Chapter 5 Transform Analysis of inear Time-Invariant systems ◆50 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 ◆5.5A- Pass system 5.6 Minimum-Phase systems 5.7 Linear Systems with generalized linear phase
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
5.iNtroduction An LTI system can be characterized in time domain by impulse response hn] Output of the Lti system ]=x*=∑xk-k] With Fourier transform and z-transform an LTI system can be characterized Cin Z-domain by system function H(z) Y(z)=H()x()y(e")=h(e)x(e") Cin frequency-domain by Frequency response H
3 5.0 Introduction ◆An LTI system can be characterized in time domain by impulse response ◆Output of the LTI system: h n =− = = − k y n x n h n x k h n k Y(z) = H(z)X(z) ◆in Z-domain by system function ◆in frequency-domain by Frequency response ( ) ( ) ( ) j w j w j w Y e = H e X e ◆With Fourier Transform and Z-transform, an LTI system can be characterized H z( ) ( ) jw H e
5.1 Frequency Response of LTT Systems Frequency response hle/ Useful input signal Yle/=hle/wintel deleterious signa ◆ Magnitude response(gain)Hle") change on to Y(en)=H(emx(ery useful signal distortions . Phase response(phase shift)H(e/") ∠y()=∠H(e")+∠x(em)
4 5.1 Frequency Response of LTI Systems ( ) ( ) ( ) j w j w j w Y e = H e X e ( ) ( ) ( ) j w j w j w Y e = H e X e ◆Phase response (phase shift) ( ) jw H e ( ) ( ) ( ) j w j w j w Y e = H e +X e ( ) jw ◆Frequency response H e ( ) jw ◆Magnitude response (gain) H e distortions change on useful signal system Useful input signal + deleterious signal
5.1.1 Ideal Frequency-Selective Filters ◆ Ideal lowpass filter 少<W e -0<n<O ◆ Noncausal,not computationally H realizable ◆ no phase distortion 2丌 丌-W W,丌 2丌 C C
5 5.1.1 Ideal Frequency-Selective Filters ◆Ideal lowpass filter ( ) 1, 0, jw c lp c w w H e w w = ( ) jw H e 0 − 2 − − wc wc 2 1 sin , = − c lp w n n h n n ◆Noncausal, not computationally realizable ◆no phase distortion
5.1.1 Ideal Frequency-Selective Filters ◆ Ideal highpass filter 1-H(e)=b(a") 0 <W v<w|≤丌 sin w n n=on 0<n<0 元n H 2丌 2丌 6
6 5.1.1 Ideal Frequency-Selective Filters ◆Ideal highpass filter ( ) 0, 1, jw c hp c w w H e w w = sin , c hp w n h n n n n = − − − 2 − − wc 0 wc 2 ( ) jw H e 1 1 ( ) jw − H e lp =
5.1.1 Ideal Frequency-Selective Fiters ◆ Ideal bandpass filter W<w<1 Hb J 0. others H(e") 丌-W w0 w
7 5.1.1 Ideal Frequency-Selective Filters ◆Ideal bandpass filter ( ) 1 2 1, 0, jw c c bp w w w H e others = 0 1 wc 1 − − wc ( ) jw H e 1 2 wc 2 − wc
5.1.1 Ideal Frequency-Selective Filters ◆ Ideal bandstop filter 0,w<|w<w H others H W。0W 丌
8 5.1.1 Ideal Frequency-Selective Filters ◆Ideal bandstop filter ( ) 1 2 0, 1, jw c c bs w w w H e others = − 0 ( ) jw H e 1 1 wc 1 − wc 2 wc 2 − wc
5.1.2 Phase Distortion and Delay To understand the effect of the phase and the group delay of a linear system, first consider the ideal delay system yin=xn-n ◆ The impulse response The frequency response hide) Jwn e J ∠H d s <丌 9
9 5.1.2 Phase Distortion and Delay hid n = n − nd n nd y n = x − ( ) d j w jwn i d H e e − = ( ) =1 jw id H e H (e )= −wnd w j w i d , ◆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:
Group Delay(群延迟,grd) t W=grd [H(e) delang heip) H(en)]=-d-Wnly then [(w)=na ◆ For ideal delay system argH(e)=-(arg[e n] W d The group delay represents a convenient measure of the linearity of the phase 10
10 Group Delay(群延迟，grd ) ◆For ideal delay system ( ) ( ) arg ( ) jw jw d w H e H g d e w r d = = − d d d wn n dw = − − = ( ) 0 arg jw d If H e wn = − − ( ) d then w n = The group delay represents a convenient measure of the linearity of the phase. ( ) arg arg ( ) d d d jw jwn w H e e dw dw − = − = −
Group Delay(群延迟,grd) o Given a narrowband input xnl=s(n]cos(won) for a system with frequency response H(ejiw), it Is assumed that X(e ew)is nonzero only around W=0 Group Delay I argH(en)=-do-winas then [(w)=nd it can be shown(see Problem 5.57)that the response y(n to x(n is =(2)s-noyn-9-) the time delay of the envelope sIn) is nd
◆Given a narrowband input x[n]=s[n]cos(w0n) for a system with frequency response H(ejw), it is assumed that X(ejw) is nonzero only around w =w0 11 Group Delay(群延迟，grd ) ( ) 0 0 0 0 [ ] cos( ) d j d w y n H e s n w = − − − n w n n ( ) 0 arg , jw d If H e n = − − w ( ) d then w = n 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 . d n Group Delay
