The z-transform is a mathematical tool used to convert discrete-time signals from the time domain to the complex frequency domain. It is particularly useful in analyzing and designing digital filters and discrete-time control systems. We will now derive the z-transform of the unit step function.

Unit Step Function Definition

The unit step function, denoted as \(u[n]\) or \(u(t)\) in discrete time, is a fundamental discrete-time signal. It is defined as follows:

$$ u[n] = \begin{cases} 1 & \text{for } n \ge 0 \\ 0 & \text{for } n < 0 \end{cases} $$

This means that the function has a value of 1 for all non-negative integer values of \(n\) and 0 for all negative integer values of \(n\). This is a crucial signal for understanding system responses and initial conditions in discrete systems.

Z-Transform Definition

The unilateral (one-sided) z-transform of a discrete-time signal \(x[n]\) is defined by the following summation formula:

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

Here, \(z\) is a complex variable. The z-transform converts a sequence of numbers into a function of \(z\), which can simplify the analysis of linear time-invariant (LTI) discrete systems.

Deriving Z-Transform of Unit Step Function

To find the z-transform of the unit step function \(u[n]\), we substitute its definition into the z-transform formula:

We start with the general z-transform formula:

$$ U(z) = \sum_{n=0}^{\infty} u[n]z^{-n} $$

Since \(u[n] = 1\) for \(n \ge 0\), we can substitute this value into the summation:

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

This summation can be rewritten as:

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

This is a standard infinite geometric series of the form \(\sum_{n=0}^{\infty} r^n\), where \(r = z^{-1}\). The sum of an infinite geometric series is given by the formula:

$$ \sum_{n=0}^{\infty} r^n = \frac{1}{1 - r} \quad \text{for } |r| < 1 $$

Applying this formula to our expression, with \(r = z^{-1}\):

$$ U(z) = \frac{1}{1 - z^{-1}} $$

To express this in a more common form without negative powers of \(z\), we multiply the numerator and the denominator by \(z\):

$$ U(z) = \frac{1 \cdot z}{(1 - z^{-1}) \cdot z} $$

$$ U(z) = \frac{z}{z - z^{-1} \cdot z} $$

$$ U(z) = \frac{z}{z - 1} $$

This result is valid for \(|z^{-1}| < 1\), which implies \(1/|z| < 1\), or \(|z| > 1\). This region of convergence (ROC) is the exterior of a circle with radius 1 centered at the origin in the z-plane.

Comparing with Options

Let's compare our derived z-transform of the unit step function with the given options:

• Option 1: \(u(t) = 2z / (z - 1)\)
• Option 2: \(u(t) = 1 / (z - 1)\)
• Option 3: \(u(t) = z^2 / (z - 1)\)
• Option 4: \(u(t) = z / (z - 1)\)

Our derived z-transform, \(U(z) = \frac{z}{z - 1}\), directly matches Option 4. This demonstrates the standard result for the z-transform of a unit step function, which is frequently used in discrete-time signal processing and control systems.