Genel
YouTube Playlist:
Playlist Link: EE 306 - Spring 2021-2022
Haftalık özet
7 Mart - 13 Mart
Lecture Content:
0:00 Intro
8:14 Random Experiment
10:53 Sample Space
13:58 Events
17:06 Fundamentals of Set Theory
20:36 Properties of Set Operations
30:13 Probability AxiomsLecture Content:
0:00 Conditional Probability
10:30 Total Probability Theorem
18:20 Bayes' Rule
26:08 Independence of Events
29:12 Counting MethodsLecture Content:
0:00 Recap of Previous Lecture
1:13 Random Variables
4:38 Expectation, Variance and Moments
8:14 Important Distributions (Discrete Random Variables)
11:39 Important Distributions (Discrete Random Variables) Example 1: Expectation of Geometric Random Variable
16:04 Important Distributions (Discrete Random Variables) Example 2: Binomial and Poisson Random Variables
22:11 Important Distributions (Continuous Random Variables)
36:03 Additional Notes
41:40 Example: Football Game Tickets
14 Mart - 20 Mart
Lecture Content:
0:00 Recap of Previous Lecture
1:12 Functions of Random Variables
7:12 Functions of Random Variables Example 1: Squared Random Variable
14:46 Functions of Random Variables Example 2: Computer Generation of Random Variables
26:22 Transform Methods
26:52 Characteristic Functions (Continuous Random Variables)
31:17 Probability Generating Functions (Discrete Random Variables)
37:17 Transform Methods Example 1: Characteristic Function of Gaussian Random VariableLecture Content:
0:00 Probability Bounds
2:20 Prove Markov Bound
4:18 Probability Bounds Example 1: Comparison of Markov Bound to Exact Probability of a Binomial Variable
12:21 Probability Bounds Example 2: Application of Chebyshev Bound to Bernoulli Trials
22:44 Recursion
24:30 Recursion Example 1: Recursion to Derive Probability
36:58 Recursion Example 2: Recursion to Derive ExpectationLecture Content:
0:00 Recap of Previous Lecture
0:41 Pairs of Random Variables
4:41 Useful CDF Relations
9:47 Example 1: Pair of Discrete Random Variables
22:30 Example 2: Pair of Continuous Random Variables
35:52 Jointly Gaussian Random Variables
38:42 Conditional PMF/PDF and Expectation
21 Mart - 27 Mart
Lecture Content:
0:00 Covariance and Correlation Coefficient
5:48 Example 1: Range of Correlation Coefficient (Proof)
11:07 Example 2: Expected Value of Function of Two Random Variables
14:42 Functions of Pairs of Random Variables
26:40 Example 3: PDF of Sum of Two Independent Random VariablesLecture Content:
0:00 Vectors of Random Variables
9:33 Example 1: Three Jointly Gaussian Random Variables
16:47 Example 2: Deterministic Matrix and Random Vector Multiplication
25:27 Sums of Random Variables
33:59 Example 3: Law of Large Numbers
37:43 Example 4: Central Limit TheoremLecture Content:
0:00 Recap of Previous Lecture
1:38 Estimation
5:15 Parameter Estimation
11:03 Example 1: Sample Mean Estimator
21:16 Example 2: Maximum Estimator
32:06 Minimum MSE Linear Estimator (Random Variable Estimation)
28 Mart - 3 Nisan
Lecture Content:
0:00 Recap of Previous Lecture
0:48 MMSE Linear Estimator (continued)
4:31 Gradient and Hessian
14:11 MMSE Generalized Estimator
22:06 Example 1: Estimators of Two Joint Random Variables
32:21 Linear Least Squares (LLS)
39:38 Example 2: Ordinary Least Squares (OLS)Lecture Content:
0:00 Recap of Previous Lecture
3:21 Linear Programming
6:11 Example 1: 2D Optimization on a Pentagonal Region
14:16 Convex-Concave Functions
22:37 Example 2: Convexity/Concavity of Quadratic and Logarithmic Functions
25:07 Duality in Optimization
36:53 Example 3: Minimization of Vector Squared Norm with Equality Constraint
41:02 Example 4: Minimization of Linear Function with Equality ConstraintLecture Content:
0:00 Recap of Previous Lecture
1:32 Notes on Convexity and Duality
4:42 Lagrange Dual Problem
11:22 Example 1: Dual Problem of Linear Programming Problem
15:08 Karush-Kuhn-Tucker (KKT) Conditions
24:27 Example 2: Water Filling on Communication Channels
39:25 Descent Methods
44:18 Gradient Descent Algorithm
4 Nisan - 10 Nisan
Lecture Content:
0:00 Introduction to Module 2
0:58 Discrete Stochastic Processes Examples
10:28 Definition of a Discrete Stochastic Process
15:16 Introduction to Markov Chains
17:32 Example 1: Two State Markov Chain (Rain/no rain)
28:17 Example 2: Two State Markov Chain (Passing Calculus I)
37:06 Transition Probabilities
42:52 Example 3: Random Walk with Infinite Number of States
48:58 Example 4: Random Walk with Finite Number of States (Spider and Insect)Lecture Content:
0:00 n-step Transition Probabilities
12:54 Example 1: 4-State Markov Chain with 2 Steady State Outcomes (Spider Web)
27:11 Example 2: 3-State Markov Chain with 2 Absorbing States (Passing Calculus II)
30:48 Example 3: Finding a Markov Chain Model (Raining)
37:57 Example 4: 2 Umbrella ProblemLecture Content:
0:00 Example 1: Solution of the “2 Umbrella Problem” from Previous Lecture
15:47 Example 2: Solution of the “Passing EE202 Problem”
30:57 Example 3: First passage probability in a Random Walk
43:48 Example 4: Expected first passage time for a Random Walk in a Finite State Space
11 Nisan - 17 Nisan
Lecture Content:
0:00 Classification of States
0:35 Accessibility
5:40 Communicating States
14:08 Recurrent and Transient States
24:20 The number of visits to a Transient State
26:50 Further Discussion on Recurrent States
33:18 Recurrent ClassesLecture Content:
0:00 Periodic States
13:22 Example 1: Periodic Markov Chain with 3x3 Transition Matrix
17:35 Example 2: Aperiodic Markov Chain with 2x2 Transition Matrix
26:36 Example 3: MC with 2 absorbing states (“Passing EE202” example from previous lectures)
30:30 Discussion of examples 2 and 3: ergodic and non-ergodic processes
36:00 Relation Between Convergence and Periodicity
42:00 Example 4: Markov Chain with Multiple Recurrent Classes
51:25 Example 5: Markov Chain with a Single Recurrent ClassLecture Content:
0:00 Steady State Probabilities
2:07 Example 1: Steady State for 3-state Markov Chain (“2 LED”s)
17:00 Theorem: Global Balance Equations for Stationary Probabilities
26:08 Long Term Frequency of Occurence
32:28 Example 2: Long Term Averages in the 2 Umbrella Problem
40:26 Example 3: Long Term Averages in the Rain/No Rain Problem
18 Nisan - 24 Nisan
Lecture Content:
0:00 Birth-Death Chains
3:40 Example 1: Random Walk with Reflecting Barriers
8:27 Example 2: Geo/Geo/1 Queue
20:40 Mean First Passage and Recurrence Times
28:00 Example 3: Expected Number of Rainy Days (Rain/No Rain)
33:40 Example 4: Mean First Passage and Recurrence Times for MC with 1 Recurrent Class and 2 Transient StatesLecture Content:
0:00 Example 4: Mean First Passage and Recurrence Times for MC with 1 Recurrent Class and 2 Transient States (continued)
Lecture Content:
0:00 Exponential Random Variable
4:28 Memorylessness Property of Exponential Distribution
8:21 Example 1: Waiting Time with Exponential Distribution (Bus Inter-Arrival Times)
13:39 Racing Exponentials
18:17 Example 2: Service Times at Bank with Exponential DistributionsLecture Content:
0:00 Counting Processes
2:02 Sample Path of a Counting Process
4:40 Equivalent Characterization of Counting Process by Arrival Times
10:12 Counting Process Properties: Property 1
14:20 Counting Process Properties: Property 2
15:30 Counting Process Properties: Property 3
19:50 Counting Process Properties: Property 4
25 Nisan - 1 Mayıs
Lecture Content:
0:00 Definition 1 of Poisson Process
2:15 Example 1: Arrival Times of 3 Buses
4:43 Residual Time
8:17 Fresh Start and Independent Increments Properties
11:35 Stationary Increments Property
14:50 Example 2: Probability of Arrival in an Interval of 𝛿
22:24 Little-o Notation for Linear Decay
27:09 Definition 2 of Poisson Process (Based on Little-o Notation)
33:43 Moment Generating Functions and Poisson DistributionLecture Content:
0:00 Recap of Previous Lecture
2:08 Definition 3 of Poisson Process (Based on Poisson Distribution)
3:57 Equivalence of Definitions 2 and 3 of Poisson Process
9:06 Equivalence of Definitions 1 and 3 of Poisson Process
13:30 Example 1: Reception of Emails Modelled as a Poisson Process
20:36 Waiting Times in a Poisson Process
32:06 Example 2: Entering Building A
38:20 Time Reversed Poisson Process
41:13 Example 3: Number of People in the Bus before BoardingLecture Content:
0:00 Recap of Previous Lecture
4:05 Example 1: 3 Buses Departing from METU
11:52 Splitting Poisson Processes (Proof by Using Baby Bernoulli)
19:48 Splitting Poisson Processes (Proof by Using Definition 1 of Poisson Process)
30:30 Splitting Poisson Processes (Proof of Two Split Processes Being Indpendent)
41:30 Merging Independent Poisson Processes (Proof by Racing Exponentials)
9 Mayıs - 15 Mayıs
Lecture Content:
0:00 Summary of Poisson Process
9:11 Example 1: Bus Departure from METU (long example)
12:03 Example 1, Part a: Expected Waiting Time after Bus Leaves
13:29 Example 1, Part b: Expected Waiting Time at Random Arrival
15:05 Example 1, Part c: 5 Bus Departure Probability
17:50 Example 1, Part d: 3 Buses out of 5 Going to Specific Location
20:25 Example 1, Part e: Expected Value of People Getting on Bus
31:50 Example 1, Part f: Expected Value of People on Board at Random Time
37:44 Example 1, Part g: Probability of 6 Bus Departures in First 10 MinutesLecture Content:
0:00 Example 1: Random Telegraph Signal
18:00 Example 2: Shot Noise
16 Mayıs - 22 Mayıs
Lecture Content:
0:00 Example 1: Deterministic Modelling (Projectile Motion of a Cannonball)
11:35 Example 2: Deterministic Modelling (Reconstruction of a Bandlimited Signal from the Samples)
16:10 Stochastic Modelling
19:06 Example 3: Random Slope Signal
24:55 Description of Random Processes (1st Order p.d.f Description)
28:28 Example 4: Uniformly Distributed Random Slope Signal
40:40 Description of Random Processes (2nd Order Joint p.d.f Description)
42:20 Example 5: Uniformly Distributed Random Slope Signal at Two Time InstantsLecture Content:
0:00 Example 1: Uniformly Distributed Random Slope Signal at Two Time Instants (continued from previous lecture)
12:47 Comments on Nth Order Joint p.d.f. Description
23:18 Example 2: Calculation of 1st Order p.d.f from 2nd Order p.d.f.
35:23 Partial Description of Random Processes with Moment Descriptions
40:07 Example 3: Mean and Autocorrelation of Random Slope Signal (1st Order Description)Lecture Content:
0:00 Recap of Previous Lecture
4:44 Example 1: Mean and Autocorrelation of Random Slope Signal (2nd Order Description)
10:00 Example 2: Random Phase Cosine Signal
16:40 Example 2: Random Phase Cosine Signal (1st Order p.d.f. Description)
42:30 Example 2: Random Phase Cosine Signal (2nd Order p.d.f. Description)
23 Mayıs - 29 Mayıs
Lecture Content:
0:00 Example 1: Moment Description of Random Phase Cosine (Mean Function)
5:57 Example 1: Moment Description of Random Phase Cosine (Autocorrelation Function)
11:44 Example 1: Moment Description of Random Phase Cosine (Covariance Function)
12:34 Notes on Moment Descriptions
23:55 Stationary Processes
29:00 1st Order Stationarity
33:10 Example 2: 1st Order Stationarity of Random Phase Cosine and Random Slope Signals
38:13 2nd Order Stationarity
52:04 Nth Order StationarityLecture Content:
0:00 Nth Order Stationarity Continued
0:59 Notes on Stationarity
6:02 Example 1: Stationarity of Random Phase Cosine Signal
11:00 Strict Sense Stationarity
12:30 Brief Summary of Stationarity
21:25 Stationarity in Moment Descriptions (Mean Function)
25:34 Stationarity in Moment Descriptions (Autocorrelation Function)
30:00 Example 2: Stationarity in Autocorrelation for Random Phase Cosine Signal
35:13 Stationarity in Covariance Function
38:15 Wide Sense Stationarity
39:52 Comments on StationarityLecture Content:
0:00 Recap of Previous Lecture
0:59 Example 1: Wide Sense Stationarity of Some Processes
23:50 Gaussian Processes
32:29 Notes on Gaussian Processes
30 Mayıs - 5 Haziran
Lecture Content:
0:00 Example 1: Gaussian Process with ‘sinc’ Autocorrelation
4:41 Example 1: Gaussian Process with ‘sinc’ Autocorrelation (Sampling Rate for Independent Samples)
17:23 Example 1: Gaussian Process with ‘sinc’ Autocorrelation (Joint Density of Independent Samples)
24:38 LTI Processing of WSS Random Processes
26:20 Jointly WSS Processes
29:14 Steps of Showing Output is WSS for WSS Input
31:03 Steps of Showing Output is WSS for WSS Input (Mean Function)
36:13 Steps of Showing Output is WSS for WSS Input (Autocorrelation Function)Lecture Content:
0:00 Example 1: Moment Characterization of Output Process for WSS Input
16:50 Comments on the Previous Example for Input as Gaussian Process
29:01 Comments on the Previous Example for Input with Impulsive Autocorrelation
38:45 White NoiseLecture Content:
0:00 Properties of Autocorrelation Function of WSS Processes
1:55 Properties of Autocorrelation Function of WSS Processes (Hermitian Symmetry)
4:50 Properties of Autocorrelation Function of WSS Processes (Positive Semi-definiteness)
14:43 Properties of Autocorrelation Function of WSS Processes (Non-negativity of Zero-shift Term)
18:20 Properties of Autocorrelation Function of WSS Processes (Zero-shift Term is Absolutely Largest)
31:45 Example 1: Autocorrelation Function Candidates
35:57 Comments on Necessity and Sufficieny of Autocorrelation Properties
41:07 Example 2: Autocorrelation and Crosscorrelation of a Discerete Process Passed Through an LTI System
6 Haziran - 12 Haziran
Lecture Content:
0:00 Power Spectral Density Brief Introduction
2:43 Power Spectral Density Definition
9:25 Power Spectral Density Properties (Area under the PSD Function)
14:46 Power Spectral Density Properties (Real-valued Function)
18:29 Power Spectral Density Properties (Nonnegativity)
28:30 Power Spectral Density Conclusion (Why it is Called ‘Density’?)
33:37 Example 1: Power Spectral Density of Random Phase Cosine
44:23 Example 2: Power Spectral Density of Sum of Complex Exponentials with Random CoefficientsLecture Content:
0:00 Recap of Previous Lecture
1:45 Notes on Power Spectral Density (Processing of White Noise with an LTI System)
9:57 Notes on Power Spectral Density (Flatness of White Noise in Frequency Domain)
12:50 Notes on Power Spectral Density (Sufficient Condition on Power Spectral Density for Valid Autocorrelation Functions)
21:20 Notes on Power Spectral Density (Classification of Processes by Power Spectral Density Based on Passband Properties)
23:35 Representation of Bandpass Signals/Processes (Hilbert Transform)
35:07 Note on Stability of Hilbert Transform
37:53 Example 1: Hilbert Transform of a SinusoidLecture Content:
0:00 Single Sideband Modulations as a Hilbert Transform Application
10:30 Bandpass Signal Representation
14:27 Bandpass Signal Representation (Lowpass Equivalent of a Bandpass Signal-Finding Analytic/Pre-envelope Signal)
19:04 Bandpass Signal Representation (Lowpass Equivalent of a Bandpass Signal-Shifting Analytic/Pre-envelope Signal to DC)
22:50 Notes on Lowpass Equivalent Signal (Complex Valued Signal)
25:04 Notes on Lowpass Equivalent Signal (Location of the Center Frequency)
26:36 Obtaining Bandpass Signal from Lowpass Equivalent
38:20 Example 1: Lowpass Equivalent of a Sinusoid (Center Frequency as the Sinusoid Frequency)
43:41 Example 1: Lowpass Equivalent of a Sinusoid (Center Frequency Different from the Sinusoid Frequency)
50:28 Generation of Bandpass Signals
55:15 Generation of I/Q Signals
13 Haziran - 19 Haziran
Lecture Content:
0:00 Recap of Previous Lecture
2:54 Representation of Bandpass Systems
12:28 Representation of Bandpass WSS Processes
14:34 Obtaining Zero-Mean Information from Power Spectral Density
20:18 Steps to Represent a Bandpass Process with an Equivalent Lowpass Process
23:08 Step 1: Analytic Process Generation
36:06 Step 2: Lowpass Equivalent Process Generation
43:50 Notes on Lowpass Equavialent of a Bandpass ProcessLecture Content:
0:00 Finding I/Q Components of a Bandpass Process
6:42 Statistics of I/Q Components
17:17 Statistics of I/Q Components (Relations Between Correlations of the Bandpass Process and Its Hilbert Transform)
26:25 Statistics of I/Q Components (Final Trigonometric Calculations)Lecture Content:
0:00 Recap of Previous Lecture
13:20 Power Spectral Density of I/Q Components of a Bandpass Process
23:00 Cross Power Spectral Density of I/Q Components of a Bandpass Process
30:06 Example 1: Cross and Power Spectral Densities of I/Q Componets of a Process with Triangular Spectrum
42:35 Notes on Spectral Density of I/Q Components of a Bandpass Process
53:35 Some Results Based on Practical Assumptions