4.1: Introduction to the Frequency Domain

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Learning Objectives

This module serves as an introduction to working in the frequency domain and thinking of signals in terms of their spectral components.
The Fourier transform can be used to represent any signal in terms of frequency instead of time and facilitates the computation of the transfer function of a system.

In developing ways of analyzing linear circuits, we invented the impedance method because it made solving circuits easier. Along the way, we developed the notion of a circuit's frequency response or transfer function. This notion, which also applies to all linear, time-invariant systems, describes how the circuit responds to a sinusoidal input when we express it in terms of a complex exponential. We also learned the Superposition Principle for linear systems: The system's output to an input consisting of a sum of two signals is the sum of the system's outputs to each individual component.

The study of the frequency domain combines these two notions--a system's sinusoidal response is easy to find and a linear system's output to a sum of inputs is the sum of the individual outputs--to develop the crucial idea of a signal's spectrum. We begin by finding that those signals that can be represented as a sum of sinusoids is very large. In fact, all signals can be expressed as a superposition of sinusoids.

As this story unfolds, we'll see that information systems rely heavily on spectral ideas. For example, radio, television, and cellular telephones transmit over different portions of the spectrum. In fact, spectrum is so important that communications systems are regulated as to which portions of the spectrum they can use by the Federal Communications Commission in the United States and by International Treaty for the world (see Frequency Allocations). Calculating the spectrum is easy: The Fourier transform defines how we can find a signal's spectrum.