### General

**YouTube Playlist:****Playlist Link:**EE 306 - Spring 2021-2022

### General

**YouTube Playlist:****Playlist Link:**EE 306 - Spring 2021-2022

### 7 March - 13 March

**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 Axioms**Lecture Content:****0:00**Conditional Probability**10:30**Total Probability Theorem**18:20**Bayes' Rule**26:08**Independence of Events**29:12**Counting Methods**Lecture 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 March - 20 March

**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 Variable**Lecture 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 Expectation**Lecture 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 March - 27 March

**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 Variables**Lecture 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 Theorem**Lecture 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 March - 3 April

**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 Constraint**Lecture 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 April - 10 April

**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 Problem**Lecture 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 April - 17 April

**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 Classes**Lecture 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 Class**Lecture 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 April - 24 April

**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 States**Lecture 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 Distributions**Lecture 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 April - 1 May

**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 Distribution**Lecture 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 Boarding**Lecture 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 May - 15 May

**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 Minutes**Lecture Content:****0:00**Example 1: Random Telegraph Signal**18:00**Example 2: Shot Noise

### 16 May - 22 May

**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 Instants**Lecture 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 May - 29 May

**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 Stationarity**Lecture 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 Stationarity**Lecture 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 May - 5 June

**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 Noise**Lecture 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 June - 12 June

**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 Coefficients**Lecture 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 Sinusoid**Lecture 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 June - 19 June

**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 Process**Lecture 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