Digital Communication Systems ECS 452

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Digital Communication Systems ECS 452 Asst. Prof. Dr. Prapun Suksompong prapun@siit.

1 Digital Communication Systems ECS 452 Asst. Prof. Dr. Prapun Suksompong Discrete Channel Office Hours: BKD Monday 4:00-6:00 Wednesday 4:40-6:00

2 Noise & Interference Elements of digital commu. sys. Message Transmitter Information Source Source Encoder Remove redundancy (compression) Channel Encoder Add systematic redundancy to combat errors introduced by the channel Digital Modulator Channel Transmitted Signal Recovered Message Destination Source Decoder Receiver Channel Decoder Digital Demodulator Received Signal 2

3 Noise & Interference System considered previously Message Transmitter Information Source Source Encoder Remove redundancy Channel Encoder Add systematic redundancy Digital Modulator Transmitted Signal Channel Recovered Message Destination Source Decoder Receiver Channel Decoder Digital Demodulator Received Signal 3

Noise & Interference System considered in this section Message Transmitter Information Source Source Encoder Remove redundancy Channel Encoder Add systematic redundancy Digital Modulator

4 Noise & Interference System considered in this section Message Transmitter Information Source Source Encoder Remove redundancy Channel Encoder Add systematic redundancy Digital Modulator Equivalent Channel Channel Transmitted Signal X: channel input Recovered Message Destination Source Decoder Receiver Channel Decoder Digital Demodulator Received Signal Y: channel output 4

5 Ex: BSC >> BSC_demo ans = ans = %% Simulation parameters % The number of symbols to be transmitted n = 20; % Channel Input S_X = [0 ]; S_Y = [0 ]; p_x = [ ]; % Channel Characteristics p = 0.; Q = [-p p; p -p]; p_x = p_x_sim = q = q_sim = Q = Q_sim = PE_sim = PE_theretical = Elapsed time is seconds. [BSC_demo.m]

6 MATLAB %% Generating the channel input x x = randsrc(,n,[s_x;p_x]); % channel input %% Applying the effect of the channel to create the channel output y y = DMC_Channel_sim(x,S_X,S_Y,Q); % channel output function y = DMC_Channel_sim(x,S_X,S_Y,Q) %% Applying the effect of the channel to create the channel output y y = zeros(size(x)); % preallocation for k = :length(x) % Look at the channel input one by one. Choose the corresponding row % from the Q matrix to generate the channel output. y(k) = randsrc(,,[s_y;q(find(s_x == x(k)),:)]); end [DMC_Channel_sim.m] 6 [DMC_Analysis_demo.m]

7 Rel. freq. from the simulation %% Statistical Analysis % The probability values for the channel inputs p_x % Theoretical probability p_x_sim = hist(x,s_x)/n % Relative frequencies from the simulation % The probability values for the channel outputs q = p_x*q % Theoretical probability q_sim = hist(y,s_y)/n % Relative frequencies from the simulation % The channel transition probabilities from the simulation Q_sim = []; for k = :length(s_x) I = find(x==s_x(k)); LI = length(i); rel_freq_xk = LI/n; yc = y(i); cond_rel_freq = hist(yc,s_y)/li; Q_sim = [Q_sim; cond_rel_freq]; end Q % Theoretical probability Q_sim % Relative frequencies from the simulation 7 [DMC_Analysis_demo.m]

8 MATLAB %% Naive Decoder x_hat = y; %% Error Probability PE_sim = -sum(x==x_hat)/n % Error probability from the simulation % Calculation of the theoretical error probability PC = 0; for k = :length(s_x) t = S_X(k); i = find(s_y == t); if length(i) == PC = PC+ p_x(k)*q(k,i); end end PE_theretical = -PC 8 [DMC_Analysis_demo.m]

9 Ex: BSC >> BSC_demo %% Simulation parameters % The number of symbols to be transmitted n = e4; % Channel Input S_X = [0 ]; S_Y = [0 ]; p_x = [ ]; % Channel Characteristics p = 0.; Q = [-p p; p -p]; p_x = p_x_sim = q = q_sim = Q = Q_sim = PE_sim = PE_theretical = Elapsed time is seconds. 9 [BSC_demo.m]

10 Binary Symmetric Channel (BSC) Three equivalent descriptions of BSC: P Y 0 X 0 Q 0 0 p P Y X 0 Q 0 p P Y 0 X Q 0 p P Y X Q p X 0 p p -p -p 0 Y Q y 0 x 0 p p p p 0

11 Tree Diagram and Conditional Probability P A B A P A B =P A B P B B P B P A c B A c P A c B =P A c B P B P B c P A B c A P A B c =P A B c P B c B c P A c B c A c P A c B c =P A c B c P B c

12 Tree Diagram and Total Probability Theorem = = P A = P A B P B + P A B c P B c = = 2

13 Binary Asymmetric Channel (BAC) Three equivalent descriptions of BAC: P Y 0 X 0 Q 0 0 P Y X 0 Q 0 P Y 0 X Q 0 P Y X Q X Y Q y 0 x 0 3

14 4 Ex: BAC %% Simulation parameters % The number of symbols to be transmitted n = 20; % Binary Assymmetric Channel (BAC) % Ex 3.8 in lecture note (.3 in [Z&T, 200]) % Channel Input S_X = [0 ]; S_Y = [0 ]; p_x = [ ]; % Channel Characteristics Q = [ ; ]; x: y: >> BAC_demo ans = ans = p 0 = 0. 5 X p = Y p_x = p_x_sim = q = q_sim = Q = Q_sim = PE_sim = PE_theretical = [BAC_demo.m]

15 MATLAB %% Naive Decoder x_hat = y; %% Error Probability PE_sim = -sum(x==x_hat)/n % Error probability from the simulation % Calculation of the theoretical error probability PC = 0; for k = :length(s_x) t = S_X(k); i = find(s_y == t); if length(i) == PC = PC+ p_x(k)*q(k,i); end end PE_theretical = -PC 5 [DMC_Analysis_demo.m]

6]; 0 20 30 40 50 60 70 80 90 00 0 20 30 40 50 60 70 80 90 00 p 0 = 0. 5 X p = 0. 5 0 0.3 0.4 0 20 30 40 50 60 70 80 90 00 0.7 0.

16 6 Ex: BAC %% Simulation parameters % The number of symbols to be transmitted n = e4; % Binary Assymmetric Channel (BAC) % Ex 3.8 in lecture note (.3 in [Z&T, 200]) % Channel Input S_X = [0 ]; S_Y = [0 ]; p_x = [ ]; % Channel Characteristics Q = [ ; ]; p 0 = 0. 5 X p = Y p_x = p_x_sim = q = q_sim = Q = Q_sim = PE_sim = PE_theretical = [BAC_demo.m]

17 7 x: y: Ex: DMC %% Simulation parameters % The number of symbols to be transmitted n = 20; % General DMC % Ex. 3.6 in lecture note % Channel Input S_X = [0 ]; S_Y = [ 2 3]; p_x = [ ]; % Channel Characteristics Q = [ ; ]; >> DMC_demo ans = ans = p 0 = 0. 2 X p = Y p_x = p_x_sim = q = q_sim = Q = Q_sim = PE_sim = PE_theretical = [DMC_demo.m]

3]; 0 20 30 40 50 60 70 80 90 00 p 0 = 0. 2 X 0 20 30 40 50 60 70 80 90 00 p = 0. 8 0 0 20 30 40 50 60 70 80 90 00 0 20 30 40 50 60 70 80 90 00 0.5 0.

18 Ex: DMC 8 %% Simulation parameters % The number of symbols to be transmitted n = e4; % General DMC % Ex. 3.6 in lecture note % Channel Input S_X = [0 ]; S_Y = [ 2 3]; p_x = [ ]; % Channel Characteristics Q = [ ; ]; p 0 = 0. 2 X p = Y p_x = p_x_sim = q = q_sim = Q = Q_sim = PE_sim = PE_theretical = [DMC_demo.m]

19 Evaluation of Probability Consider two random variables X and Y. Suppose their joint pmf matrix is P XY, x y Find P X + Y < Step : Find the pairs (x,y) that satisfy the condition x+y < 7 One way to do this is to first construct the matrix of x+y. x y x y

20 Evaluation of Probability Consider two random variables X and Y. Suppose their joint pmf matrix is P XY, x y Find P X + Y < Step 2: Add the corresponding probabilities from the joint pmf (matrix) P X + Y < 7 = = 0.3 x y x y

21 Evaluation of Probability Consider two random variables X and Y. Suppose their joint pmf matrix is P XY, x y Find P X = Y P X = Y = = 0.5 2

22 Sum of two discrete RVs Consider two random variables X and Y. Suppose their joint pmf matrix is P XY, x y Find P X + Y = P X + Y = 7 = 0. x y x y

23 Optimal Decoder for BSC close all; clear all; tic 23 >> BSC_decoder_ALL_demo Decoder_Table_ALL = ans = x 0 0 x.0000 P E Optimal_Detector = 0 Min_PE = Elapsed time is seconds. [BSC_decoder_ALL_demo.m] %% Simulation parameters % Channel Input S_X = [0 ]; S_Y = [0 ]; p0 = 0.8; p = -p0; p_x = [p0 p]; % Channel Characteristics p = 0.; Q = [-p p; p -p]; %% All possible "reasonable" decoders % X_hat = Y; X_hat = -Y; X_hat = ; X_hat = 0 Decoder_Table_ALL = [0 ; 0; ; 0 0] %% Calculate the error probability for each of the decoder PE_ALL = []; for k = :size(decoder_table_all,) Decoder_Table = Decoder_Table_ALL(k,:); PC = 0; for k = :length(s_x) I = (Decoder_Table == S_X(k)); Q_row = Q(k,:); PC = PC + p_x(k)*sum(q_row(i)); end PE_theretical = -PC; PE_ALL = [PE_ALL; PE_theretical]; end %% Display the results [Decoder_Table_ALL PE_ALL] %% Find the optimal detectors [V I] = min(pe_all); Optimal_Detector = Decoder_Table_ALL(I,:) Min_PE = V toc

24 DIY Decoder >> DMC_decoder_DIY_demo ans = ans = ans = PE_sim = PE_theretical = Elapsed time is seconds. %% Simulation parameters % The number of symbols to be transmitted n = 20; % General DMC % Ex. 3.6 in lecture note % Channel Input S_X = [0 ]; S_Y = [ 2 3]; p_x = [ ]; % Channel Characteristics Q = [ ; ]; %% DIY Decoder Decoder_Table = [0 0]; % The decoded values corresponding to the received Y 24 [DMC_decoder_DIY_demo.m]

25 DIY Decoder %% DIY Decoder Decoder_Table = [0 0]; % The decoded values corresponding to the received Y % Decode according to the decoder table x_hat = y; % preallocation for k = :length(s_y) I = (y==s_y(k)); x_hat(i) = Decoder_Table(k); end PE_sim = -sum(x==x_hat)/n % Error probability from the simulation % Calculation of the theoretical error probability PC = 0; for k = :length(s_x) I = (Decoder_Table == S_X(k)); q = Q(k,:); PC = PC+ p_x(k)*sum(q(i)); end PE_theretical = -PC 25 [DMC_decoder_DIY_demo.m]

DIY Decoder >> DMC_decoder_DIY_demo PE_sim = 0.523 PE_theretical = 0.5200 Elapsed time is 2.54024 seconds.

6 in lecture note % Channel Input S_X = [0 ]; S_Y = [ 2 3]; p_x = [0.2 0.

3]; %% DIY Decoder Decoder_Table = [0 0]; % The decoded values corresponding to the received Y 0 0 0 20

26 DIY Decoder >> DMC_decoder_DIY_demo PE_sim = PE_theretical = Elapsed time is seconds. %% Simulation parameters % The number of symbols to be transmitted n = e4; % General DMC % Ex. 3.6 in lecture note % Channel Input S_X = [0 ]; S_Y = [ 2 3]; p_x = [ ]; % Channel Characteristics Q = [ ; ]; %% DIY Decoder Decoder_Table = [0 0]; % The decoded values corresponding to the received Y [DMC_decoder_DIY_demo.m]

27 Searching for the Optimal Detector >> DMC_decoder_ALL_demo ans = x x 2 x 3 P E Min_PE = Optimal_Detector = Elapsed time is seconds. 27

28 Guessing Game Consider a random variable X whose pmf is / 4, x,2, p X x / 2, x 3, 0, otherwise. You have to guess the value of X. This game is repeated many times. (Say, 00 times) The goodness of your guess is determined by how often you guess correctly. For example, get 0 each time that you guess correctly. How should you guess? 28

29 Guessing Game 3000 xˆ xˆ xˆ

30 Guessing Game: Version 2 Consider a random variable X whose pmf is / 4, x,2, p X x / 2, x 3, 0, otherwise. You have to guess the value of X. This game is repeated many times. (Say, 00 times) The goodness of your guess is determined by how close you are to the value of X. For example, you pay nothing if your guess is correct but you have to pay X x 2 if your guess is wrong. How should you guess? 30

31 Guessing Game: Version xˆ xˆ xˆ [GuessingGame_2.m]

32 Guessing Game: Version Average Return g [GuessingGame_2_2.m]

33 MAP Decoder %% MAP Decoder P = diag(p_x)*q; % Weight the channel transition probability by the % corresponding prior probability. [V I] = max(p); % For I, the default MATLAB behavior is that when there are % multiple max, the index of the first one is returned. Decoder_Table = S_X(I) % The decoded values corresponding to the received Y %% Decode according to the decoder table x_hat = y; % preallocation for k = :length(s_y) I = (y==s_y(k)); x_hat(i) = Decoder_Table(k); end PE_sim = -sum(x==x_hat)/n % Error probability from the simulation %% Calculation of the theoretical error probability PC = 0; for k = :length(s_x) I = (Decoder_Table == S_X(k)); Q_row = Q(k,:); PC = PC+ p_x(k)*sum(q_row(i)); end PE_theretical = -PC 33 [DMC_decoder_MAP_demo.m]

34 ML Decoder %% ML Decoder [V I] = max(q); % For I, the default MATLAB behavior is that when there are % multiple max, the index of the first one is returned. Decoder_Table = S_X(I) % The decoded values corresponding to the received Y %% Decode according to the decoder table x_hat = y; % preallocation for k = :length(s_y) I = (y==s_y(k)); x_hat(i) = Decoder_Table(k); end PE_sim = -sum(x==x_hat)/n % Error probability from the simulation %% Calculation of the theoretical error probability PC = 0; for k = :length(s_x) I = (Decoder_Table == S_X(k)); Q_row = Q(k,:); PC = PC+ p_x(k)*sum(q_row(i)); end PE_theretical = -PC 34 [DMC_decoder_ML_demo.m]

Digital Communication Systems ECS 452

Digital Communication Systems ECS 452 Asst. Prof. Dr. Prapun Suksompong prapun@siit.tu.ac.th 3 Discrete Memoryless Channel (DMC) Office Hours: BKD, 6th floor of Sirindhralai building Tuesday 4:20-5:20