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probset-two - EE533 Problem Set 2 1 Determine the DTFT of...

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EE533 Problem Set 2 1. Determine the DTFT of the “rectangular window” sequence w ( n ) = 1 , 0 n M, 0 , otherwise . Express your answer in the form W ( e ) = e - jωM/ 2 A ( ω ) where A ( · ) is a real, even function of its argument. 2. Determine the inverse DTFT of X ( e ) specified as X ( e ) = j (4 + 2 cos ω + 3 cos 2 ω ) sin( ω/ 2) e jω/ 2 . What is the period of sin( ω/ 2)? Is X ( e ) periodic with period 2 π ? 3. A periodic function F ( ω ) is defined in terms of a positive real number A for - π ω π : F ( ω ) = A + Aω/π, - π ω < 0 , - A + Aω/π, 0 ω π. Determine the inverse DTFT of F ( ω ). Using this result, derive the DTFT of the sequence: x ( n ) = 1 /n, n = 0 and x (0) = 0. Apply Parseval’s theorem to this transform pair to obtain the value of n =1 1 /n 2 . 4. Evaluate n = -∞ sin( πn/ 4) sin( πn/ 6) π 2 n 2 . The value of the summand at n = 0 is 1 / 24. 5. Consider the real, positive function G ( ω ) = 1 A - B cos ω where 0 < B < A . We wish to find an absolutely summable sequence x ( n ) = 1 α c n u ( n ), where α > 0, such that G ( ω ) = | X ( e ) | 2 . (a) By setting up a pair of equations, solve for α and c in terms of A and B . (b) Using Parseval’s theorem, obtain an expression for (1
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