Read the following statements and select the correct option pertaining to Z-transform.

Statement 1: Inverse Z-transforms can be obtained by partial fraction expansion methods.

Statement 2: Z-transform can be used for the solution of linear difference equations.  

Explanation:

Z-Transform

Definition: The Z-transform is a mathematical tool used in the field of signal processing and control systems to analyze discrete-time signals and systems. It transforms a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation.

Working Principle: The Z-transform of a discrete-time signal \( x[n] \) is defined as:

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

where \( z \) is a complex variable. This transformation allows the analysis of the signal in the z-domain, providing insights into its frequency components and system behavior.

Inverse Z-Transform: The inverse Z-transform is used to convert a signal back from the z-domain to the time domain. One common method to find the inverse Z-transform is the partial fraction expansion method. This involves expressing the Z-transform as a sum of simpler fractions, each of which can be inverted individually.

Advantages:

• Facilitates the analysis of discrete-time systems and signals in the frequency domain.
• Helps in the design and analysis of digital filters and control systems.
• Provides a convenient way to solve linear difference equations.

Disadvantages:

• Complexity in mathematical manipulation, especially for large systems.
• Requires a good understanding of complex variable theory.

Applications: The Z-transform is widely used in digital signal processing, control system design, and telecommunications to analyze and design discrete-time systems and filters.

Correct Option Analysis:

The correct option is:

Option 1: Both Statement 1 and Statement 2 are true.

This option correctly identifies the validity of both statements. Let's analyze the given statements in detail:

Statement 1: Inverse Z-transforms can be obtained by partial fraction expansion methods.

This statement is true. One of the common methods to find the inverse Z-transform is by using partial fraction expansion. By expressing the Z-transform as a sum of simpler fractions, the inverse can be determined for each fraction, which are then summed to obtain the original discrete-time signal.

Statement 2: Z-transform can be used for the solution of linear difference equations.

This statement is also true. The Z-transform is a powerful tool for solving linear difference equations. By transforming the difference equation into the z-domain, it becomes an algebraic equation that is easier to solve. The solution can then be transformed back to the time domain using the inverse Z-transform.

Additional Information

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

Option 2: Statement 1 is true and Statement 2 is false.

This option is incorrect because both statements are true. The inverse Z-transform can indeed be obtained by partial fraction expansion methods, and the Z-transform is also used for solving linear difference equations.

Option 3: Both Statement 1 and Statement 2 are false.

This option is incorrect because both statements are true. The inverse Z-transform can be obtained by partial fraction expansion, and the Z-transform is used for solving linear difference equations.

Option 4: Statement 2 is true and Statement 1 is false.

This option is incorrect because both statements are true. The Z-transform is used for solving linear difference equations, and the inverse Z-transform can be obtained by partial fraction expansion methods.

Conclusion:

Understanding the Z-transform and its applications is crucial in the fields of signal processing and control systems. The Z-transform provides a powerful method for analyzing discrete-time signals and systems, and its inverse can be determined using partial fraction expansion. It is also a valuable tool for solving linear difference equations, making it an essential concept for engineers and researchers in related fields.