This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ECE 421  Sum 2010 Detailed Solutions  Set 2: Filter Design Using ODEs 1 24. Let an ODE for an analog filter be given by d dt g t 10 g t = 20 f t . Using the Backward Difference method, the digital filter producing g [ m ℄ can be written as g [ m ℄= K 1 g [ m 1 ℄ K 2 f [ m ℄ Compute K 1 and K 2 if T s has the following values: (a) T s = 0.2 sec, (b) T s = 0.1 sec, (c) T s = 0.01 sec DETAILED SOLUTION: The continuous differential equation for this problem is given in the problem statement by d dt g t 10 g t = 20 f t a The backward differece approximation for the first derivative is given from the Notes by d dt g t g [ m ℄ g [ m 1 ℄ T s b Make the substitution b into equation a , substitute g [ m ℄ for g t into equation a , and substitute f [ m ℄ for f t into equation a . This provides the digital (or discretetime) equation g [ m ℄ g [ m 1 ℄ T s 10 g [ m ℄= 20 f [ m ℄ c Multiply both sides of c by T s to obtain g [ m ℄ g [ m 1 ℄ 10 T s g [ m ℄= 20 T s f [ m ℄ d Solve equation d for g [ m ℄ , producing g [ m ℄= 1 1 10 T s g [ m 1 ℄ 20 T s 1 10 T s f [ m ℄ e Comparing e with the form for g [ m ℄ given in the problem statement shows that K 1 = 1 1 10 T s ; K 2 = 20 T s 1 10 T s f ; g Now we can work parts (a), (b), and (c) by substituting the specified values of T s into equations f and g . ECE 421  Sum 2010 Detailed Solutions  Set 2: Filter Design Using ODEs 2 24 (cont.) (a) : T s = 0.2 sec The constants K 1 and K 2 for this part are given by substituting T s = 0.2 into equations f and g . This gives K 1 = 1 1 10 : 2 ; K 2 = 20 : 2 1 10 : 2 h ; i Simplifiying h and i then gives K 1 = : 333 ; K 2 = 1 : 333 j ; k (b) : T s = 0.1 sec The constants K 1 and K 2 for this part are given by substituting T s = 0.1 into equations f and g . This gives K 1 = 1 1 10 : 1 ; K 2 = 20 : 1 1 10 : 1 l ; m Simplifiying l and m then gives K 1 = : 500 ; K 2 = 1 : 000 n ; o (c) : T s = 0.01 sec The constants K 1 and K 2 for this part are given by substituting...
View
Full
Document
This note was uploaded on 09/07/2010 for the course ECE 421 taught by Professor Hallen during the Summer '08 term at N.C. State.
 Summer '08
 HALLEN
 Signal Processing

Click to edit the document details