Mathematical Description of
Continuous-Time Signals
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Edited by Dr. Robert Akl
Typical Continuous-Time Signals
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2
Continuous vs Continuous-Time
Signals
All continuous signals that are functions of time are
continuous-time but not all continuous-time signals are
continuous
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3
Continuous-Time Sinusoids
()
(
)
(
)
(
g t = Acos 2p t / T0 + q = Acos 2p f0t + q = Acos w 0t + q
- Amplitude
Period Phase Shift
(s)
(radians)
Cyclic
Radian
Frequency
Frequency
( Hz)
(radians/s)
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)
4
Continuous-Time Exponentials
g ( t ) = Ae-t /t
- Amplitude Time Constant (s)
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5
Complex Sinusoids
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6
The Signum Function
ì 1 , t > 0ü
ï
ï
sgn ( t ) = í 0 , t = 0 ý
ï-1 , t < 0 ï
î
þ
Precise Graph
Commonly-Used Graph
The signum function, in a sense, returns an indication of
the sign of its argument.
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7
The Unit Step Function
ì1 , t > 0
ï
u ( t ) = í1 / 2 , t = 0
ï0 , t < 0
î
The product signal g ( t ) u ( t ) can be thought of as the signal g ( t )
“turned on” at time t = 0.
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8
The Unit Step Function
The unit step function can mathematically describe a
signal that is zero up to some point in time and nonzero after that.
()
()
i ( t ) = (V / R ) e
u (t )
v ( t ) = V (1- e
) u (t )
v RC t = Vb u t
-t / RC
b
-t / RC
C
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b
9
The Unit Ramp Function
ìt , t > 0 ü t
ramp ( t ) = í
ý = ò u (l ) dl = t u (t )
î0 , t £ 0 þ -¥
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10
The Unit Ramp Function
Product of a sine wave and a ramp function.
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11
Introduction to the Impulse
ì1 / a , t < a / 2
Define a function D ( t ) = í
, t >a/2
î0
Let another function g ( t ) be finite and continuous at t = 0.
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12
Introduction to the Impulse
The area under the product of the two functions is
a/2
1
A = ò g ( t ) dt
a - a/2
As the width of D ( t ) approaches zero,
a/2
1
1
lim A = g ( 0 ) lim ò dt = g ( 0 ) lim ( a ) = g ( 0 )
a®0 a
a®0 a
a®0
- a/2
The continuous-time unit impulse is implicitly defined by
g (0) =
¥
ò d ( t ) g ( t ) dt
-¥
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13
The Unit Step and Unit Impulse
As a approaches zero, g ( t ) approaches a unit
step and g¢ ( t ) approaches a unit impulse.
The unit step is the integral of the unit impulse and
the unit impulse is the generalized derivative of the
unit step.
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14
Graphical Representation of the
Impulse
The impulse is not a function in the ordinary sense because its
value at the time of its occurrence is not defined. It is represented
graphically by a vertical arrow. Its strength is either written beside
it or is represented by its length.
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15
Properties of the Impulse
The Sampling Property
¥
ò g ( t )d ( t - t ) dt = g ( t )
0
0
-¥
The sampling property “extracts” the value of a function at
a point.
The Scaling Property
1
d ( a ( t - t 0 )) = d ( t - t 0 )
a
This property illustrates that the impulse is different from
ordinary mathematical functions.
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16
The Unit Periodic Impulse
The unit periodic impulse is defined by
dT (t ) =
¥
å d ( t - nT )
, n an integer
n=-¥
The periodic impulse is a sum of infinitely many uniformlyspaced impulses.
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The Periodic Impulse
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18
The Unit Rectangle Function
ì 1 , t <1/ 2ü
ï
ï
rect ( t ) = í1 / 2 , t = 1 / 2 ý = u ( t + 1 / 2 ) - u ( t - 1 / 2 )
ï 0 , t > 1 / 2ï
î
þ
The product signal g ( t ) rect ( t ) can be thought of as the signal g ( t )
“turned on” at time t = - 1 / 2 and “turned back off” at time t = + 1 / 2.
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19
Combinations of Functions
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20
Shifting and Scaling Functions
Let a function be defined graphically by
and let g ( t ) = 0 , t > 5
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Shifting and Scaling Functions
Amplitude Scaling, g(t ) ® Ag(t )
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22
Shifting and Scaling Functions
Time shifting, t ® t - t 0
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23
Shifting and Scaling Functions
Time scaling,
t ®t /a
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24
Shifting and Scaling Functions
Multiple transformations
æ t - t0 ö
g ( t ) ® Ag ç
è a ÷ø
A multiple transformation can be done in steps
ætö
æ t - t0 ö
-t 0
/a
g ( t ) ¾ ¾¾¾
® Ag ( t ) ¾t®t
¾¾
® Ag ç ÷ ¾t®t
¾¾
® Ag ç
è aø
è a ÷ø
amplitude
scaling, A
The sequence of the steps is significant
æt
ö
æ t - t0 ö
-t 0
/a
g ( t ) ¾ ¾¾¾
® Ag ( t ) ¾t®t
¾¾
® Ag ( t - t 0 ) ¾t®t
¾¾
® Ag ç - t 0 ÷ ¹ Ag ç
èa
ø
è a ÷ø
amplitude
scaling, A
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Shifting and Scaling Functions
æ t - t0 ö
Simultaneous scaling and shifting g ( t ) ® Ag ç
è a ÷ø
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26
Shifting and Scaling Functions
Simultaneous scaling
and shifting, Ag ( bt - t0 )
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27
Shifting and Scaling Functions
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Shifting and Scaling Functions
()
((
) )
If g 2 t = Ag1 t - t0 / w what are A, t0 and w?
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29
Shifting and Scaling Functions
()
()
Height +5 ® -2 Þ A = -0.4 , g1 t ® -0.4 g1 t
()
( )
= -5 / 3 Þ -0.4 g ( 3t ) ® -0.4 g ( 3( t + 5 / 3))
Width +6 ® +2 Þ w = 1 / 3 Þ -0.4 g1 t ® -0.4 g1 3t
Shift left by 5/3 Þ t0
1
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30
Shifting and Scaling Functions
()
(
)
If g 2 t = Ag1 wt - t0 what are A, t0 and w?
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31
Shifting and Scaling Functions
()
()
Shift left 5 Þ t = -5 Þ -0.4 g ( t ) ® -0.4 g ( t + 5)
Width +6 to +2 Þ w = 3 Þ -0.4 g ( t + 5) ® -0.4 g ( 3t + 5)
Height +5 ® -2 Þ A = -0.4 Þ g1 t ® -0.4 g1 t
0
1
1
1
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1
32
Shifting and Scaling Functions
()
( (
If g 2 t = Ag1 w t - t0
)) what are A, t
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0
and w?
33
Shifting and Scaling Functions
()
()
Width +6 ® -3 Þ w = -2 Þ -0.6 g ( t ) ® -0.6 g ( -2t )
Shift Right 1/2 Þ t = 1 / 2 Þ -0.6 g ( -2t ) ® -0.6 g ( -2 ( t - 1 / 2 ))
Height +5 ® -3 Þ A = -0.6 Þ g1 t ® -0.6 g1 t
1
0
1
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1
1
34
Shifting and Scaling Functions
()
(
)
If g 2 t = Ag1 t / w - t0 what are A, t0 and w?
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35
Shifting and Scaling Functions
()
()
Shift Left 1 Þ t = -1 Þ -0.6 g ( t ) ® -0.6 g ( t + 1)
Width +6 ® -3 Þ w = -1 / 2 Þ -0.6 g ( t + 1) ® -0.6 g ( -2t + 1)
Height +5 ® -3 Þ A = -0.6 Þ g1 t ® -0.6 g1 t
0
1
1
1
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1
36
Differentiation
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37
Integration
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38
Even and Odd Signals
Even Functions
Odd Functions
g ( t ) = g ( -t )
g ( t ) = - g ( -t )
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39
Even and Odd Parts of Functions
g ( t ) + g ( -t )
The even part of a function is g e ( t ) =
.
2
g ( t ) - g ( -t )
The odd part of a function is g o ( t ) =
.
2
A function whose even part is zero is odd and a function
whose odd part is zero is even.
The derivative of an even function is odd and the derivative
of an odd function is even.
The integral of an even function is an odd function, plus a
constant, and the integral of an odd function is even.
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40
Even and Odd Parts of Functions
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41
Products of Even and Odd Functions
Two Even Functions
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42
Products of Even and Odd Functions
An Even Function and an Odd Function
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Products of Even and Odd Functions
An Even Function and an Odd Function
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Products of Even and Odd Functions
Two Odd Functions
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Integrals of Even and Odd Functions
a
a
-a
0
ò g ( t ) dt = 2 ò g ( t ) dt
a
ò g ( t ) dt = 0
-a
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Integrals of Even and Odd Functions
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Periodic Signals
If a function g(t) is periodic, g ( t ) = g ( t + nT ) where n is any integer
and T is a period of the function. The minimum positive value of T
for which g ( t ) = g ( t + T ) is called the fundamental period T0 of the
function. The reciprocal of the fundamental period is the fundamental
frequency f0 = 1 / T0 .
A function that is not periodic is aperiodic.
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48
Sums of Periodic Functions
The period of the sum of periodic functions is the least common
multiple of the periods of the individual functions summed. If the
least common multiple is infinite, the sum function is aperiodic.
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49
ADC Waveforms
Examples of waveforms which
may appear in analog-to-digital
converters. They can be
described by a periodic repetition
of a ramp returned to zero by a
negative step or by a periodic
repetition of a triangle-shaped
function.
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50
Signal Energy and Power
The signal energy of a signal x ( t ) is
¥
Ex =
ò x (t )
2
dt
-¥
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51
Signal Energy and Power
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52
Signal Energy and Power
é
æ t + 1ö ù
Find the signal energy of x ( t ) = ê 2 rect ( t / 2 ) - 4 rect ç
÷ø ú u ( t + 2 )
è
4 û
ë
¥
Ex =
ò x (t )
¥
2
dt =
-¥
ò
-¥
2
é
æ t + 1ö ù
ê 2 rect ( t / 2 ) - 4 rect çè 4 ÷ø ú u ( t + 2 ) dt
ë
û
¥
2
é
æ t + 1ö ù
Ex = ò ê 2 rect ( t / 2 ) - 4 rect ç
÷ø ú dt
è
4 û
-2 ë
¥
é
æ t + 1ö ù
2
2 æ t + 1ö
Ex = ò ê 4 rect ( t / 2 ) + 16 rect ç
÷ø - 16 rect ( t / 2 ) rect çè
÷ø ú dt
è
4
4 û
-2 ë
¥
¥
¥
æ t + 1ö
æ t + 1ö
Ex = 4 ò rect ( t / 2 ) dt + 16 ò rect ç
dt
16
rect
t
/
2
rect
dt
(
)
÷
ç
÷
ò
è 4 ø
è 4 ø
-2
-2
-2
1
1
1
-1
-2
-1
Ex = 4 ò dt + 16 ò dt - 16 ò dt = 8 + 48 - 32 = 24
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53
Signal Energy and Power
Some signals have infinite signal energy. In that case
It is more convenient to deal with average signal power.
The average signal power of a signal x ( t ) is
T /2
1
2
Px = lim
x ( t ) dt
T ®¥ T ò
-T /2
For a periodic signal x ( t ) the average signal power is
1
2
x
t
dt
(
)
ò
T T
where T is any period of the signal.
Px =
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54
Signal Energy and Power
A signal with finite signal energy is
called an energy signal.
A signal with infinite signal energy and
finite average signal power is called a
power signal.
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55
Signal Energy and Power
Find the average signal power of a signal x ( t ) with fundamental
period 12, one period of which is described by
x ( t ) = ramp ( -t / 5 ) , - 4 < t < 8
1
Px =
T
ò x (t )
T
8
2
0
1
1
2
2
ramp ( -t / 5 ) dt =
dt =
( -t / 5 ) dt
ò
ò
12 -4
12 -4
0
0 - ( -64 / 3) 16
1
1 t2
3
éët / 3ùû =
@ 0.0711
=
dt =
Px =
ò
-4
225
300
300
12 -4 25
0
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56