Aliasing in the Impulse Invariance H(e)=∑h1+ 2丌 k jH(g2)≠0,g2≥/k then h(e)≠H V≤兀 丌/ H(el) 2丌 2丌 27
27 − Td Td − Aliasing in the Impulse Invariance ( ) =− = + k d d c j w k T j T w H e H j 2 ( ) 0, c d if H j T ( ) , jw c d w then H e H j T w
discrete-time filter design by impulse invariance VIf input is bandlimited and f>2m H(2) H(e0),2<z/T Q>T/T CD H(e/o) D/C ya(t) =m(m)(2)=∑|1y+12xk =97frw<xH(1)=0,2≥xz/ then h(e/=h <兀 28
28 ◆If input is bandlimited and f s>2fmax , : ( ) ( ), 0, j T eff H e T H j T = discrete-time filter design by impulse invariance w f Td = or w ( ) 2 d jw k d c T T w H e H j j k =− = + ( ) 0, c d if H j = T ( ) , j d w c w then H e H j T w = h n h =Td d c (nT ) T T
relation between frequencies Q=9,-x<O<丌,00<9<o Relationship of frequencies n(")=∑(1+1k H()=0.p2x“)=m上万 S plane 3/ Z plane
29 = Td ,− ,− relation between frequencies S plane Z plane - 3 / d T j / d T / d − T ( ) =− = + k d d c j w k T j T w H e H j Relationship of 2 frequencies ( ) c Td if H j = 0, ( ) = w T w then H e H j d c j w
discrete-time filter design by impulse invariance x[n]=x (tlen=x(nr) h{1m]=h(n7) H(e/@)_I ∑X C2兀k TT X(e)=xct H(e0)=∑H 2兀k T H(ejo FTHcJT =nu)e")=∑y+1k k= jH(g2)=0,922m/ then h w<丌 30
30 discrete-time filter design by impulse invariance ( ) =− = + k d d c j w k T j T w H e H j 2 ( ) c Td if H j = 0, ( ) , jw c d w then H e H j w T = ( ) h n = Td hc nTd 1 2 ( ) c k j k X X j T T T e =− = − 1 ( ) c j X X j T T e = 1 2 ( ) c k j k H H j T T T e =− = − 1 ( ) c j H H j T T e = [ ] ( ) | ( ) c t nT c x n x t x nT = = = [ ] ( ) c h n h nT =
Steps of DT filter design by transforming a prototype continuous-time filter Obtain the specifications for continuous time filter by transforming the specifications for the desired discrete-time filter Find the system function of the continuous time filter Transform the continuous-time filter to derive the system function of the discrete time filter
31 Steps of DT filter design by transforming a prototype continuous-time filter ◆Obtain the specifications for continuoustime filter by transforming the specifications for the desired discrete-time filter. ◆Find the system function of the continuoustime filter. ◆Transform the continuous-time filter to derive the system function of the discretetime filter