Signal Processing Chapter 1

1. BEE2143
SIGNALS &
NETWORKS
Chapter 1
Introduction to Signals & Systems
Prepared by Nurul Wahidah Arshad, August 2019

2. 1
2
3
4
5
6
Classifications of Signals & System
Signal Characteristic
Time and Frequency domains
Elementary signals
Signals Operations
Convolution
Introduction
to
Signals &
Systems

3. At the end of Chapter 1, student should be
able to:
• Identify each signal based on its characteristic.
• Write equation of elementary signal and draw
its graph.
• Perform operation between signals.
• Solve signals convolution in time domain.
Course outcome
#1
Identify the different types &
operations of signal, and
suitable Fourier techniques.
3
Learning Outcomes Chapter 1

4. SIGNAL
• Variables That Carry Information
4
• Example:
– Electric signals
• V & I in electric circuit
– Acoustic signals
• Audio, speech, voice, sounds etc.
– Video signals
• Intensity variations in an image (e.g. CAT scan)
– Biological signals
• Sequence of bases in a gene

5. SYSTEM
Processing the signals to become other signals or information
5
Systems
Input signal
(excitation)
Output signal
(response)

6. 1.1 CLASSIFICATIONS OF SIGNALS &
SYSTEM
• Continuous-time
• Discrete-time
• Continuous-value
• Discrete-value
• Random
• Nonrandom
6
(classified according to how a system interacts with the input
signal applied to the system)
• Memory & memoryless systems
• Causal & non-causal systems
• Linear & nonlinear systems
• Time-invariant & time-variant
systems
• Linear & time-invariant (LTI) systems
1 2
Signal System

7. 1.1.1 Classifications of Signals 7
Continuous-
Time Signals
(CT)
• For CT, we use x(t) if time is the domain
• Example of CT signals: voltage, current,
temperature, velocity, etc.
• Many physical systems operate in
continuous time such as mass and spring,
and leaky tank

8. Discrete-Time
Signals (DT)
• For DT, we use x[n] for n is a number where
time varies discretely (samples)
• Examples of DT signals: population, DNA
based sequence
• Digital computations are done in discrete
time such as state machines: given the
current input and current state, what is the
next output and next state.
8

9. Continuous-time vs.
Discrete-time Signals
• Continuous-time signals
– Defined values at every instant of
time over time interval
– Real world (analog)
• Discrete-time signals
– Defined values only at discrete
points in time (not between them)
– Set of samples
– Usually transmitted as digital signal

10. 1.1.2 Classifications of Systems 10
Memoryless
Systems
• The output at time t0 depends only on the input at the
same time to. For example:
• Therefore, vo(to) depends upon the value of vi(to) and not
on vi(t) for t ≠ to.

11. Systems
With
Memory
• The output at time t0 depends on the input at the
some range of time t. For example:
• Therefore the systems that relates v to i exhibits
memory

12. Causal
Systems
• For a causal system the output at time to
depends only on the input for ti ≤ to
- the system cannot anticipate the input.
• output only exist after the input applied to the
system

13. Linear
Systems
• The output is proportional to the input.
• Linear systems satisfy the properties of
addition, superposition and scaling.
• Addition
Given x1(t)  y1(t) and x2(t)  y2(t)
 x1(t) + x2(t)  y1(t) + y2(t)
• Scaling
Given x(t)  y(t)
 kx(t)  ky(t)
• Superposition
Given x1(t)  y1(t) and x2(t)  y2(t)
 k1 x1(t) + k2 x2(t)  k1 y1(t) + k2 y2(t)

14. Nonlinear
Systems – The homogeneous (superposition & scaling)
and additive properties doesn’t apply
– Nonlinear equations are usually complex
– Example of nonlinear systems equations:
- For x = 0, but y ≠ 0 is also a nonlinear system

15. Time-
invariant &
Time-variant
Systems
Time-invariant
– Delaying the time in input will delayed the
time of output in same amount
where τ is
delayed time
y(t) = x(t-2)
Time-variant systems
- The opposite of time-invariant systems.
Simple example:

16. Linear &
Time-
Invariant
(LTI)
Systems
• Combination of linear systems and time-
invariant systems
• Thus, both system’s properties apply
• Will be focused more in this course
• Many powerful analysis tools will be covered
for calculating and transforming LTI systems
(Fourier, Laplace transform)

17. 1.2 SIGNAL CHARACTERISTICS 17
Characteristic Example Characteristic Example
1. Periodic
• x(t) = x(t + T)
for all t
• E.g.: Sine,
cosine, square
signal
3. Even symmetry
x(t) = x(-t)
2. Aperiodic
• Random
signals
• E.g.: speech,
all type of
noise signals
4. Odd symmetry
x(t) = -x(-t)

18. 2
T

 
Periodic function is defined as
If T in second, f in hertz (oscillation per second), the periodic function is
   ,
time
period
f t f t T
t
T
 


1
T
f

If angular frequency, in radians per second is defined by , then
 2 f
 

Periodic Function

19. 4

3

2



0

2
3
4
t
 
f t
1
1

A
f(t) is a waveform with an amplitude A = 1, period and the angular
frequency, . This waveform represented analytically by
2
T 

1
 
 
   
sin ,
2
f t t t
f t f t
 

   
 

20. From the graph, find the period T, the angular frequency , and the
amplitude A, for 𝑓 𝑡 = 3 sin 2𝑡

Example 1
2



0

2
t
 
f t
3
3

 
) 3sin 2
a f t t

answer
3
2
2
A
T 





 

21. From each of the following waveform, find the analytical description.
Example 2
 
   
4, 0 5
0
a
,
nswe
5 7
7
r
t
f t
t
f t f t
 

 
 

 
 
   
, 2
an w r
2
e
4
s
f t t t
f t f t
   
 

22. Even & Odd Function
4

3

2



0

2
3
4
t
 
cos t
   
For function, the function is inverted on the other
side of the . That is say :
eve
for all
n
.
y axis
f t f t t

  
4

3

2



0

2
3 t
 
sin t
   
For function, the function is symmetric about
the . That is say :
for all .
odd
origin
f t f t t
   

23. Sketch the graph of each of these periodic functions and determine
whether its is even, odd or neither.
Example 3

24. Solution
f(t)
t
2
-2
2
 3
2
 

3

a)
The graph is
symmetric about the
origin. Hence, the
periodic function is
odd.
The periodic function is
even since the graph is
symmetric about the
vertical axis.
f(t)
t
3
3


2


2
 3
2
 5
2

b)
The periodic function is
neither even nor odd
function since the graph
is not symmetric about
the both the origin and
the vertical axis.
f(t)
t
3
2
 3


c)

25. 1.3 TIME & FREQUENCY
REPRESENTATION
25
• The most common representation
of signals is time domain
• However, most signal analysis
techniques work only in frequency
domain
• Solutions can be more easily found
in the frequency domain
• The frequency domain is simply
another way of representing a
signal

26. Example of
Laplace transform:
Relationship between
time &
frequency domains
26

27. 1
2
3
4
5
6
Unit step
Impulse response
Sinusoid & exponential complex
Unit Ramp
Sync function
Rectangle
Triangle
Signum
1.4
ELEMENTARY
SIGNAL
7
8

28. 28
Unit-Step Function
• The unit step function u(t), also known
as the Heaviside unit function.






0
,
1
0
,
0
)
(
t
t
t
u

29. • Derivative of the unit step function
• Also known as Dirac delta function
• The unit impulse (t) is zero anywhere except
at t=0
Unit Impulse Function
0
0
0
,
0
,
,
0
)
(
)
(










t
t
t
undefined
t
u
dt
d
t


30. 30
• A sinusoid is a signal that has the form of the sine or cosine function.
• A general expression for the sinusoid,
Sinusoids
)
sin(
)
( 
 
 t
A
t
f )
cos(
)
( 
 
 t
A
t
f
Where;
A = the amplitude of the sinusoid
ω = the angular frequency in
radians/s = 2/T
 = the phase

31. Complex Exponential Signal
• Review of Complex Numbers:
𝑥 = 𝑟 cos 𝜃
𝑦 = 𝑟 sin 𝜃
𝑧 = 𝑟 cos 𝜃 + 𝑗𝑟 sin 𝜃
𝑟 = 𝑥2 + 𝑦2 𝜃 = 𝑡𝑎𝑛−1 𝑦
𝑥
• Using Euler’s formula for the complex exponential:
𝑧 𝑡 = 𝐴𝑒𝑗𝜃
= 𝐴(cos 𝜃 + 𝑗 sin 𝜃); 𝑤ℎ𝑒𝑟𝑒 𝜃 = 𝜔0𝑡 + ∅
Real
part
Imaginary
part

32. Example 4
Complex Exponential Signal 32

33. • Integration of the unit step function
Unit Ramp
𝑟 𝑡 =
𝑡, 𝑡 > 0
0, 𝑡 < 0
𝑟 𝑡 =
−∞
𝑡
𝑢 𝑡 𝑑𝑡 = 𝑡 𝑢(𝑡)
Unit Ramp

34. Sinc Function
𝑓 𝑡 =
sin 𝑡
𝑡

35. Rectangle function
• Functions as a switch to turn on and turn off any equipment
in a specific time period
2
1
others
2
1
,
0
,
1
)
(









t
t
t
rect
rect(t)

36. Triangle Function







1
|
|
1
|
|
,
0
|,
|
1
)
(
t
t
t
t
tri

37. Signum Function
1
-1
Derivation of Signum function
using Unit step function:

38. Recall
Common
Function
38

40. 1
2
3
4
Riversal
Scaling
Shifting
Addition & Multiplication
1.5 SIGNAL
OPERATIONS

41. Signals Operation
1
Time
(operation will
affect the x-axis)
2
Amplitude
(operation will
affect the y-axis)
41

42. Reversal
𝑦 𝑡 = 𝑓(−𝑡)
1
Time Riversal (Time Folding)
2
Amplitude Riversal
𝑦 𝑡 = −𝑓 (𝑡)
Note: For impulse response, 𝛿 −𝑡 = 𝛿(𝑡)

43. Compresses or dilates a signal by multiplying the time variable by
some quantity, a.
• If a> 1, the signal becomes narrower and the operation is called
compression,
• if a<1, the signal becomes wider and is called dilation.
Scaling
1 Time Scaling
𝑦 𝑡 = 𝑓 𝑎𝑡 ; Where a is a real constant

44. Example 5
)
2
/
(
)
(
)
2
(
)
(
,
any time
for
0
0
0
0
0
1
1
t
y
t
x
t
x
t
y
t
t




)
10
(
)
(
)
1
.
0
(
)
(
,
any time
for
0
2
0
0
0
2
0
t
y
t
x
t
x
t
y
t
t




Draw the signal if time
scaling is applied, y1(t)=x(2t)
& y2(t)=x(0.1t)

45. Amplitude is scaled by a factor ‘a’
Scaling 45
2 Amplitude Scaling
𝑦 𝑡 = 𝑎𝑓 𝑡 ;

46. Example 6
Draw the signal if amplitude
scaling is applied,
y1(t)=2.5x(t) & y2(t)=10x(t)

47. • The shifting of a signal in time.
• When we add a constant to the time, we obtain the advanced
signal, & when we decrease the time, we get the delayed
signal.
– y(t) = f (t - a)
Shifting
1 Time Shifting

48. Shifting
1 Time Shifting - in elementary signals







o
o
o
t
t
t
t
t
t
u
,
1
,
0
)
(









o
o
o
t
t
t
t
t
t
u
,
1
,
0
)
(
0
0
0
0
0
0
0
0
,
0
,
)
(
t
t
t
t
t
t
t
t
t
t
t
t
ramp










 


• Shifted unit step
• Shifted ramp function

49. • Delayed 3 unit of t: • Advanced 3 unit of t:
Example 7

50. x(t) is an original signal, find y(t)=x(-2t-1)
• Method 1: Shift x(t) to get x(t-t0) and scaling x(t-t0) with a
Example 8
If a signal experiencing simultaneous time scale and shift y(t)=x(at-t0)
- If a is negative: time reversal is apply

51. • Method 2:
Scaling x(t) with a to get x(at) and shift x(at) with t0/a  x[a(t-t0/a)]= x(at-t0)
• Method 3: Replace t with . Let =at-t0 ,  =-2t-1. Then draw y(t) using the new t.

52. Upward & downward shifting.
Shifting 52
2 Amplitude Shifting
𝑦 𝑡 = 𝑎 + 𝑓 𝑡 ;

53. Example 9
Can you write
both equation for
x(t) and y(t)?

54. Scaling & amplitude shift y(t)=Ax(t)+B, where A & B are constants. Given
x(t) as shown in figure, find y(t) = -2x(t)+1.
Example 10

55. 





































3
1
1
0
0
2
;
8
3
1
9
3
;
5
1
6
;
3
1
2
)
(
3
1
1
0
0
2
;
9
3
;
6
;
2
)
(
2
t
t
t
-
t
t
t
y
t
t
t
-
t
t
x

56. Find y1(t) and y2(t)
a. y1(t) = x(t)+x(-t)
b. y2(t) = x(t) x [(t+1/2)-(t-1/2)]
solution: a. y1(t) = x(t)+x(-t)
Addition & Multiplication
+ =
y1(t)
Example 11

57. b. y2(t) = x(t) x [(t+1/2)-(t-1/2)]
x =

58. Example 12
𝑥 𝑡 =
0 − 0 + 0 + 0 = 0; 𝑡 < −3
3 − 0 + 0 + 0 = 3; −3 < 𝑡 < 0
3 − 1 + 0 + 0 = 2; 0 < 𝑡 < 3
3 − 1 + 3 + 0 = 5; 3 < 𝑡 < 6
3 − 1 + 3 + 1 = 6; 𝑡 > 6

59. 1.6 Convolution

60. Convolution Integral
• Convolution means “folding”
• The formula equation:
• Useful application
– Finding response function y(t):
– Alternative method:
 



t
d
t
h
x
t
x
t
h
t
y
0
)
(
)
(
)
(
)
(
)
( 


)
(
)
(
)
( t
x
t
h
t
y 
 In time domain
)
(
)
(
)
( s
X
s
H
s
Y  In frequency domain
(Laplace Transform)

61. Example 13:
Two
rectangular
pulses
Change t to λ.
- Fold & shift one of the signal, let choose x1(t):
1
)
(
1
)
(
1
)
(
1
)
(
1









t
h
h
h
t
x
t
t
t













1
,
0
1
,
1
0
,
1
0
,
Fold: Shift:

62. 62
• Change t to λ fot x2(t):
• For t < 0: no overlapping between two signals, y(t) = 0
• For 0 < t < 1:
𝑥2 𝑡 = 1 , 0 < 𝑡 <3
𝑥 𝜆 = 1 , 0 < 𝑡 <3
𝑦 𝑡 =
0
𝑡
1 1 𝑑𝜆
𝑦 𝑡 = 𝜆|0
𝑡
= 𝑡
Table of Basic Integral

63. 63
• For 1 < t < 3 :
• For 3 < t < 4:
• For t > 4: no overlapping between
two signals, y(t) = 0
𝑦 𝑡 =
𝑡−1
𝑡
1 1 𝑑𝜆
𝑦 𝑡 = 𝜆|𝑡−1
𝑡
= 𝑡 − 𝑡 − 1 = 1
𝑦 𝑡 =
𝑡−1
3
1 1 𝑑𝜆
𝑦 𝑡 = 𝜆|𝑡−1
3
= 3 − 𝑡 − 1 = −t + 4

64. 𝑦 𝑡 = 𝑥1 𝑡 ∗ 𝑥2 𝑡 =
0 ; 𝑡 < 0
𝑡 ; 0 < 𝑡 < 1
1 ; 1 < 𝑡 < 3
−𝑡 + 4 ; 3 < 𝑡 < 4
0 ; 𝑡 > 4

65. GROUP
ACTIVITIES
Fold & shift
the given
function.
65
1.
2.
3.

66. Change t to λ:
Practice Problem
15.12
1
)
(
1
)
(
1
)
(
1
)
(
1









t
h
h
h
t
x
t
t
t













1
,
0
1
,
1
0
,
1
0
,








2
1
)
(
2
1
)
(
2

x
t
x
2
1
,
1
0
,
2
1
,
1
0
,










t
t

67. • For 0 < t < 1:
• For 1 < t < 2:
 
t
d
d
x
t
h
t
y
t
t
t







0
0
0
)
1
)(
1
(
)
(
)
(
)
(





   
t
d
d
t
y
t
t
t
t






 

1
1
1
1
1
1
2
)
2
)(
1
(
)
1
)(
1
(
)
(





68.  For 2 < t < 3:
 For t < 0 & t > 3: no overlap, so y(t) = 0
 Thus,
 
t
d
t
y
t
t
2
6
2
)
2
)(
1
(
)
(
2
1
2
1








 2
1
0 2
1 t-1 t λ









0
2
6
0
)
(
t
t
t
y
3
,
3
2
,
2
0
,
0
,






t
t
t
t
2
0 3
1 2 t
y(t)

69. Change t to λ:
Practice
Problem
15.13
1
)
(
1
)
(
1
)
(
1
)
(









t
h
h
h
t
g
t
t
t













1
,
0
1
,
1
0
,
1
0
,

 



e
x
e
t
f t
3
)
(
3
)
(
0
,
0
,



t

70. • For 0 < t < 1:
• For t > 1:
  
 
 
t
t
t
t
e
e
e
d
e
t
y











 
1
3
3
3
3
3
1
)
(
0
0



  
 
)
1
(
3
3
3
3
3
1
)
(
1
1
















e
e
e
e
e
e
d
e
t
y
t
t
t
t
t
t
t




71. For t < 0: no overlap.
Thus,










)
1
(
3
)
1
(
3
0
)
(
e
e
e
t
y
t
t
1
,
1
0
,
0
,




t
t
t

72. Thank you
72
1
2
3
4
5
6
Classifications of Signals & System
Signal Characteristic
Time and Frequency domains
Elementary signals
Signals Operations
Convolution
Conclusion:
Nurul Wahidah Arshad, FKEE
+09-424 6090
wahidah@ump.edu.my