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Determine the z-transform for each of the following sequences. Sketch the polezero plot

Chapter 10, Problem 10.21

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QUESTION:

Determine the z-transform for each of the following sequences. Sketch the polezero plot and indicate the region of convergence. Indicate whether or not the Fourier transform of the sequence exists. (a) S[n + 5] (b) S[n- 5] (c) ( -l)nu[n] (d) C4Y+ 1 u[n + 3] (e) (-~)nu[-n- 2] (0 Ci)nu[3- n] (g) 2nu[ -n] + (i)nu[n- 1] (h) (~)n- u[n- 2]

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QUESTION:

Determine the z-transform for each of the following sequences. Sketch the polezero plot and indicate the region of convergence. Indicate whether or not the Fourier transform of the sequence exists. (a) S[n + 5] (b) S[n- 5] (c) ( -l)nu[n] (d) C4Y+ 1 u[n + 3] (e) (-~)nu[-n- 2] (0 Ci)nu[3- n] (g) 2nu[ -n] + (i)nu[n- 1] (h) (~)n- u[n- 2]

ANSWER:

Step 1 of 8

(a)

Consider the following sequence:

                                                                 

Apply z-transform to the sequence.

                                                      

 is impulse sequence defined at  

Hence

                                                            

Hence the z-transform of  is,

                                                           

The sequence converges for all values of  . Hence region of convergence (ROC) is all z, except .

Since ROC includes a unit circle  , hence, has a Fourier transform.

The pole-zero plot is,

                                                         

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