# hw4 - 1(b x 2 n = ± n 2-N ≤ n ≤ N otherwise x 2 n]is...

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

hw4 - 1(b x 2 n = ± n 2-N ≤ n ≤ N otherwise x 2 n]is...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online