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Lecture #1
Introduction to signals
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signals
Signals are functions of independent variables that
carry information.
• Electrical signals
– Voltages and currents in a CKT
• Acoustic signals
– Audio or speech signals
• Video signals
– Intensity variations in an image
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Independent variables
• Can be continuous
• Can be discrete
• Can be 1-dimension, 2-D, …N-D
For this course we focus on a signal 1-D independent
variable which we called “time”
Continuous time x(t) , t– continuous values
Discrete time x[k], k– Integer values only
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Continuous time (CT) signal
Most of the signals in the physical world are CT signals—
E.g. voltage & current, pressure, temperature, velocity, etc.
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Discrete time (DT) signal
Examples of DT signals in nature:
• DNA base sequence
• Population of the nth generation of certain species
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Continuous-time analog signal
Sampling
Discrete-time analog signal
Quantizing
&
encoding
Discrete-time digital signal
0001
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Classification of signals
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Continuous-time and discrete-time signals
Even and odd signals
Periodic and nonperiodic signals
Deterministic and random signals
Periodic signals
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f (t )  f (t  T p )
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Even signal
f (t )  f (t )
t
odd signal
f (t )
f (t )   f (t )
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t
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Elementary signals
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Exponential signals
Sinusoidal signals
Step function
Rectangular pulse
Impulse function
Derivatives of the impulse
Ramp function
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Exponential signals
x(t )  Be , a  0
at
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x(t )  Be at , a  0
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x(t )  Aet sin( t   ) ,   0
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Step function
1, t  0
u (t )  
0 , t  0
Shift a
1
1, t  a
u (t  a)  
0 , t  a
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Rectangular pulse
 A ,  0.5  t  0.5
x(t )  
 0 , otherwise
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Impulse function



 (t )
(1)
 (t )dt  1
 (t )  0
Amplitude
for t  0

t
width  0
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Derivatives of the impulse
t
 (t )
(1)
t
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r (t )
u (t )

d
dt
 (t )
 (t )
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Ramp function
t, t  0
r (t )  
0, t  0
dr (t )
u (t ) 
dt
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or
t
r (t )   u ( )d

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simple operation
f (t )
f (t )  u (t )  r (t )  r (t  1)
u (t )
r (t  1)
r (t )
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Basic operations on signals
• Operations performed on dependent variables
– Amplitude scaling, Addition, Multiplication,
differentiation
• Operations performed on independent variables
– Time scaling
– Reflection
– Time shifting
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Time scaling
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Reflection
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Time shifting
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Precedence rule for time shifting and time scaling
y(t )  x(at  b)
 y(0)  x(b)
 x(0)  y ( ba )
Example 1.5 find y(t)=x(2t+3)
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Precedence rule for discrete-time signal
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