Midterm Solution 2005 Spring

Midterm Solution 2005 Spring - Problem 1(a): n = - DTFT: X...

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Problem 1(a): DTFT: −∞ = = n n j j e n x e X ω ] [ ) ( IDTFT: = π d e e X n x n j j ) ( 2 1 ] [ (Note: The limits of the integral can be changed to 0 and 2pi.) Problem 1(b): The DTFT is uniformly convergent if x[n] is absolutely summable, i.e., −∞ = < n n x | ] [ | . On the other hand, it is mean-square convergent if x[n] is not absolutely summable but square summable (has finite energy): −∞ = < n n x 2 | ] [ |. Problem 1(c): Problem 1(d): Problem 1(e): i. False. A left-sided sequence x[n] is anti-causal if and only if x[n]=0 for n 0. ii. True. An absolutely summable sequence is also mean-sequare summable and, therefore, has finite energy. iii. True. For example, an accumulator h[n]=u[n] is IIR with bounded impulse response for all n, but is not stable.
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Midterm Solution 2005 Spring - Problem 1(a): n = - DTFT: X...

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