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

Outline

1. Basic definitions

2. Continuous-time and discrete-time signals

3. Elementary signals• Unit impulse signal

• Unit step signal

• Unit ramp signal

4. Relations among elementary signals

5. Signal operations

6. Composite signals

7. Practice problem

8. Question (Exam point of view)

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 𝑓 𝑡 has infinite values corresponding to infinite

time 𝑡 values. Mathematically,

𝑓 𝑡 = 𝑢(𝑡 − 1)

Discrete Time (DT) Signals

It is signal that is obtained after sampling of Continuous Time at equal

intervals. Mathematically,

𝑓 𝑛𝑇 = 𝑢 𝑛𝑇 − 1 or 𝑓 𝑛 = 𝑢 𝑛 − 1

Continuous Time and Discrete Time – Graphically

𝑡

𝑥(𝑡)

Continuous Signal

𝑛

𝑥(𝑛)

0 1 2 3 4 5 6 7

Discrete Time Signal

𝑛

𝑥(𝑛)

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Digital Signal

Unit I: Classification of Signals and Systems

Signals and Systems http://DrSatvir.in

Elementary Signals

Unit Impulse Signal

Continuous-Time Signal

𝛿 𝑡 = ቊ1 𝑡 = 00 𝑡 ≠ 0

Discrete-Time Signal

𝛿 𝑛 = ቊ1 𝑛 = 00 𝑛 ≠ 0

𝛿(𝑡)

𝑡0

𝛿(𝑛)

𝑛0 1 2 3−1−2−3

An impulse signal has zero value except at 𝑡 = 0. It has infinitelyhigh value 𝑡 = 0.

Unit Step Signal

Continuous-Time Signal

𝑢 𝑡 = ቊ1 𝑡 ≥ 00 𝑡 < 0

Discrete-Time Signal

𝑢 𝑛 = ቊ1 𝑛 ≥ 00 𝑛 < 0

A unit step signal has unity value for 𝑡 ≥ 0 else zero value.

𝑢(𝑡)

𝑡0

𝑢(𝑛)

𝑛0 1 2 3−1−2−3

Unit Ramp Signal

Continuous-Time Signal

𝑟 𝑡 = ቊ𝑡 𝑡 ≥ 00 𝑡 < 0

Discrete-Time Signal

𝑟 𝑛 = ቊ𝑛 𝑛 ≥ 00 𝑛 < 0

A ramp step signal has unity slop value for 𝑡 ≥ 0, otherwise it haszero value.

𝑟(𝑡)

𝑡0

𝑟(𝑛)

𝑛0 1 2 3−1−2−3

Rectangular Pulse Signal

Continuous-Time Signal

𝑔 𝑡 = ቐ1 −

𝜏

2≤ 𝑡 ≤ +

𝜏

20 Otherwise

Discrete-Time Signal

𝑔 𝑛 = ቊ1 −𝑚 ≤ 𝑛 ≤ +𝑚0 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.

𝑔(𝑡)

𝑡0 𝜏

2−𝜏

2

1𝑔(𝑛)

𝑛0 1 2 3−1−2−3

1

Exponential Signal

Continuous-Time Signal

𝑔 𝑡 = 𝐴𝑒𝑏𝑡 𝐴 > 0

Discrete-Time Signal

𝑔 𝑛 = 𝐴𝑒𝑏𝑛 𝐴 > 0

An exponential signal can either have exponentially rising orfalling amplitude depending upon its exponent value.

𝑔(𝑛)

𝑛0 1 2 3−1−2−3

𝐴

𝑔 𝑡

𝑡0

𝐴

𝑏 > 0𝑔 𝑡

𝑡0

𝐴

𝑔(𝑛)

𝑛0 1 2 3−1−2−3

𝐴

𝑏 < 0

Unit I: Classification of Signals and Systems

Signals and Systems http://DrSatvir.in

Relationships Impulse, Step & Ramp Signals

Relations - Integration & Differentiation

𝑢(𝑡)

𝑡0

𝛿(𝑡)

𝑡0

𝑟(𝑡)

𝑡0

𝑑𝑢 𝑡

𝑑𝑡= 𝛿 𝑡

𝑑𝑟 𝑡

𝑑𝑡= 𝑢 𝑡

𝑟 𝑡 = 𝑡

Inte

gra

tio

n

Dif

fere

nti

ati

on

𝛿 𝑡

න𝛿 𝑡 𝑑𝑡 = 𝑢 𝑡

න𝑢 𝑡 𝑑𝑡 = 𝑡 = 𝑟 𝑡

Unit I: Classification of Signals and Systems

Signals and Systems http://DrSatvir.in

Signal Operations

Signal Operations - Right Shifting

𝛿 𝑡 − 𝜏 = ቊ1 𝑡 = 𝜏0 𝑡 ≠ 𝜏

𝑢 𝑡 − 𝜏 = ቊ1 𝑡 ≥ 𝜏0 𝑡 < 𝜏

𝑟 𝑡 − 𝜏 = ቊ𝑡 𝑡 ≥ 𝜏0 𝑡 < 𝜏

Impulse Signal

Step Signal

Ramp Signal

𝛿(𝑡 − 𝜏)

𝑡0 𝜏

𝑢(𝑡 − 𝜏)

𝑡0 𝜏

𝑟(𝑡 − 𝜏)

𝑡0 𝜏

Signal Operations - Left Shifting

𝛿 𝑡 + 𝜏 = ቊ1 𝑡 = −𝜏0 𝑡 ≠ −𝜏

𝑢 𝑡 + 𝜏 = ቊ1 𝑡 ≥ −𝜏0 𝑡 < −𝜏

𝑟 𝑡 + 𝜏 = ቊ𝑡 𝑡 ≥ −𝜏0 𝑡 < −𝜏

Impulse Signal

Step Signal

Ramp Signal

𝛿(𝑡 + 𝜏)

𝑡0−𝜏

𝑢(𝑡 + 𝜏)

𝑡0−𝜏

𝑟(𝑡 + 𝜏)

𝑡0−𝜏

Signal Time Operations

Time Reversal

Right Shifting

Left Shifting

𝑥(𝑡)

𝑡0−𝜏1 +𝜏2

𝑥(−𝑡)

𝑡0−𝜏2 +𝜏1

𝑥(𝑡 − 𝜏)

𝑡0−𝜏1 +𝜏2𝜏

𝑥(𝑡 + 𝜏)

𝑡0−𝜏1 +𝜏2−𝜏

Expansion 𝑎 < 1𝑥

𝑡

2

𝑡0−𝑎𝜏1 +𝑎𝜏2

Compression 𝑎 > 1

𝑥 2𝑡

𝑡0−𝑎𝜏1 +𝑎𝜏2

Time Scaling 𝑥 𝑎𝑡

Unit I: Classification of Signals and Systems

Signals and Systems http://DrSatvir.in

Composite Signal

Composite Signals

𝑥 𝑡 = 𝑟 𝑡 − 𝑢 𝑡 − 4 + 𝑟 𝑡 − 4

𝑟(𝑡 − 4)

𝑡0 1 2 3 4 75 6

𝑟(𝑡)

𝑡0 1 2 3 4 75 6

𝑢(𝑡 − 4)

𝑡0 1 2 3 4 75 6

𝑥(𝑡)

𝑡0 1 2 3 4

𝑥(𝑡)

𝑡0 1 2 3 4 75 6

−𝑟(𝑡 − 4)−𝑢(𝑡 − 4)

𝑟(𝑡)

1

2

3

1 -2 -3+ +

Unit I: Classification of Signals and Systems

Signals and Systems http://DrSatvir.in

Practice Problem

Signal Operation – Practice Problem

Given a Continuous-Time signal 𝒙 𝒕 is shown in the following figure.

Sketch following signals: a) 𝒚𝟏(𝒕) = 𝒙 𝒕 + 𝟏 (Right Shifting)

b) 𝒚𝟐(𝒕) = 𝒙 𝒕 − 𝟏 (Left Shifting)

c) 𝒚𝟑(𝒕) = 𝒙 −𝒕 (Time Reversal)

d) 𝒚𝟒(𝒕) = 𝒙𝒕

𝟐(Time Expansion)

e) 𝒚𝟓(𝒕) = 𝒙 𝟐𝒕 (Time Compression)

𝑥(𝑡)

𝑡0 1 2 3 4−1−2−3−4

Signal Operation – Solution

𝑥(𝑡)

𝑡0 1 2 3 4−1−2−3−4

Right Shifting 𝒚𝟏(𝒕) = 𝒙 𝒕 + 𝟏

Left Shifting𝒚𝟐 𝒕 = 𝒙 𝒕 − 𝟏

𝑦1(𝑡)

𝑡0 1 2 3 4−1−2−3−4 5

𝑦2(𝑡)

𝑡0 1 2 3 4−1−2−3−4

Signal Operation – Solution

Time Compression𝒚𝟓(𝒕) = 𝒙 𝟐𝒕

Time Reversal𝒚𝟑(𝒕) = 𝒙 −𝒕

Time Expansion

𝒚𝟒(𝒕) = 𝒙𝒕

𝟐

𝑦3(𝑡)

𝑡0 1 2−1−2−3−4

𝑥(𝑡)

𝑡0 1 2 3 4−1−2−3−4

𝑦4(𝑡)

𝑡3 4 5 6 7210−1 8

𝑦5(𝑡)

𝑡0 1 2 3−1−2−3

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.

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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