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EE 205 1
EE 205
Signals and
Systems
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Lecture 1
Basic Continuous-Time
Signals
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CT Signals: Rectangular Pulse
Centered rectangular (square) pulse with
time duration
General rectangular pulse centered at
)()2
()2
()(
ttututp
)()2
()2
()( 000
ttttuttutp
0t
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)(
t
t2
2
)( 0
tt
t2
0
t
20
t
0t
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)3()(cos)( tututtx
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Energy Signals
Signal is an energy signal if
Example: One-sided exponential signal
)(tx
dttxxE )(2
)()exp()( tuttx
2
10|
2
)2exp(
0
)2exp(
tdttxE
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Power Signals
Energy of power signals is infinite
Average power over an interval is
Periodic signals are power signals
2
1
)(2
12
1T
T
dttxTT
xP
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Lecture 2
Linear Transformation
of Time
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Time Transformation:
1. Shifting (Delay)
Time-shifted transform of signal by
time constant is
Example: One-sided exponential signal
)()(0ttxty
)(tx
0t
)()exp()( tuttx
)()](exp[)(00ttuttty
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Time Transformation:
2a. Positive Scaling
Time-scaled transform of signal by
a constant is
Example: Rectangular pulse
)()( atxty
)(tx0a
)()(
ttx
)/
()()(a
tatty
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)(tx
1:)( aty
t
t
t
1:)( aty
2
a2
2
a2
a2
a2
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Time Transformation:
2b. Negative Scaling
Time-scaled transform of signal by
a constant is
This results in posiitve scaling and
reflection of the signal
)()( atxty
)(tx0a
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)(tx
1||:)( aty
t
t
t
1||:)( aty
2
||2 a
2
||2 a
||2 a
||2 a
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Time Transformation:
3. Shift and Scaling
Time-scaled then shifted transform of
signal is
Time-shifted then scaled transform of
signal is
)()(0tatxty
)(tx
)()]([)(00
attaxttaxty
)(tx
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t2
2
ta
t2
0
at
20
0t
)(tx
)]([)(0ttaxty
taa
t
20
a
t0
aa
t
20
)()(0tatxty
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Lecture 3
Basic Continuous-Time
Systems
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d
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Input
output
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Input
output
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Lecture 4
Linear Time-Invariant
Systems
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Test for Linearity
a
b
)(1 tx
)(2 tx
)(tx
a
b
)(1 ty
)(2 ty
)(tz
)(ty
System
ABC
System
ABC
System
ABC
)(1 tx
)(2 txSystem ABC
is linear if
)()( tytz
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)( tx
Test for Time-Invariance
)(tx
)(tz
)(tySystem
ABC
System ABC is
time-invariant if
)()( tytz
System
ABC
Delay
by
Delay
by
)(tx
)( ty
)(tw
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Problem
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Solution
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Lecture 5
Convolution Integral for Linear Time-Invariant
Systems
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Lecture 6
Sinusoidal Signals
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Lecture 7
Periodic Signals and
Fourier Series
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Lecture 8
Output of LTI Systems
for Sinusoidal Inputs
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Lecture 9
Continuous-Time Signals
and Fourier Transform
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Inverse Fourier
Transform
Forward Fourier
Transform
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Lecture 10
Sampling
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