Understanding Engineering Mathematics

Answers

1. (i)

10

3

e−^3 t+

2

3

(ii) et− 1 −t

2. 10e−t− 20 e−^2 t+ 10 e−^3 t

17.7 Linear systems and the principle of superposition

Before we move on to a different type of transform, we will look briefly at the sort of
systems for which it is used. Consider a system whose response to a time dependent
inputx(t)is an outputy(t). The system is calledlinearif the output of the sum of two
inputs is the sum of the separate outputs to the two inputs. That is, ifx 1 (t)→y 1 (t)and
y 2 (t)→y 2 (t)impliesx 1 (t)+x 2 (t)→y 1 (t)+y 2 (t).
Many, but not all, systems behave in this way – any that do not are callednon-linear
and are very difficult to analyse. This additivity of outputs of linear systems is called the
principle of superposition.
If a complicated signal can be split up into a linear combination of simpler signals then
the effect of a linear processing system on the total signal can be analysed by adding up
the separate effects on the component input signals. This is the philosophy behindFourier
analysis.AFourier seriessplits a periodic input (such as a square wave) into a sum of
sinusoidal components of different amplitude and frequency. The effect of a linear system
on each separate sinusoid is easy to determine in general and the separate outputs can
be added up to synthesize the total output corresponding to the original periodic interval.
This is illustrated in Figure 17.5.

System

Input

Input F.S.

Linear

system

Output

Output F.S.

Figure 17.5Fourier decomposition of input and output.

We will now summarise some terminology for a general sinusoidal function:

f(t)=Asin(ωt+α)

Ais called theamplitude,ωtheangular frequency(radians per second) andαis the
phaserelative toAsinωt. The period of such a sinusoid isT= 2 π/ω. The inverse ofT,
ω/ 2 π,isthefrequencyin cycles per second, or Hertz (Hz). See Figure 17.6.
Linear operations such as differentiation, integration, addition, etc. change the amplitude
and phase of such a sinusoid,but not its frequency. For example

d

dt

[Asin(ωt+α)]=Aωcos(ωt+α)

=Aωsin(ωt+α+π/ 2 )

which has the same frequency as the original, but a modified amplitude and phase.