Probability Theory...

Probability Theory ( 機率學 )

ICRM & TRUs

ICPICP

MTX

輔仁大學電子工程系所

林昇洲博士

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Department of Electronic EngineeringFu-Jen Catholic University


Probability Theory

Chapter 1 - Introduction Dr. Sheng-Chou Lin

Personal Profile學歷 : Ph. D. — University of Texas at Arlington, Texas, USA.

經歷 :

電子工業研究所，數位積體電路設計部 - 助理工程師

中山科學院系統發展中心，飛彈模擬室，天弓飛彈微波模擬組 - 助理工程師

北方貝爾研究室 Bell Northern Research (BNR) TX, USA - 短期研究生

北電網路 NORTEL Networks, TX, USA，無線網路系統部 - 資深工程師

輔仁大學電子工程系所 - 訪問助理教授

專長 :

Wireless, Mobile Communications ( 無線，行動通訊系統 )

Radar System Design ( 雷達系統設計 )

Fiber Optics Communications( 光纖通訊 )

Statistical Signal Processing ( 統計信號處理 )

Microwave Theory and Engineering ( 微波原理與工程 )

Semiconductor Devices and Manufacture ( 半導體元件及製程 )

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Probability Theory


Personal Profile (Offer Courses)•Probability Theory ( 機率學 )

•Complex Function ( 複變函數 )

•Communication Engineering ( 通訊系統 )

•Digital Communications ( 數位通訊 )

•Random Signal and Process( 隨機信號與過程 )

•Wireless Communications ( 無線通訊 )

•Mobile Radio Communications ( 行動通訊 )

•Radar System Desigh( 雷達系統設計 )

•Fiber-Optics Communications ( 光纖通訊 )

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Probability Theory


Objectives•Covers essentials of probability and statistics and certain

of their engineering applications.

•Provide a required course in probability, since many theaspects of modern engineering analysis and designrequire the tools of probability as well as statistics.

Text book:• “Elements of Engineering Probability & Statistics”, Ziemer Tranter.

Reference Books:• “Probability and Random Processes for Electrical Engineering°®, Alberto Leon-Garcia.

Office HoursThursday: 2:00 ~4:00 pm at office ( 聖言樓 SF726B)

Phone number: (02)29031111~3798, email:[email protected]

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Probability Theory


Continuous Courses

•通訊工程 - 數位通訊 - 通訊模擬 - 消息與編碼理論

•數位訊號處理 - 適應性訊號處理 - 估計理論

•無線通訊 - 行動通訊 - 展頻通訊 - 雷達系統設計

•隨機過程 - 統計學

•影像處理 - 影像圖形識別

•隨機控制 - 類神經網路

•排隊理論 - 計算機網路

•VLSI 可靠性分析

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Probability Theory


Grading

1. Attendance (Not Mandatory) 10%

• -20 for each absence

2. Quizs 30%

•1 hour, Twice in one semester

3. Midterm Exam 30%4. Final Test 30%

•2 hour, inclass/proctored

•Close book, Close notes

5. MATLAB Project (Optional) 10%

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Probability Theory


Applications of Probability andStatistics

•Communications

•Electrical and ElectronicEngineering

•Acoustics and DigitalAudio

•Geoscience ( 地球科學 )

• Instrumentation

•Systems Reliability

•Ocean and EnvironmentSystem

•Computers

•Physics

•Computer Networks

•Chemical and MechanicalEngineering

•Radio Astronomy

•Numerous othersDisciplines

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Probability Theory


Example 1 (Speech Recognition)

Computer Recognition of Speech

• Correlate test with each stored template

• largest correlation is the best match

• A complicated task due to some factors

Interference from surroundings, variability in theamplitudeand duration,changes inothercharacteristics (speaker’s pitch...)

• Variability may be described by theoryof Probability and Random Process

used to develop models design system forrecognition

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Probability Theory


Example 2

Image System

• Improve the S/N through filtering

• Maximize the S/N

CD player

• Errors caused by imperfections inCD (dirt, scratch)

• Errors are found and correctedusing powerful Reed-Solomoncoding and decoding schemes.

• The design of codes anddecoding algorithms is based onprobability theory and correlation

Communications, Audio, and imaging systems,noise or adjacent-channel interference maycause errors in transmitted data

Probability, statistics and randomprocesses may model, thereby assistingdevelopment of design solutions

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Probability Theory


Ex 3: Image Processing

Image Enhancement - Histogram Equalization

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Probability Theory


Ex 3-1: Image Processing

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Probability Theory


Ex 4: Basic Radar Operation

Size -> strengthRange -> time delayAngular -> Antenna beamVelocity -> Doppler

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Probability Theory


Radar Applications (1)

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Probability Theory


Radar Applications (2)

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Probability Theory


Radar Detection

Statistical Process

• Atomopheric, thermal noise,electromagnetic interference,background clutter, propagationmedium anomalies, electroniccountermeasures.

• Target’s RCS

Detection

• Single Pulse Probability of False Alarm

• Probability of Detection

• Output SNR

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Probability Theory


Radar Detection (No Target)

False Alarm

• SNR = 0, no Target is present

• Rayleigh Distribution Approach

• Threshold

• Due to Random Signals

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Probability Theory


Radar Detection (Target)

Probability of Detection

• SNR= 1, 2, 3 ... with target

• Rican Distribution -> Direct ray + Rayleigh

• Threshold

Maximum Detection Range

• Single Pulse Probability of FalseAlarm (Pfa)

• Probability of Detection (Pd)

• Output SNR

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Probability Theory


Ex 5: Digital Communications

Noise Effect

• Position

• Width

• Error Rate

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Probability Theory


Ex 6: Signal Models

Deterministic

Random

Periodic

Aperiodic

•Deterministic and Random

•Periodic and Aperiodic

Correlator

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Probability Theory


Performance FM v.s. DSM, SSB, AM

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Probability Theory


Ex7: Mobile Radio (Multiple-path effect)

Signal levels vary as user movesSlow variations come from

blockage and shadowing by largeobjects such as hills and buildings

Rapid Fading comes as signalsreceived from many paths drift intoand out of phase•phase cancellation occurs, causing

rapid fades that are occasionallydeep

•the fades are roughly /2 apart:7 inches apart at 800 MHz.3 inches apart at 1900 MHz

•called Rayleigh fading, after thestatistical model that describes it

A

t

10-15 dB

Rayleigh Fading

A

t

10-15 dB

A

t

10-15 dB

Rayleigh Fading

Multi-path PropagationMulti-path Propagation

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Probability Theory


Mobile Radio(Statistical Propagation Models)

Prediction of Signal Strength as a function of distance withoutregard to obstructions or features of a specific propagation path

RSSI, dBm

-120

-110

-100

-90

-80

-70

-60

-50

0 3 6 9 12 15 18 21 24 27 30 33

Distance from Cell Site, km

measured signalstrength

signal strength predicted by Okumura-Hata propagation model

FieldStrength,dBuV/m

+90

+80

+70

+60

+50

+40

+30

+20

RSSI, dBm

-120

-110

-100

-90

-80

-70

-60

-50

0 3 6 9 12 15 18 21 24 27 30 33

Distance from Cell Site, km

measured signalstrength

signal strength predicted by Okumura-Hata propagation model

FieldStrength,dBuV/m

+90

+80

+70

+60

+50

+40

+30

+20

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Probability Theory


Outline1. Fundamental Concepts of Probability•Approaches to Probability, Set Theory

2. Random Variables and Probability Distributions

•Discrete Random Variables and distributions,Continuous Random Variables anddistributions, Mean and Discrete Variance of Random Variables

•Transformations of Functions of a Single Random Variable,Computer Generationof Random Variables

3. Bivariate Random Variables and Distributions

•Bivariate CDF and PDFl, Statistically Independent Random Variables,Functions of Two Random Variables

•Transformation of a pair of Random Variables,Central Limit Theorem

4. Elementary Statistics•Sample Mean and Sample Variancel, Regression Techniquesl, Estimate Theory,

Engineering Decisions

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Probability Theory


Outline

1. Fundamental Concepts of Probability•Approaches to Probability, Set Theory

2. Random Variables and Probability Distributions

•Discrete Random Variables and distributions,Continuous Random Variables anddistributions, Mean and Discrete Variance of Random Variables

•Transformations of Functions of a Single Random Variable,Computer Generationof Random Variables

3. Bivariate Random Variables and Distributions

•Bivariate CDF and PDFl, Statistically Independent Random Variables,Functions of Two Random Variables

•Transformation of a pair of Random Variables,Central Limit Theorem

4. Elementary Statistics•Sample Mean and Sample Variancel, Regression Techniquesl, Estimate Theory,

Engineering Decisions

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Probability Theory


Fundamental Concepts of Probability

Approaches to Probability

• Subjective Probability

• Equally Likely outcomes

• Relativee Frequency

Set Theory

• Outcomes

• Events

• Sample Space

Set Operation

• Union

• Intersection

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Probability Theory


Random Variables and Distribution

RandomVariable

Discrete R.V. (PDF)

Continuous R.V. (CDF)

Continuous R.V. (PDF)

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Probability Theory


Random Variable and Prob. distribution

fx x

Random variable& Distribution

• Discrete

• Continuous

Description

• cumulative probabilitydistribution (CDF)

• Probability densityfunction (PDF)

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Probability Theory


Discrete R.V. Distribution

Binomial pdf Poisson pdf

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Probability Theory


Discrete R.V. Distribution

Geometric pdf Poisson pdf

Discrete R.V.Distribution• Bernoulli

• Binomial

• Geometric

• Hypergeometric

• Negative Binomial

• Poisson

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Probability Theory


Continuous R.V. Distribution

Continuous R.V. Distribution• Uniform

• Gaussian (Normal)

• Exponential

• Gamma

• Cauchy

• Rician

• Rayleigh

• Weibull

• Laplacian

Uniform pdf

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Probability Theory


Gaussian Distribution

Standard Gaussian

• Mean = 0, variance = 1

• Q-Function: Q(x)

• Error Function: erf(x), erfc(x)

x

Q(x)

standardGaussian

m 5= 2=

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Probability Theory


Q-Function

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Probability Theory


Continuous R. V. Distribution (Cont.)

fx x

Exponential

Rayleigh

Rician

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Probability Theory


Continuous R. V. Distribution (Cont.)

Gamma Distribution

Two special cases

• Chi-square

• Erlang

Chi-square

Erlang

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Probability Theory


Distribution for two Random Variables

Equal Probability density contours 3-D of Bi-variate pdf

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Probability Theory


Transform of Random Variable

Y ax b+=

a 0

a 0

Y X 2=

X N 0 1

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Probability Theory


MATLAB Simulation

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Probability Theory


Computer Simulation

Uniform Exponential

Probability Theory...

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