 能做LP, HP filter (b) Type II z 0=-1, 不能做HP filter Fig. 5. 38 Typical plots of zeros for linear-phase systems Type III z 0=± 1, 不能做 LP, HP filter Type IV z 0=1, 不能做LP filter 179

5. 7. 4 Relation of FIR Linear-Phase Systems to Minimum-Phase Systems u. FIR linear-phase systems with real impulse responses have zeros at conjugate reciprocal locations, unit circle 或写成 where have the same magnitude u has all zeros inside the unit circle. u has all zeros outside the unit circle u has all zeros on the unit circle. u The order of H(z): 2 Mi+M 0 180

Ex. 5. 19 Decomposition of a Linear-Phase System For Minimum-Phase System of Page 155, Eq. 5. 99 Determine the frequecny response of Maximum-Phase System and the system cascaded by two. Solution: 已知FIR最小相位系统 → 相同幅频响应的线性相位系统 181