Practical Signals Theory with MATLAB Applications

The Laplace Transform

LEARNING OBJECTIVES

By the end of this chapter, the reader will be able to:

• Explain the relationship between the Laplace transform and the Fourier transform
• List the properties of the Laplace transform
• Compute the Laplace transform of common functions
• Identify the region of convergence
• Use the Laplace transform to analyze simple network circuits
• Derive a pole-zero diagram from a transfer function
• Interpret the significance of a pole-zero diagram
• Create a Bode plot by hand to estimate frequency and phase response curves
• Apply the Laplace transform to the design of filters with predictable properties

The Fourier transform has been studied as a way to represent a signal s(t) as a linear decomposition of orthogonal sinusoidal components. By transforming a time-domain signal s(t), calculations are often simpler, and the new perspective of a different domain can give a better understanding of signals behavior. In many applications, a different transform technique is required, one that goes beyond sinusoidal components and incorporates the ability to manipulate the derivative and integral response functions that are typically found in real systems. The Laplace transform is sometimes described as a general-purpose Fourier transform, although the two transforms usually serve in different applications.

While the Fourier transform relates the time and frequency domains, ...