Topic 1: LTI Systems Overview: Introduction to Signals Types of - PowerPoint PPT Presentation

Signal Power and Energy � Signal: A function of a time-varying amplitude � Signal: Many different physical entities � No unit for energy/power � Often, a signal is a function of varying amplitude over time � A good measurement of the strength of a signal would be the area under the function � But this area may have a negative part which does not have less strength than a positive signal of the same size � This suggests either squaring the signal or taking its absolute value, then finding the area under that curve � Energy/power: strength of the signal 50

CT and DT Signals Energy of 51

Signal Energy and Power � A signal with finite signal energy is called an energy signal � If the signal does not decay � infinite energy � A signal with infinite signal energy and finite average signal power is called a power signal 52

DT Signal Energy of 53

DT Signal Energy of 54

CT Signal: Example Energyof 55

CT Signal: Example Energy of 56

DT Signal: Example Energy of 57

Signal Power 58

Signal Power 59

Signal Power: Example 60

Outline Introduction to Signals � Types of Signals: CT/DT, analog/digital, periodic/aperiodic � Periodic Signals � Special signals: Unit Impulse, Unit Step, … � Signal Energy and Power � Transformations of the Independent Variable � Even and Odd Signals � Introduction to Systems � Basic System Properties � Summary � 61

Transformations of CT signals + τ x ( t ) x : a function of t t : is the independen t variable of x τ : is a parameter of x + τ t : is the argument of x ⇒ τ + τ for a fixed what is x ( t ) 62

Transformations of CT signals � Transform x(t): � Transform the independent variable • e.g., x(t/2) � Combine Signals: � z(t) = x(t) y(t) � z(t) = x(t)/y(t) 63

Transformations of CT signals: Examples 64

Transformations of CT signals: Combination of Signals 65

Transformation of CT Signals 66

Transformation of CT Signals 67

Transformation of CT Signals 68

Transformations of CT signals:Time Shifting � Time-shifting occurs in many real physical systems: � Listening to someone talking 2m away � Received signal will be delayed, but the delay won’t be noticeable � Satellite communication systems (delay can be noticeable if ground stations are not directly below the satellite) � Radar systems: • Transmitted signal Ax(t) • Received signal Bx(t-t o ), with B<A, due to attenuation 69

Transformation of CT Signals 70

Transformations of CT signals: Time Scaling � Examples: � Playing an audio tape at a faster or slower speed � Doppler effect: standing by the side of a road while a fire truck approaches and then passes by 71

Multiple Transformation of CT Signals 72

Transformations of CT signals: Examples 73

Transformations of CT signals: Examples 74

Transformation of CT Signals: Examples 75

Transformation of CT Signals: Differentiation 76

Transformation of CT Signals: Integration 77

Transformation of DT Signals 78

Transformations of DT signals: Time Shifting 79

Transformations of DT signals: Time Scaling 80

Transformations of DT signals: Time Scaling 81

Transformations of DT signals: Differencing 82

Transformations of DT signals: Accumulation 83

Outline Introduction to Signals � Types of Signals: CT/DT, analog/digital, periodic/aperiodic � Periodic Signals � Special signals: Unit Impulse, Unit Step, … � Signal Energy and Power � Transformations of the Independent Variable � Even and Odd Signals � Introduction to Systems � Basic System Properties � Summary � 84

Even and Odd Signals � An even signal is identical to its time reversed � Example: � An odd signal has the property � Example : 85

Even and Odd CT Signals 86

Even and Odd Parts of CT Signals � The even part of a CT function is � The odd part of a CT function is � A function whose even part is zero is odd and a function whose odd part is zero is even � The derivative of an even CT function is odd and the derivative of an odd CT function is even � The integral of an even CT function is an odd CT function, plus a constant, and the integral of an odd CT function is even 87

Even and Odd Signals: Example 88

Products of Even and Odd CT Functions 89

Product of 2 Odd Functions 90

Integrals of Even and Odd CT Functions 91

Even and Odd DT Signals 92

Outline Introduction to Signals � Types of Signals: CT/DT, analog/digital, periodic/aperiodic � Periodic Signals � Special signals: Unit Impulse, Unit Step, … � Signal Energy and Power � Transformations of the Independent Variable � Even and Odd Signals � Introduction to Systems � Basic System Properties � Summary � 93

Introduction to Systems � To get the output y[n] � Apply the system S {} on input x(n) � y[n] is the response of S{} to x[n] � A system: An integrated whole composed of diverse, interacting, specialized parts � System performs a function not possible with any of the individual parts � Any system has objectives � Systems respond to particular signals by producing other signals or some desired behavior 94

Introduction to Systems: Examples 95

Introduction to Systems: a Communication System � A communication system has an information signal plus noise signals � This is an example of a system that consists of an interconnection of smaller systems � Cellphones are based on such systems 96

Introduction to Systems: Image System to Aid Perception 97

Introduction to Systems: Sound Recording 98

Introduction to Systems: CT and DT Systems � CT system: CT input signals are applied and result in CT output signals � The input-output relation of a CT system is � DT system: Transforms DT inputs into DT outputs. � The input-output relation of a DT system is 99

Introduction to Systems: Response of Systems � Systems respond to signals and produce new signals � Real signals are applied at system inputs and response signals are produced at system outputs � Example: What is the response of a system to a unit impulse? 100