Fourier Analysis in Systems Theory

In the most general sense, any process wherein a stimulus generates a corresponding response can be dubbed a system. For a temporal system with single input, f (t), and single output, g(t), the relation can be written as . . . g(t) = S{ f (t)} (3.1) . . . where S{·} is the system operator. This is illustrated in Figure 3.1. There exist numerous system types. We define them here in terms of continuous signals. The equivalents in discrete time are given as an exercise. For homogeneous systems, amplifying or attenuating the input likewise amplifying or attenuating the output. For any constant, a,. . . S{a f(t)} = aS{ f (t)} (3.2) If the response of the sum is the sum of the responses, the system is said to be additive. Specifically,. . . S{ f1(t) + f2(t)} = S{ f1(t)} + S{ f2(t)} (3.3) . . . Systems that are both homogeneous and additive are said to be linear. The criteria in (3.2) and (3.3) can be combined into a single necessary and sufficient condition for linearity.. . . S{a f1(t) + bf2(t)} = aS{ f1(t)} + bS{ f2(t)} (3.4) . . . where a and b are constants. All linear systems produce a zero output when the input is zero. . . . S{0} = 0. (3.5). . . To show this, we use (3.4) with a = −b and f1(t) = f2(t). Note that, because of (3.5), the system defined by . . . g(t) = b f(t) + c . . . where b and c¹ 0 are constants, is not linear. It is not homogeneous since . . . S{a f} = b f + c ≠aS{ f} = a (b f + c) .