Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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 ) 》

2 Chapter 5 Transform Analysis of Linear Time-Invariant Systems  5.0 Introduction  5.1 Frequency Response of LTI Systems  5.2 System Functions For Systems Characterized by Linear Constant-coefficient Difference equation  5.3 Frequency Response for Rational System Functions  5.4 Relationship Between Magnitude and Phase  5.5 All-Pass System  5.6 Minimum-Phase Systems  5.7 Linear Systems with Generalized Linear Phase

3 5.0 Introduction  An LTI system can be characterized in time domain by impulse response  Output of the LTI system:  in Z-domain by system function  in frequency-domain by Frequency response  With Fourier Transform and Z-transform, an LTI system can be characterized

4 5.1 Frequency Response of LTI Systems  Phase response (phase shift)  Frequency response  Magnitude response (gain) distortions change on useful signal system Useful input signal + deleterious signal

Ideal Frequency-Selective Filters  Ideal lowpass filter  Noncausal, not computationally realizable  no phase distortion

Ideal Frequency-Selective Filters  Ideal highpass filter

Ideal Frequency-Selective Filters  Ideal bandpass filter

Ideal Frequency-Selective Filters  Ideal bandstop filter

Phase Distortion and Delay  The frequency response  The impulse response  To understand the effect of the phase and the group delay of a linear system, first consider the ideal delay system:

10 Group Delay( 群延迟， grd )  For ideal delay system The group delay represents a convenient measure of the linearity of the phase.

 Given a narrowband input x[n]=s[n]cos(w 0 n) for a system with frequency response H(e jw ), it is assumed that X(e jw ) is nonzero only around w =w 0 11 Group Delay( 群延迟， grd ) it can be shown (see Problem 5.57) that the response y[n] to x[n] is the time delay of the envelope s[n] is. Group Delay

12 Example 5.1 Effect of Attenuation and Group Delay Three consecutive narrowband pulses is applied to a filter 连贯的

13 Ex. 5.1 Effect of Attenuation and Group Delay Filter frequency response Group Delay magnitude

14 Group Delay 50 Group Delay 200

Phase Distortion and Delay  Ideal lowpass filter with linear phase delay  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.  The impulse response (delayed by time n d )

System Functions For LTI Systems Characterized by Linear Constant- coefficient Difference equation  Linear Constant-coefficient Difference equation

LTI System Characterized by Linear Constant-coefficient Difference equation If a system is not LTI, then the following Z-transform cannot be derived. (see P37, example 2.16, for x[n]=kδ[n], y[n]=a n+1 c + Ka n u[n], for all n, x[n] have z-transform K, y[n] have no z-transform.)

18  For an LTI system:  its poles and zeros: 5.2 System Functions For Systems Characterized by Linear Constant- coefficient Difference equation

19 Ex. 5.2 find difference equation for second-order System function Solution:

Stability and Causality  The difference equation does not uniquely specify the impulse response of a linear time-invariant system.  Each possible choice for the ROC of the system function will lead to a different impulse response, but they will all correspond to the same difference equation.

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

22 Stability  ROC of includes the unit circle  For a stable system The impulse response must be absolutely summable, i.e.,

23 Ex. 5.3 Determine the ROC, Stability and causality for LTI system:  poles: 1/2, 2; zeros(two) ： 0 Solution:

24 Example 5.3 Determining the ROC 1) 2) 3)

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

Inverse Systems  Time domain:  Not all systems have an inverse. Ideal LPF hasn’t  For a LTI system, the inverse system which cascaded with satisfies:  Frequency response x[n]x[n]y [n]y [n]

Inverse Systems  ROC of and ROC of must overlap, for convolution theorem to hold: systems with rational system functions:

28 Ex. 5.4 analyse Inverse System for First-Order System Solution:

29 Ex. 5.5 find Inverse for System with a Zero in the ROC Solution: 1) 2)

30 Minimum-phase Systems  A 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  Such systems are referred as minimum- phase systems

Impulse Response for Rational System Functions  For a LTI system  can be infinite impulse response (IIR)  or finite impulse response (FIR) If causal,

32 FIR System

33 Ex.5.6 A First-Order IIR System Determine System function, condition of stability, h[n] for stable, causal System. Solution: ∵ it is causal condition of stability:

34 Ex.5.7 A Simple FIR System Determine System function, zero-pole plot, stability, difference equation For h[n]: Solution:

35 Example 5.7 A Simple FIR System  Difference equation

Frequency Response for Rational System Functions  If a stable LTI system has a rational system function  Its frequency response is

Frequency Response for Rational System Functions magnitude-squared function:

38 Log magnitude

39 Output: Log magnitude, phase

40 Phase Response for a rational system function group delay:

41 Phase Response

42 Phase Response

43 Principal Value （主值）  Principal Value of the phase of

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

Frequency Response of a Single Zero or Pole 1. formular method 2. Geometrical method magnitude-squared function:

Frequency Response of a Single Zero or Pole group delay: Phase Response :

47 Log Magnitude response for a single zero with r=0.9

48 for a single zero with r=0.9 Group Delay Phase response

49 frequency response is associated with vector diagrams in the complex plane and pole-zero plots 2. Geometrical method

50 frequency response is associated with vector diagrams in the complex plane and pole-zero plots 2. Geometrical method

51 Magnitude response for a single zero with

52 Phase response for a single zero with

53 Group Delay for a single zero with

54

55 for a single zero outside the unit circle, with

56 Magnitude response for a single zero outside the unit circle, with

57 Phase response for a single zero outside the unit circle, with

58 Group Delay for a single zero outside the unit circle, with

Examples with Multiple Poles and Zeros （ self study ）

Relationship Between Magnitude and Phase  In general, knowledge about the magnitude provides no information about the phase, and vice versa.  If the magnitude of the frequency response and the number of poles and zeros are known, then there are only a finite number of choices for the associated phase.  For frequency response of LTI system

Relationship Between Magnitude and Phase  Under a constraint referred to as minimum phase, the frequency-response magnitude specifies the phase uniquely, and the frequency-response phase specifies the magnitude to within a scale factor.  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.

Relationship Between Magnitude and Phase 共轭倒数对 conjugate reciprocal pairs magnitude-squared system function

Relationship Between Magnitude and Phase 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).  The poles and zeros of occur in conjugate reciprocal pairs, with one element of each pair associated with and one element of each pair associated with  but its zeros are not uniquely identified by C(z)

64 Example 5.11  Two systems with have same magnitude-squared system function

65 Example

66 Ex given Determine zeros and poles of stable,causal system H(z), if coefficients are real. number of poles, zeros is known(3) H(z): conjugate pole,zero pairs

67 Ex given Determine zeros and poles of stable,causal system H(z), if coefficients are real. number of poles, zeros is known(3) H(z): conjugate pole,zero pairs 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 ),

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

69 General Form of All-Pass System  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.  all-pass system: A system for which the frequency-response magnitude is a constant.  It passes all frequency components of its input with a constant gain A (is not restricted to be unity in this text).

70 General Form of All-Pass System

All-Pass System  phase response

All-Pass System  group delay of a causal all-pass system is positive

73 nonpositivity of the unwrapped Phase of All-Pass Systems

74 Example 5.13 analyse First-Order All-Pass System Log magnitude

All-Pass System  phase response

76 Example 5.13 First-Order All- Pass System: "wrapped" phase

77 Example 5.13 First-Order All- Pass System: group delay

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

79 Ex Second-Order All-Pass System : Magnitude Phase "wrapped" Group delay

80 Fig fourth order all-pass system

81 Frequency response of Fig Magnitude Phase Group delay "wrapped"

82 Application of All-Pass Systems  Used as compensators for phase or group delay (Chapter 7)  Be useful in the theory of minimum- phase systems (Section 5.6)  Be useful in transforming frequency- selective lowpass filters into other frequency-selective forms and in obtaining variable-cutoff frequency- selective filters (chapter 7)

Minimum-Phase Systems  For an LTI system:  its poles and zeros:  Its inverse:

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

Minimum-Phase Systems  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

Minimum-Phase and All- Pass Decomposition  Any rational system function can be expressed as  Suppose has one zero outside the unit circle at,, and the remaining poles and zeros are inside the unit circle. stable, causal in the text, but it applies more generally. reflect the zero to conjugate reciprocal locations inside the unit circle:

87 Example 5.14 Minimum- Phase/All-Pass Decomposition reflect this zero to conjugate reciprocal locations inside the unit circle: (1)

88 Example 5.14 Minimum-Phase/ All-Pass Decomposition reflect two zeros to conjugate reciprocal locations inside the unit circle: (2)

89 Example 5.14 Minimum-Phase/All- Pass Decomposition

Frequency-Response Compensation  When a signal has been distorted by an LTI system with an undesirable frequency response, perfect compensation: If poles and zeros of H d (z) are inside the unit circle :

 assume that the distorting system is stable and causal and require the compensating system to be stable and causal, Frequency-Response Compensation  then perfect compensation is possible only if is a minimum-phase system.

Frequency-Response Compensation  If isn’t minimum-phase, its inverse then isn’t stable, so we decompose stable

93 Example 5.15 Compensation of an FIR System zeros: outside the unit circle, Solution: decomposition is needed:

94 frequency response of Magnitude Phase Group delay "wrapped"

95 Example 5.15 Compensation of an FIR System

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

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

Properties of Minimum-Phase Systems  1. Minimum Phase-Lag Property For all systems that have a given magnitude response, minimum-phase system has the Minimum Phase-Lag.

99  1. Minimum Phase-Lag Property  to make the interpretation of Minimum Phase-Lag systems more precise, it is necessary to impose the additional constraint that be positive at  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.  since

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

Properties of Minimum-Phase Systems  3. Minimum Energy-Delay Property  For any causal, stable, LTI systems  HW 5.65 If then

102 Minimum Energy-Delay Property  For any causal LTI systems, define the partial energy of the impulse response  HW 5.66 For all systems that have a given magnitude response, minimum-phase system has the Minimum Energy-Delay.

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

104 Minimum-Phase System and Maximum-Phase System  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)  Maximum-Phase System: poles are all in the unit circle, zeros are all outside the unit circle. It’s causal and stable.  (noncausal)Maximum-Phase System: anti-causal, stable System whose System function has all its poles and zeros outside the unit circle. (problem5.63).

Maximum energy-delay systems are also often called maximum-phase systems. 105 Sequences corresponding to the pole-zero plots of Fig minimum-phase sequence h a [n] maximum-phase sequence h b [n].

106 Fig.5.32 Partial energies for the four sequences of Fig (Note that E a [n] is for the minimum-phase sequence h a [n] and E b [n] is for the maximum-phase sequence h b [n]. 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.

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

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

System with Linear Phase

System with Linear Phase  Specially, if

111 Interpretation of comes from sampling a continuous-time signal, if

112 Interpretation of comes from sampling a continuous-time signal, if

113 General frequency response with linear phase For nonconstant magnitude response multiplication in frequency domain

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

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

116 (2) Ex symmetry of impulse response of Ideal Lowpass with Linear Phase is an integer

117 (3) is not an integer Ex symmetry of impulse response of Ideal Lowpass with Linear Phase

118  Ex symmetry of impulse response of Ideal Lowpass with Linear Phase

Generalized Linear phase For moving average system (Ex.2.20, Page 45) 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

120 If a system with h[n] has Linear phase

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

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

123 Another set of condition : odd symmetry 0 M M/2 satisfy Shown inType III, IV FIR 0 M/2 M=3 M even M odd

Causal Generalized Linear-Phase Systems

Causal Generalized Linear-Phase Systems  Causal FIR systems have generalized linear phase if they have impulse response length and satisfy

Causal Generalized Linear-Phase Systems  If  then It’s sufficient condition, not necessary condition

Causal Generalized Linear-Phase Systems  If  then It’s sufficient condition, not necessary condition

Causal Generalized Linear-Phase Systems  The above two FIR conditions are sufficient to guarantee a causal system with generalized linear phase.  Clements (1989) showed that causal IIR can also have Fourier transforms with generalized linear phase.  The corresponding system function, however, are not rational, and thus, the systems cannot be implemented with difference equations.

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

130 Type I FIR Linear-Phase Systems  Symmetric impulse response 0 M/2 M=10 K=012 M/2 1 2 h[n]  M: even integer,  M/2 : integer.

131 Type I FIR Linear-Phase Systems

132 Type I FIR Linear-Phase Systems  Symmetric impulse response 0 M/2 M=10 K=012 M/2 1 2  M: even integer,  M/2 : integer.

133 Ex determine H(e jw ) of Type I FIR Linear-Phase Systems Solution:

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

 M: odd integer. integer plus one-half. Type II FIR Linear-Phase Systems  Symmetric impulse response 0 M/2 M=9 K121 2 (M+1)/2

Type II FIR Linear-Phase Systems

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

138 Ex determine H(e jw ) of Type II FIR Linear-Phase Systems Solution:

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

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

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

142 Ex determine H(e jw ) of Type III FIR Linear-Phase Systems Solution:

143 Ex Frequency response Type III Magnitude Group delay Phase "wrapped"

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

145 Ex determine H(e jw ) of Type IV FIR Linear-Phase Systems Solution:

146 Ex Frequency response Type IV Magnitude Phase "wrapped" Group delay

M is evenM is odd h[M  n] = h[n] h[M  n] =  h[n] Causal FIR Linear-Phase Systems  satisfies: h[M  n] = ± h[n] for n = 0,1,…,M Type I Type III Type II Type VI

148 Type I ， I I FIR Linear-Phase Systems  Symmetric impulse response 0 M/2 M=10  M: odd integer. : integer plus one-half. 0 M/2 M=7  M: even integer.  integer.

149 Type III, IV FIR Linear-Phase Systems  Antisymmetric impulse response  M: even integer.  integer. 0 M/2 M=4  M: odd integer.  :integer plus one-half. 0 M/2 M=3

150 Locations of Zeros for FIR Linear-Phase Systems  For Type I and II,  For Type III and IV,

151 Type I and II  If is a zero of,  This implies that if is a zero of, then is also a zero of  then  The same result for Type III and IV factor has factor

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

153 Type I and II,Type III and IV  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  complex zeros on the unit circle Type II Type I

154 Type I and II,Type III and IV  if a zero of is real, and not on the unit circle, the reciprocal is also a zero, and have the factors of the form Type II Type I

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

156 Type I and II Type II Type I cannot be HP filter can be HP,LP filter can be LP filter

157 Type III and IV Type IV Type III

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

159 Type III and IV  The case of  must be zero of Type III generalized linear-phase systems.  If M is even, Type IV Type III  If M is odd ( Type IV), cannot be HP filter can be HP filter

160 Fig.5.41Typical plots of zeros for linear-phase systems Type IV (a) Type I Type III (b) Type II z 0 =-1, 不能做 HP filter z 0 =±1, 不能做 LP, HP filter z 0 =1, 不能做 LP filter 能做 LP, HP filter

Relation of FIR Linear-Phase Systems to Minimum- Systems  All FIR linear-phase systems have zero of  has all zeros inside the unit circle.  has all zeros on the unit circle.  has all zeros outside the unit circle Same magnitude

162 Example 5.21 Decomposition of a Linear-Phase System For Minimum-Phase System of Page 287, Eq Determine the frequecny response of Maximum-Phase System and the system cascaded by two. Solution:

163 Example 5.21 Decomposition of a Linear-Phase System

164 Example 5.21 Decomposition of a Linear-Phase System

165 Frequency response of Magnitude Phase Group delay "wrapped"

166 Frequency response of Magnitude Phase Group delay "wrapped"

167 Frequency response of Magnitude Phase Group delay "wrapped"

Review 168 Minimum-Phase System and Maximum-Phase System  a LTI bsystem is said to be minimum- phase if the system and its inverse are causal and stable.  Minimum-Phase System: all the zeros and poles are in the unit circle.  Properties:  The Minimum Phase-Lag Property  The Minimum Group-Delay Property  The Minimum Energy-Delay Property

Review 169 Minimum-Phase System and Maximum-Phase System  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)  Maximum-Phase System: poles are all in the unit circle, zeros are all outside the unit circle. It’s causal and stable.  (noncausal)Maximum-Phase System: anti-causal, stable System whose System function has all its poles and zeros outside the unit circle. (problem5.63).

170 Rational System Function has an equal number of poles and zeros Review z= ∞, z=0 may be poles or zeros

171  If L=0 and M>N, then M−N extra poles at z = 0 are induced by the numerator.  If L=0 and M<N, then N−M zeros at z = 0 appear from the denominator.  if L N, ↓ for M<N].  If L>0, then H(z) has L poles at z= ∞, L zeros at z = 0 [-(M-N), ↓ for M>N, ↑ for M<N]. Review Rational System Function has an equal number of poles and zeros

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

173 Type I FIR Linear-Phase Systems  Symmetric impulse response 0 M/2 M=10 K=012 M/2 1 2 h[n]  M: even integer,  M/2 : integer. Review

174 Type I FIR Linear-Phase Systems Review

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Chapter 5 HW  5.3, 5.4, 5.32,  5.12, 5.15, 5.19,  5.22, 5.43, 5.65, 5.66, 上一页下一页 返 回