Signals and Systems - Electrical Engineering

242 C H A P T E R 4: Frequency Analysis: The Fourier Series

FIGURE 4.1

RC circuit and corresponding phasor

circuit.

vs(t)

1 Ω

1F vc(t)

+

− +

−

1

−j

+

−

Vc

+

−

Vs

corresponding tovc(t):

Vc

Vs

=

−j

1 −j

=

−j( 1 +j)

2

=

√

2

2

∠−π/ 4

SinceVs= 4 ∠π/4, then

Vc= 2

√

2 ∠ 0

so that in the steady state,

vc(t)= 2

√

2 cos(t)

n Eigenfunction approach.Considering the output is the voltage across the capacitor and the input

is the voltage source, the transfer function is obtained using voltage division as

H(s)=

Vc(s)

Vs(s)

=

1 /s

1 + 1 /s

=

1

s+ 1

so that the system frequency response at the input frequency 0 =1 is

H(j 1 )=

√

2

2

∠−π/ 4

According to the eigenfunction property the steady-state response of the capacitor is

vc(t)= 4 |H(j 1 )|cos(t+π/ 4 +∠H(j 1 ))

= 2

√

2 cos(t)

which coincides with the solution found using phasors. n

nExample 4.2

An ideal communication system provides as output the input signal with only a possible delay in

the transmission. Such an ideal system does not cause any distortion to the input signal beyond