Wodefiltering

Wodefiltering - ECE 421 - Sum 2010 Detailed Solutions - Set...

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Unformatted text preview: ECE 421 - Sum 2010 Detailed Solutions - Set 2: Filter Design Using ODEs 1 2-4. 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 discrete-time) 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 2-4 (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...
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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.

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Wodefiltering - ECE 421 - Sum 2010 Detailed Solutions - Set...

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