Signals and Systems
Lecture 7
Frequency domain Models:
To think about signals and systems in terms of frequency rather than time
Use of the complex exponential function
as the basic signal component.
t
j
e
)
e

e
(
2j
1
sin
)
e
e
(
2
1
cos
jsin

cos
e
jsin
cos
e
j

j
j

j
j

j
Signal models:
If we put θ = ωt
)
e

e
(
2j
1
t
sin
)
e
e
(
2
1
t
cos
t
j

t
j
t
j

t
j
Ex.
t
t
t
t
A
t
o
o
o
o
j

j
j

j
o
e
2
e
2
A
)
e
e
(
2
A
Acos
x(t)
Amplitude
2
A
2
A
o
o

0
We represent the signal in terms of amplitude, phase and
frequency of its complex exponential components.
Ex.
A voltage signal is modelled as the sinusoid.
0.5)
5cos(3t
V(t)
Express the signal in terms of exponential frequency component
and sketch the frequency domain representation of the signal.
Sol.
)
e
e
(
2
5
v(t)
0.5)
j(3t

0.5)
j(3t
j0.5

j3t

0.50
j(3t

j0.5
j3t
0.5)
j(3t
e
e
e
e
e
e
j3t

j0.5

j3t
j0.5
e
]
e
2
5
[
e
]
e
2
5
[
v(t)
2
5
3
3
ω
Amplitude
0.5
0.5
phase
ω
System models:
The response of a linear system to a sinusoidal input
component
is itself a sinusoid
of the same frequency as the input but,
changed amplitude and phase.
This means that we can develop a model for the
behavior
of a system
in terms of
the effect the system has on the
amplitude
and
phase
of
sinusoid.
We know that.
d
)

x(t
)
(
h
y(t)

Where h(t) is the unit impulse response of the system
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 Spring '18
 Exponential Function, Complex number, j