This document provides an introduction to signals and systems. It discusses the classification of signals as continuous-time or discrete-time, periodic or aperiodic, deterministic or random, energy or power signals. It also discusses the classification of systems as continuous-time or discrete-time, linear or nonlinear, time-variant or time-invariant, causal or non-causal, stable or unstable. It then introduces some basic standard signals including step, ramp, impulse, sinusoidal, and exponential signals. It describes the properties and applications of these signals.

1. Welcome you all!
NIMITHA N
Assistant Professor.
Department of ECE
RMK College of Engineering
and Technology
1

2. Signals
and
Systems

3. 3
Objective:
Understand the
operations and
classifications of
signals & systems
UNIT I - CLASSIFICATION OF SIGNALS AND SYSTEMS
Standard signals- Step,
Ramp, Pulse, Impulse, Real and
complex exponentials and
Sinusoids
Classification of Signals-
Continuous time (CT) and
Discrete Time (DT) signals,
Periodic & Aperiodic signals,
Deterministic & Random
signals, Energy & Power signals
Classification of systems-
CT systems and DT systems- –
Linear & Nonlinear, Time-
variant & Time-invariant, Causal
& Non-causal, Stable &
Unstable.

4. Introduction of Signals
Types of Signals
Basic Standard signals
Applications
4
Contents

5. Introduction
of signals

6. SIGNALS
INFORMATION
PHYSICAL PHENOMENON
INDEPENDENT VARIABLE
DEPENDENT VARIABLE
DIGITAL
DISCRETE
ENERGY
POWER
MECHANICAL
AUDIO
ELECTRICALVISUAL
ANALOG/CONTINUOUS
FUNCTION
WAVEFORM

7. Signals
Signals are mathematical function
• Dependent variable = voltage, temperature,
flow rate, pressure etc..
• Independent variable = time

8. A signal is the function of one or more
independent variables that carriers some
information to represent a physical
phenomenon
Pressure Electrical
signal

9. Types of
signals

10. Categories of Signals
Based on dimension of signal
• 1 Dimensional – Speech, ECG, EEG
• 2 Dimensional - Image
• 3 Dimensional – Colour Image, Video

11. Based on nature of signal
• Analog Signal
• Digital Signal
• Continuous Time Signal (CT)
• Discrete Time Signal (DT)
Based on
amplitude
Based on
time

12. Analog or digital?
Combination
of sinusoidal
signal

13. Analog Signals
• The amplitude of an analog signal can have any
value (including fractions) at any point in time.
• Analog signal have infinite values.
13

14. Digital Signals
14
> The amplitude of signal can have only finite and
discrete values
> The special case of Digital signal having two
discrete values is known as Binary signal

15. Analog and Digital signal
15
Comparison
Element
Analog Signal Digital Signal
Representation Continuous Discontinuous
Analysis Difficult Easy
Recording
Techniques
Original Signal is
preserved
Samples of the signal are
taken and preserved
Storage Infinite Memory Easily Stored

16. Continuous Time Signals
• Signal that has a value for all points in time (Finite ,real
valued, Smooth function)
• Written as x(t) because the signal “x” is a function of
time
• Commonly found in the physical world/ system
ex. Human speech, ECG, EEG etc..
• Displayed graphically as a line
16
A lot of physics lives in Continuous time

17. Continuous Time Signals
17
All continuous(Analog) signals that are
functions of time are
continuous-time
but not all continuous-time signals are
continuous(Analog)

18. Whether these are CT signals?
Point of
discontinuity
Point of
discontinuity

19. Discrete Time Signals
• Signal that has a value for only specific points in time (Finite ,real
valued)
• Typically formed by “sampling” a continuous-time signal
• Represented as x(n). Also called as sequence
• Commonly found in all Computational systems (digital world)
ex. Audio signal mp3
Image JPEG
• Displayed graphically as individual values
Called a “stem” plot
19
DT signals – smoothness is not applicable

20. Sampling
> Discrete-time signals are often obtained by sampling
continuous-time signals
20
Where, T = Time between
samples

21. How to convert Analog to Digital
21
SAMPLING
QUANTIZING
ENCODING
Analog signal
Discrete signal
Discrete signal
Digital signal
Continuous
Amplitude
Continuous Time
Continuous
Amplitude
Discrete Time
Discrete Amplitude
Discrete Time
DIGITIZING

22. 22

23. 23
Theoretical
Signals are not constrained
• Real - this is often violated (complex numbers)
• Finite/bounded energy - violated ALL the time.
• Smoothness - this is often violated by many of the
continuous time signals.

24. 24
Question 1
Which tool is used to
convert CT signal to
DT signal
SAMPLING

25. 25
Question 2
Which type of signal
Discrete Time Signal

26. Basic standard signals

27. Engineering Design Cycle
Necessity of standard signals

28. Basic (Elementary or Standard) signals
• Step Signal
• Ramp signal
• Impulse signal
• Parabolic Signal
• Sinusoidal and exponential signal
• Sinc signal
• Rectangular signal
• Signum signal
• Triangular signal
28
CT and DT
Standard signals are called singularity function
Test Signals
Input Signals
for process

29. 29
STEP Signal
UNIT STEP
Signal
STEP SIGNAL
• Heaviside function
• Represented by U(t) and
U(n)
• It is a Power signal
• Neither Even signal nor
Odd signal
Application:
• DC Generator(Switching
on and off of a device)
• Communication appln

30. 30
RAMP
Signal
UNIT RAMP
Signal
RAMP SIGNAL
• Represented by r(t) and
r(n)
• Neither Even nor odd
signal
• Neither Energy nor
power signal
Application:
Current and Voltage
relation circuits

31. 31
IMPULSE Signal
Unit sample
sequence
IMPULSE SIGNAL
• Delta function
• CT - Dirac delta function
• DT- Kronecker delta
function
• Unit Area signal
• Even Signal
• Neither Energy nor
Power signal
Application
• Thunderbolt
• ECG function

32. 32
PROPERTIES OF IMPULSE SIGNAL
Property 1
Proof

33. 33
PROPERTIES OF IMPULSE SIGNAL
Delayed signal
Property 2
Proof

34. 34
PARABOLIC
Signal
Unit Parabolic
parabolic SIGNAL
Acceleration
function
Represented as p(t)
and p(n)
Application:
The Bike responds
to acceleration

35. 35

36. Unit Step Signal
36
Unit ramp Signal Unit Impulse Signal
Continuous Time signals
Slope
Unit Area

37. Unit Step Signal
37
Unit ramp Signal Unit Impulse Signal
Discrete Time signals
෍
𝑘=−∞
∞
𝛿(𝑛) = 1r(n) = n u(n)
u(n) = δ(n)+ δ(n-1)+ δ(n-2)+…..
Step and ramp
Step and Impulse

38. Relation between step and impulse
38
time

39. Relationship between standard signals
39
A=1

40. 40
SIN Signal.
COS SignalSINUSOIDAL SIGNAL
COS signal
SIN Signal
•Smooth
•Finite power signal.
•Violates none of our
criteria for real-
world signals.
Application
•Any sound signal
•The light signal
•A tornedo –
Complex sinusoidal
signal

41. 41
CT Signal
DT Signal
Sinc SIGNAL
• Sine Cardinal
function
• Normalized Sinc
function
• Represented by
sinc(t)
• Energy Signal
Used in
•Digital Signal
Processing
•Information Theory
, t = 0
1 , t = 0

42. 42
Exponential SIGNAL
Exponential signal is
defined as
where
Apply Euler’s Identity
Complex number, S= a+ jω
= A e(a+jω)t
=A eat e jωt
= A eat (cosωt + j sinωt)

43. 43
Growing Sine
Decaying Sine
Exponential SIGNAL
Where
Where
Growing Cosine
Decaying Cosine
Rising
Falling
x(t)= A eat (cosωt + j sinωt)
xr(t) = A eat (cosωt)
xj(t) = A eat (sinωt)
When , a= +ve
When , a= -ve
x(t)= A e-at (cosωt + j sinωt)
xr(t) = A e-at (cosωt)
xj(t) = A e-at (sinωt)

44. Real Exponential
signal is defined as
For CT
44
Growing signal
Exponential SIGNAL
Consider,
Amplitude A= 1
a>0 = Growing/Rising
signal
a<0 = Decaying / Falling
signal
Decaying SignalConsider ω = 0

45. 45
Consider a=1 (a>0)
x(t) = eat
t= 0, e0 = 1
t= 1, e1 = 2.7
t= 2, e2 = 7.3
Consider a=-1 (a<0)
x(t) = eat
t= 0, e0 = 1
t= 1, e-1 = 0.36
t= 2, e-2 = 0.13
Rising
Falling

46. Real Exponential signal is
define as
For DT
46
Exponential SIGNAL
x(n) = αn
α>1 = rising exponential
α<1 = falling exponential
Consider α= 2 (α>1)
n= 0, 20 = 1
n= 1, 21 = 2
n= 2, 22 = 4

47. 47
Exponential SIGNAL
Consider α= ½ (α<1)
n= 0, (½)0 = 1
n= 1, (½)1 = ½ = 0.5
n= 2, (½)2 = ¼ = 0.25
Practise
α= -½
α= -2

48. 48
Practise
α= -2
α= -½
Consider α= -½
n= -2, (-½)-2 = 4
n= -1, (-½)-1 = -2
n= 0, (-½)0 = 1
n= 1, (-½)1 = -½ = -0.5
n= 2, (-½)2 = ¼ = 0.25
Consider α = - 2
n= -2, (-2)-2 = 1/4
n= -1, (-2)-1 = -1/2
n= 0, (-2)0 = 1
n= 1, (-2)1 = -2
n= 2, (-2)2 = 4
n= 3, (-2)3 = -8
-2 -1 0 1 2 3
x(n)= (-2)n
1/4
-1/2
1
-2
4
-8
n
-2 -1 0 1 2 3
x(n)= (-2)n
1/4
1/2 1
2
4
8
n
w.k.t X(n) = αn

49. 49
Exponential SIGNAL
Complex Exponential
signal is defined as
Euler’s Identity
Case (i)
ejω t = cos(ωt) + j sin(ωt)
(Counterclockwise)
Case (i)
e-jω t = cos(-ωt)+ jsin(-ωt)
e-jω t = cos(ωt) - jsin(ωt)
(Clockwise)
Where
cos(ωt) = Real part
sin(ωt) = Imaginary part
Consider ω = 0

50. 50
sin(ωt)
cos(ωt)
Case (i) ejω t = cos(ωt) + j sin(ωt)

51. 51
3D View
Case (ii) e-jω t = cos(ωt) - jsin(ωt)

52. 52
Rect CT Signal
Rect DT Signal
Rectangular SIGNAL
•Pi Function, Gate
function
•Represented by rect(t)
•Even Signal
•Energy Signal
•Combination of 2
shifted step signal
rect(t)= u(t-a)-u(t-a)
Application
•Basic element for all
digital signals

53. 53
V2(t)= -3u(t-2)
P(t)= V1(t)+ V2(t)
V1(t)= 3u(t-1)

54. 54
CT Signal
DT Signal
Signum SIGNAL
• Sign function
• Represented by
sgn(t)
• Energy Signal
• Fourier Transform
of Rectangular
signal
Used in
•Communication

55. 55
CT Signal
DT Signal
Triangular SIGNAL
• Triangular function
is the convolution of
two rectangular
functions
• It’s a rising and
falling ramp function
• Represented by tri(t)
• Even Signal
• Energy Signal
Application
• Analog to Digital
conversion circuits
tri(t) = 1+t , -1≤ t ≤ 0
1-t , 0 ≤ t ≤ 1

56. 56

57. 57
NEVER STOPS LEARNING
BECAUSE
LIFE NEVER STOPS TEACHING

58. 58
Thanks!
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