COMM 602: Digital Signal Processing. Lecture 8. Digital Filter Design
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1 COMM 60: Digital Signal Proeing Leture 8 Digital Filter Deign
2 Remember: Filter Type
3 Filter Band
4 Pratial Filter peifiation
5 Pratial Filter peifiation H ellipti H
6 Pratial Filter peifiation p p
7 IIR Filter M M
8 IIR Filter Deign Deign mean alulation of the oeffiient of the differene equation or the tranfer funtion. We ue Analog filter deign and then we onvert the deign into digital domain beaue: Analog filter deign i a well developed field and highly advaned. Analog filter deign uually give loed form olution. Extenive table are available for analog filter deign.
9 IIR Filter Deign Method of IIR Filter Deign: - Impule invariant - Step invariant - Bilinear tranformation.
10 Impule Invariant Method Step of Deign:
11 Example: Convert the following analog filter ytem funtion into a digital IIR filter by mean of the impule invariane method: H a h a t H a 3e 0. 5t a 0.5nT n t 3e a h 3 tnt where 3 a H e 0.5T n n n Zh a nt hn 3a n0 n0 3 a
12 Example (Contd. H 0. It i lear that ha a pole at 5 a 3 b 0, a y(n But H ha a pole at a e 0.5T
13 Example: Convert the following analog filter ytem funtion into a digital IIR filter by mean of the impule invariane method:
14 Charateriti of Analog Filter Butterworth Filter: All pole filter (no ero, with no ripple in both the paband and top band. The normalied Butterworth tranfer funtion are: Filter Order Tranfer Funtion 3 3 : ( Every Cut off of i frequeny the ana log divided by filter for LPF
15 Example: Uing the impule invariant method deign a LP digital filter atifying the following requirement: The analog angular ut off frequeny i 000 (rad/e, Sampling rate=000 H, t order Butterworth filter. Solution H a h a ( ( t / e t h a ( nt 000 e h( n 000 e (000 nt n / H( 000 /( a 000 a 000 e n (000 n/ 000 where a e /
16 Example: Uing the impule invariant method deign a LP digital filter atifying the following requirement: The digital ut off angular frequeny i 0. (rad/e, Sampling rate=500 H, nd order Butterworth filter. Solution / in / in ( / ( / ( / ( / ( / / ( / (00 / ( t e t e t h H t t a a F
17 n e n h F T ubtitute nt e n h n nt in ( / in ( 0. / ( o in(0. 00 ( o in( ( o in( in( ( ( e e e H e a e a e H Then a a an and e X e n x Sine a a a tranform Z a tranform Z an
18 Step Invariant Method Step of Deign: ( Find H( of the analog filter. u( t ( (3 Find tep repone Find a(nt by a(t ampling H ( (3 Find H(Z a( nt (4 Realiation
19 Example: T T nt t e e H nt u e nt a t u e t a Then / ( / ( ( H a(t ( H( H( ( ( 5 5 ( ( ( 5 5 ( ( ( ( 5 Solution filter onvert it to digital 5 T.F: Given the analog
20 Bilinear Tranformation See Text book For the derivation Of thi equation tan T Page 74
21 Bilinear Tranformation Derivation of: (
22
23 Frequeny Warping Effet tan T
24 Propertie of Bilinear Tranformation Let T. Putting σ j in equation (, we may repreent a : r e and arg( jθ tan where the radiu and angle are defined ( tan ( σ r ( σ for 0 From ( and (3 we have : r for 0 r for 0 r for 0 tan / by : ( (3
25 Propertie of Bilinear Tranformation Aordingly, we may tate the propertie of bilinear tranformation: ( The left-half -plane i mapped onto the interior of the unit irle in the -plane. ( The entire j axi of the -plane i mapped onto one omplete revolution of the unit irle in the -plane (3 The right-half of the -plane i mapped onto the exterior of the unit irle in the -plane. Then if the analog filter i table and aual, the reulting digital filter i alo aual and table.
26 BILINEAR Z TRANSFORMATION (Contd. S-Plane Z-Plane = - Unit Cirle = LHS J-axi RHS
27 Step of Bilinear Tranformation H( tan T H(
28 Example: H( tan T For T=
29 H H(
30 H xn yn
31 Example: H T T H T H Convert the following analog filter ytem funtion into a digital IIR filter by mean of the bilinear tranformation method: Solution
32 Example (Contd T H T H T T H T T T H
33 Example (Contd. For T= H 3 T T 0.5 T T T 0.5 T H 0.333
34 Butterworth Filter Order A A For H j and H j, the analog filter order N i determined by the equation: A log A N log A A
35 Tranformation of LPF to Other Filter 0 ˆ 0 u l ˆ 0 ˆ u l
36 Example: Ue the bilinear Tranformation method to: T= tan T =0.55
37 Example (ont.:
38 Example: Deign a 3 rd order Butterworth LPF to have a utoff frequeny of.53 rad/e uing bilinear tranformation. olution T=
39
40
41 Example: Uing bilinear tranformation, deign IIR filter that atifie thee peifiation: LPF with maximum ripple allowed in the paband =0.93, the maximum ripple allowed in the topband =0., paband edge frequeny= 0.5*pi, and topband edge frequeny=0.5*pi. The pre-warped analog frequenie are: olution p p 0.5 tan tan tan tan ( rad ( rad The required amplitude are: A A p p A A p p
42 Example: The required order i: p p A A N log log log log N The utoff frequeny i the frequeny at whih the amplitude i Let N =. p N p A A p p
43 Example:
44 Example:
45 Example:
46
47 Example:
48 Remember: -3 db Cut-off frequeny
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