Hansung University Dept. of Information and Communication Eng.1 1Dept. Electronics and Information Engineering

Signal and System (Chapter 1. Fundamentals of Signals)

Prof. Kwang-Chun Hokwangho@hansung.ac.krTel: 02-760-4253 Fax:02-760-4435

Hansung University Dept. of Information and Communication Eng.2 2Dept. Electronics and Information Engineering

Basic Concepts

Signals play an important role in our daily lifeExamples of signals encountered often:speechmusicpicturevideo signals

Definition of signalA signal is a function of independent variables such

as time, distance, position, temperature, and pressure

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Basic Concepts

ExamplesSpeech and music signals

Represent air pressure as a function of time at a point in space

Black-and-white pictureRepresentation of light intensity as a function of two

spatial coordinatesVideo signal in television

Consists of a sequence of images, called frames, and is a function of three variables, two spatial coordinates and time

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Basic Concepts

Most signals we encounter are generated naturally

However, a signal can also be generated synthetically or by a computer

A signal carries informationObjective of signal processing

Extract the useful information carried by the signalMethod of information extraction from system

Depends on the type of signal and the nature of the information being carried by the signal

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Basic Concepts

(Demonstration of different types of signals)

(Music) (Male Voice) (Female Voice)

time t

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MatLab Demo of Speech(Read and play a wave file by using MatLab script)% read Microsoft WAVE (".wav") sound file[y, fs] = wavread(‘doremi.wav’);% play sound using Windows audio output deviceaudioplayer(y, fs)plot((1:length(y))/fs, y);xlabel('Time in seconds');ylabel('Amplitude');

Here, y is the sampled data, and fs is the sample rate in Hertz.

0 2 4 6 8 10 12-0.4

-0.2

0

0.2

0.4

Time in seconds

Ampl

itude

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Classification of Signals

Analog SignalsA signal whose amplitude can take on any value in a

continuous range: Continuous-Amplitude SignalsDigital SignalsA signal whose amplitude can take on only a finite

number of values: Discrete-Amplitude SignalsContinuous-time signalsWill always be treated as a function of the time

variable, tParenthesis (round brackets) will be used to denote

continuous-time functions

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Classification of Signals

Example: x(t) t is a continuous independent variable (real-valued)

Discrete-time signalsWill always be treated as a real-valued function of

the discrete variable, nSquare brackets will be used to denote discrete-time

functionsExample: x[n]n is an independent integerThis course is concerned with the continuous-time

representation of signals and their continuous-time processing

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Classification of Signals

(Analog, discrete-time) (Digital, discrete-time)

(Analog, continuous-time) (Digital, continuous-time)

Will be discussed in this course !

t

t

x(t) x(t)

n

x[n]

n

x[n]

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Classification of Signals

Ideal continuous-to-discrete converterThis can be done by sampling a continuous-time

signal at isolated and equally spaced points in time

(Sampler)(Sampling Time)

-10 0 10 20 30 40-1

-0.5

0

0.5

1

-10 0 10 20 30 40-1

-0.5

0

0.5

1

% MatLab Scriptomega = 0.05*2*pi;Ts = 1; % sampling timen = -10:Ts:40;% discrete-time signalx_n = cos(omega*n); stem(n, x_n);

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Continuous-Time Signals

Periodic signals: This is an important class of signals that we will

encounter frequently. A periodic continuous signal x(t) has property that it repeats itself over a period of time T

In the other words, a periodic signal is unchanged with a time-shift of T and T is called the period

Is the following signal periodic or not?

( )x t x t T

( )x t

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Continuous-Time Signals

Random signals:Noise is a random signalA random signal is a signal such that its value cannot be

predicted at a particular timeA random signal can exist in Continuous-time or Discrete-

time

Even & Odd signals:A signal x(t) is referred to as an even signal if it is identical

to its time-reversed counterpart: ( )x t x t

( )x t

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Continuous-Time Signals

An odd signal is symmetric about the origin, i.e., with its reflection about the origin:

Any signal can be resolved into an even and an odd components such that

Then, considering that

the odd and even parts of the signal can be expressed as

( )x t x t ( )x t

( )x t

(odd signal) (even signal)

( ) ( ) ( )e ox t x t x t

( ) ( ) ( ) ( ) ( )e o e ox t x t x t x t x t

1 1( ) ( ) ( ) , ( ) ( ) ( )2 2e ox t x t x t x t x t x t

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Continuous-Time Signals( )x t ( )ex t ( )ox t

= +

= +

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Common signals for signal processingThe important signals for signal processing:

Unit impulse and unit step Consider a short pulse Designed to have unit area

As , becomes higher and narrower, maintaining its unit area.

( )t

Continuous-Time Signals

( )t( ) 1t d t

0 ( )t

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Continuous-Time Signals Its limiting form, , can then be thought as an

idealization of an impulse as the duration becomes zero

Unit step is defined as

and is shown as

0, 0( )

1, 0t

u tt

0( ) lim ( )t t

( )t

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Continuous-Time Signals

Continuous-time complex exponential signal It is of the form , where C and a are generally

complex numbers If a signal represents an exponential that

starts at , and we want this signal to start at , it can be described as

Sinc function It is defined as

1, 0ate C a

( ) atx t Ce

t 0t ( )ate u t

sinsinc xxx

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Continuous-Time Signals

Two-sided signalsThe signals are nonzero for both positive and

negative time(non-causal signal)

(a) 1.5-Hz Sinusoid of the form

(b) Square wave with a fundamental frequency of 3Hz

(c) Two-sided decaying exponential signal

( ) 10cos 3x t t

Continuous-time signals that are infinitely long:

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Continuous-Time Signals

One-sided signalsThe signals has the property that on a semi-

infinite interval(anti-causal or causal signal)( ) 0x t

(a) Unit step signal(b) Cosine wave of the form

(c) Decaying exponential signal of the form

( ) 10cos 3 / 4 ( )x t t u t

Right-sided continuous-time signals(causal signals):

( ) 5 ( )tv t e u t

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Meiling CHEN 20

Signal Decomposition

(Square pulse)

( ) 4 1 ( 4) ( 2)s t u t u t

(Triangular pulse)

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Meiling CHEN 21

Homework Assignment #1

[Problem 1] Resolve the waveforms into odd and even components

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Meiling CHEN 22

Homework Assignment #1

[Problem 2] Write the equation for the waveforms using shifted step functions