Probability Theory ( 機率學 )
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
輔仁大學電子工程系所
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Ex 3: Image Processing
Image Enhancement - Histogram Equalization
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Ex 3-1: Image Processing
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Ex 4: Basic Radar Operation
Size -> strengthRange -> time delayAngular -> Antenna beamVelocity -> Doppler
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Radar Applications (1)
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Radar Applications (2)
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Radar Detection (No Target)
False Alarm
• SNR = 0, no Target is present
• Rayleigh Distribution Approach
• Threshold
• Due to Random Signals
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Ex 5: Digital Communications
Noise Effect
• Position
• Width
• Error Rate
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Ex 6: Signal Models
Deterministic
Random
Periodic
Aperiodic
•Deterministic and Random
•Periodic and Aperiodic
Correlator
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Performance FM v.s. DSM, SSB, AM
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Random Variables and Distribution
RandomVariable
Discrete R.V. (PDF)
Continuous R.V. (CDF)
Continuous R.V. (PDF)
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Random Variable and Prob. distribution
fx x
Random variable& Distribution
• Discrete
• Continuous
Description
• cumulative probabilitydistribution (CDF)
• Probability densityfunction (PDF)
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Discrete R.V. Distribution
Binomial pdf Poisson pdf
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Discrete R.V. Distribution
Geometric pdf Poisson pdf
Discrete R.V.Distribution• Bernoulli
• Binomial
• Geometric
• Hypergeometric
• Negative Binomial
• Poisson
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Continuous R.V. Distribution
Continuous R.V. Distribution• Uniform
• Gaussian (Normal)
• Exponential
• Gamma
• Cauchy
• Rician
• Rayleigh
• Weibull
• Laplacian
Uniform pdf
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
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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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Q-Function
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Continuous R. V. Distribution (Cont.)
fx x
Exponential
Rayleigh
Rician
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Continuous R. V. Distribution (Cont.)
Gamma Distribution
Two special cases
• Chi-square
• Erlang
Chi-square
Erlang
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Distribution for two Random Variables
Equal Probability density contours 3-D of Bi-variate pdf
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Transform of Random Variable
Y ax b+=
a 0
a 0
Y X 2=
X N 0 1
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
MATLAB Simulation
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Department of Electronic EngineeringFu-Jen Catholic University
輔仁大學電子工程系所
Probability Theory
Chapter 1 - Introduction Dr. Sheng-Chou Lin
Computer Simulation
Uniform Exponential