continuous-time and discrete-time signals: impulse, step...
TRANSCRIPT
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Unit I: Classification of Signals and Systems
Signals and Systems http://DrSatvir.in
Continuous-Time and Discrete-Time Signals: Impulse, Step, Ramp, Pulse, Exponential
1-01
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Outline
1. Basic definitions
2. Continuous-time and discrete-time signals
3. Elementary signalsβ’ Unit impulse signal
β’ Unit step signal
β’ Unit ramp signal
4. Relations among elementary signals
5. Signal operations
6. Composite signals
7. Practice problem
8. Question (Exam point of view)
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Basic Definitions
What is a Signal?
A signal is an electrical or electromagnetic current that is used for carrying
information from one device or network to another.
Continuous Time (CT) Signals
Continuous Time signal π π‘ has infinite values corresponding to infinite
time π‘ values. Mathematically,
π π‘ = π’(π‘ β 1)
Discrete Time (DT) Signals
It is signal that is obtained after sampling of Continuous Time at equal
intervals. Mathematically,
π ππ = π’ ππ β 1 or π π = π’ π β 1
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Continuous Time and Discrete Time β Graphically
π‘
π₯(π‘)
Continuous Signal
π
π₯(π)
0 1 2 3 4 5 6 7
Discrete Time Signal
π
π₯(π)
0 1 2 3 4 5 6 70
1
2
3
4
5
6
7
Digital Signal
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Unit I: Classification of Signals and Systems
Signals and Systems http://DrSatvir.in
Elementary Signals
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Unit Impulse Signal
Continuous-Time Signal
πΏ π‘ = α1 π‘ = 00 π‘ β 0
Discrete-Time Signal
πΏ π = α1 π = 00 π β 0
πΏ(π‘)
π‘0
πΏ(π)
π0 1 2 3β1β2β3
An impulse signal has zero value except at π‘ = 0. It has infinitelyhigh value π‘ = 0.
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Unit Step Signal
Continuous-Time Signal
π’ π‘ = α1 π‘ β₯ 00 π‘ < 0
Discrete-Time Signal
π’ π = α1 π β₯ 00 π < 0
A unit step signal has unity value for π‘ β₯ 0 else zero value.
π’(π‘)
π‘0
π’(π)
π0 1 2 3β1β2β3
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Unit Ramp Signal
Continuous-Time Signal
π π‘ = απ‘ π‘ β₯ 00 π‘ < 0
Discrete-Time Signal
π π = απ π β₯ 00 π < 0
A ramp step signal has unity slop value for π‘ β₯ 0, otherwise it haszero value.
π(π‘)
π‘0
π(π)
π0 1 2 3β1β2β3
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Rectangular Pulse Signal
Continuous-Time Signal
π π‘ = α1 β
π
2β€ π‘ β€ +
π
20 Otherwise
Discrete-Time Signal
π π = α1 βπ β€ π β€ +π0 Otherwise
A unit rectangular pulse has unit amplitude within a time interval,otherwise it has zero value. It is also called the Gate pulse, Pulsefunction, or Window function, etc.
π(π‘)
π‘0 π
2βπ
2
1π(π)
π0 1 2 3β1β2β3
1
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Exponential Signal
Continuous-Time Signal
π π‘ = π΄πππ‘ π΄ > 0
Discrete-Time Signal
π π = π΄πππ π΄ > 0
An exponential signal can either have exponentially rising orfalling amplitude depending upon its exponent value.
π(π)
π0 1 2 3β1β2β3
π΄
π π‘
π‘0
π΄
π > 0π π‘
π‘0
π΄
π(π)
π0 1 2 3β1β2β3
π΄
π < 0
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Unit I: Classification of Signals and Systems
Signals and Systems http://DrSatvir.in
Relationships Impulse, Step & Ramp Signals
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Relations - Integration & Differentiation
π’(π‘)
π‘0
πΏ(π‘)
π‘0
π(π‘)
π‘0
ππ’ π‘
ππ‘= πΏ π‘
ππ π‘
ππ‘= π’ π‘
π π‘ = π‘
Inte
gra
tio
n
Dif
fere
nti
ati
on
πΏ π‘
ΰΆ±πΏ π‘ ππ‘ = π’ π‘
ΰΆ±π’ π‘ ππ‘ = π‘ = π π‘
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Unit I: Classification of Signals and Systems
Signals and Systems http://DrSatvir.in
Signal Operations
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Signal Operations - Right Shifting
πΏ π‘ β π = α1 π‘ = π0 π‘ β π
π’ π‘ β π = α1 π‘ β₯ π0 π‘ < π
π π‘ β π = απ‘ π‘ β₯ π0 π‘ < π
Impulse Signal
Step Signal
Ramp Signal
πΏ(π‘ β π)
π‘0 π
π’(π‘ β π)
π‘0 π
π(π‘ β π)
π‘0 π
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Signal Operations - Left Shifting
πΏ π‘ + π = α1 π‘ = βπ0 π‘ β βπ
π’ π‘ + π = α1 π‘ β₯ βπ0 π‘ < βπ
π π‘ + π = απ‘ π‘ β₯ βπ0 π‘ < βπ
Impulse Signal
Step Signal
Ramp Signal
πΏ(π‘ + π)
π‘0βπ
π’(π‘ + π)
π‘0βπ
π(π‘ + π)
π‘0βπ
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Signal Time Operations
Time Reversal
Right Shifting
Left Shifting
π₯(π‘)
π‘0βπ1 +π2
π₯(βπ‘)
π‘0βπ2 +π1
π₯(π‘ β π)
π‘0βπ1 +π2π
π₯(π‘ + π)
π‘0βπ1 +π2βπ
Expansion π < 1π₯
π‘
2
π‘0βππ1 +ππ2
Compression π > 1
π₯ 2π‘
π‘0βππ1 +ππ2
Time Scaling π₯ ππ‘
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Unit I: Classification of Signals and Systems
Signals and Systems http://DrSatvir.in
Composite Signal
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Composite Signals
π₯ π‘ = π π‘ β π’ π‘ β 4 + π π‘ β 4
π(π‘ β 4)
π‘0 1 2 3 4 75 6
π(π‘)
π‘0 1 2 3 4 75 6
π’(π‘ β 4)
π‘0 1 2 3 4 75 6
π₯(π‘)
π‘0 1 2 3 4
π₯(π‘)
π‘0 1 2 3 4 75 6
βπ(π‘ β 4)βπ’(π‘ β 4)
π(π‘)
1
2
3
1 -2 -3+ +
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Unit I: Classification of Signals and Systems
Signals and Systems http://DrSatvir.in
Practice Problem
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Signal Operation β Practice Problem
Given a Continuous-Time signal π π is shown in the following figure.
Sketch following signals: a) ππ(π) = π π + π (Right Shifting)
b) ππ(π) = π π β π (Left Shifting)
c) ππ(π) = π βπ (Time Reversal)
d) ππ(π) = ππ
π(Time Expansion)
e) ππ(π) = π ππ (Time Compression)
π₯(π‘)
π‘0 1 2 3 4β1β2β3β4
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Signal Operation β Solution
π₯(π‘)
π‘0 1 2 3 4β1β2β3β4
Right Shifting ππ(π) = π π + π
Left Shiftingππ π = π π β π
π¦1(π‘)
π‘0 1 2 3 4β1β2β3β4 5
π¦2(π‘)
π‘0 1 2 3 4β1β2β3β4
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Signal Operation β Solution
Time Compressionππ(π) = π ππ
Time Reversalππ(π) = π βπ
Time Expansion
ππ(π) = ππ
π
π¦3(π‘)
π‘0 1 2β1β2β3β4
π₯(π‘)
π‘0 1 2 3 4β1β2β3β4
π¦4(π‘)
π‘3 4 5 6 7210β1 8
π¦5(π‘)
π‘0 1 2 3β1β2β3
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1-01 Questions (Exam Point of View)
1. What are signals?
2. What is continuous time signal?
3. What are elementary signals?
4. Give the relation among
a) Unit Impulse Signal
b) Unit Step Signal and
c) Unit Ramp Signal
5. Define the terms signal and system.
6. Write mathematical and graphical representation of unit step function.
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Unit I: Classification of Signals and Systems
Signals and Systems http://DrSatvir.in
Thank YouNext Topic: Classification of Continuous-Time and Discrete-Time Signals
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