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Review_Quiz_Slns_Fall_08

# Review_Quiz_Slns_Fall_08 - Review Quiz Solution 1(a x[n is...

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Review Quiz Solution 1. (a) x [ n ] is a stable, causal, complex sequence and X ( z ) = n = -∞ x [ n ] z - n , with ROC { z : | z | > r 1 } , with r 1 < 1 . The sequence r [ n ] = k = -∞ x * [ k ] x [ n + k ] . Then R ( z ) = X n = -∞ r [ n ] z - n = X n = -∞ X k = -∞ x * [ k ] x [ n + k ] z - n = X k = -∞ x * [ k ] X n = -∞ x [ n + k ] z - n = X k = -∞ x * [ k ] X l = -∞ x [ l ] z - ( l - k ) = X k = -∞ x * [ k ] X l = -∞ x [ l ] z - l z k = X k = -∞ x * [ k ] z k X ( z ) = X k = -∞ x * [ k ]( z - 1 ) - k X ( z ) = ˆ X k = -∞ x [ k ]( z *- 1 ) - k ! * X ( z ) = X * ( z *- 1 ) X ( z ) = X * ( 1 z * ) X ( z ) Since the ROC associated with X * ( 1 z * ) is { z : | z | < 1 r 1 } , the Region of convergence is equal to ROC = { z : | z | < r 1 } ∩ { z : | z | < 1 r 1 } = { z : r 1 < | z | < 1 r 1 } . (b) r 1 < 1 1 r 1 > 1 . Hence the ROC includes the unit circle and the Fourier transform exists. 2. (a) Since the systems is stable, the region of convergence includes the unit circle. The system has poles at 1 / 2 and - 1 / 3 . So the ROC of the transfer function is | z | > 1 / 2. Since the ROC is exterior of a circle and includes infinity, the system is causal.

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