Question

What is the z-transform of the signal x(n) = -α^nu(-n-1)?

(a) \(\frac{1}{1-\alpha z^{-1}}\);ROC |z|

(b) \(-\frac{1}{1+\alpha z^{-1}}\);ROC |z|

(c) \(-\frac{1}{1-\alpha z^{-1}}\);ROC |z|>|α|

(d) \(-\frac{1}{1-\alpha z^{-1}}\);ROC |z|

The question was posed to me in homework.

The question is from Z Transform topic in division Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

The correct choice is (d) \(-\frac{1}{1-\alpha z^{-1}}\);ROC |z|

Explanation: Given x(n)= -α^nu(-n-1)=0 for n≥0

=-α^n for n≤-1

From the definition of z-transform, we have

X(z)=\(\sum_{n=-\infty}^{-1}(-α^n)z^{-n} = -\sum_{n=-\infty}^{-1}(\alpha^{-1} z)^{-n}=-\frac{1}{1-\alpha z^{-1}}\)  and |α^-1z|<1 => |z|<|α|.