082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

23.9 Frequency response of a linear system 751
The observant reader will note that the expressions forc n
appear invalid whenn = 0,
since the denominator is then zero. We can computec 0
in either of two ways. Using an
integral expression we see that
c 0
= 1 T
∫ T/4
−T/41dt = 1 2
Also,using a Taylor series expansion itispossible toshow
[ ]
1 e
−j2nπt/T T/4
lim
= 1
n→0T
−j2nπ/T
−T/4
2
giving a consistentresult.
EXERCISES23.8
1 Find the complex Fourier seriesrepresentation of
(a) f(t) =
{
1 0 <t<2
0 2<t<4
period 4
(b) f(t)=e t −1<t<1 period2
{
Asinωt 0<t <π/ω
(c) f(t) =
0 π/ω<t<2π/ω
period 2π/ω
2 If
∞∑
f(t)= c n e j2nπt/T
−∞
show thatthe coefficients,c n ,are given by
c n = 1 ∫ T
f (t)e −j2nπt/T dt
T 0
[Hint: multiply both sidesbye −j2mπt/T and integrate
over [0,T].]
Solutions
1 (a)
(b)
∞∑ j
(cosnπ −1)ejnπt/2
2nπ
−∞
(
)
1
∞∑ e −jnπ+1 −e −1+jnπ
e jnπt
2 1−jnπ
−∞
(c)
∞∑ A(1 +e −jnπ )e jnωt
2π(1 −n
−∞
2 )
23.9 FREQUENCYRESPONSEOFALINEARSYSTEM
Linear systems have the property that the response to several inputs being applied to
thesystemcanbeobtainedbyaddingtheeffectsoftheindividualinputs.Anotheruseful
property of linear systems is that if a sinusoidal input is applied to the system then the
output will also be a sinusoid of the same frequency but with modified amplitude and
phase.This isillustratedinFigure 23.26.
InSection9.7wesawthatsinusoidalsignalscanberepresentedbycomplexnumbers
and that an a.c. electrical circuit can be analysed using complex numbers. This is true