Example 7.1 Determining Specifications for a Discrete-Time Filter H(ejw) Known Specifications of the continuous-time filter ◆1. passband1-001m()<1+001m0s952x(200 42.5obd()0109 F0() (20),2|<x/r Q>z/T=2f 2T T 2T T 27(500 CD H(e/) D/C yIn ya(t) For bandlimited input, aliasing avoided when sampling frequency ihigh 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− = max max ( ) 2 2 2 5000 2 f T 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:
Example 7.1 Determining specifications for a discrete-Time Filter Specifications of the continuous-time filter passband1-001<H(g)<1+0010709527(2000 ◆2. stopband|H()<001027(300 IHer(in) 1+61 tolerance scheme 容限图 =0011-4 Passband i transition Stopband 2 =0.001 g2=27(20092,=2(3000
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
Example 7.1 Determining Specifications for a discrete-Time Filter H Solution: Specifications of the Same 61 discrete-time filter in a tolerance 01 W limits: 5=0.0 hle 2=0.001 Passband i Transition Stopband H(iQ2 +61 62 61 O.=0.4丌 0.6丌 Passband i Transition Stopband O=9T=10-49 9p=27(200=2x(3000 T=10
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 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