Simulation of Dynamic Systems with MATLAB® and Simulink®, 3rd Edition

4.3Transfer Function

Before we introduce the transfer function, the concept of an impulse function is presented because of its relevance to the response of LTI systems.

4.3.1Impulse Function

An impulse function at t = t0, denoted δ(t − t0), is defined by its property of sifting the value of a function f(t) at t0 inside an integral, that is,

∫−∞∞δ(t−t0)f(t)dt=f(t0),−∞<t0<∞	(4.121)

The impulse function δ(t − t0) is equal to zero wherever t ≠ t0 and is not finite at t = t0. No such function exists in a physical sense; however, it can be used to approximate real signals x(t), which occur over a very short duration Δ and satisfy the condition

∫t0t0+Δx(t)dt=1	(4.122)

as illustrated in Figure 4.6.

Figure 4.6The unit impulse function ...