Z transform of continuous unit step function is: 

Explanation:

Z-Transform of a Continuous Unit Step Function

Definition: The Z-transform is a powerful mathematical tool used in the analysis and design of discrete-time control systems. It transforms a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation. The Z-transform is particularly useful in solving linear, constant-coefficient difference equations.

Continuous Unit Step Function: The continuous unit step function, often denoted as u(t), is defined as:

\[ u(t) = \begin{cases} 0 & \text{for } t < 0 \\ 1 & \text{for } t \ge 0 \end{cases} \]

However, in the context of discrete-time signals, we consider the discrete unit step function, denoted as u[n], which is defined as:

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

Z-Transform of the Unit Step Function:

The Z-transform of the discrete unit step function u[n] is obtained by applying the definition of the Z-transform:

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

Since u[n] = 1 for all n ≥ 0, the Z-transform becomes:

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

This is a geometric series with the first term a = 1 and common ratio r = z^-1. The sum of an infinite geometric series is given by:

\[ \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \]

Applying this formula, we get:

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

To express it in a more standard form, we multiply the numerator and the denominator by z:

\[ U(z) = \frac{z}{z - 1} \]

Thus, the Z-transform of the discrete unit step function u[n] is:

\[ U(z) = \frac{z}{z - 1} \]

Correct Option Analysis:

The correct option is:

Option 4: \(\rm X(t)=\frac{z}{z-1}\)

This option correctly represents the Z-transform of the discrete unit step function. The transformation and the properties of the geometric series lead to the result \(\rm \frac{z}{z-1}\), which matches the correct Z-transform of the unit step function.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \(\rm X(t)=\frac{z}{1-z}\)

This option is incorrect because it does not correctly represent the Z-transform of the unit step function. The denominator should be z - 1, not 1 - z.

Option 2: \(\rm X(t)=\frac{1}{1-z}\)

This option is also incorrect. While it resembles the form of a geometric series, it lacks the factor of z in the numerator, which is necessary for the correct Z-transform expression.

Option 3: \(\rm X(t)=\frac{1}{z-1}\)

This option is incorrect as well. It does not match the correct form of the Z-transform of the unit step function. The numerator should include the factor of z.

Conclusion:

Understanding the Z-transform and its application to discrete-time signals is crucial for analyzing and designing discrete-time control systems. The Z-transform of the discrete unit step function u[n] is correctly given by \(\rm \frac{z}{z-1}\), which matches the correct option 4. Evaluating the other options helps reinforce the proper understanding and application of the Z-transform.