Question

The z-transform of $a^n u(n)$ is:

• A. $\frac{z}{z-a}$
• B. $\frac{1}{1-az}$
• C. $\frac{1}{z-a}$
• D. $\frac{1}{1+az}$

Hint:

The Z-transform of a right-sided causal signal always has an ROC of the form $|z| > |a|$, extending outward from the outermost pole.
Conversely, left-sided anti-causal signals have an ROC of the form $|z| < |a|$, extending inward from the innermost pole.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question asks for the standard Z-transform of a causal, discrete-time exponential sequence $x[n] = a^n u[n]$.

Step 2: Key Formula or Approach:
The bilateral Z-transform of a discrete-time signal $x[n]$ is defined by the power series:
\[ X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \]

Step 3: Detailed Explanation:

• Let the input signal be:
\[ x[n] = a^n u[n] \]

• Substitute $x[n]$ into the definition of the Z-transform:
\[ X(z) = \sum_{n=-\infty}^{\infty} a^n u[n] z^{-n} \]

• The unit step function $u[n]$ restricts the non-zero terms of the summation to the range $n \geq 0$:
\[ X(z) = \sum_{n=0}^{\infty} a^n z^{-n} = \sum_{n=0}^{\infty} (a z^{-1})^n \]

• This expression is an infinite geometric series with a common ratio of $r = a z^{-1}$.

• For the infinite series to converge to a finite value, the absolute value of the common ratio must be strictly less than 1:
\[ |a z^{-1}| |a| \]

• This inequality defines the Region of Convergence (ROC) of the Z-transform.

• Under this convergence condition, the sum of the infinite geometric series is:
\[ X(z) = \frac{1}{1 - a z^{-1}} \]

• Multiplying both the numerator and the denominator by $z$ to express the result in terms of positive powers of $z$:
\[ X(z) = \frac{z}{z - a} \]

Step 4: Final Answer
Therefore, the Z-transform of $a^n u(n)$ is $\frac{z}{z-a}$ with the Region of Convergence $|z| > |a|$, which matches option (A).