Constraints of transformation In transformations to preserve the essential properties of the e frequency response, the imaginary axis of the S-plane is mapped onto the unit circle of the z-plane j2分z=e m plane lane e Re
22 Constraints of Transformation ◆In transformations, to preserve the essential properties of the frequency response, the imaginary axis of the s-plane is mapped onto the unit circle of the z-plane. jw s j z e = = s − plane z − plane Im Im Re Re
2.从5)极点分布与原函数的对应关系 几种興型情况 S+a 2
2．H(s)极点分布与原函数的对应关系 j O −α α 0 jω 0 − jω 几种典型情况 p211 1 s 1 s a + 2 2 s + 2 1 s
Constraints of transformation In order to preserve the property of stability, If the continuous system has poles only in the left half of the s-plane then the discrete-time filter must have poles only inside the unit circle e m z plane Re Re
24 Constraints of Transformation ◆In order to preserve the property of stability, If the continuous system has poles only in the left half of the s-plane, then the discrete-time filter must have poles only inside the unit circle. s − plane Im Re Im z − plane Re
7. 1.1 Filter Design by Impulse Invariance The impulse response of discrete-time system is defined by sampling the impulse response of a continuous-time system n]=Th(nta) Relationship of ∑(号 2丌 hleB k frequencies jH(A2)=0,g2 t then hel=h enId ≤丌 L e, if the continuous-time W=2Torw<丌 filter is bandlimited one period 25
25 7.1.1 Filter Design by Impulse Invariance ◆The impulse response of discrete-time system is defined by sampling the impulse response of a continuous-time system. ( ) h n = Td hc nTd ( ) c Td if H j = 0, ( ) = w T w then H e H j d c j w , w= Td for w ( ) =− = + k d d c j w k T j T w H e H j Relationship of 2 frequencies i.e. if the continuous-time filter is bandlimited, one period
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 26
26 , , = 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