signal processing fundamentals - eieenpklun/eie327/intro.pdf1 the hong kong polytechnic university...

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THE HONG KONG POLYTECHNIC UNIVERSITYDepartment of Electronic and Information Engineering

EIE 327EIE 327

Signal Processing FundamentalsSignal Processing Fundamentals

Part I: Spectrum Analysis and Filtering Part I: Spectrum Analysis and Filtering –– Dr Daniel Dr Daniel LunLun, EIE, EIE

Part II: Statistical Signal ProcessingPart II: Statistical Signal Processing–– Dr Bonnie Law, EIEDr Bonnie Law, EIE

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EIE 327EIE 327Signal Processing Fundamentals Signal Processing Fundamentals

PartPart--IISpectrum Analysis and Filtering Spectrum Analysis and Filtering

Lecturer:Lecturer: Dr. Daniel PakDr. Daniel Pak--Kong LUNKong LUN

Room:Room: DE637DE637 Tel:Tel: 2766625527666255EE--Mail:Mail: enpklunenpklun@@polyupolyu..eduedu..hkhkWeb page:Web page: www.www.eieeie..polyupolyu..eduedu..hkhk/~/~enpklunenpklun/EIE327//EIE327/

EIE327.htmlEIE327.html

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Contents• Signals and Systems• Sinusoids and Complex Numbers• Spectrum Representation• Sampling and Aliasing• Fourier Transform and Spectrum Analysis• Fast Fourier Transform• Convolution, FIR Filters and Z-transform

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References• DSP First – A Multimedia Approach

J.H. McClellan, R.W. Schafer and M.A. YoderPrentice Hall, 1998Comment: Entry level with lots of multimedia illustrations

• Introduction to Digital Signal ProcessingA.L. Paul and W. FuerstJohn Wiley and Sons Inc, 2nd Ed. 1998Comment: More formal treatment

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1. 1. Signals and SystemsSignals and Systems

Signal Processing Fundamentals – Part ISpectrum Analysis and Filtering

1. Signals and Systems

Input Output

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What is Signal?“Something” that carries information



“Something” ⇒ pattern of variations of physical quantity that can be manipulated, stored, or transmitted by physical process

e.g. Speech signals, audio signals, image signals, video signals, radar signals, etc.

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Examples of Signals

Speech Audio

Image

Video Multimedia



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• Signals can change to other physical form, e.g. electricity

wave

change of voltage

mic

Light

Lens

CCD

Object

change of voltage



Facilitate processing by electrical equipment

Facilitate processing by electrical equipment

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Continuous-time Signals• Many signals can be considered as pattern

changing with time, e.g. speech

• For every time instant, signal is found• Hence continuous-time signal (or analogue signal)• Most natural signals, such as speech, audio, etc.

are continuous-time signals

timevoltage

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Discrete-time Signals• With the advance of computer technology,

we want to process signals by computer• Computer can only handle data, but not

continuous signals• Need sampling⇒ extract signal at some time instants

timevoltage Signal is found only at some instantsDiscrete-time signal

244.2

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Typical Signal Processing Systems

Mic

Sound card with sampler(or A/D converter)

Processed signalD/A converter speaker

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Why signal processing?• Change the signal characteristic

• e.g. increase or decrease the loudness of speech• Signals in their original form may not be

manipulated easily• e.g. Speech compression – reduce the effort for

storage and transmission of speech• To understand the signal

• e.g. Speech recognition – to recognize the content of the speech



Since we often process discrete-time signals, such processing is called digital signal processing (DSP)

Since we often process discrete-time signals, such processing is called digital signal processing (DSP)

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Examples of DSP Systems

Video Object tracking system

Speech compression

system

10 times compressed

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Steps to construct a DSP system

1. Develop signal processing algorithm• Express input and output in mathematical form• Using mathematics to figure out the solution to

the problem2. Realize the signal processing algorithm

• Translate the algorithm into computer program• Execute the program with the computer

Input Output



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Mathematic Representation of Continuous-time Signals

x(t)

x: a symbol represents this signal

t: a real number variable

x(t) is defined for all value of t, hence continuouse.g. x(0) = 0; x(70.5) = -90.5; x(100.23) = 80

t = 100.23

t = 70.5t = 0

We use () to indicate that x is continuous



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Mathematic Representation of Discrete-time Signals

x[n]

x: a symbol represents this signal

n: an integer variable

We use [] to indicate that x is discrete

n = 0

n = 100

n = 70

x[n] is defined only at some instants of time, since n is integer

e.g. x[0] = 0; x[70] = -90.5; x[100] = 80



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Ts

x(t)

x[n]

x[n] = x(n*Ts)x[n] = x(n*Ts)

x[0] = x(0)x[1] = x(Ts)x[2] = x(2Ts)x[3] = x(3Ts)

: :

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Simplest DSP System - Amplifier Input

(256 samples)

1. Develop signal processing algorithm• Input – x[n]; Output – y[n]• y[n] = x[n] × 2 for all n = 1 to 256 (algorithm)

2. Realize the signal processing algorithmmain() { int n, x[], y[];

for (n=1; n<=256; n++) y[n] = x[n]*2;}

Output(256 samples) Our task:

Output = Input x 2

Our task: Output = Input x 2



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Mathematic Representation of Systems

• A signal processing system transforms signals into new signals or different signal representations

• Mathematically

y(t) = T{x(t)}y(t) = T{x(t)}Output signal Input signal

System or Operator

• T, in this case, works on continuous-time signals, hence it is a continuous-time system

e.g. y(t) = 2*x(t)

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Input Output

System T⇓

x 2

x[n] y[n]

⇒ y[n] = 2*x[n]

• T now works on discrete-time signals, hence it is a discrete-time system

y[n] = T{x[n]}y[n] = T{x[n]}

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Other Simple System Examples

System T⇓

sampling

x(t) y(t) = T{x(t)} = x(nTs)

System T⇓

square

x[n] y[n] = T{x[n]} = x[n]2

{0, 2, -1, -3, 4…} {0, 4, 1, 9, 16…}

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2. Sinusoids and Complex Numbers

2. 2. Sinusoids and Complex NumbersSinusoids and Complex Numbers

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• Sinusoidal signals, or more concisely, sinusoids, are the most basic signals in signal processing

What are Sinusoids?

)tcos(A)t(x o φω +=

A continuous-time signal

Amplitudecosine function

radian frequency = 2π f

phase-shift

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)tcos(A)t(x o φω += ⇒ x(t) = 10 cos (2π1000t + 0)

• The signal above represents a cosine function with amplitude 10 and frequency 1000 Hz with no phase shift

e.g. x(0) = 10 cos(2π0) = 10x(0.5*10-3) = 10 cos(2π0.5) = -10x(1*10-3) = 10 cos(2π) = 10x(2*10-3) = 10 cos(2π2) = 10x(3*10-3) = 10 cos(2π3) = 10



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x(t) = 10 cos (2π1000t + 0)

x(t) = 5 cos (2π1000t + 0)

Change of Amplitude

Change of Amplitude



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Change of FrequencyChange of Frequency

x(t) = 10 cos (2π1000t + 0)

x(t) = 10 cos (2π500t + 0)

1 period = 2ms

1 period = 1ms



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Change of Phase shiftChange of Phase shift

x(t) = 10 cos (2π1000t + 0)

x(t) = 10 cos (2π1000t + pi/2)

Signal will shift 1 period if the cosine function advances by 2πHence shifting pi/2 means shifting ¼period



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Sinusoids are Real Signals• Although cosine are mathematical functions,

sinusoids can be generated by real instrument• e.g. Tuning fork

• Tuning fork vibrates in air at 440Hz and generates wave at the same frequency

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Why Sinusoids are so Important?• In 1807, a famous mathematician, Fourier, showed

that Almost all signals can be constructed by the summation of sinusoids of different frequencies, amplitudes and phase shifts

• The larger is the number of sinusoids, the closer is the square wave

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• The larger is the number of sinusoids, the closer is the waveform to square wave

A0+A1cos(ωot )

A0 + A1cos(ωot ) + A3cos(3ωot )

A0 + A1cos(ωot ) + A3cos(3ωot ) + A5cos(5ωot ) + A7cos(7ωot )



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• Sometimes, calculation involving cosine functions is not that convenient

1.7cos(2π10t + 70π /180) + 1.9cos (2π10t + 200π/180) = ???

• Solution: use complex numbers

jbaA +=

Complex number A The real

part of A

The imaginary part of A

1−=j

Phasor additionPhasor addition

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• Euler’s formula

θθθ sincos je j += { }θθ jeRecos =or

( ) ( )2211 coscos φωφω +++ tAtA• Consider

{ } { }( ){ } ( ){ }

{ } { }{ } ( )φωφω

ωφω

ωφφω

φωφω

+====

+=+=

+=

+

++

tAAeXeAee

XXeeAeAeeAeA

tj

tjjtj

tjjjtj

tjtj

cosReReRe

ReRe

ReRe

)(

2121

)(2

)(1

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Phasors

X = X1 + X2

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1.7cos(2π10t + 70π /180) + 1.9cos (2π10t + 200π/180) = ???

597.15814.07.1 180/70

1

jeX j

+== π

6498.0785.19.1 180/200

2

jeX j

−−== π

180/79.14121

532.1

9476.0204.1πje

jXXX=

+−=+=

( )180/79.141102cos532.1: ππ +tSolution



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Exercise



( )( )4/31002cos4)(

4/1002cos8)(

2

1

ππππ

−=+=ttxttx

What is the sum of x1 and x2?

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Solution

)7071.07071.0(88 4/

1

jeX j

+== π

)7071.07071.0(44 4/3

2

jeX j

−−== − π

4/21

4

)7071.07071.0(4πje

jXXX

=

+=+=

( )4/1002cos4: ππ +tSolution

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