Signals and Systems

Department of Electrical & Electronic Engineering

Southern University of Science and Technology

Spring 2020

Signals and Systems

2

WANG RuiUSTC - CSE - BEng

HKUST - ECE - PhD

Huawei - Senior Research Engineer

SUSTech EEE - Associate Professor

Research Interests:Wireless communications: 5G, VLC, mmWave and etc.

Cloud and edge computing

Stochastic optimization, Reinforcement learning, convex

optimization and etc.

Office: A7 1107

Email: wang.r@sustech.edu.cn

Website:

http://eee.sustc.edu.cn/p/wangrui/

mailto:wang.r@sustc.edu.cn

Signals and Systems

, Willsky

and Nawab, 2nd Edition, 1997, Prentice-Hall.

This course teaches Chapters 1 to 10.

Roughly two weeks for one chapter

Middle-term exam for Chapters 1 to 4

Chapters 6, 8, 9, 10 in a short manner

Final exam for all

3

Textbook reading is crucial, as I cannot cover every

detail in slides

Scope of Lecture

Signals and Systems

Lectures

(Tutorial)Matlab Labs

Assignment/Quiz 20%

Mid-term Exam 25%

Final Exam 25%

Lab Reports

Project Report &

Presentation

30%

4

Three Pillars

YOU

100%

Signals and Systems

Class Schedules

Lab Session Starts from the first week

Instructor: Dr. Guang Wu ( )

Tutorials TBD

Every week (no for week 1)

TA

Jie LI, lij2019@mail.sustech.edu.cn

Xinrao LI, lixr2019@mail.sustech.edu.cn

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Signals and Systems

Which taste of do you like? Salty or sweet

How can a southern Chinese get used to sweet ?

Assignment: Every week (no for week 1)

Submit assignment in softcopy after one week.

6

Practice is Important

Signals and Systems

Signals and Systems

Signals: everything which carries information

Systems: everything which processes input

signal and generate output signal

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09

Transmitter ReceiverChannel

Transmitted signal

noise

Received signal

Estimate of message signal

(information)

Messagesignal

(information) speech, text, image, video

r(t)s(t)

Communication Signals & Systems

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Can you find any example of signals and

systems when making a phone call?

Transmitter, channel and receiver are

all systems.

Each system has one input signal and

one output signal.

Signals and Systems

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Image Processing

Signals and Systems

More examples of signals

Electrical signals voltages and currents in a circuit

Acoustic signals audio or speech signals

Video signals movie

Biological signals sequence of bases in a gene

We will treat noise as unwanted signals.

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Signals and Systems

Signals and Systems from Our Point of

View

Signals are variables that carry information, like function.

Systems process input signals to produce output signals.

The course is about using mathematical techniques to

analyze and synthesize systems which process signals.

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Signals and Systems

Independent Variable of Signals

Time is often the independent variable.

Example: the electrical activity of the heart recorded

with chest electrodes the electrocardiogram

(ECG).

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file://upload.wikimedia.org/wikipedia/commons/c/c1/ECGcolor.svg

Signals and Systems

Signal Classification 1:

Dimension of Independent Variable

An independent variable can be 1-D (t in the ECG),

2-D (x, y in an image), or 3-D (x, y, t in an video).

We focus on 1-D for mathematical simplicity but the

results can be extended to 2-D or even higher

dimensions.

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Signals and Systems

Signal Classification 2:

Continuous-time (CT) Signals

Independent variable is continuous

Most of the signals in the physical world are CT signals.

E.g. voltage & current, pressure, temperature, velocity, etc.

Notation: x(t)

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Signals and Systems

Discrete-time (DT) Signals

Independent variable is integer

Examples of DT signals: DNA sequence, population of

the n-th generation of certain species

Notation: x[n]

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Signals and Systems

Many Human-made Signals are DT

Why DT? Can be processed by modern digital

computers and digital signal processors (DSPs).

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Weekly Dow-Jones

industrial averageDigital image

Signals and Systems

Signal Classification 3:

Deterministic Signal

t

T0½T0-T0 -½T0

x(t)

Each value of the signal is fixed, and can be determined by a

mathematical expression, rule, or table.

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Signals and Systems

Signal Classification 3:

Random Signal

x(t)

t

Signal value at any time instance is a random variable.

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Signals and Systems

Periodic Signals

CT: x(t) = x(t + T) T : period

x(t) = x(t + mT) m: integer

DT: x[n] = x[n + N]=x[n + mN], N: period

Fundamental period: the smallest positive period

Aperiodic: NOT period

19Signal Classification 4:

Periodic / Aperiodic

Signals and Systems

Signal Classification 5: Even / Odd

Even and Odd Signals

Even x(t) = x t) or x[n] = x n]

Odd x(t x t) or x[n x n]

x(0)=0, and x[0]=0

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Example: cos(t)

Example: sin(t)

Signals and Systems

Any signals can be expressed as a sum of Even and

Odd signals. That is:

x(t) = xeven(t) + xodd(t),

where:

xeven(t) = [x(t) + x(-t)]/2,

xodd(t) = [x(t) - x(-t)]/2.

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Signals and Systems

A right-sided signal is zero for t < T, and

A left-sided signal is zero for t > T, where T can be

positive or negative.

22Signal Classification 6:

Right- and Left-Sided

Signals and Systems

Bounded signal: the absolute value of signal is bounded.

Unbounded signal: otherwise

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Classification 7: Bounded and Unbounded

Signals and Systems

0 4

x(t)

t

2 6

y(t)

t

Transformation of a Signal

System

x(t)

y(t)

Signals and Systems

Transformation of a Signal25

Time Shift

Time Reversal

Time Scaling

Combination

][][,)()( 00 nnxnxttxtx

][][,)()( nxnxtxtx

?][,)()( nxatxtx

?][,)()( nxbatxtx

Signals and Systems

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Time Scaling

Transformation of a Signal

Signals and Systems

0 4

2

x(t)

x(-2t+2) ?

t

Class problem27

Signals and Systems

A very important class of signals is presented as:

CT signals of the form x(t) = ej t

DT signals of the form x[n] = ej n

For both exponential CT and DT signals, x is a

complex quantity and has:

a real and imaginary part [i.e., Cartesian form], or

equivalently

a magnitude and a phase angle [i.e., polar form].

We will use whichever form that is convenient.

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Exponential Signals

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is defined as phase

t

t

Real and imaginary parts are periodic

signals with the same period, but out

of phase (90o phase difference)

Signals and Systems

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0

0 0( ) cos( ) sin( )j t

x t e t j t

-Fundamental (angular) frequency:

-Fundamental period:

-In CT, always periodic

-larger => higher frequency

0

0

0

2T

0j te

0

Signals and Systems

( ) ( 1)j n j n ne e

2 2( ) (1) 1j n j n ne e

Is it periodic?

Larger => higher frequency?

Signals and Systems

Periodicity Properties of DT Complex

Exponentials

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0 0 0( 2 ) 2j m n j n j njm ne e e e

However,

Signals and Systems

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0 0 0/ 8

0 / 4

0 / 20

0 3 / 2

0 7 / 4 015 / 8 0 2

Signals and Systems

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- We need only consider a frequency interval of length 2, and

on most cases, we use the interval: or

- does not have a continually increasing rate of oscillation

as is increased.lowest-frequency (slowly varying): near 0, 2

highest-frequency (rapid variation): near ±

00 2 , 0

0j ne

0

0

0

(2 1) ( ) ( 1)j k n j n j n ne e e2 2( ) (1) 1j n j n ne e

Signals and Systems

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A set of periodic exponentials which have a

common period.

Harmonically Related Signal Sets

Fundamental (Angular) Frequency : |

Fundamental Period:

Common Period:

Signals and Systems

General Complex Exponential Signals

- CT

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r > 0

r < 0

e.g., damped sinusoids

General format (C and a are complex numbers)

Signals and Systems

General Complex Exponential Signals

- DT

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General format (C and are complex numbers)

0 0( )[ ]n nj n j nn jx n C C e e C e

| | < 1

| | > 1