5.1 the frequency response of LTI system 5.2 system function 5.3 frequency response for rational system function 5.4 relationship between magnitude and phase 5.5 all-pass system 5.6 minimum-phase system 5.7 linear system with generalized linear phase Chapter 5 transform analysis of linear time-invariant system

5.1 the frequency response of LTI system magnitude response or gain magnitude square function log magnitude magnitude attenuation magnitude-frequency characteristic ：

log magnitude linear magnitude transform curve from linear to log magnitude

phase-frequency characteristic ： phase response principal phase continuous phase group delay

Figure 5.7

Figure 5.1 EXAMPLE understand group delay

Figure 5.2

5.2 system function Characteristics of zeros and poles ： （ 1 ） take origin and zeros and poles at infinite into consideration, the numbers of zeros and poles are the same. （ 2 ） for real coefficient, complex zeros and poles are conjugated, respectively. （ 3 ） if causal and stable, poles are all in the unit circle. （ 4 ） FIR ： have no nonzero poles, called all-zeros type, steady IIR ： have nonzero pole; if no nonzero zeros, called all-poles type EXAMPLE Difference about zeros and poles in FIR and IIR

5.3 frequency response for rational system function 1.formular method

2. Geometrical method

EXAMPLE magnitude response in w near zeros is minimum, there are zeros in unit circle, then the magnitude is 0 ； magnitude response in w near poles is maximum ； zeros and poles counteracted each other and in origin does not influence the magnitude.

EXAMPLE

B=1 A=[1,-0.5] figure(1) zplane(B,A) figure(2) freqz(B,A) figure(3) grpdelay(B,A,10) EXAMPLE 3.matlab method

5.4 relationship between magnitude and phase

Figure 5.20 EXAMPLE Pole-zero plot for ， H(z): causal and stable ， Confirm the poles and zeros

5.5 all-pass system Zeros and poles are conjugate reciprocal For real coefficient, zeros are conjugated, poles are conjugated.

EXAMPLE Y Y Y N

Characteristics of causal and stable all-pass system: application ： 1. compensate the phase distortion 2. compensate the magnitude distortion together with minimum-phase system

5.6 minimum-phase system inverse system:

explanation ： （ 1 ） not all the systems have inverse system 。 （ 2 ） inverse system may be nonuniform 。 （ 3 ） the inverse system of causal and stable system may not be causal and stable 。 the condition of both original and its inverse system causal and stable ： zeros and poles are all in the unit circle ， such system is called minimum- phase system, corresponding h[n] is minimum-phase sequence 。 poles are all in the unit circle, zeros are all outside the unit circle, such system is called maximum-phase system 。

zeros outside the unit circle poles outside the unit circle minimum-phase system: conjugate reciprocal zeros and poles all-pass system: counteracted zeros and poles, zeros and poles outside the circle minimum-phase and all-pass decomposition ： If H(z) is rational, then :

Figure 5.25 Application of minimum-phase and all-pass decomposition ： Compensate for amplitude distortion

Properties of minimum-phase systems: （ 1 ） minimum phase-delay （ 2 ） minimum group-delay Minimum-phase system and some all-pass system in cascade can make up of another system having the same magnitude response, so there are infinite systems having the same magnitude response.

（ 3 ） minimum energy-delay （ i.e. the partial energy is most concentrated around n=0 ）

Figure 5.30 最小相位 maximum phase EXAMPLE minimum phase Systems having the same magnitude response

Figure 5.31 minimum phase

Figure 5.32

5.7 linear system with generalized linear phase definition conditions of generalized linear phase system causal generalized linear phase (FIR)system

5.7.1 definition Strict: Generalized: Systems having constant group delay phase

EXAMPLE ideal delay system differentiator ： magnitude and phase are all linear EXAMPLE physical meaning ： all components of input signal are delayed by the same amount in strict linear phase system ， then there is only magnitude distortion, no phase distortion. it is very important for image signal and high-fidelity audio signal to have no phase distortion. when B=0, for generalized linear phase, the phase in the whole band is not linear, but is linear in the pass band, because the phase +PI only occurs when magnitude is 0, and the magnitude in the pass band is not 0.

square wave with fundamental frequency 100 Hz linear phase filter ： lowpass filter with cut-off frequency 400Hz nonlinear phase filter ： lowpass filter with cut-off frequency 400Hz EXAMPLE

Generalized linear phase in the pass band is strict linear phase

5.7.2 conditions of generalized linear phase system Or:

Figure 5.35 M:even M:odd M:not integer

EXAMPLE M:not integer

determine whether these system is linear phase,generalized or strict?a and ß=? EXAMPLE (1)(2) (3) (4)

5.7.3 causal generalized linear phase (FIR)system

Magnitude and phase characteristics of the 4 types ：

III

IIIIV

Characteristic of zeros: commonness

Figure 5.41 Characteristic of every type ：

type I ： type II ： type III ： type IV ： characteristic of magnitude get from characteristic of zeros ：

M is evenM is odd low high band pass band stop h[n] is even (I) Y Y Y Y Y N Y N (II) h[n] is odd (III) N N Y N N Y Y N (IV) Application of 4 types of linear phase system:

5.1 the frequency response of LTI system : 5.2 system function 5.3 frequency response for rational system function: 5.4 relationship between magnitude and phase : 5.5 all-pass system 5.6 minimum-phase system 5.7 linear system with generalized linear phase （ FIR) definition: conditions : h[n] is symmetrical causal generalized linear phase system 1.condition 2.classification 3.characteristics of magnitude and phase, filters in point respectively 4.analyse of characteristic of magnitude from the zeros of system function summary

requirement: concept of magnitude and phase response, group delay; transformation among system function, phase response and difference equation; concept of all-pass, minimum-phase and linear phase system and characteristic of zeros and poles; minimum-phase and all-pass decomposition; conditions of linear phase system, restriction of using as filters key and difficulty ： linear phase system

exercises 5.17 complementarity ： minimum-phase and all-pass decomposition

the first experiment problem 1 （ D ） problem 11 problem 13 （ C ） problem 22 （ A ） problem 24 （ A ）（ C ） Get subjects from the experiment instruction book or downloading from network ：