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# hw21 - ENEE 241 02 HOMEWORK ASSIGNMENT 21 Due Thu 05/07...

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ENEE 241 02 * HOMEWORK ASSIGNMENT 21 Due Thu 05/07 Consider the FIR filter with input-output relationship y [ n ] = x [ n ] - 3 x [ n - 1] + 3 x [ n - 3] - x [ n - 4] , n Z (Note the zero coefficient.) (i) Determine the response y ( i ) [ · ] of the filter to each of the input signals given by the equations below (valid for all n Z ). x (1) [ n ] = 2 - n (2 pts.) x (2) [ n ] = ( - 2) n (2 pts.) x (3) [ n ] = 3 · 2 - n + ( - 2) n (2 pts.) x (4) [ n ] = cos( πn/ 4 - 0 . 4) (3 pts.) x (5) [ n ] = 3 - n · cos( πn/ 4) (4 pts.) (ii) (5 pts.) The filter above is connected in series (cascade) with the filter of the previous assignment, i.e., with input-output relationship y [ n ] = x [ n ] + x [ n - 2] - x [ n - 6] - x [ n - 8] , n Z Determine the system function H ( z ) of the two-filter cascade. (iii) (2 pts.) Write out the input-output relationship of the two-filter cascade. Solved Examples S 21.1 ( P 4.4 in textbook). Consider the FIR filter whose input x and output y are related by y [ n ] = x [ n ] - x [ n - 1] - x [ n - 2] + x [ n - 3] (i) Write out an expression for the system function H ( z ). (ii) Express | H ( e ) | 2 in terms of cosines only. Plot | H ( e ) | as a function of ω . (iii) Determine the output y [ n ] when the input sequence x is given by each of the following expres- sions (where n Z ): x [ n ] = 1 x [ n ] = ( - 1) n x [ n ] = e jπn/ 4 x [ n ] = cos(

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