ASAP PLEASE Derive the formula for the Zee-transform of a periodic discrete-timesequence defined by f[k] = f[k+ N], where N is the discrete-timeperiod.B)A)Using the formula in A) above or otherwise, find the Zee transformof the periodic discrete-time sequence defined byf[k] = f[k + N] = 1, for kf[k] =0, 1, 2, 6, 7, 8, ...and3, 4, 5, 9, 10, 11, ...(N = 6)f[k + N] = -1, for k =.....

According to question: A) For Periodic…

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Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

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ASAP PLEASE

Derive the formula for the Zee-transform of a periodic discrete-time
sequence defined by f[k] = f[k+ N], where N is the discrete-time
period.
B)
A)
Using the formula in A) above or otherwise, find the Zee transform
of the periodic discrete-time sequence defined by
f[k] = f[k + N] = 1, for k
f[k] =
0, 1, 2, 6, 7, 8, ...and
3, 4, 5, 9, 10, 11, ...(N = 6)
f[k + N] = -1, for k =
.....

Expert Solution

Step 1

According to question:

$For Periodic function f [k] = f [k + N] ∴ f [k] = f_{1} [k] + f_{1} [k + N] + f_{2} [k + 2 N] . . . . . . . . Applying Z-transform on both sides ∴ F [Z] = F_{1} [Z] [1 + Z^{N} + Z^{2 N} + . . . . .] \Rightarrow F [Z] = F_{1} [Z] [1 + (Z^{N}) + {(Z^{N})}^{2} + . . . . .] \Rightarrow F [Z] = F_{1} [Z] [\frac{1}{1 - Z^{N}}]; |Z^{N}| < 1$

Step 2

$f_{1} [k] = [\underset{↑}{1} 1 1 - 1 - 1 - 1] N = 6 Applying Z transform ∴ F_{1} [Z] = [1 + Z^{1} + Z^{2} - Z^{3} - Z^{4} - Z^{5}] \Rightarrow F [Z] = [1 + Z^{1} + Z^{2} - Z^{3} - Z^{4} - Z^{5}] [\frac{1}{1 - Z^{N}}] ∴ Z transform is \Rightarrow F [Z] = [1 + Z^{1} + Z^{2} - Z^{3} - Z^{4} - Z^{5}] [\frac{1}{1 - Z^{6}}] ROC is entire Z plane except 0 and ∞$

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