1. SIGNALS & SYSTEMS

2. Can you believe it?

3. Examples of System

4. 1. INTRODUCTION

5. What is a Signal?
(DEF) Signal : A signal is formally defined as a function of one or more variables, which conveys information on the nature of physical phenomenon.
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6. What is a System?
(DEF) System : A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.
system
output signal
input signal

7. Some Interesting Systems
Communication system
Control systems
Remote sensing system
Biomedical system(biomedical signal processing)
Auditory system

8. Some Interesting Systems
Communication system

9. Some Interesting Systems
Control systems

10. Some Interesting Systems
Papero

11. Some Interesting Systems
Remote sensing system
Perspectival view of Mount Shasta (California), derived from a pair of stereo radar images acquired from orbit with the shuttle Imaging Radar (SIR-B). (Courtesy of Jet Propulsion Laboratory.)

12. Some Interesting Systems
Biomedical system(biomedical signal processing)

13. Some Interesting Systems
Auditory system

14. Classification of Signals
Continuous and discrete-time signals
Continuous and discrete-valued signals
Even and odd signals
Periodic signals, non-periodic signals
Deterministic signals, random signals
Causal and anticausal signals
Right-handed and left-handed signals
Finite and infinite length

15. Continuous and discrete-time signals
Continuous signal
- It is defined for all time t : x(t)
Discrete-time signal
- It is defined only at discrete instants of time : x[n]=x(nT)

16. Continuous and Discrete valued singals
CV corresponds to a continuous y-axis
DV corresponds to a discrete y-axis
Digital signal

17. Even and odd signals Even signals : x(-t)=x(t)
Odd signals : x(-t)=-x(t)
Even and odd signal decomposition
xe(t)= 1/2·(x(t)+x(-t)) xo(t)= 1/2·(x(t)-x(-t))

19. Periodic signals, non-periodic signals
- A function that satisfies the condition
x(t)=x(t+T) for all t
- Fundamental frequency : f=1/T
- Angular frequency : = 2/T
Non-periodic signals

20. Deterministic signals, random signals
-There is no uncertainty with respect to its value at any time. (ex) sin(3t)
Random signals
- There is uncertainty before its actual occurrence.

21. Causal and anticausal Signals
Causal signals : zero for all negative time
Anticausal signals : zero for all positive time
Noncausal : nozero values in both positive and negative time
causal signal
anticausal signal
noncausal signal

22. Right-handed and left-handed Signals
Right-handed and left handed-signal : zero between a given variable and positive or negative infinity

23. Finite and infinite length
Finite-length signal : nonzero over a finite interval tmin< t< tmax
Infinite-length singal : nonzero over all real numbers

24. Basic Operations on Signals
Operations performed on dependent signals
Operations performed on the independent signals

25. Operations performed on dependent signals
Amplitude scaling
Addition
Multiplication
Differentiation
Integration

26. Operations performed on the independent signals
Time scaling
a>1 : compressed
0<a<1 : expanded

27. Operations performed on the independent signals
Reflection

28. Operations performed on the independent signals
Time shifting
- Precedence Rule for time shifting & time scaling

29. The incorrect way of applying the precedence rule. (a) Signal x(t)
The incorrect way of applying the precedence rule. (a) Signal x(t). (b) Time-scaled signal v(t) = x(2t). (c) Signal y(t) obtained by shifting v(t) = x(2t) by 3 time units, which yields y(t) = x(2(t + 3)).
The proper order in which the operations of time scaling and time shifting (a) Rectangular pulse x(t) of amplitude 1.0 and duration 2.0, symmetric about the origin. (b) Intermediate pulse v(t), representing a time-shifted version of x(t). (c) Desired signal y(t), resulting from the compression of v(t) by a factor of 2.

30. Elementary Signals Exponential signals Sinusoidal signals
Exponentially damped sinusoidal signals

31. Elementary Signals
Step function

32. (a) Rectangular pulse x(t) of amplitude A and duration of 1 s, symmetric about the origin. (b) Representation of x(t) as the difference of two step functions of amplitude A, with one step function shifted to the left by ½ and the other shifted to the right by ½; the two shifted signals are denoted by x1(t) and x2(t), respectively. Note that x(t) = x1(t) – x2(t).

33. Elementary Signals Impulse function
(a) Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e., unit impulse). (b) Graphical symbol for unit impulse. (c) Representation of an impulse of strength a that results from allowing the duration Δ of a rectangular pulse of area a to approach zero.

34. Elementary Signals
Ramp function

35. Systems Viewed as Interconnection of Operations
output signal
input signal

36. Properties of Systems Stability Memory Invertibility Time Invariance
Linearity

37. Stability(1)
BIBO stable : A system is said to be bounded-input bounded-output stable iff every bounded input results in a bounded output.
Its Importance : the collapse of Tacoma Narrows suspension bridge, pp.45

38. Dramatic photographs showing the collapse of the Tacoma Narrows suspension bridge on November 7, (a) Photograph showing the twisting motion of the bridge’s center span just before failure. (b) A few minutes after the first piece of concrete fell, this second photograph shows a 600-ft section of the bridge breaking out of the suspension span and turning upside down as it crashed in Puget Sound, Washington. Note the car in the top right-hand corner of the photograph.

39. Stability(2) Example pp.46 - y[n]=1/3(x[n]+x[n-1]+x[n-2])
- y[n]=rnx[n], where r>1

40. Memory
Memory system : A system is said to possess memory if its output signal depends on past values of the input signal
Memoryless system
(example)

41. Memory or memoryless?

42. Causality
Causal system : A system is said to be causal if the present value of the output signal depends only on the present and/or past values of the input signal.
Non-causal system
(example)
y[n]=x[n]+1/2x[n-1]
y[n]=x[n+1]+1/2x[n-1]

43. Invertiblity(1)
Invertible system : A system is said to be invertible if the input of the system can be recovered from the system output.
H:xy, H-1:yx
H-1{y(t)}= H-1{H{x(t)}}, H-1H=I H
H-1
x(t)
y(t)

44. Invertiblity(2)
(Example) -

45. Time Invariance (1)
Time invariant system : A system is said to be time invariant if a time delay or time advance of the input signal leads to a identical time shift in the output signal.

46. Time Invariance (2) Are following two systems equivalent? St0 H x(t)
yi(t)
x(t-t0)
y0(t)

47. Time Invariance (3)
Examples

48. Linearity(1)
Linear system : A system is said to be linear if it satisfies the principle of superposition.

49. Linearity(2) a1 a2 aN .  H x1(t) x2(t) xN(t) y(t) H . a1 a2 aN 

50. Linearity(3)
Examples -
Check superposition with simple two inputs.

51. Theme Examples
Example of multiple propagation paths in a wireless communication environment.

52. Tapped-delay-line model of a linear communication channel, assumed to be time-invariant

53. Stock Price : filtering
(a) Fluctuations in the closing stock price of Intel over a three-year period.
(b) Output of a four-point moving-average system.

54. Chapter 1: Signals & Systems Concepts
(1) Systems, signals, mathematical models. Continuous-time and discrete-time signals and systems. Energy and power signals. Linear systems. Examples for use throughout the course, introduction to Matlab and Simulink tools
Specific Objectives:
Introduce, using examples, what is a signal and what is a system
Why mathematical models are appropriate
What are continuous-time and discrete-time representations and how are they related
Brief introduction to Matlab and Simulink
EE-2027 SaS, L1

55. Recommended Reading Material
Signals and Systems, Oppenheim & Willsky, Section 1
Signals and Systems, Haykin & Van Veen, Section 1
MIT Chapter 1
Mastering Matlab 6
Mastering Simulink 4
Many other introductory sources available. Some background reading at the start of the course will pay dividends when things get more difficult.
EE-2027 SaS, L1

56. What is a Signal? A signal is a pattern of variation of some form
Signals are variables that carry information
Examples of signal include:
Electrical signals
Voltages and currents in a circuit
Acoustic signals
Acoustic pressure (sound) over time
Mechanical signals
Velocity of a car over time
Video signals
Intensity level of a pixel (camera, video) over time
EE-2027 SaS, L1

57. How is a Signal Represented?
Mathematically, signals are represented as a function of one or more independent variables.
For instance a black & white video signal intensity is dependent on x, y coordinates and time t f(x,y,t)
On this course, we shall be exclusively concerned with signals that are a function of a single variable: time t
f(t)
EE-2027 SaS, L1

58. Example: Signals in an Electrical Circuitvs + - C vc
The signals vc and vs are patterns of variation over time
Note, we could also have considered the voltage across the resistor or the current as signals
Step (signal) vs at t=1
RC = 1
First order (exponential) response for vc
vs, vc t
EE-2027 SaS, L1

59. Continuous & Discrete-Time Signals
Continuous-Time Signals
Most signals in the real world are continuous time, as the scale is infinitesimally fine.
Eg voltage, velocity,
Denote by x(t), where the time interval may be bounded (finite) or infinite
Discrete-Time Signals
Some real world and many digital signals are discrete time, as they are sampled
E.g. pixels, daily stock price (anything that a digital computer processes)
Denote by x[n], where n is an integer value that varies discretely
Sampled continuous signal x[n] =x(nk) – k is sample time
x(t) t
x[n] n
EE-2027 SaS, L1

60. Signal Properties
On this course, we shall be particularly interested in signals with certain properties:
Periodic signals: a signal is periodic if it repeats itself after a fixed period T, i.e. x(t) = x(t+T) for all t. A sin(t) signal is periodic.
Even and odd signals: a signal is even if x(-t) = x(t) (i.e. it can be reflected in the axis at zero). A signal is odd if x(-t) = -x(t). Examples are cos(t) and sin(t) signals, respectively.
Exponential and sinusoidal signals: a signal is (real) exponential if it can be represented as x(t) = Ceat. A signal is (complex) exponential if it can be represented in the same form but C and a are complex numbers.
Step and pulse signals: A pulse signal is one which is nearly completely zero, apart from a short spike, d(t). A step signal is zero up to a certain time, and then a constant value after that time, u(t).
These properties define a large class of tractable, useful signals and will be further considered in the coming Chapters
EE-2027 SaS, L1

61. What is a System?
Systems process input signals to produce output signals
Examples:
A circuit involving a capacitor can be viewed as a system that transforms the source voltage (signal) to the voltage (signal) across the capacitor
A CD player takes the signal on the CD and transforms it into a signal sent to the loud speaker
A communication system is generally composed of three sub-systems, the transmitter, the channel and the receiver. The channel typically attenuates and adds noise to the transmitted signal which must be processed by the receiver
EE-2027 SaS, L1

62. How is a System Represented?
A system takes a signal as an input and transforms it into another signal
In a very broad sense, a system can be represented as the ratio of the output signal over the input signal
That way, when we “multiply” the system by the input signal, we get the output signal
This concept will be firmed up in the coming weeks
System
Input signal
x(t)
Output signal
y(t)
EE-2027 SaS, L1

63. Example: An Electrical Circuit Systemvs + - C vc
Simulink representation of the electrical circuit
vs(t)
vc(t)
first order
system
vs, vc t
EE-2027 SaS, L1

64. Continuous & Discrete-Time Mathematical Models of Systems
Continuous-Time Systems
Most continuous time systems represent how continuous signals are transformed via differential equations.
E.g. circuit, car velocity
Discrete-Time Systems
Most discrete time systems represent how discrete signals are transformed via difference equations
E.g. bank account, discrete car velocity system
First order differential equations
First order difference equations
EE-2027 SaS, L1

65. Properties of a System
On this course, we shall be particularly interested in signals with certain properties:
Causal: a system is causal if the output at a time, only depends on input values up to that time.
Linear: a system is linear if the output of the scaled sum of two input signals is the equivalent scaled sum of outputs
Time-invariance: a system is time invariant if the system’s output is the same, given the same input signal, regardless of time.
These properties define a large class of tractable, useful systems and will be further considered in the coming Chapters
EE-2027 SaS, L1

66. Introduction to Matlab/Simulink (1)
Command
window
Click on the Matlab icon/start menu initialises the Matlab environment:
The main window is the dynamic command interpreter which allows the user to issue Matlab commands
The variable browser shows which variables currently exist in the workspace
Variable
browser
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67. Introduction to Matlab/Simulink (2)
Type the following at the Matlab command prompt
>> simulink
The following Simulink library should appear
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68. Introduction to Matlab/Simulink (3)
Click File-New to create a new workspace, and drag and drop objects from the library onto the workspace.
Selecting Simulation-Start from the pull down menu will run the dynamic simulation. Click on the blocks to view the data or alter the run-time parameters
EE-2027 SaS, L1

69. How Are Signal & Systems Related (i)?
How to design a system to process a signal in particular ways?
Design a system to restore or enhance a particular signal
Remove high frequency background communication noise
Enhance noisy images from spacecraft
Assume a signal is represented as
x(t) = d(t) + n(t)
Design a system to remove the unknown “noise” component n(t), so that y(t)  d(t)
x(t) = d(t) + n(t)
y(t)  d(t)
System ?
EE-2027 SaS, L1

70. How Are Signal & Systems Related (ii)?
How to design a system to extract specific pieces of information from signals
Estimate the heart rate from an electrocardiogram
Estimate economic indicators (bear, bull) from stock market values
Assume a signal is represented as
x(t) = g(d(t))
Design a system to “invert” the transformation g(), so that y(t) = d(t)
x(t) = g(d(t))
y(t) = d(t) = g-1(x(t))
System ?
EE-2027 SaS, L1

71. How Are Signal & Systems Related (iii)?
How to design a (dynamic) system to modify or control the output of another (dynamic) system
Control an aircraft’s altitude, velocity, heading by adjusting throttle, rudder, ailerons
Control the temperature of a building by adjusting the heating/cooling energy flow.
Assume a signal is represented as
x(t) = g(d(t))
Design a system to “invert” the transformation g(), so that y(t) = d(t)
x(t)
y(t) = d(t)
dynamic
system ?
EE-2027 SaS, L1

72. Chapter 1: Summary
Signals and systems are pervasive in modern engineering courses:
Electrical circuits
Physical models and control systems
Digital media (music, voice, photos, video)
In studying the general properties of signals and systems, you can:
Design systems to remove noise/enhance measurement from audio and picture/video data
Investigate stability of physical structures
Control the performance mechanical and electrical devices
This will be the foundation for studying systems and signals as a generic subject on this course.
EE-2027 SaS, L1

73. Chapter 1: Exercises
Read SaS OW, Chapter 1. This contains most of the material in the first three Chapters, a bit of pre-reading will be extremely useful!
SaS OW:
Q1.1
Q1.2
Q1.4
Q1.5
Q1.6
In Chapter 2, we’ll be looking at signals in more depth and look at how they can be represented in Matlab/Simulink
EE-2027 SaS, L1