GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 4(b) Solution

Explain the Differentiation property of Z-Transform.

GTU Sem 4 | Signals and Systems | Summer 2021 | Question 4(b) OR

Table of Contents

• Introduction
• Statement of the Question
• Theory Part Related to the Question
• Solution in Detail
• Key Takeaways
• Final Words
• Related Posts

Introduction

In this post, we will discuss the Z-transform's Differentiation property, a key concept in signal processing and system analysis. The Z-transform is widely used to analyze discrete-time signals and systems, and understanding this property is crucial for solving problems involving signal transformation.

Statement of the Question

The Differentiation property of the Z-Transform states the following:

If X(z) is the Z-Transform of the discrete-time signal x[n], then the Z-Transform of n · x[n] is given by:

    Z{n · x[n]} = -z * d/dz (X(z))

Where:

• X(z) is the Z-Transform of x[n],
• d/dz X(z) is the derivative of X(z) with respect to z,
• n is the time index of the discrete-time signal.

This property helps analyze the effect of multiplying a signal by n, which can appear in various system characteristics.

Theory Part Related to the Question

The Z-transform's Differentiation property provides a method for finding the Z-transform of a signal multiplied by its time index, n. This property simplifies the process of calculating the Z-transform of such signals, which is often required in control systems, digital signal processing, and communications.

This property works under the assumption that the Z-Transform exists for the given signal, meaning the Region of Convergence (ROC) is valid for the manipulation.

Derivation

To understand this property in detail, let's consider the Z-Transform of the signal x[n]:

    X(z) = Σ (from n=-∞ to ∞) x[n] * z^(-n)

Now, if we multiply the signal x[n] by n, we obtain:

    Z{n · x[n]} = Σ (from n=-∞ to ∞) n · x[n] · z^(-n)

Using the derivative with respect to z, we can show that:

    Z{n · x[n]} = -z * d/dz (X(z))

This result can be interpreted as the Z-Transform of the signal n · x[n], which is simply the negative of the derivative of the original Z-Transform, scaled by z.

Solution in Detail

In this section, we will walk through an example to demonstrate the use of the Differentiation property.

Example:

Let’s consider a simple signal x[n] = a^n, where a is a constant.

The Z-Transform of x[n] is:

    X(z) = 1 / (1 - a * z^(-1)) for |z| > |a|

Now, using the Differentiation property, the Z-Transform of n · a^n can be found by differentiating X(z):

    Z{n · a^n} = -z * d/dz (1 / (1 - a * z^(-1)))

By calculating the derivative and simplifying, we can find the Z-Transform of n · a^n, which would be useful in various signal processing applications.

Key Takeaways

• The Z-transform's Differentiation property is essential for finding the Z-transform of signals multiplied by their time index, n.
• It allows us to compute the Z-Transform of more complex signals efficiently.
• This property is widely used in discrete-time signal processing and system analysis.

Final Words

The Differentiation property is a powerful tool in signal analysis. By understanding and applying this property, we can simplify the calculation of Z-Transforms for signals that involve the time index. This is particularly useful when working with discrete-time systems in engineering and communications.