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Lecture #1
Introduction to signals
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signals
• Electrical signals– Voltages and currents in a CKT
• Acoustic signals– Audio or speech signals
• Video signals– Intensity variations in an image
Signals are functions of independent variables that carry information.
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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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Sampling
Quantizing&
encoding
Continuous-time analog signal
Discrete-time analog signal
Discrete-time digital signal
0001
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Classification of signals
• Continuous-time and discrete-time signals
• Even and odd signals
• Periodic and nonperiodic signals
• Deterministic and random signals
Periodic signals )()( pTtftf
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Even signal
odd signal
)()( tftf
)()( tftf
t
t
)(tf
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Elementary signals
• Exponential signals
• Sinusoidal signals
• Step function
• Rectangular pulse
• Impulse function
• Derivatives of the impulse
• Ramp function
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Exponential signals
0,)( aBetx at0,)( aBetx at
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0,)sin()( tAetx t
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Step function
0,0
0,1)(
t
ttu
at
atatu
,0
,1)( t
1
a
Shift a
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Rectangular pulse
otherwise
tAtx
,0
5.05.0,)(
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Impulse function
00)(
1)(
tfort
dtt
t
)1()(t
Amplitude
width 0
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t
)1()(t
Derivatives of the impulse
t
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)(tr
)(tu
)(t
)(t
dt
d
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Ramp function
0,0
0,)(
t
tttr
tdutror
dt
tdrtu )()(
)()(
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simple operation
)(tf
)1()()()( trtrtutf
)(tu)1( tr
)(tr
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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
Example 1.5 find y(t)=x(2t+3)
)()0(
)()0(
)()(
abyx
bxy
batxty
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Precedence rule for discrete-time signal
