No Aliasing in the Impulse Invariance H(e)=∑h1+ 2丌 k jH(A2)=0,g2≥7 k W then hle =h d 1≤丌 T H(e/ 27 2丌 2
27 No Aliasing in the 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 T w = −
Aliasing in the Impulse Invariance H(e)=∑h1+ 2丌 k jH(g2)≠0,g2≥/k then h(e")≠B/; p≤z 2x/-xn/2 H(el) 2丌 2丌 28
28 − 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 th n H e H j T e w − 2− Td 2 Td
discrete-time filter design by impulse invariance O If input is bandlimited and f> fma to make Hf(02) H(en), 2<7/T work the relation is needed 0 >兀 CD H(e/a D/C ya(t) 川=mh(m)H(-)=∑互1+ 2丌 k =97frw<xH(1)=0,2≥xz/ then h(e/=h <兀 29
29 ◆If input is bandlimited and f s > 2fmax , to make ( ) ( ), 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 = ( ). d d c h n h n = T TT T work, the relation is needed:
relation between frequencies Q=gT,-x<0<丌,-00<9<o Relationship of 12兀 k frequencies Hlem) C UH(Q)=0, o>/T the H(e")=Hl I bosm No Aliasing H(1)=0 S plane 3/ ane 2≥/7 丌 one period 丌 30
30 , , = 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 ( ) 0, c d if H j T = ( ) = w T w then H e H j d c j w , one period ( ) 0, H j c = No Aliasing Td
discrete-time filter design by impulse invariance In=x(t) =n7= 1 hn]=h(nt) JO C2兀k Y(eJo ∑X TT X(e/) C H(e JO )=∑H 2丌k H(eo) k= TT 小]=mh(n)H(")=∑ /;p 2丌 k= jH(g2)=0,922m/ then h w<丌
31 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 h = Td c (nTd ) 1 2 ( ) c k j k X X j T T T e =− = − ( ) 1 c j X j T T e X = 1 2 ( ) c k j k H H j T T T e =− = − ( ) 1 c j H j T T e H = [ ] ( ) | ( ) c t nT c x n x t x nT = = = [ ] ( ) c h n h nT =