Transcript Slide 1

Signals and Systems
(Part one: Continuous)
General Information

Lecturer
– Dr. Yasmine Fahmy

Reference
– Signals and systems, Oppenheim A.V.,
Wilski A.S., Prentice Hall, 1997
Dr. Yasmine Fahmy
Course Contents
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
Signals (definition, properties , important signals) 
Systems (definition, properties)
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Linear Time Invariant (LTI) Systems

Fourier Series

Fourier Transform

LTI Systems in Frequency Domain

Applications (Filters, Sampling, Modulation)
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
Dr. Yasmine Fahmy
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9
Signals
variables carrying information
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Electrical signals
– Voltages and currents in a circuit
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Acoustic signals
– Acoustic pressure (sound) over time
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Mechanical signals
– Velocity of a car over time
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Video signals
– Intensity level of a pixel (camera, video) over
time
Dr. Yasmine Fahmy
Continuous /
Discrete
x(t)
x[n]
t
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Velocity
Voltage
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-2
-1
0
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2
3
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Pixels
Daily stock price
n
Analog
/
Digital

Continuous Analog Signal

Discrete Analog Signal
t
-3
-2
-1
0
1
2
3
4
n

-3
-2
-1
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0
1
2
3
4
Continuous Quantized
Analog Signal
Analog
/
Digital
01001111010011010
t
-3
-2
-1
0
1
2
3
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n
-3
-2
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-3
-2
-1
0
1
2
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4
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Dr. Yasmine Fahmy
-1
0
1
2
3
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Sampling
Quantization
Coding
n
Properties of Signals
1. Signal Energy and Power
2. Transformation in Time
(Shift, Reverse, Scaling)
3. Periodic Signals
4. Even and Odd Signals
Dr. Yasmine Fahmy
Signal Energy and Power
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Energy over time interval
2
t2
Et1 t2   x(t ) dt
t1

Average Power over time interval
Pt1 t2
Dr. Yasmine Fahmy
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
t 2  t1
t2
 x(t )
t1
2
dt 
Et1 t2
t 2  t1
Signal Energy and Power
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Total Energy
2
T
E   lim
 x(t )
dt
T   T
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Average Power
P  lim
T 
Dr. Yasmine Fahmy
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2T
T
 x(t )
T
2
dt  lim
T 
E
2T
Transformation in Time
x(t)
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Time Shift
t
x(t-to)
x(t+to)
-to
+to
Advance
Dr. Yasmine Fahmy
t
t
+to
-to
Delay
Transformation in Time
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Time Reverse
x(t)
x(-t)
t
Dr. Yasmine Fahmy
t
Transformation in Time
x(t)
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Time Scaling
t
x(‫׀‬α‫ ׀‬t)
x(‫׀‬α‫ ׀‬t)
t
‫׀‬α ‫ < ׀‬1
Dr. Yasmine Fahmy
Compressed
t
‫׀‬α ‫ > ׀‬1
Stretched
Example 1
X(t)
1
0
Find:
1. The equation of x(t)
2. The values of E0 2 P0 2 E P
3. x(t+1)
4. x(-t+1)
5. x(-3/2t+1)
6. x(-3/2t-1)
Dr. Yasmine Fahmy
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2
t
,
,
,
NOTE
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Energy signals:
– Finite Energy
– Zero Power
Dr. Yasmine Fahmy
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Power signals:
– Infinite Energy
– Finite Power
Periodic Signals
x(t) = x(t+T)
Where
 Period := T
 Fundamental Period := To
(is the minimum value of T)
Dr. Yasmine Fahmy
t
Example 2

Find the period of the following signals:
 t
– x 1 (t )  5  cos 
 3
–

 2 t 
  sin 


 9 
x 2 (t )  cos  2t 1  sin 5t  2
Dr. Yasmine Fahmy
Even & Odd Signals
Even
x(t) = x(-t)
Odd
x(t) = -x(-t)
t
Symmetric around the axis
Dr. Yasmine Fahmy
t
Symmetric around the origin
Even & Odd Signals
For any signal x(t)
x(t) = xe(t)+ xo(t)
Where
 xe(t)=1/2 [x(t)+x(-t)]
 xo(t)=1/2 [x(t) -x(-t)]
Dr. Yasmine Fahmy
Example 3
X(t)
1
-1
,
0
1
,
,
Find and Sketch
The Even and Odd components of x(t)
Dr. Yasmine Fahmy
Example 3
Dr. Yasmine Fahmy
Lecture Overview
 Signal
(continuous/discrete/analog/digital)
 Signal Properties
1.
2.
3.
4.
Signal Energy and Power
Transformation in Time (Shift, Reverse, Scaling)
Periodic Signals
Even and Odd Signals
Dr. Yasmine Fahmy