4.7 Derivation of the fourier transform  (Page 2/3)

Properties of the Fourier transform and some useful transform pairs are provided in the accompanying tables ( [link] and [link] ). Especiallyimportant among these properties is Parseval's Theorem , which states that power computed in either domain equals the power in the other.

Note that the mathematical relationships between the time domain and frequency domain versions of the same signal are termed transforms . We are transforming (in the nontechnical meaning of the word) a signal from onerepresentation to another. We express Fourier transform pairs as $\leftrightarrow (s(t), S(f))$ . A signal's time and frequency domain representations are uniquely related to each other. A signal thus "exists" inboth the time and frequency domains, with the Fourier transform bridging between the two. We can define an information carryingsignal in either the time or frequency domains; it behooves the wise engineer to use the simpler of the two.

A common misunderstanding is that while a signal exists in both the time and frequency domains, a single formula expressing asignal must contain only time or frequency: Both cannot be present simultaneously. This situation mirrorswhat happens with complex amplitudes in circuits: As we reveal how communications systems work and are designed, we will definesignals entirely in the frequency domain without explicitly finding their time domain variants. This idea is shown in another module where we define Fourier series coefficients according to letter to betransmitted. Thus, a signal, though most familiarly defined in the time-domain, really can be defined equally as well (andsometimes more easily) in the frequency domain. For example, impedances depend on frequency and the time variable cannotappear.

We will learn that finding a linear, time-invariant system's output in the timedomain can be most easily calculated by determining the input signal's spectrum, performing a simple calculation in thefrequency domain, and inverse transforming the result. Furthermore, understanding communications and informationprocessing systems requires a thorough understanding of signal structure and of how systems work in both the time and frequency domains.