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Slide 1 ELT2035 Signals & Systems Hoang Gia Hung Faculty of Electronics and Telecommunications University of Engineering and Technology, VNU Hanoi Lesson 1 Introduction to signals ❑ Started to be in E[.]

ELT2035 Signals & Systems

Hoang Gia Hung Faculty of Electronics and Telecommunications University of Engineering and Technology, VNU Hanoi

Lesson 1: Introduction to signals

❑ 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

❑ 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?

❑ 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

❑ 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

❑ 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

− 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

❑ 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→∞ ׬−𝑇𝑇 sin 𝑡 2 𝑑𝑡 = lim

𝑇→∞ ׬−𝑇𝑇 sin2 𝑡 𝑑𝑡 = lim

𝑇→∞

1

2 ቂ ቃ

׬−𝑇𝑇 𝑑𝑡 −

׬−𝑇𝑇 cos 2𝑡 𝑑𝑡 = lim

2 − sin 2𝑡

4 −𝑇

𝑇

= lim 𝑇→∞

𝑇−(−𝑇)

2 − sin 2𝑇−sin(−2𝑇)

➢ 𝑃𝑓 = lim

𝑇→∞

1 2𝑇 ׬−𝑇𝑇 sin 𝑡 2𝑑𝑡 = lim

𝑇→∞

1 2𝑇

𝑇−(−𝑇)

2 − sin 2𝑇−sin(−2𝑇)

2

❑ The energy and power classifications of signals are mutually

exclusive

➢ There are signals that are neither energy nor power signals

❑ 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.

Examples of signals multiplication

❑ Sketch the signal 𝑥 𝑡 = 4𝑒−2𝑡 cos(6𝑡 − 60°)

Examples of time shifting a signal

❑ Given 𝑓 𝑡 = ቊ𝑒−2𝑡, 𝑡 ≥ 0

0, 𝑡 < 0 , sketch the signals 𝑓 𝑡 − 1 & 𝑓 𝑡 + 1

Examples of time scaling a signal

❑ A signal 𝑓 𝑡 is depicted below Sketch 𝑓 2𝑡 & 𝑓 𝑡

2

❑ Unit step signal: 𝑢(𝑡) = ቊ1, 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

❑ 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

𝑎𝛿(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

Example of constructing a complex

signal from elementary signals

❑ 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