Understanding Zeros and Poles of an LTI System Through its Transfer Function

Linear Time-Invariant (LTI) systems play a crucial role in various fields, including signal processing, control systems, and communications. One of the fundamental aspects of these systems is their transfer function, which describes their behavior in the frequency domain. This article aims to explain how to find the zeros and poles of an LTI system from its transfer function, providing a detailed, step-by-step guide.

Introduction to Transfer Functions

The transfer function, denoted as (H(s)), is a mathematical representation of an LTI system. It is the ratio of the Laplace transform of the output to the Laplace transform of the input, given by:

H(s) frac{N(s)}{D(s)}

where (N(s)) is the numerator polynomial and (D(s)) is the denominator polynomial.

Steps to Find Zeros and Poles

1. Identify the Transfer Function

To find the zeros and poles, start by writing down the given transfer function (H(s)).

2. Find Zeros

Zeros are the values of (s) that make the numerator polynomial equal to zero.

Definition: Zeros are the roots of the numerator polynomial (N(s)). Procedure: Set the numerator (N(s)) to zero and solve for (s). Example: If the numerator (N(s) s^2 - 3s - 2), then set (s^2 - 3s - 2 0). Solving this quadratic equation using the quadratic formula or factoring.

3. Find Poles

Poles are the values of (s) that make the denominator polynomial equal to zero.

Definition: Poles are the roots of the denominator polynomial (D(s)). Procedure: Set the denominator (D(s)) to zero and solve for (s). Example: If the denominator (D(s) s^2 - 4s - 4), then set (s^2 - 4s - 4 0). Solving this quadratic equation using the quadratic formula or factoring.

Example

Consider the transfer function:

H(s) frac{s^2 - 3s - 2}{s^2 - 4s - 4}

Find Zeros: Solve (s^2 - 3s - 2 0). Factoring gives (s - 1)(s 2) 0). Zeros are (s -1) and (s 2). Find Poles: Solve (s^2 - 4s - 4 0). Factoring gives (s - 2)(s 2) 0). Pole at (s -2) with multiplicity 2.

Summary

Zeros: (s -1, 2) Poles: (s -2) with multiplicity 2

Conclusion

By following these steps, one can systematically determine the zeros and poles of any LTI system from its transfer function. Consulting these zeros and poles is essential for analyzing the system's stability and frequency response.

Addition: Zeros and Poles in the Z-Transform

When dealing with the Z-transform, the concept of zeros and poles becomes analogous to the Laplace domain but within the finite limit of the Z-domain.

Poles: Zeros of the denominator polynomial in the Z-domain, represented by the notation (times). Zeros: Zeros of the numerator polynomial in the Z-domain, represented by the notation (0).

In the region of convergence (ROC) of the Z-transform, zeros can be present, but poles are generally not. This region defines where the Z-transform converges, ensuring the system behaves predictably in the frequency domain.