EERI 222 – Signal Theory 1
Study Unit 2 – Continuous-Time Signals and
Systems
Prof ASJ Helberg
2026
Class Program
July
14
Introduction to Course, Matlab
16
SU 1.1 - 1.3
17
SU 2.1 - 2.2
21
TUT SU1
23
SU 2.3-2.4
24
August
Sept
TUT SU2
30
Semester Test 2
July
Date
August
Event
Date
Saturday
1
Sunday
2
Monday
3
Tuesday
4
RECESS
September
Event
Date
Event
October
Date
Event
November
Date
2
TUTSU 3 + miniprac handout
3
6
SU 3.3
3
TUTSU5 + Hand out Prac
Friday
3
7
SU 3.4 - 3.5
4
17
SU 6
Saturday
4
8
5
3
7
18
SU 6
Sunday
5
9
6
4
8
Monday
6
10
Public Holiday
7
5
Tuesday
7
11
TUTSU 3
8
Wednesday
8
Thursday
9
13
SU4
10
Semester test 1
6
TUTPrac
SU 3.1
8
Tut Prac
31
SU 3.2
9
Semester Test 3
4
TUTSU 3 + mini-prac handout
13
Demo Practical
6
SU 3.3
15
Demo Practical
7
SU 3.4 - 3.5
16
Demo Practical
11
TUTSU 3
20
Demo Practical
13
SU4
22
Exam Review
14
Semester test 1
23
Demo Practical (ReDo)
27
Demo Practical (ReDo)
2
Mid Semester
assessments
2
RECESS
5
1
Thursday
TUTSU 6
12
9
Semester Test 2
Mid Semester
assessments
4
1
2
6
RECESS
6
9
TUTPrac
7
10
11
8
Tut Prac
12
Semester Test 3
13
Friday
10
14
11
9
11
15
12
10
14
Sunday
12
16
13
11
15
Monday
13
RECESS
17
14
12
16
Introduction to
Course, Matlab
Matlab Mini-Prac
Tuesday
14
Wednesday
15
Thursday
16
SU 1.1 - 1.3
20
SU4
17
SU 6
15
Demo Practical
19
Friday
17
SU 2.1 - 2.2
21
SU4
18
SU 6
16
Demo Practical
20
Saturday
18
22
19
17
21
22
19
15
TUTSU5 + Hand
out Prac
16
13
Sunday
19
23
20
18
20
24
21
19
Tuesday
21
Wednesday
22
Thursday
23
SU 2.3-2.4
27
SU 5
24
Public Holiday
22
SU 2.5
28
SU 5
25
RECESS
23
TUT SU1
25
TUTSU 4
26
22
Demo Practical
14
Monday
TUTSU 6
23
20
17
25
Exam Review
Demo Practical
(ReDo)
SU4
Friday
24
21
SU4
Saturday
25
29
26
24
28
Sunday
26
30
27
25
29
31
28
26
27
SU 5
Tuesday
28
28
SU 5
Wednesday
29
Thursday
30
SU 3.1
29
SU 3.2
30
Friday
31
Saturday
1
Sunday
2
TUT SU2
29
30
RECESS
27
27
30
Demo Practical
(ReDo)
2nd OPP
28
31
2nd OPP
26
20
27
2nd OPP
24
Matlab Mini-Prac
Monday
1st OPP
23
Demo Practical
18
TUTSU 4
1st OPP
18
21
25
1st OPP
5
Saturday
18
Event
1
1
22
Oct
Days
Wednesday
15
SU 2.5
28
1 - 11
Months
1st OPP
Agenda
• SS 2.1 – Transformations of Signals
• SS 2.2 – Signal Characteristics
• SS 2.3 – Basic Signals
• SS 2.4 – Block Diagrams
• SS 2.5 – Characteristics of Systems
• Resources:
• Phillips et al, Chapter 2
SS 2.1 – Transformations of Signals
Study Outcomes:
• Formulate Continuous-Time Signals by Means of
Mathematical Functions.
• Apply the Following Time-Transformations to OneDimensional Continuous-Time Signals: Reversal;
Scaling, and Translation (Shifting).
• Apply the Following Amplitude-Transformations
on One-Dimensional Continuous-Time Signals:
Reversal; Scaling and Shifting.
Continuous-Time Signals and Systems
Transformations of Signals
Original Signal ⇒ 𝑥(𝑡)
Transformed Signal ⇒ 𝑦 𝑡
Two Types of Signal Transformations:
1. Transformations in Time
2. Transformations in Amplitude
Transformations of Signals: Transformations in Time
Three Sub-Types of Signal Transformations in Time:
𝑦 𝑡 = 𝑥 𝑎𝑡 + 𝑏
1. Time Reversal
𝑦(𝑡) = 𝑥(−𝑡)
2. Time Scaling
𝑦 𝑡 = 𝑥(𝑎𝑡)
3. Time Shifting
𝑦 𝑡 = 𝑥 𝑡 − 𝑡0
Transformations in Time
Time Reversal
Formula:
𝑦(𝑡) = 𝑥(−𝑡)
Description:
Mirror Reflection about the Vertical Axis
Transformations in Time
Time Scaling
Formula:
𝑦 𝑡 = 𝑥 𝑎𝑡
Description:
• If 𝑎 > 1, Time is Sped-Up
• If 𝑎 < 1, Time is Slowed-Down
Transformations in Time
Time Shifting
Formula:
𝑦 𝑡 = 𝑥 𝑡 − 𝑡0 , Where 𝑡0 is Constant
Description:
• If 𝑡0 > 0, Time is delayed or shifted to the right (Going back in time)
• If 𝑡0 < 0, Time is advanced or shifted to the Left (Going forward in time)
Transformations in Time
General Approach to Formulating/Plotting Independent-Variable Transformations
Formula:
𝑦 𝑡 = 𝑥 𝑎𝑡 + 𝑏
Where 𝑎 and 𝑏 are Real Constants
If 𝜏 denotes the time variable in the original signal, then
𝜏−𝑏
𝜏 = 𝑎𝑡 + 𝑏 ⇒ 𝑡 =
𝑎
Transformations in Time
Example:
Plot the transformed signal
𝑦 𝑡 =𝑥 1−
𝑡
2
Answer:
1. Replace 𝑡 with 𝜏, since 𝜏 denotes the time variable in the original signal
2. Identify 𝑎 and 𝑏:
If 𝑦 𝑡 = 𝑥 𝑎𝑡 + 𝑏 , then
1
2
𝑏=1
𝑎=−
Transformations in Time
Example:
Plot the transformed signal
𝑦 𝑡 = 𝑥 1−
𝑡
2
Answer (Continues):
3. Solve 𝑡 for each timestamp from the original time axis
𝜏−𝑏
1
𝑡=
, Where 𝑎 = − and 𝑏 = 1
𝑎
2
𝑡=
For 𝜏 = −1, 0, 1,
2
𝑡 = 4 , 2, 0, −2
𝜏− 1
1
−
2
Transformations in Time
Example:
Plot the transformed signal
𝑦 𝑡 =𝑥 1−
𝑡
2
Answer (Continues):
4. Plot 𝑡 on the x-axis (From small to large)
5. Match the values of 𝑡 with the corresponding 𝜏 values
Transformations of Signals: Transformations in Amplitude
Three sub-types of signal transformations in amplitude:
𝑦(𝑡) = 𝐴𝑥(𝑡) + 𝐵
1. Amplitude Reversal
𝑦 𝑡 = −𝑥(𝑡)
2. Amplitude Scaling
𝑦(𝑡) = 𝐴𝑥(𝑡)
3. Amplitude Shifting
𝑦(𝑡) = 𝑥(𝑡) + 𝐵
Transformations in Amplitude
Amplitude Reversal
Formula:
𝑦 𝑡 = −𝑥(𝑡)
Description:
Mirror reflection about the horizontal axis
Transformations in Amplitude
Amplitude Scaling
Formula:
𝑦 𝑡 = 𝐴𝑥 𝑡
Description:
• If 𝐴 > 1, Signal is Amplified
• If 𝐴 < 1, Signal is Attenuated
Transformations in Amplitude
Amplitude Shifting
Formula:
𝑦 𝑡 = 𝑥 𝑡 + 𝐵, where 𝐵 is Constant
Description:
• If 𝐵 > 0, signal is shifted upward (Raising DC)
• If 𝐵 < 0, signal is shifted downward (Lowering DC)
Transformations in Amplitude
Example:
Plot the transformed signal
𝑦 𝑡 = 3𝑥 𝑡 − 1
Answer:
1. Solve 𝑦(𝑡) for each value from the original signal axis
For 𝑥 𝑡 = 0, 1, 1, 0, 1, 1, 0
𝑦 𝑡 = −1, 2, 2, −1, 2, 2, −1
2. Plot 𝑦(𝑡)
Transformations in Time & Amplitude
Example:
Plot the transformed signal
𝑦 𝑡 = 3𝑥 1 −
𝑡
−1
2
Answer:
1. Start with the amplitude transformation
2. Perform the time transformation
3. Plot 𝑦(𝑡)
Summary: Signal Transformations
Individual Activity 2.1
1. Tackle Problem 2·1(a) on p. 78 of Phillips. (Purely
to Make Sure that You are Able to Execute TimeTransformation of Signals Independently.
2. Do the Following Problems at the End of Chapter 2
in Phillips (4th and 5th edition) as Preparation for
the Tutorial Session: 2·1(b); 2·2(c); 2·3(a)
Questions?
SS 2.2 – Signal Characteristics
Study Outcomes:
• Know the definitions of even and uneven functions and
describe it in your own words.
• Deduce expressions with which the even and/or uneven
components of a signal can be calculated from the signal and
a mirror image thereof.
• Prove the characteristics of even and uneven functions.
• Determine the even and uneven components of an arbitrary
signal.
• Determine the average value of a signal.
• Write down the definition of periodic signals.
• Determine whether an arbitrary signal is periodic or not
(and calculate the period of the signal if it is a periodic
signal).
Signal with Even Symmetry
Formula:
𝑥𝑒 (𝑡) = 𝑥𝑒 −𝑡
Description:
Symmetry with respect to the vertical axis
Example:
𝑥 𝑡 = cos 𝜔𝑡
Signal with Uneven Symmetry
Formula:
𝑥𝑜 𝑡 = −𝑥𝑜 −𝑡
Description:
Symmetry with respect to the origin
Example:
𝑥 𝑡 = sin 𝜔𝑡
Signal Characteristic Expressions
Defining Any Signal
𝑥 𝑡 = 𝑥𝑒 𝑡 + 𝑥𝑜 (𝑡)
Finding the Even Part of a Signal
𝑥𝑒 𝑡 =
1
𝑥 𝑡 + 𝑥(−𝑡)
2
𝑥𝑜 𝑡 =
1
𝑥 𝑡 − 𝑥(−𝑡)
2
Finding the Odd Part of a Signal
Finding the Average Value of a Signal
𝐴𝑥 = 0
For Uneven Signals
1 𝑇
1 𝑇
𝐴𝑥 = lim
𝑥 𝑡 𝑑𝑡 = lim
𝑥𝑒 𝑡 𝑑𝑡
𝑇→∞ 2𝑇 −𝑇
𝑇→∞ 2𝑇 −𝑇
Characteristics of Even and Odd Functions
1. The sum of two even functions is even
2. The sum of two odd functions is odd
3. The sum of an even function and an odd function is neither even nor odd
4. The product of two even functions is even
5. The product of two odd functions is even
6. The product of an even function and an odd function is odd
Periodic Signals
Formula:
𝑥 𝑡 = 𝑥 𝑡 + 𝑛𝑇 ,
𝑇 > 0 and 𝑛 ∈ ℤ
The Fundamental Period (𝑻𝟎 in s):
The minimum value of 𝑇 > 0 that satisfies 𝑥 𝑡 = 𝑥(𝑡 + 𝑇)
The Fundamental Frequency:
𝒇𝟎 (in Hz)
1
𝑓0 = 𝑇
0
or
𝝎𝟎 (in rad/s)
2𝜋
𝜔0 = 2𝜋𝑓0 = 𝑇
0
Test Signals for Periodicity
Sum of periodic signals is periodic ⟺ Ratios of periods are ratios of integers
(i.e., Rational - ℚ)
What is the fundamental period for a sum of periodic signals?
Given a set of periods of relevant signals (𝑇01 , 𝑇02 , … , 𝑇0𝑁 ),
𝑇01
,2 ≤ 𝑖 ≤ 𝑁
𝑇0𝑖
1.
Convert each period ratio
to a ratio of integers (ℤ).
2.
Eliminate common factors from the numerator and denominator for each ratio (Simplify).
3.
Then the common fundamental period is:
𝑇0 = 𝑘0 𝑇01
Where 𝑘0 is the least common multiple of the denominators.
Test Signals for Periodicity
Example:
Given three periodical signals:
7𝑡
6
The sum of which equals 𝑣 𝑡 . To determine whether 𝑣 𝑡 is periodic, we must see if
the ratios of the periods of 𝑥1 𝑡 , 𝑥2 𝑡 and 𝑥3 𝑡 are ratios of integers:
𝑥1 𝑡 = 𝑐𝑜𝑠 3.5𝑡 ,
We use:
𝑥2 = 𝑠𝑖𝑛 2𝑡 ,
𝑎𝑛𝑑
𝑥3 𝑡 = 2 cos
1
2𝜋
0
0
𝜔0 = 2𝜋𝑓0 ⇒ 𝑇0 = 𝑓 = 𝜔
To first determine all the periods:
2𝜋 2𝜋
2𝜋 2𝜋
𝑇01 =
=
,
𝑇02 =
=
,
𝜔1 3.5
𝜔2
2
2𝜋 2𝜋
𝑇03 =
=
.
𝜔3 7
6
Then, the ratios of periods are:
2𝜋
𝑇01
2
4
3.5
=
=
=
2𝜋
𝑇02
3.5
7
2
and
𝑇01
=
𝑇03
Both are ratios of integers and therefore, 𝑣 𝑡 is periodic.
2𝜋
2𝜋
3.5
7 6
7
7
= 3.56 = 21 .
Finding The Fundamental Frequency/Period
The period of 𝑣 𝑡 is determined by first eliminating any common factors from the
numerator and denominator of each ratio of periods.
𝑇
4
• In this case, we have no common factor in the ratio 𝑇01 = 7,
𝑇
7
But the ratio 𝑇01 = 21 has 7 as a common factor.
03
𝑇
𝑇03
02
1
3
• After eliminating the common factor, we have 01 = .
• The next step is to find the least common multiple of the denominators of the ratios.
in this case, the least common multiple is 𝑛1 = 3 × 7 = 21.
• Therefore, the fundamental period of 𝑣 𝑡 is 𝑇0 = 𝑛1 𝑇01 = 21 ×
• This period can be confirmed by examination of Figure 2.11(a).
2𝜋
= 12𝜋
3.5
𝒔 .
Test Signals for Aperiodicity
Another signal, 𝑥4 𝑡 = 3𝑠𝑖𝑛 5𝜋𝑡 , is added to 𝑣 𝑡 to give:
𝑤 𝑡 = 𝑥1 𝑡 + 𝑥2 𝑡 + 𝑥3 𝑡 + 𝑥4 𝑡 .
• This sum of four periodic signals is shown in Figure 2.11(b). The periodicity of
this new sum of signals must be determined as was done previously for 𝑣 𝑡 .
The addition of 𝑥4 𝑡 gives the additional ratio of periods,
2𝜋
𝑇01
3.5 = 5𝜋 .
=
2𝜋
𝑇04
3.5
5𝜋
• Since 𝜋 is an irrational number, this Ratio is NOT Rational.
Therefore, the signal 𝑤(𝑡) is NOT Periodic/Aperiodic.
Individual Activity 2.2
1. Do the following problems at the end of chapter 2
in Phillips as preparation for the tutorial session:
• Phillips et al: 4th edition 2·13(a)(b); 2·14
• Phillips et al: 5th edition 2·11(a)(b); 2·12
Questions?
SS 2.3 – Basic Signals
Study Outcomes:
• Know and explain Euler’s relationship.
• Write a sine or cosine function in terms of exponential functions (and
deduce these formulae).
• Present a sine or cosine function in phasor notation.
• Know and describe in your own words the definition of harmonic-related
complex exponential signals.
• Know the definitions of time-constants and enveloping and practically work
with these magnitudes.
• Know and write down the definitions of the unit step function, rectangularshaped pulse function, unit impulse function and unit gradient function.
• Present functions in terms of linear combinations of unit step functions,
rectangular-shaped pulse functions and straight-line segments.
• Know and be able to name the characteristics of the unit impulse function.
• Plot any signal graphically.
• Describe any signal with the aid of a mathematical function.
Euler’s Relation
Formula:
𝑒 𝑗𝜃 = cos 𝜃 + 𝑗 sin 𝜃
Further Expressions:
𝑒 −𝑗𝜃 = cos 𝜃 − 𝑗 sin 𝜃
𝑒 𝑗𝜃 + 𝑒 −𝑗𝜃
cos 𝜃 =
2
𝑒 𝑗𝜃 − 𝑒 −𝑗𝜃
sin 𝜃 =
2𝑗
Euler’s Relation in Polar Form
Formula:
𝑅𝑒 𝑗𝜃 = 𝑅∠𝜃
Determining the Magnitude:
𝑒 𝑗𝜃
=
2 1 2
2
cos 𝜃 + sin 𝜃
=1
Determining the Angle:
arg 𝑒
𝑗𝜃
= tan
−1
sin 𝜃
=𝜃
cos 𝜃
Euler’s Relation Special Cases
Euler Case 1
𝑥 𝑡 = 𝐶𝑒 𝑎𝑡 = 𝐶𝑒 −𝑡 𝜏 ,
𝜏 > 0 and 𝐶 ∈ ℝ and 𝑎 ∈ ℝ
Examples
Transient response of RL and RC circuits
τ - Time Constant
Number of Time
Constants [𝝉]
Level of Decay
[%]
Level of Increase
[%]
1
36.79
63.21
2
13.53
86.47
3
4.98
95.02
4
1.83
98.17
5
0.67
99.33
Euler’s Relation Special Cases
Euler Case 2
𝑥 𝑡 = 𝐶𝑒 𝑎𝑡 , where 𝐶 = 𝐴𝑒 𝑗𝜙 = 𝐴∠𝜙 and 𝑎 = 𝑗𝜔0
𝜏 > 0 and 𝐶 ∈ ℂ and 𝑎 ∈ 𝕀
Therefore,
𝑥 𝑡 = 𝐴𝑒 𝑗𝜙 𝑒 𝑗𝜔0 𝑡 = 𝐴𝑒 𝑗(𝜔0 𝑡+𝜙)
= 𝐴 cos 𝜔0 𝑡 + 𝜙 + 𝑗𝐴 sin 𝜔0 𝑡 + 𝜙
𝜔0 is the Frequency And
2𝜋
= 𝑇0 is the Period
𝜔0
Examples
Transient response of LC circuits
Harmonic Related Complex Exponentials
A set of functions with frequencies related by integers
𝑥𝑘 𝑡 = 𝐴𝑘 𝑒 𝑗𝑘𝜔0 𝑡 ,
𝑘 = ±1, ±2, … .
Euler’s Relation Special Cases
Euler Case 3
𝑎𝑡
𝑗𝜙
𝑥 𝑡 = 𝐶𝑒 , where 𝐶 = 𝐴𝑒
and 𝑎 = 𝜎0 + 𝑗𝜔0
𝜏 > 0 and 𝐶 ∈ ℂ and 𝑎 ∈ ℂ
Therefore,
Envelope:
± 𝐴𝑒 𝜎0 𝑡
Undamped Sinusoid
(Case 2)
If 𝜎0 = 0
𝑥 𝑡 = 𝐴𝑒 𝑗𝜙 𝑒 (𝜎0 +𝑗𝜔0 )𝑡 = 𝐴𝑒 𝜎0 𝑡 𝑒 𝑗(𝜔0 𝑡+𝜙)
= 𝐴𝑒 𝜎0 𝑡 cos 𝜔0 𝑡 + 𝜙 + 𝑗𝐴𝑒 𝜎0 𝑡 sin 𝜔0 𝑡 + 𝜙
= 𝑥𝑟 𝑡 + 𝑗𝑥𝑖 𝑡
Underdamped
Sinusoid
If 𝜎0 < 0
Examples
Transient response of RLC circuits
Singularity Functions
Two Singularity Functions Will Be Discussed:
1. Unit Step Function
2. Unit Impulse Function
Unit Step Function (Heaviside) – 𝒖 𝒕
𝑢 𝜏 =
1,
0,
𝑢(𝑡 − 𝑡0 ) =
𝜏>0
𝜏<0
1,
0,
𝑡 > 𝑡0
𝑡 < 𝑡0
Used to switch functions (Multiply the unit step function with the function)
Singularity Functions – Unit Step Function
Example:
𝜏= 𝑡−5
𝑢(𝑡 − 5) =
1,
0,
𝑡−5>0⇒𝑡 >5
𝑡−5<0⇒𝑡 <5
Properties:
𝑢(𝑡 − 𝑡0 ) =
1,
0,
𝑡 > 𝑡0
𝑡 < 𝑡0
𝑢 𝑡 − 𝑡0 = 𝑢(𝑡 − 𝑡0 ) 𝑘
𝑢 𝑎𝑡 − 𝑡0 = 𝑢 𝑡 − 𝑡0 /𝑎 , 𝑎 ≠ 0
𝑢(𝑡0 − 𝑡) =
1,
0,
𝑡 < 𝑡0
𝑡 > 𝑡0
Singularity Functions – Unit Step Function
Extra: another useful switching function
Rectangular Pulse Function
rect 𝑡 𝑇 =
𝑇 is the period
1,
0,
−𝑇 2 < 𝑡 < 𝑇 2
otherwise
Singularity Functions – Unit Step Function
Rectangular pulse function can be defined with unit step functions in 3 ways:
𝑢 𝑡+𝑇 2 −𝑢 𝑡−𝑇 2
rect 𝑡 𝑇 = 𝑢 𝑇 2 − 𝑡 − 𝑢 −𝑇 2 − 𝑡
𝑢 𝑡+𝑇 2 ×𝑢 𝑇 2−𝑡
Singularity Functions – Unit Step Function
Time-Shifted rectangular pulse function
rect 𝑡 − 𝑡0
𝑇 =
1,
0,
𝑡0 − 𝑇 2 < 𝑡 < 𝑡0 + 𝑇 2
otherwise
Singularity Functions – Unit Step Function
Example: Use switching functions
𝑥 𝑡 = cos 𝑡 × 𝑢 𝑡 − 𝑢 𝑡 − 2𝜋
=
cos 𝑡 ,
0,
0 < 𝑡 < 2𝜋
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Or
𝑥 𝑡 = cos 𝑡 × rect 𝑡 − 𝜋 2𝜋
Periodic signal representation for pulse functions:
𝑣 𝑡 = 𝑣1 𝑡 + 𝑣1 𝑡 − 𝑇0 + 𝑣1 𝑡 − 2𝑇0 + ⋯
∞
=
𝑣1 𝑡 − 𝑘𝑇0
𝑘=0
Singularity Functions – Unit Impulse Function
Unit Impulse Function (Dirac Delta) – 𝜹(𝒕)
Integrate Unit Step Function
𝑡
𝑓 𝑡 =
𝑢 𝜏 − 𝑡0 𝑑𝜏 =
0
𝑡
𝑡
𝑑𝜏 = 𝜏
𝑡0
𝑡0
= 𝑡 − 𝑡0 𝒖 𝒕 − 𝒕𝟎
Unit Ramp Function
𝑥 𝑡 = 𝑡 − 𝑡0 𝑢 𝑡 − 𝑡0
Singularity Functions – Unit Impulse Function
What is the Unit Impulse Function?
• Energy in a physical system cannot be transferred instantaneously.
Therefore, when zoomed in, a more practical unit step function looks like
𝑓 𝑡 , with 𝜖 → 0.
• Deriving 𝑓(𝑡), results in the rectangular pulse 𝑔(𝑡) with width 𝜖 and
amplitude → ∞. The integral = 1 (from 𝑓(𝑡)), and thus, the area under
𝑔 𝑡 = 1.
• We call the limit of this function (When 𝜖 → 0) The Unit Impulse
Function.
lim 𝑔(𝑡) = 𝛿 𝑡 − 𝑡0
𝜖→0
Therefore, The unit impulse function can be described by
𝛿 𝑡 − 𝑡0 = 0, 𝑡 ≠ 𝑡0 ;
∞
𝛿(𝑡 − 𝑡0 ) 𝑑𝑡 = 1
−∞
Singularity Functions – Unit Impulse Function
Shifting Properties of The Unit Impulse Function
∞
𝑓(𝑡)𝛿(𝑡 − 𝑡0 ) 𝑑𝑡 = 𝑓(𝑡0 )
−∞
And thus,
𝑓 𝑡 𝛿 𝑡 − 𝑡0 = 𝑓(𝑡0 )𝛿(𝑡 − 𝑡0 )
Singularity Functions – Unit Impulse Function
Summary of the Unit Impulse Function’s Properties
Mathematical Functions for Signals
Example: Function to Graph
Mathematical Functions for Signals
Graph to Function Procedure
When the slope of a signal changes, a ramp function is added at that point, with the slope of this ramp function
equal to the new slope minus the previous slope (𝑚1 − 𝑚0 ). At any point that a step occurs in the signal, a step
function is added. An example using this procedure is now given.
𝑥 𝑡 = 𝑚0 𝑡 − 𝑡0 𝑢 𝑡 − 𝑡0 + 𝑚1 − 𝑚0 𝑡 − 𝑡1 𝑢 𝑡 − 𝑡1 + ⋯
Example: Graph to Function
𝑥 𝑡 = 3 𝑡 + 2 𝑢 𝑡 + 2 − 6 𝑡 + 1 𝑢 𝑡 + 1 + 3 𝑡 − 1 𝑢 𝑡 − 1 + 3𝑢 𝑡 − 3
Another method is discussed in the study guide, which takes much longer
Mathematical Functions for Signals
Example: Graph to Function
𝑉
𝑣 𝑡 = 𝑡,
𝑇
0<𝑡<𝑇
Or
𝑉
𝑉
𝑣 𝑡 = 𝑡 𝑢 𝑡 − 𝑢(𝑡 − 𝑇) = 𝑡 rect
𝑇
𝑇
𝑇
𝑡−
2
𝑇
Mathematical Functions for Signals
Example: Periodic signal representation for pulse functions (Pulse Train)
From the previous example, 𝑣1 𝑡 can be written As:
𝑉
𝑉
𝑇
𝑣1 𝑡 = 𝑡 𝑢 𝑡 − 𝑢(𝑡 − 𝑇) = 𝑡 rect 𝑡 −
𝑇
𝑇
𝑇
2
If shifted one period to the right;
𝑉
𝑣1 𝑡 − 𝑇 = 𝑡 − 𝑇 𝑢 𝑡 − 𝑇 − 𝑢(𝑡 − 2𝑇)
𝑇
𝑉
3𝑇
= 𝑡 − 𝑇 rect 𝑡 −
𝑇
𝑇
2
Therefore, for 𝑘𝑇 < 𝑡 < (𝑘 + 1)𝑇 or 𝑣1 𝑡 − 𝑘𝑇
𝑉
𝑣1 𝑡 − 𝑘𝑇 = 𝑡 − 𝑘𝑇 𝑢 𝑡 − 𝑘𝑇 − 𝑢(𝑡 − 𝑘𝑇 − 𝑇) ,
𝑇
The Sawtooth wave can be represented as:
∞
𝑣 𝑡 =
𝑣1 𝑡 − 𝑘𝑇
𝑘=−∞
𝑘∈ℤ
Individual Activity 2.3
Do the following problems at the end of
Chapter 2 in Phillips as preparation for
the tutorial session:
• Phillips et al: 4th edition:
• 2·8
• 2·12(c)(e)
• 2·7(a)(d)
• 2·11
• 2·20(c)
• 2·21(a)(c)
• 2·19(a)(b)(c)
• Phillips et al: 5th edition:
• 2.6
• 2.1 (c)(e)
• 2·5(a)(d)
• 2.9
• 2.19 (c)
• 2·20(a)(c)
• 2·26(a)(b)(c)
Questions?
SS 2.4 – Block Diagrams
Study Outcomes:
• Formulate mathematical equations from system
block diagrams.
• Draw up block diagrams representing system
equations.
Block Diagrams
System – A process for which cause-and-effect relations exist
𝑦 𝑡 = 𝑇[𝑥 𝑡 ]
𝑇[𝑥 𝑡 ] is referred to as the mathematical model. This is usually differential
equations for continuous-time systems.
Block Diagram Components
1. System Component – Block
2. Junction Components – Circle
• Summing Junction
• Product Junction
Block Diagram Components
3. System Connections
• Parallel Connection
𝑦 𝑡 = 𝑦1 𝑡 + 𝑦2 𝑡 = 𝑇1 𝑥 𝑡
+ 𝑇2 𝑥 𝑡
=𝑇𝑥 𝑡
• Series or Cascade Connection
𝑦 𝑡 = 𝑇2 𝑦1 𝑡
= 𝑇2 𝑇1 𝑥 𝑡
= 𝑇[𝑥 𝑡 ]
Block Diagrams Example
Example:
𝑦1 𝑡 = 𝑇1 𝑥 𝑡
𝑦2 𝑡 = 𝑇2 𝑥 𝑡
𝑦3 𝑡 = 𝑇1 𝑥 𝑡
𝑦4 𝑡 = 𝑇3 𝑦3 𝑡
+ 𝑇2 𝑥 𝑡
= 𝑇3 𝑇1 𝑥 𝑡
+ 𝑇2 𝑥 𝑡
𝑦5 𝑡 = 𝑇4 𝑥 𝑡
𝑦 𝑡 = 𝑦4 𝑡 × 𝑦5 𝑡 = 𝑇3 𝑇1 𝑥 𝑡
+ 𝑇2 𝑥 𝑡
× 𝑇4 𝑥 𝑡
Control Theory – Feedback-Control System (Automatic Control)
• The plant is the physical system to be controlled.
• The controller is a physical system inserted by the design engineers to give the
total system certain desired characteristics.
• The sensor measures the signal to be controlled, and the input signal represents
the desired output.
• The Error Signal (𝑒 𝑡 ) is a measure of the difference between the desired
output (𝑥 𝑡 ) and the measurement of the output (𝑦 𝑡 ).
𝑒 𝑡 = 𝑥 𝑡 − 𝑇3 𝑦 𝑡
𝑦 𝑡 = 𝑇2 𝑚 𝑡
= 𝑇2 𝑇1 𝑒 𝑡
𝑦 𝑡 = 𝑇2 𝑇1 𝑥 𝑡 − 𝑇3 𝑦 𝑡
Individual Activity 2.4
1. Do the following problems at the end of Chapter 2
in Phillips as preparation for the tutorial session:
• Phillips et al: 4th edition: 2·23(a)(b); 2·24; 2·25
• Phillips et al: 5th edition: 2.30; 2.28; 2.29
Questions?
SS 2.5 – Characteristics of Systems
Study Outcomes:
• know the definitions for the following characteristics
of continuous-time systems: memory; reversibility;
system-inverse; causality; stability; timeindependency and linearity.
• Determine which of the above characteristics are
displayed by an arbitrary system.
Characteristics of Systems
𝒙 𝒕 Produces 𝒚 𝒕
𝑥 𝑡 →𝑦 𝑡
𝑦 𝑡 =𝑇 𝑥 𝑡
The Six Properties of Continuous-Time Systems
1.
2.
3.
4.
5.
6.
Memory
Invertibility
Causality
Stability
Time Invariance
Linearity
Properties of Continuous-Time Systems - Memory
Memory (Dynamic System)
A system has memory if its output at time 𝑡0 (i.e. 𝑦 𝑡0 ) depends on input values
other than 𝑥 𝑡0 . Otherwise, the system is memoryless.
System that Has Memory: Dynamic System
Examples: Integrating Amplifier, Voltage of Capacitor
System that Does not Have Memory: Static System
Examples: Ideal Amplifier, Resistance, Squaring Circuit
Properties of Continuous-Time Systems - Invertibility
Invertibility
A system is said to be invertible if distinct inputs result in distinct outputs. (For an invertible
system, the system input can be determined uniquely from its output.)
System that Is Invertible: Invertible System
Examples: Ideal Amplifier
System that Is Not Invertible: Non-Invertible System
Examples: Squaring Circuit
Special Case: Identity System – Output is equal to its input
Examples: Ideal Amplifier with gain of unity
Inverse of a System
The inverse of a system (Denoted by 𝑇) is a second system (Denoted by 𝑇𝑖 ) that, when
cascaded with the system 𝑇, yields the identity system.
𝑦 𝑡 =𝑇 𝑥 𝑡
⇒ 𝑥 𝑡 = 𝑇𝑖 𝑦 𝑡
Transducers must have invertible systems
Properties of Continuous-Time Systems – Causality
Causality
A system is causal if the output at any time 𝑡0 is dependent on the input only for 𝑡 ≤ 𝑡0 .
System that is Causal: Non-Anticipatory System
Examples: Any Physical System, Ideal Time Delay
System that is Not Causal: Non-Real Time System
Examples: Ideal Time Advance
Properties of Continuous-Time Systems – Stability
Bounded-Input-Bounded-Output (BIBO) Stability
A system is stable if the output remains bounded for any bounded input. (i.e.,
for all time, the input signal (𝑥 𝑡 )) and output signal (𝑦 𝑡 ) are both bounded.
System that is Stable: Stable System
Examples: Ideal Amplifier, Squaring System
System that is Not Stable: Unstable System
Examples: Ramp Function (e.g., Integrating Unit Step)
Special Case: Marginally Stable – Undefined Bounds
Examples: Exponential Functions
Properties of Continuous-Time Systems – Time Invariance
Time Invariance
A system is said to be time invariant if a time shift in the input signal results only in the
same time shift in the output signal.
𝑦 𝑡 − 𝑡0 = 𝑇 𝑥 𝑡 − 𝑡0
𝑦 𝑡 − 𝑡0 = 𝑦𝑑 𝑡
System that is Time-Invariant: Fixed System
Examples: Recording
System that is Time Varying:
Examples: Time Reversal Operation
Properties of Continuous-Time Systems – Linearity
Linearity
A System is linear if it meets the following two criteria:
1. Additivity:
2. Homogeneity:
If 𝑥1 𝑡 → 𝑦1 𝑡 and 𝑥2 𝑡 → 𝑦2 𝑡
𝑥1 𝑡 + 𝑥2 𝑡 → 𝑦1 𝑡 + 𝑦2 𝑡 .
If 𝑥1 𝑡 → 𝑦1 𝑡 , then
𝑎𝑥1 𝑡 → 𝑎𝑦1 𝑡 ; ∀ 𝑥1 𝑡 ,
𝑥2 𝑡 and 𝑡.
I.E, Principle of Superposition Holds:
𝑎1 𝑥1 𝑡 + 𝑎2 𝑥2 𝑡 → 𝑎1 𝑦1 𝑡 + 𝑎2 𝑦2 𝑡
System that Has Linearity: Linear System
Examples: Ideal Amplifier
System that Does Not Have Linearity: Non-Linear System
Examples: Squaring Circuit, Systems with Non-Zero Initial Conditions
Examples
Examples
Individual Activity 2.5
1. Do the Following Problems at the End of Chapter 2
in Phillips as Preparation for the Tutorial Session:
2·31(a)(d)
• Phillips et al: 4th edition: 2·31(a)(d)
• Phillips et al: 5th edition: 2·35(a)(d)
Questions?