Three Reasons for designing DT filter by transformation of CT filter (designed) 2.Many useful continuous-time IIR design method have relatively simple closed form design formulas. Therefore,discrete-time IIR filter design methods based on standard continuous-time design formulas are rather simple to carry out. 23
23 Three Reasons for designing DT filter by transformation of CT filter (designed) 2. Many useful continuous-time IIR design method have relatively simple closed form design formulas. Therefore, discrete-time IIR filter design methods based on standard continuous-time design formulas are rather simple to carry out
Three Reasons for designing DT filter by transformation of CT filter(designed) 3.The standard approximation methods that work well for continuous-time IIR filters do not lead to simple closed- form design formulas when these methods are applied directly to the discrete-time IIR case, because the frequency response of a discrete-time filter is periodic,and that of a continuous-time filter is not. 24
24 Three Reasons for designing DT filter by transformation of CT filter (designed) 3.The standard approximation methods that work well for continuous-time IIR filters do not lead to simple closedform design formulas , when these methods are applied directly to the discrete-time IIR case, ➢ because the frequency response of a discrete-time filter is periodic, and that of a continuous-time filter is not
Steps of DT IIR filter design by transforming a prototype continuous-time filter 1Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter.Impulse Invariance,=27 Bilinear transformation 2Determine the system function H(s)or h(t) of the continuous-time filter. 3System function H(z)or h[n] of discrete-time filter is obtained by applying transformation to He(s)or he(t)._Impulse Invariance,h[n]=Th.(nT) Bilinear transformation required that the essential properties of the continuous- time system be preserved: stable,causal
25 Steps of DT IIR filter design by transforming a prototype continuous-time filter ①Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter. ②Determine the system function Hc (s) or hc (t) of the continuous-time filter. ③System function H(z) or h[n] of discrete-time filter is obtained by applying transformation to Hc (s) or hc (t). required that the essential properties of the continuoustime system be preserved: Impulse Invariance, Bilinear transformation stable, causal Impulse Invariance, Bilinear transformation = T [ ] ( ) c h n Th nT =
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. stable, S=jS2←→z=ew causal Im Im z-plane s-plane Re Re 27
27 Constraints of Transformation ◆In transformations, to preserve the essential properties of the frequency response, j 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. stable, causal
Constraints of Transformation In order to preserve the property of stability and causality,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. Im s-plane Im z-plane Re Re 28
28 Constraints of Transformation s − plane Im Re Im z − plane Re ◆In order to preserve the property of stability and causality, 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