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ENEE241hw12 - ENEE 241 HOMEWORK ASSIGNMENT 12 Due Tue 05/03...

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ENEE 241 HOMEWORK ASSIGNMENT 12 Due Tue 05/03 Problem 12A Consider the FIR filter given by the following input-output relationship (note the missing coe - cient): y [ n ] = x [ n ] + 3 x [ n 1] 3 x [ n 3] x [ n 4] , n Z (i) (3 pts.) Show that the input sequences defined for all n by x (1) [ n ] = 1 and x (2) [ n ] = ( 1) n both result in output sequences which are identically equal to zero. (ii) (3 pts.) Write MATLAB code which computes and plots the amplitude and phase response of the filter at 1024 equally spaced frequencies in [0 , 2 π ). Submit the plots, properly labeled. (iii) (4 pts.) Express the filter’s frequency response in the form H ( e j ω ) = je j ( ω M/ 2) F ( ω ) where F ( ω ) is a real-valued sum of sines. (iv) (5 pts.) The amplitude response plotted in (ii) above has six zeros at frequencies other than ω = 0 and ω = π . Determine the values of these frequencies analytically, using the result of part (iii) and the identity sin 2 θ = 2 sin θ cos θ . (v) (5 pts.) Determine analytically the exact locations of the local maxima of the amplitude response | H ( e j ω ) | in the frequency range [0 , π / 2]. Do so by di ff erentiating F ( ω
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