Transcript Lectures
Mathematical Description of Continuous-Time Signals
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Typical Continuous-Time Signals
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
3
Continuous-Time Sinusoids g ( ) =
A
( p
t
/
T
0 + q ) =
A
( p
f
0
t
+ q ) =
A
cos ( w 0
t
+ q ) Amplitude Period Phase Shift Cyclic Radian (s) (radians) Frequency Frequency ( Hz) (radians/s)
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
4
Continuous-Time Exponentials g =
Ae
-
t
/ t Amplitude Time Constant (s)
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
5
Complex Sinusoids
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
6
The Signum Function
sgn = ì 1 , ì 0 , ìì 1 ,
t t t
< > = 0 0 ì ì 0 ìì Precise Graph Commonly-Used Graph The signum function, in a sense, returns an indication of the sign of its argument.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
7
The Unit Step Function
u = ì 1 , ì 1 / 2 ,
t t
ìì 0 ,
t
> 0 = < 0 0 The product signal g “turned on” at time
t
( ) u = ( ) 0. can be thought of as the signal g
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
8
The Unit Step Function
The unit step function can mathematically describe a signal that is zero up to some point in time and non zero after that.
i v
RC
( ) = v
C
(
V b
= =
V b
/
V b
u
R
)
e
-
t
/
RC
( 1 -
e
-
t
/
RC
u ) u ( ) ( )
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
9
The Unit Ramp Function
ramp = £ £ £
t
, 0 ,
t t
> 0 £ 0 £ £ £ =
t
-£ £ u ( )
d
l =
t
u
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
10
The Unit Ramp Function Product of a sine wave and a ramp function.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
11
Introduction to the Impulse
Define a function D = £ £ £ 1 / 0
a
, ,
t t
<
a
/ 2 >
a
/ 2 Let another function g ( ) be finite and continuous at
t
= 0.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
12
Introduction to the Impulse
The area under the product of the two functions is
A
As the width of D = 1
a
/2 £ g ( )
dt a t
-
a
/2 ( ) approaches zero, lim
a
® 0
A
= ( ) lim
a
® 0 1
a a
/2 £ -
a
/2
dt
= ( ) lim
a
® 0 1
a
= The continuous-time unit impulse is implicitly defined by = £ -£ £ d ( ) g ( )
dt M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
13
The Unit Step and Unit Impulse
As
a
approaches zero, g step and g ¢ ( ) approaches a unit ( ) 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.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
15
Properties of the Impulse
The Sampling Property
¥ -¥ ò g ( ) d (
t
-
t
0 )
dt
= g The sampling property “extracts” the value of a function at a point.
The Scaling Property
d ( -
t
0 ) ) = 1
a
d (
t
-
t
0 ) This property illustrates that the impulse is different from ordinary mathematical functions.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
16
The Unit Periodic Impulse
The unit periodic impulse is defined by d
T
= ¥ å
n
=-¥ d (
t
-
nT
) ,
n
an integer The periodic impulse is a sum of infinitely many uniformly spaced impulses.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
17
The Periodic Impulse
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
18
The Unit Rectangle Function
rect = ì 1 , ì ì 1 / 2 , ì 0 ,
t t t
< = 1 / 2 1 / 2 > 1 / 2 ì ì ì = u (
t
+ 1 / 2 ) u (
t
1 / 2 ) The product signal g ( ) rect ( ) can be thought of as the signal g “turned on” at time
t
= 1 / 2 and “turned back off” at time
t
= + 1 / 2.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
19
Combinations of Functions
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
20
Shifting and Scaling Functions
Let a function be defined graphically by and let g = 0 ,
t
> 5
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
21
Shifting and Scaling Functions
Amplitude Scaling
, g ( ) ®
A
g ( )
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
22
Shifting and Scaling Functions
Time shifting
,
t
®
t
-
t
0
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
23
Shifting and Scaling Functions
Time scaling
,
t
®
t
/
a M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
24
Shifting and Scaling Functions
Multiple transformations g ( ) ®
A
g ì ìì
t
-
t
0
a
ì ìì A multiple transformation can be done in steps g ì amplitude ì ì ì
A
g ì
t
ì ì ®
A
g ì ìì
t a
ì ìì ì
t
ì ì 0 ®
A
g ì ìì
t
-
t
0
a
ì ìì The sequence of the steps is significant g ì amplitude ì ì ì
A
g ì
t
ì ì 0 ®
A
g (
t
-
t
0 ) ì
t
ì ì ®
A
g ì ìì
t a
-
t
0 ì ìì ì
A
g ì ìì
t
-
t
0
a
ì ìì
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
25
Shifting and Scaling Functions
Simultaneous scaling and shifting g ®
A
g ì ìì
t
-
t
0
a
ì ìì
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
26
Shifting and Scaling Functions
Simultaneous scaling and shifting,
A
g (
bt
-
t
0 )
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
27
Shifting and Scaling Functions
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
28
Shifting and Scaling Functions If g 2 =
A
g 1 ( -
t
0 /
w
) what are
A
,
t
0 and
w
?
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
29
Shifting and Scaling Functions Height +5 ® Width +6 ® 2 + 2 Þ Shift left by 5/3 Þ Þ
t
0
A
= 0.4 , g 1
w
= 1 / 3 Þ = 5 / 3 Þ 0.4 g 1 ® 0.4 g 1 ( ) 0.4 g 1 3
t
® 0.4 g 1 ® 0.4 g 1 ( 3 (
t
+ 5 / 3 ) )
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
30
Shifting and Scaling Functions If g 2
t
( ) =
A
g 1 (
wt
-
t
0 ) what are
A
,
t
0 and
w
?
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
31
Shifting and Scaling Functions Height +5 Shift left 5 ® Þ
t
0 2 Width +6 to +2 Þ = 5 Þ
A
= 0.4
Þ 0.4 g 1 Þ ( ) g 1 ®
w
= 3 Þ 0.4 g 1 ® 0.4 g 1
t
0.4 g 1 5 ® ( ) 0.4 g 1 ( 3
t
+ 5 )
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
32
Shifting and Scaling Functions If g 2
t
( ) =
A
g 1 ( -
t
0 ) what are
A
,
t
0 and
w
?
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
33
Shifting and Scaling Functions Height +5 Width +6 ® ® 3 Þ 3 Þ Shift Right 1/2 Þ
t
0
A
= 0.6
w
= 2 = 1 / 2 Þ Þ Þ g 1 0.6 g 1 0.6 g 1 ( ) ( ) ® 0.6 g 1 2 ® 0.6 g 1 ® 0.6 g 1 ( 2 (
t
1 / 2 ) )
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
34
Shifting and Scaling Functions If g 2
t
( ) =
A
g 1 (
t
/
w
-
t
0 ) what are
A
,
t
0 and
w
?
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
35
Shifting and Scaling Functions Height +5 ® 3 Þ Shift Left 1 Þ Width +6 ® 3 Þ
A
= 0.6
Þ
t
0 = 1 Þ 0.6 g 1
w
= 1 / 2 Þ
t
g 1 ® ® 0.6 g 1 0.6 g 1 0.6 g 1 ( ) ( ) ® 0.6 g 1 ( 2
t
+ 1 )
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
36
Differentiation
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
37
Integration
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
38
Even and Odd Signals
Even Functions Odd Functions g = g ( ) g = g
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
39
Even and Odd Parts of Functions
The
even part
of a function is g
e
= g + g .
The
odd part
of a function is g
o
= g 2 g .
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.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
40
Even and Odd Parts of Functions
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
41
Products of Even and Odd Functions
Two Even Functions
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
42
Products of Even and Odd Functions
An Even Function and an Odd Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
43
Products of Even and Odd Functions
An Even Function and an Odd Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
44
Products of Even and Odd Functions
Two Odd Functions
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
45
Integrals of Even and Odd Functions
a
ò -
a
g ( )
dt
= 0
a
( )
dt a
ò -
a
g ( )
dt
= 0
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
46
Integrals of Even and Odd Functions
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
47
Periodic Signals
If a function g(
t
) is
periodic
, g = 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 = g (
t
+
T
) is called the
fundamental period
T
0 of the function. The reciprocal of the fundamental period is the
fundamental frequency
f
0 = 1 /
T
0 .
A function that is not periodic is
aperiodic
.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
50
Signal Energy and Power The signal energy of a signal x ( ) is
E
x = ¥ -¥ ò x 2
dt M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
51
Signal Energy and Power
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
52
Signal Energy and Power Find the signal energy of x = ì 2 rect ìì ( ) 4 rect ì ìì
t
+ 1 4 ì ìì ì ìì u (
t
+ 2 )
E
x = ì -ì ì x 2
dt
= ì -ì ì ì 2 rect ìì ( ) 4 rect ì ìì
t
+ 1 4 ì ìì ì ìì u (
t
+ 2 ) 2
dt E
x
E
x
E
x = = = ì 2 ì ì 2 rect ìì ( ) 4 rect ì ìì
t
+ 1 4 ì ìì ì ìì 2
dt
ì 2 ì ì 4 rect 2 ìì ì 2 ( ) + 16 rect 2 ( )
dt
ì + ì 2 ì ìì
t
ì ìì
t
+ 1 4 ì ìì 16 rect ( ) rect ì ìì
t
+ 1 4 ì ìì
dt
ì ì 2 + 1 4 ì ìì ì ìì
dt
( ) rect ì ìì
t
+ 1 4 ì ìì
dt E
x = 4 1 1 ì
dt
+ 16 1 2 ì
dt
16 1 1 ì
dt
= 8 + 48 32 = 24
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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 ( ) is
P
x For a periodic signal x = lim
T
®Þ 1
T T
/2 Þ x 2
dt t
-
T
/2 ( ) the average signal power is
P
x = 1
T
Þ
T
x where
T
is any period of the signal.
2
dt M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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
.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
55
Signal Energy and Power Find the average signal power of a signal x ( ) with fundamental period 12, one period of which is described by x = ramp ( -
t
/ 5 ) , 4 <
t
< 8
P
x
P
x = 1
T
= 1 12 ì
T
x 0 4 ì
t
2 25
dt
2
dt
= = 1 12 1 300 ìì
t
3 8 4 ì ramp ( -
t
/ 3 ìì 0 4 = 0 / 5 ) 2
dt
( 64 / 3 ) 300 = 1 12 0 4 ì ( -
t
= 16 225 / 5 ) 2
dt
@ 0.0711
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
56