Frequency-Selective Filters ◆ [deal highpass filter 0 W<<兀 h(n)=8[n 0<1<00 H 2丌 2丌
7 Frequency-Selective Filters ◆Ideal highpass filter ( ) 0, 1, jw c hp c w w H e w w = 0 wc − 2 − − wc 2 ( ) jw H e 1 ( ) sin , c hp w n h n n n n = − −
Frequency-Selective Filters Ideal bandpass filter ww<w H 0. others H - 8
8 Frequency-Selective Filters ◆Ideal bandpass filter ( ) = others w w w H e j w c c bp 0, 1, 1 2 0 1 wc 1 − − wc ( ) jw H e 1 2 wc 2 − wc
Frequency-Selective Filters ◆ Ideal bandstop filter H,(en) <1 others H -
9 Frequency-Selective Filters ◆Ideal bandstop filter ( ) = others w w w H e j w c c bs 1, 0, 1 2 0 1 wc 1 − − wc ( ) jw H e 1 2 wc 2 − wc
Linear time-invariant discrete-time system If input is bandlimited and sampling frequency is high enough to avoid aliasing then overall system behave as a continuous-time system H T Ω<丌/T H(x2) g2|>m/T7 H(ejo) DIC xa(t x yIn ya(o) continuous-time specifications are converted to discrete time specifications by H H f <丌
10 ◆If input is bandlimited and sampling frequency is high enough to avoid aliasing, then overall system behave as a continuous-time system: ( ) ( ) = T H e T H j j T eff 0, , Linear time-invariant discrete-time system ( ) , eff jw w H H j w T e = continuous-time specifications are converted to discrete time specifications by: w= T
Example 7.1 Determining Specifications for a Discrete-Time Filter H(e? Known Specifications of the continuous-time filter ◆1. passband1-001-()<1+001/0r0c9522000 ◆2. stopband()010009 (e0),|2<z/r 2丌丌 hiQ 2兀fn =2z(5000 >/T maX 2T T CD H(e/) D/C yIn ya(t) For bandlimited input aliasing avoided when sampling frequency s high enough
11 Example 7.1 ◆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) 4 T s 10− = max ( ) 2 2 2 5000 2 f T T ( ) = = = ( ), , 0, j T eff H e T H j 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: