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EEL 3135: DiscreteTime Signals and Systems
Dr. Fred J. Taylor, Professor
Lesson Title: Sinusoids
Lesson Number: 03
Introduction:
Sinusoids are a fundamentally important class of signals.
They are an essential element of
modern electronic communication and control system operation. Collections of sinusoids can
also be combined to synthesize complex signals.
Sinusoidal waves can be produced by a
number of mechanisms.
Both electronic and mechanical constant frequency sinusoidal signals
of frequency
ϖ
0
can be produced by systems that can be modeled as a 2
nd
order ordinary
differential equation (ODE) of the form:
(
29
(
29
t
x
dt
t
x
d
0
2
2
ϖ
=
1.
In general, a sinusoidal signal can be mathematically modeled as:
x(t)=Acos(
φ
(t))
2.
which logically begs the question, what is meant by frequency?
Frequency is formally defined
to be:
(
29
(
29
dt
t
d
t
φ
=
3.
A familiar instance is the constant frequency case:
φ
(t)=
ϖ
0
t+
θ
0
; d
2
φ
(t)/dt
2
=
ϖ
0
4.
Another timedependent phase defined signal is the frequency modulation (FM) and phase
modulation (PM).
For example, and FM signal is given by
φ
(t)=
ϖ
0
t+
∫
x(t)dt and d
2
φ
(t)/dt
2
=
ϖ
0
+
x(t).
Interpreting Equation 2 as a constant frequency process yields the familiar equation:
x(t)=Acos(
ϖ
0
t+
θ
0
)
5.
where A is called the amplitude,
ϖ
0
is the sinusoid’s frequency in radians/second (r/s), and
θ
0
is
the phase shift in radians or degrees. The period of the constant frequency signal, T
0
, is given
by:
T
0
=1/f
0
=2
π
/
ϖ
0
6.
It should be realized that a sinusoidal time delay corresponds to a phase shift.
This can be
illustrated in the next example.
1
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View Full Document EEL 3135: DiscreteTime Signals and Systems
Dr. Fred J. Taylor, Professor
Example:
Phase Shift
Suppose x(t)=Acos(
ϖ
0
t) is delayed by T
0
/4 seconds producing a signal y(t) = Acos(
ϖ
0(
t  T
0
/4))=
Acos(
ϖ
0
t
ϖ
0
T
0
/4).
Note that
ϖ
0
T
0
/4
= ϖ
0
/4f
0
= 2
π
f
0
/4f
0
=
π
/2 which corresponds to a 90
°
phase
producing y(t) = Acos(
ϖ
0(
t 90
°
) or y(t) = Acos(
ϖ
0(
t
π
/2) as illustrated in Figure 1.
°
There are other parameters that characterize a sinusoidal signal such a wavelength.
This is
explored in the next example.
Example:
A fixed frequency sinusoid signal is assumed to have the from x(t)=
A
sin(
ϖ
0
t+
φ
0
). Such signals
are fundamental to the design and analysis of many modern systems. To illustrate, a sinusoid of
the form x(t)=
A
sin(
ϖ
0
t+
φ
0
) can be used to represent a 1 GHz (Lband) class radio transmission
at a frequency
ϖ
0
= 2
π
f
0
=2
π
*10
9
r/s, as well as seismic signal have extremely low frequencies.
The radio
transceiver
problem is motivated in Figure 2. The electromagnetic radio wave
propagation velocity (called
phase velocity
) is the speed of light, or v
p
~3x10
8
m/s.
Figure 2: Wireless communication link.
•
What is the period of the signal x(t) in seconds?
Figure 1:
x(t)=cos(
ϖ
0
t) (top) and y(t)=cos(
ϖ
0(
t  90
°
) (bottom).
2
Radio Transmitter
Antenna
Radio
Receiver
999.9375
~
1 km
T
0
/4
EEL 3135: DiscreteTime Signals and Systems
Dr. Fred J. Taylor, Professor
The sinusoid’s period
T
, measured in seconds, is given by
T
=1/
f
0
=10
9
=1ns.
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This note was uploaded on 08/21/2010 for the course EEL 3135 taught by Professor ? during the Spring '08 term at University of Florida.
 Spring '08
 ?

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