5.1.1 Ideal Frequency-Selective Fiters ◆ Ideal bandpass filter W<w<1 Hb J 0. others H(e") 丌-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
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=xin-na ◆ The impulse response 1- The frequency response hide) Jwn e ∠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 xinl=s n cos(won) for a system with frequency response H(eiw),it is assumed that X(e/w)is nonzero only around W=0 Group Delay If arg H(e =--Wmn,加henr(w)=mz it can be shown(see Problem 5.57)that the response y(n to x(n is 1=()=n%=-wn) the time delay of the envelope sn is n
◆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