山东大学：《生物医学信号处理 Biomedical Signal Processing》精品课程教学资源（PPT课件讲稿）Chapter 07 Filter Design Techniques

从S极点分布与原函数的对应关系(稳定性) 几种典型情况 $+O C s+a 2

H(s)极点分布与原函数的对应关系(稳定性)  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 oIn 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 g plane s-plane Re Re 23

23 Constraints of Transformation ◆In transformations, to preserve the essential properties of the frequency response, jw s = j  z = e s − plane z − plane Im Im Re Re the imaginary axis of the s-plane is mapped onto the unit circle of the z-plane

Constraints of transformation o In order to preserve the property of stabilit 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 ane Re Re 24

24 Constraints of Transformation s − plane Im Re Im z − plane Re ◆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

7. 1. 1 Filter Design by Impulse Invariance The impulse response of discrete-time system is defined by sampling the impulse esponse of a continuous-time system h[]=th(nta Relationship O2丌 between two H(e)=>h +jk = C systems: k f H(2)=0,((2r/Ta, then H(e o)=Hoj,<T d i. e the continuous -time filter is bandlimited, and 2>22 in one period nmax 2

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. ( ) 0, , c d if H j T  =    ( ) , j c d then H e H j T       =      Relationship between two systems: i.e. the continuous-time filter is bandlimited, and Ωs >2Ωmax in one period ( ) 2 , d j k d c T T H e H j j k     =−   = +      . Td   = h n T h nT   = d c d ( )  = Td

relation between frequencies Relationship 2T2-丌<O<x,-0<<∝ between two hejin )=∑ w x]ta 2丌 k systems: (1)=012x7,mm(e)H(xx j No Aliasing ane ane 丌 z pla 丌/T one period 丌 28

28 , ,  = 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 2 ( ) 0, , c d if H j T  =    ( ) , jw c d w th n H e H j T e w    =      one period No Aliasing Relationship between two systems: