Transform Analysis of LTI systems 主講人：虞台文

Content The Frequency Response of LTI systems Systems Characterized by Constant- Coefficient Difference Equations Frequency Response for Rational System Functions Relationship btw Magnitude and Phase Allpass Systems Minimum-Phase Systems Generalized Linear-Phase Systems

Transform Analysis of LTI systems Frequency Response of LTI systems

Time-Invariant System h(n)h(n) h(n)h(n) x(n)x(n) y(n)=x(n)*h(n) X(z)X(z)Y(z)=X(z)H(z) H(z)H(z)

Frequency Response Magnitude Phase

Ideal Frequency-Selective Filters Ideal Lowpass Filter Computationally Unrealizable Computationally Unrealizable

Ideal Frequency-Selective Filters Ideal Highpass Filter Computationally Unrealizable Computationally Unrealizable

Such filters are – Noncausal – Zero phase – Not Computationally realizable Causal approximation of ideal frequency- selective filters must have nonzero phase response. Ideal Frequency-Selective Filters

Phase Distortion and Delay --- Ideal Delay Delay Distortion Linear Phase Delay Distortion would be considered a rather mild form of phase distortion.

Phase Distortion and Delay --- A Linear Phase Ideal Filter Still a noncausal one. Not computationally realizable. Still a noncausal one. Not computationally realizable.

A convenient measure of the linearity of phase. Definition: Phase Distortion and Delay --- Group Delay Linear Phase   (  )=constant The deviation of  (  ) away from a constant indicates the degree of nonlinearity of the phase.

Transform Analysis of LTI systems Systems Characterized by Constant-Coefficient Difference Equations

N th -Order Difference Equation

Representation in Factored Form Contributes poles at 0 and zeros at c r Contributes zeros at 0 and poles at d r

Example Two zeros at z =  1 poles at z =1/2 and z =  3/4 poles at z =1/2 and z =  3/4

For a given ration of polynomials, different choice of ROC will lead to different impulse response. We want to find the proper one to build a causal and stable system. How? Stability and Causality

For Causality: – ROC of H(z) must be outside the outermost pole For Stability: – ROC includes the unit circle For both – All poles are inside the unit circle Stability and Causality

Example: Stability and Causality Re Im 1 Discuss its stability and causality

Inverse Systems H(z)H(z) X(z)X(z)Y(z)Y(z) Hi(z)Hi(z) X(z)X(z) G ( z )= H ( z ) H i ( z )=1 g(n) = h(n)* h i (n) =  (n)

Inverse Systems H(z)H(z) X(z)X(z)Y(z)Y(z) Hi(z)Hi(z) X(z)X(z) G ( z )= H ( z ) H i ( z )=1 g(n) = h(n)* h i (n) =  (n) Does every system have an inverse system? Give an example.

Inverse Systems Zeros Poles Zeros Poles

Minimum-Phase Systems A LTI system is stable and causal and also has a stable and causal inverse iff both poles and zeros of H(z) are inside the unit circle. Such systems are referred to as minimum-phase systems.

Impulse Response for Rational System Functions By partial fraction expansion:

FIR and IIR Zero poles nonzero poles

FIR and IIR Zero poles nonzero poles FIR: The system contains only zero poles. FIR: The system contains only zero poles.

FIR and IIR Zero poles nonzero poles IIR: The system contains nonzero poles (not canceled by zeros). IIR: The system contains nonzero poles (not canceled by zeros).

FIR

Example:FIR Does this system have nonzero pole? 7th-order pole M=7 One pole is canceled by zero here.

Example:FIR Write its system function. 7th-order pole M=7

Example:IIR

Transform Analysis of LTI systems Frequency Response of For Rational System Functions

Rational Systems

Log Magnitude of H ( e j  ) --- Decibels (dBs) Gain in dB = 20log 10 |H(e j  )| Scaling Contributed by zerosContributed by poles

Advantages of Representing the magnitude in dB The magnitude of Output FT The Magnitude Of Impulse Response The magnitude of Input FT

Phase for Rational Systems

Systems with a Single Zero or Pole r  r 

Frequency Response of a Single Zero or Pole

Frequency Response of a Single Zero

|H(e j  )| 2 : Its maximum is at  = . max |H(e j  )| 2 =(1+r) 2 Its minimum is at  =0. min |H(e j  )| 2 =(1  r) 2 |H(e j  )| 2 : Its maximum is at  = . max |H(e j  )| 2 =(1+r) 2 Its minimum is at  =0. min |H(e j  )| 2 =(1  r) 2

Frequency Response of a Single Zero

r = 0.9  = 0 r = 0.9  =  /2 r = 0.9  = 

r = 0.9  = 0 r = 0.9  =  /2 r = 0.9  =  Frequency Response of a Single Zero 於  處有最大凹陷 (1  r) 2

r = 0.9  = 0 r = 0.9  =  /2 r = 0.9  =  Frequency Response of a Single Zero 於 |  |  處最高 (1+r) 2

r = 0.9  = 0 r = 0.9  =  /2 r = 0.9  =  Frequency Response of a Single Zero 於  處 phase 直轉急上

Frequency Response of a Single Zero r=1/0.9 r=1.25 r=1.5 r=2  = 0  =  Zero outside the unit circle Note that the group delay is always positive when r>1

Frequency Response of a Single Zero Some zeros inside the unit circle And some outside

Frequency Response of a Single Pole The converse of the single-zero case. Why? A stable system: r < 1 Exercise: Use matlab to plot the frequency responses for various cases.

Frequency Response of Multiple Zeros and Poles Using additive method to compute – Magnitude – Phase – Group Delay

Example Multiple Zeros and Poles zeros Radius Angle 1  1  (59.45  ) poles Radius Angle  (35.85  )

Example Multiple Zeros and Poles zeros Radius Angle 1  1  (59.45  ) poles Radius Angle  (35.85  )

Transform Analysis of LTI systems Relationship btw Magnitude and Phase

Know magnitudeKnow Phase? Know PhaseKnow Magnitude? In general, knowledge about the magnitude provides no information about the phase, and vice versa. Except when … In general, knowledge about the magnitude provides no information about the phase, and vice versa. Except when …

Magnitude

Zeros of H(z): Poles of H(z): Zeros of C(z): Poles of C(z): Conjugate reciprocal pairs

Magnitude Given C(z), H(z)=? How many choices if the numbers of zeros and poles are fixed?

Allpass Factors a 1/a* Pole at a Zero at 1/a*

Allpass Factors

H1(z)H1(z) H1(z)H1(z) H1(z)H1(z) H1(z)H1(z) H ap (z) There are infinite many systems to have the same frequency-response magnitude?

Transform Analysis of LTI systems Allpass Systems

General Form Real Poles Complex Poles |H ap (e j  )|=1  |H ap (e j  )|=? grd[H ap (e j  )]=?

AllPass Factor Consider a=re j  Always positive for a stable and causal system.

Example: AllPass Factor Real Poles 0.9  0.9

Example: AllPass Factor Real Poles Phase is nonpositive for 0<  < . Group delay is positive 0.9  0.9

Example: AllPass Factor Complex Poles  /4 0.9 Continuous phase is nonpositive for 0<  < . Group delay is positive

Example: AllPass Factor Complex Poles  / /2  3/4  4/3 Continuous phase is nonpositive for 0<  < . Group delay is positive

Transform Analysis of LTI systems Minimum-Phase Systems

Properties of Minimum-Phase Systems To have a stable and causal inverse systems Minimum phase delay Minimum group delay Minimum energy delay

Rational Systems vs. Minimum-Phase Systems H(z)H(z) H(z)H(z) H min (z) H ap (z) How?

Rational Systems vs. Minimum-Phase Systems H(z)H(z) H min (z) H ap (z)

Rational Systems vs. Minimum-Phase Systems H(z)H(z) H min (z) H ap (z) Pole/zero Canceled

Frequency-Response Compensation s(n)s(n) Distorting System H d (z) Distorting System H d (z) sd(n)sd(n) Compensatiing System H c (z) Compensatiing System H c (z) s(n)s(n) The system of H d (z) is invertible iff it is a minimum-phase system.

Frequency-Response Compensation s(n)s(n) Distorting System H d (z) Distorting System H d (z) sd(n)sd(n) Compensatiing System H c (z) Compensatiing System H c (z) s(n)s(n) Distorting System H dmin (z) Distorting System H dmin (z) Allpass System H ap (z) Allpass System H ap (z) s(n)s(n)sd(n)sd(n) Compensatiing System 1 H dmin (z) Compensatiing System 1 H dmin (z)

Frequency-Response Compensation Distorting System H dmin (z) Distorting System H dmin (z) Allpass System H ap (z) Allpass System H ap (z) s(n)s(n)sd(n)sd(n) Compensatiing System 1 H dmin (z) Compensatiing System 1 H dmin (z) Hd(z)Hd(z) Hc(z)Hc(z)

Example: Frequency-Response Compensation 4 th order pole

Example: Frequency-Response Compensation 4 th order pole

Example: Frequency-Response Compensation 4 th order pole

Example: Frequency-Response Compensation 4 th order pole

Example: Frequency-Response Compensation Minimum PhaseNonminimum Phase

Minimum Phase-Lag Nonpositive For 0  more negative than

Minimum Group-Delay Nonnegative For 0  more positive than

Minimum-Energy Delay Apply initial value theorem

Transform Analysis of LTI systems Generalized Linear-Phase Systems

Linear Phase Linear phase with integer (negative slope)  simple delay Generalization: constant group delay

Example: Ideal Delay

  1   |H(e j  )| H(ej)H(ej) slope = 

Example: Ideal Delay If  =n d (e.g.,  =5) is an integer, h(n)=  (n  n d ). Impulse response is symmetric about n = n d, i.e., h(2n d  n)=h(n).

Example: Ideal Delay The case for 2  (e.g.,  =4.5) is an integer. h(2  n)=h(n).

Example: Ideal Delay  as an arbitrary number (e.g.,  =4.3). Asymmetry

More General Frequency Response with Linear Phase |H(e j  )| e  j  Zero-phase filter Ideal delay

  1   |H(e j  )| H(ej)H(ej) slope =  cc  c Example: Ideal Lowpass Filter

Example: Ideal Lowpass Filter Show that If 2  is an interger, h(2   n)=h(n). It has the same symmetric property as an ideal delay.

Generalized Linear Phase Systems Real function. Possibly bipolar.  and  are constants constant group delay

h(n) vs.  and 

Necessary Condition for Generalized Linear Phase Systems Let’s consider special cases.

Necessary Condition for Generalized Linear Phase Systems  =0 or  2  = M = an integer  =0 or  Such a condition must hold for all  and 

Necessary Condition for Generalized Linear Phase Systems  =0 or  2  = M = an integer   is an integer  2  is an integer

Necessary Condition for Generalized Linear Phase Systems  =  /2 or 3  /2 2  = M = an integer  =  /2 or 3  /2 Such a condition must hold for all  and 

Necessary Condition for Generalized Linear Phase Systems  =  /2 or 3  /2 2  = M = an integer   is an integer  2  is an integer

Causal Generalized Linear Phase Systems Generalized Linear Phase System Causal Generalized Linear Phase System

Causal Generalized Linear Phase Systems … M Type I FIR linear phase system M is even Type II FIR linear phase system M is odd

Causal Generalized Linear Phase Systems Type III FIR linear phase system M is even Type IV FIR linear phase system M is odd … … M … … M

Type I FIR Linear Phase Systems … M Type I FIR linear phase system M is even

Example: Type I FIR Linear Phase Systems

Example: Type II FIR Linear Phase Systems

Example: Type III FIR Linear Phase Systems 11

Example: Type IV FIR Linear Phase Systems 11

Zeros Locations for FIR Linear Phase Systems (Type I and II) Let z 0 be a zero of H(z) 1/z 0 is a zero If h(n) is real z 0 * and 1/ z 0 * are zeros

Zeros Locations for FIR Linear Phase Systems (Type I and II) Let z 0 be a zero of H(z) 1/z 0 is a zero If h(n) is real z 0 * and 1/ z 0 * are zeros

Zeros Locations for FIR Linear Phase Systems (Type I and II) Consider z =  1 if M is odd, z =  1 must be a zero. if M is odd, z =  1 must be a zero.

Zeros Locations for FIR Linear Phase Systems (Type III and IV) Let z 0 be a zero of H(z) 1/z 0 is a zero If h(n) is real z 0 * and 1/ z 0 * are zeros

Zeros Locations for FIR Linear Phase Systems (Type III and IV) Consider z = 1 z = 1 must be a zero. Consider z =  1 if M is even, z =  1 must be a zero. if M is even, z =  1 must be a zero.