Question

Find the Z-transform of a^n u(n);a>0.

(a) \(\frac{z}{z-a}\)

(b) \(\frac{z}{z+a}\)

(c) \(\frac{1}{1-az}\)

(d) \(\frac{1}{1+az}\)

This question was addressed to me at a job interview.

The question is from The Z-Transform in section Z-Transform and Digital Filtering of Signals and Systems

Answer 1

Right choice is (a) \(\frac{z}{z-a}\)

The explanation is: Given x(n) = a^n u(n)

We know that \(

u(n)=\begin{cases}

1 &\text{\(n≥0\)} \\

0 &\text{\(n<0\)} \\

\end{cases}\)

X(Z) = \(\sum\limits_{n=-\infty}^{\infty} x(n) z^{-n} = \sum\limits_{n=-\infty}^{\infty} a^n u(n) z^{-n}\)

= \(\sum\limits_{n=0}^{∞} a^n (1) z^{-n} = \sum\limits_{n=0}^{∞} (az^{-1})^n = (1-az^{-1})^{-1}\)

= \(\frac{1}{1-az^{-1}} = \frac{z}{z-a}\).