Example 7.1 Determining Specifications for a Discrete-Time Filter H(ejw) Known Specifications of the continuous-time filter: l◆1. passband1-001<1m()<1+00100≤9527(200 ◆2. stopband|fo(9)<0001027(30059 Her(jQ2) H(e),|p</T 2=2丌 2"max max =27(5000 CID H(e/) D/C x(t xIn yIn ya(t) T=10s 2兀=2 0000 For bandlimited input, aliasing avoided when sampling frequency is,high enough
12 Example 7.1 ◆Specifications of the continuous-time filter: ◆1. passband ◆2. stopband 1 0.01 1 0 − + H j for eff ( ) .01 0 2 20 ( 00) H j for eff ( ) 0.001 2 3000 ( ) 4 T s 10 , − = ( ) 2 2 10000 S T = = ( ) ( ), , 0, j T eff H e H j T T = For bandlimited input, aliasing avoided when sampling frequency is high enough Example 7.1 Determining Specifications for a Discrete-Time Filter H(ejw) . Known: max max 2 2 5000 ( ) 2 S f T = = =
Example 7.1 Determining Specifications for a discrete-Time filter Specifications of the continuous-time filter ◆1. passband1-001<H(2)1+001070≤9527(200 ◆2 stopband|(x)<0001m27(300059 Hem(j川 1+61 tolerance scheme =01-6 容限图 Passband i Transition Stopband max 2n=22000g9,=2x(30002x(5000 62=0.00182 0
13 Example 7.1 Determining Specifications for a Discrete-Time Filter ◆Specifications of the continuous-time filter: ◆1. passband ◆2. stopband 1− 0.01 H ( j) 1+ 0.01 for 0 2 (2000) eff Heff ( j) 0.001 for 2 (3000) 1 = 0.01 2 = 0.001 2 (2000) = p 2 (3000) = s tolerance scheme 容限图 max 2 (5000) =
Example 7.1 Determining Specifications for a discrete-Time Filter H Solution: Specifications of the Same 1+a1 discrete-time filter in a tolerance 1-81 limits: 8,=0.0T H() &=0.001 Passband I Transition Stopband Hm(Q +61 0.4丌 .=0.6 Passband transition Stopband O=9T=10-g 2n=2(200092=2x(3000 T=10-4s 0
14 Example 7.1 Determining Specifications for a Discrete-Time Filter 4 T s 10− = 4 T 10− = = Specifications of the discrete-time filter in 1 = 0.01 2 = 0.001 2 (2000) = p 2 (3000) = s 0.4 p = 0.6 s = Solution: Same tolerance limits: H j eff ( ) ( ) j H e ( ) eff jw w H H j T e =
Filter Design Constraints Designing iir filters is to find the approximation by a rational function of Z The poles of the system function must lie inside the unit circle(stability, causality) Designing fir filters is to find the polynomial approximation FIR filters are often required to be linear phase. 15
15 Filter Design Constraints ◆Designing IIR filters is to find the approximation by a rational function of z. ◆The poles of the system function must lie inside the unit circle(stability, causality). ◆Designing FIR filters is to find the polynomial approximation. ◆FIR filters are often required to be linearphase
Filter Design Techniques 7. 1 Design of Discrete-Time IIR Filters from continuous-Time filters 16
16 Filter Design Techniques 7.1 Design of Discrete-Time IIR Filters From Continuous-Time Filters