hw4 - 1 (b) x 2 [ n ] = ± n 2 ,-N ≤ n ≤ N , otherwise...

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EE 3318 Homework 4 Solution 3.18 Determine the DTFT of each of the following sequences: (b) x b [ n ] = α n ( μ [ n ] - μ [ n - 8]) , | α | < 1 The DTFT of this sequence is: X ( e ) = X n = -∞ x b [ n ] e - jωn = X n = -∞ α n ( μ [ n ] - μ [ n - 8]) e - jωn = 8 X n =0 α n e - jωn = 8 X n =0 ( αe - ) n Since it is the summation of a Geometric Sequence X ( e ) = 1 - ( αe - ) 9 1 - ( αe - ) (c) x c [ n ] = ( n + 1) α n μ [ n ] , | α | < 1 The DTFT of this sequence is: X ( e ) = X n = -∞ x c [ n ] e - jωn = X n = -∞ ( n + 1) α n μ [ n ] e - jωn From Table3.3 we have X ( e ) = 1 (1 - αe - ) 2 3.31 Without computing the DTFT, determine which of the following swquences have real- valued DTFTs and which have imaginary-valued DTFTs: (a) x 1 [ n ] = ± n, - N n N 0 , otherwise x 1 [ n ]is an odd sequence or antisymmetric sequence, since x 1 [ - n ] = - n = - x 1 [ n ]. Hence, x 1 [ n ] cos ( jωn ) is odd and x 1 [ n ] sin ( jωn ) is even. Therefore, the DTFT of this sequence has only imaginary value.
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Unformatted text preview: 1 (b) x 2 [ n ] = ± n 2 ,-N ≤ n ≤ N , otherwise x 2 [ n ]is an even sequence or symmetric sequence, since x 2 [-n ] = (-n ) 2 = ( n ) 2 = x 2 [ n ]. Hence, x 2 [ n ] cos ( jωn ) is even and x 2 [ n ] sin ( jωn ) is odd. Therefore, the DTFT of this sequence has only real value. (c) x 3 [ n ] = sinω c n πn Since x 3 [-n ] =-sinω c n-πn = sinω c n πn = x 3 [ n ], it is an even or symmetric sequence. Hence, x 3 [ n ] cos ( jωn ) is even and x 3 [ n ] sin ( jωn ) is odd. Therefore, the DTFT of this sequence has only real value. 2...
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hw4 - 1 (b) x 2 [ n ] = ± n 2 ,-N ≤ n ≤ N , otherwise...

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