The z-transform is a powerful mathematical tool widely used in digital signal processing and discrete-time control systems. It converts a discrete-time signal, which is a sequence of numbers, into a complex frequency domain representation. This transformation helps in analyzing and designing discrete-time systems, similar to how the Laplace transform is used for continuous-time systems.

Z-Transform Definition

For a discrete-time sequence \(x(n)\), its z-transform, denoted as \(X(z)\), is defined by the following unilateral sum:

$$X(z) = \sum_{n=-\infty}^{\infty} x(n)z^{-n}$$

Here, \(z\) is a complex variable. The set of all values of \(z\) for which the sum converges is known as the Region of Convergence (ROC).

Exponential Sequence Analysis

The given real exponential sequence \(x(n)\) is defined as:

• \(x(n) = a^n\) for \(n \geq 0\)
• \(x(n) = 0\) for \(n < 0\)

This sequence is a right-sided sequence because it is non-zero only for non-negative values of \(n\). We are also given that \(a > 0\).

Applying the Z-Transform to the Sequence

To find the z-transform of this sequence, we substitute the definition of \(x(n)\) into the z-transform formula. Since \(x(n)\) is zero for \(n < 0\), the summation limits change:

$$X(z) = \sum_{n=0}^{\infty} x(n)z^{-n}$$

Substitute \(x(n) = a^n\) for \(n \geq 0\):

$$X(z) = \sum_{n=0}^{\infty} (a^n)z^{-n}$$

We can rewrite the term \((a^n)z^{-n}\) as \((az^{-1})^n\):

$$X(z) = \sum_{n=0}^{\infty} (az^{-1})^n$$

Geometric Series Convergence

The expression above is a standard form of an infinite geometric series. A geometric series of the form \(\sum_{k=0}^{\infty} r^k\) converges to \(\dfrac{1}{1-r}\) if and only if the common ratio \(|r| < 1\).

In our case, the common ratio \(r = az^{-1}\).

Therefore, the sum converges to:

$$X(z) = \dfrac{1}{1 - az^{-1}}$$

The condition for convergence is:

$$|az^{-1}| < 1$$

This inequality can be broken down as:

$$|a| |z^{-1}| < 1$$

Since \(a > 0\), \(|a| = a\):

$$a |z^{-1}| < 1$$

Recall that \(|z^{-1}| = \left|\dfrac{1}{z}\right| = \dfrac{1}{|z|}\):

$$a \dfrac{1}{|z|} < 1$$

Multiplying both sides by \(|z|\) (which is positive):

$$a < |z|$$

Or, equivalently:

$$|z| > a$$

Resulting Z-Transform and ROC

So, the z-transform of the given real exponential sequence \(x(n) = a^n u(n)\) is:

$$X(z) = \dfrac{1}{1 - az^{-1}}$$

with the Region of Convergence (ROC) being \(|z| > a\). This ROC indicates that the convergence region is the exterior of a circle centered at the origin with radius \(a\) in the z-plane. For right-sided sequences, the ROC is always the exterior of a circle.

Comparing this result with the given options, the correct one matches this derived form.