# CHAPTER 16FOURIER AND LAPLACE TRANSFORMS

Using Fourier series, periodic signals can be expressed as infinite sums of sines and cosines. For a general nonperiodic signal, we use the Fourier transforms. It turns out that most of the signals encountered in applications can be broken down into linear combinations of sines and cosines. This process is called spectral analysis. In this chapter, Fourier transforms are introduced along with the basics of signal analysis. Apart from signal analysis, Fourier transforms are also very useful in quantum mechanics and in the study of scattering phenomenon. Another widely used integral transform is the Laplace transform, which will be introduced with its basic properties and applications to differential equations and to transfer functions.

## 16.1 TYPES OF SIGNALS

Signals are essentially modulated forms of energy. They exist in many types and shapes. Gravitational waves emitted by neutron star mergers and the electromagnetic pulses emitted from a neutron star are natural signals. In quantum mechanics, particles can be viewed as signals in terms of probability waves. In technology, electromagnetic waves and sound waves are two predominant sources of signals. They are basically used to transmit and receive information. Signals exist in two basic forms as **analog** and **digital**. Analog signals are given as continuous functions of time. Digital signals are obtained from analog signals either by **sampling** or **binary coding**.

In sampling (Figure 16.1

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