Fourier Transforms

CHAPTER 9 LAPLACE, Z, AND HILBERT TRANSFORMS

In this chapter we look at three other important transforms that are closely related to the Fourier transform. The venerable Laplace transform,

is widely used in the analysis and design of dynamic systems. In addition to its “operational” properties, we will explore some of its connections with complex analysis and the Fourier transform. The Z transform,

fulfills the same role for discrete-time systems that the Laplace transform does for continuous-time systems and has a similar relationship to the discrete-time Fourier transform. Finally, we consider some important properties of one-sided functions, such as the impulse responses of causal LTI systems. These relationships, which exist for all four Fourier transforms, are collectively known as Hilbert transforms.

9.1 THE LAPLACE TRANSFORM

We have earlier seen how to define and calculate the Fourier transform for functions which are absolutely integrable (L¹), square integrable (L²), or slowly growing (e.g., polynomials), and for generalized functions (e.g., δ(x)). The Laplace transform extends the capability of the Fourier transform to certain important functions of rapid growth which are not Fourier transformable.

9.1.1 Definition, Basic Properties

The Laplace transform is typically ...