To learn about signal and system

1. ELT2035 Signals & Systems
Hoang Gia Hung
Faculty of Electronics and Telecommunications
University of Engineering and Technology, VNU Hanoi
Lesson 1: Introduction to signals

2. ❑ Started to be in ECE curricula in the 1980s (the first
textbook was Signals and Systems, by A. V. Oppenheim
and A. S. Willsky, published in 1983)
❑ Concepts of signals and systems
❑ Mathematical descriptions of signals and systems
❑ Analysis of Linear Time Invariant Systems
Course overview
Provide the necessary background for follow-up courses at UET:
ELT3051 – Control Engineering
ELT3144E – Digital Signal Processing
ELT3057 – Digital Communications and Coding Theory
ELT3094 – Introduction to Signal Processing for Multimedia Systems
ELT3281 – Microprocessor and embedded system

3. ❑ Roughly speaking, anything that carries information can
be considered a signal: speech, ECG, VN index, …
❑ Plotted against time, which is called an independent
variable
❑ A signal may have more independent variables: pictures,
videos, …
What is signal?

4. ❑ Roughly speaking, any physical device or computer
program can be considered a system if the application of a
signal to the device or program generates a new signal.
What is system?
System
Input signal Output signal
Design/build a system to obtain desirable outputs from the input

5. ❑ Continuous – Discrete time signals
❑ What is time?
❑ Periodic – Nonperiodic signals
❑ Causal – Anticausal – Noncausal signals
❑ Odd – Even signals
❑ Deterministic – Random signals
❑ Finite – infinite length signals
❑ Multichannel – multidimensional signals
Classification of signals
Conversion of a CT signal to a DT signal by sampling

6. ❑ The total energy of a continuous time signal f(t) is
𝐸𝑓 = න
−∞
∞
𝑓(𝑡) 2𝑑𝑡
❑ And its average power is
𝑃𝑓 = lim
𝑇→∞
1
𝑇
න
− Τ
𝑇 2
Τ
𝑇 2
𝑓(𝑡) 2𝑑𝑡
❑ Similarly, for a discrete time signal f(n)
𝐸𝑓 = ෍
𝑛=−∞
∞
𝑓[𝑛] 2
𝑃𝑓 = lim
𝑁→∞
1
2𝑁 + 1
෍
𝑛=−𝑁
𝑁
𝑓[𝑛] 2
Energy and power of signals

7. ❑ A signal is referred to as an energy signal iff the total energy
of the signal is bounded
❑ A signal is referred to as a power signal iff the average
power of the signal is bounded
❑ Quiz: Find the energy and power of 𝑓 𝑡 = sin 𝑡
Energy and power signals
❑ Solution:
➢ 𝐸𝑓 = lim
𝑇→∞
‫׬‬−𝑇
𝑇
sin 𝑡 2 𝑑𝑡 = lim
𝑇→∞
‫׬‬−𝑇
𝑇
sin2 𝑡 𝑑𝑡 = lim
𝑇→∞
1
2
ቂ
ቃ
‫׬‬−𝑇
𝑇
𝑑𝑡 −
‫׬‬−𝑇
𝑇
cos 2𝑡 𝑑𝑡 = lim
𝑇→∞
ቚ
𝑡
2
−
sin 2𝑡
4 −𝑇
𝑇
= lim
𝑇→∞
𝑇−(−𝑇)
2
−
sin 2𝑇−sin(−2𝑇)
4
= ∞.
➢ 𝑃𝑓 = lim
𝑇→∞
1
2𝑇
‫׬‬−𝑇
𝑇
sin 𝑡 2
𝑑𝑡 = lim
𝑇→∞
1
2𝑇
𝑇−(−𝑇)
2
−
sin 2𝑇−sin(−2𝑇)
4
=
1
2
.
❑ The energy and power classifications of signals are mutually
exclusive
➢ There are signals that are neither energy nor power signals

8. ❑ Objective: design/built a system to manipulate signals.
How are signals be manipulated inside a system?
❑ Operations performed on dependent variables: amplitude
scaling, addition, multiplication, differentiation, integration.
❑ Operations performed on the independent variable:
Basic operations on signals
➢ Time scaling
➢ Reflection
➢ Time shifting
A system is usually built by combining multiple basic operations on
input signals to obtain the desirable output signals.
The product is called an exponentially damped signal.

9. Examples of signals multiplication
❑ Sketch the signal 𝑥 𝑡 = 4𝑒−2𝑡 cos(6𝑡 − 60°)

10. Examples of time shifting a signal
❑ Given 𝑓 𝑡 = ቊ
𝑒−2𝑡
, 𝑡 ≥ 0
0, 𝑡 < 0
, sketch the signals 𝑓 𝑡 − 1 & 𝑓 𝑡 + 1

11. Examples of time scaling a signal
❑ A signal 𝑓 𝑡 is depicted below. Sketch 𝑓 2𝑡 & 𝑓
𝑡
2
.

12. ❑ Unit step signal: 𝑢(𝑡) = ቊ
1, 𝑡 ≥ 0
0, 𝑡 < 0
❑ Unit impulse signal (a.k.a. Dirac delta function): 𝛿 𝑡 = 0 ∀𝑡 ≠ 0
and ‫׬‬
−∞
∞
𝛿 𝑡 𝑑𝑡 = 1. Notice that 𝛿 𝑡 is undefined at 𝑡 = 0.
❑ Unit ramp signal: 𝑡𝑢(𝑡)
❑ Sinusoidal signal: 𝐴 cos(𝜔𝑡 + 𝜑)
❑ (Real) exponential signal: 𝐵𝑒𝛼𝑡
❑ (Complex) exponential signal: 𝑒𝑠𝑡 where 𝑠 = 𝜎 + 𝑗𝜔.
Elementary signals
Why do we need elementary signals?
Modelling natural signals
Construct more complex signals
System identification

13. ❑ The important of the unit impulse is not its shape but the fact
that its width approaches zero while its area remains unity.
❑ Multiplication of a unit impulse 𝛿 𝑡 by a function 𝑥(𝑡) that is
known to be continuous at 𝑡 = 0:
➢ 𝑥 𝑡 𝛿 𝑡 = 𝑥(0)𝛿(𝑡).
➢ Similarly, 𝑥 𝑡 𝛿 𝑡 − 𝑇 = 𝑥(𝑇)𝛿(𝑡 − 𝑇), provided 𝑥(𝑡) is continuous at 𝑡 = 𝑇.
❑ Sampling/sifting property:
➢ ‫׬‬
−∞
∞
𝑥 𝑡 𝛿 𝑡 𝑑𝑡 = ‫׬‬
−∞
∞
𝑥 0 𝛿 𝑡 𝑑𝑡 = 𝑥 0 ‫׬‬
−∞
∞
𝛿 𝑡 𝑑𝑡 = 𝑥 0 . Similarly,
‫׬‬
−∞
∞
𝑥 𝑡 𝛿 𝑡 − 𝑇 𝑑𝑡 = 𝑥(𝑇).
➢ The area under the product of a function with an unit impulse equals the
value of that function where the unit impulse is located.
❑ Time-scaling property: 𝛿 𝑎𝑡 =
1
𝑎
𝛿(𝑡). Proof: HW.
❑ Unit impulse is not an ordinary function but rather a generalized
function.
➢ In this approach, 𝛿 𝑡 is defined by its effect on other functions at every
instant of time (i.e. the sampling property).
Impulse properties

14. Example of constructing a complex
signal from elementary signals

15. ❑ Classification of signal: determining the type of a given
signal
❑ Calculation of the total energy and power of a given signal
❑ Performing basic operations, especially a combination of
time scaling and time shifting, on a given signal
❑ Construction of a complex signal from several elementary
signals
Practice