디지털통신 Random Process 임 민 중 동국대학교 정보통신공학과 1

Random Processes

Probability Definitions Axioms Sample space S Event Elementary event the set of all possible outcomes of the experiment Event a single sample point or a set of sample points in the space S Elementary event a single sample point Mutually exclusive the occurrence of one events precludes the occurrence of the other event Axioms 0  P(A)  1 for an event A P(S) = 1 If A and B are two mutually exclusive events, then P(AB) = P(A) + P(B)

Conditional Probability P(B|A) The probability of event B, given that A has occurred Bayes’ rule Example: binary data transmission P(X = 0) = 0.6 P(Y = 0 | X = 0) = 0.9 P(Y = 1 | X = 1) = 0.8 P(error) = P(X  Y) = P(X = 0  Y = 1) + P(X = 1  Y = 0) = P(X = 0) P(Y = 1 | X = 0) + P(X = 1) P (Y = 0 | X = 1) P(A  B) = P(A) P(B|A) Transmitted Data X Y Received Data 1 1 P(X = 1) = 0.4 P(Y = 1 | X = 0) = 0.1 P(Y = 0 | X = 1) = 0.2 = 0.6 0.1 + 0.4 0.2 = 0.14

Random Variables - 1 Random variable, X(A) the function relationship between a random event, A, and a real number Cumulative distribution function (CDF) Properties of CDF Probability density function (PDF) Properties of PDF

Random Variables - 2 Mean (Ensemble average) Variance Relationship between variance and mean-square value E(X)

Rolling a dice repeatedly Random Variables - 3 Example: Rolling a dice pdf, pmf (probability mass function) cdf E(X) = V(X) = The sum should be one 1/6 1 2 3 4 5 6 time average Rolling a dice repeatedly increasing from 0 to 1 ... 3 1 5 4 5 2 6 3 1 1 4 6 2 ... 6 2 3 1 1 2 3 4 5 6 1/6 ensemble average 1 (1/6) + 2 (1/6) + 3 (1/6) + 4 (1/6) + 5 (1/6) + 6 (1/6) = 3.5 12 (1/6) + 22 (1/6) + 32 (1/6) + 42 (1/6) + 52 (1/6) + 62 (1/6) - 3.52 = 2.9167

Random Variables - 4 Example: Uniform random variable pdf cdf E(X) = V(X) = f(x) 1/(b-a) area = 1 a b a b F(x) 1

Random Variables - 5 Average Symbol Energy 1 1 1 1 A t -A 1 1 1 E(0) = E(1) = 1/2 A t -A T = symbol duration 10 11 01 10 00 01 11 00 10 E(00) = E(01) = E(10) = E(11) = 1/4 3A A t -A T = symbol duration -3A

Random Variables - 6 Average Symbol Energy 1 1 A -B B assuming E(0) = E(1) = 1/2 E(00) = E(01) = E(10) = E(11) = 1/4 0  0 1  A 1 0  B 1  -B 1 A -B B Symbol Energy = (0 + A2) / 2 = A2 / 2 Symbol Energy = ((-B)2 + B2) / 2 = B2 00  D (1 + j) 01  D (-1 + j) 10  D (1 - j) 11  D (-1 - j) 01 D 00 00  C 01  3C 10  -C 11  -3C 11 10 00 01 -D D -3C -C C 3C Symbol Energy = ((-3C)2 + (-C)2 + C2 + (3C)2) / 4 = 5C2 11 -D 10 Symbol Energy = 2D2

Gaussian Random Variables - 1 2-dimensional Gaussian Random Variable 2-dimensional amplitude Rayleigh = Amplitude of Zero-Mean Complex Gaussian Complex Gaussian Rayleigh distribution in amplitude

Gaussian Random Variables - 2 Example Central Limit Theorem Probability distribution of the sum of j statistically independent random variables approaches the Gaussian distribution as j   x y = (x1 + x2) / sqrt(2) y = (x1 + x2 + x3) / sqrt(3) y = (x1 + x2 + ... + x100) / sqrt(100)

Gaussian Random Variables - 3 x y = (x1 + x2) / sqrt(2) y = (x1 + x2 + x3) / sqrt(3) y = (x1 + x2 + x3 + x4) / sqrt(4) y = (x1 + x2 + x3 + x4 + x5) / sqrt(5) y = (x1 + x2 + ... + x100) / sqrt(100)

Functions of Random Variables - 1 Expected Value Properties Example Y = 2X + 1, E(X) = 1, V(X) = 1 E(Y) = ? V(Y) = ? X: random variable a, b: constants 1 1

Functions of Random Variables - 2 Q function X: Gaussian RV with zero mean and unit variance Q(x)  P(X > x) Table is given Q(0.5) = 0.309, Q(1) = 0.159, Q(1.5) = 0.0668, Q(2) = 0.0228, Q(3) = 0.00135 Q(0) = ?, Q() = ?, Q(-) = ?, Q(-1) = ? Example Y: Gaussian random variable with mean = b and variance = a2 P(Y > T) = Q((T – b) / a) Random Variable mean = 0 variance = 1 -1 b T (T-b)/a

Functions of Random Variables - 3 Example P(Y > 0) where Y is a Gaussian RV with mean = -1 and variance = 22 ? Example P(Y < 0) where Y is a Gaussian RV with mean = 1 and variance = (1/2)2 ? 1 -1 1 -1 -2 0.5 Q(0.5) = 0.309, Q(1) = 0.159, Q(1.5) = 0.0668, Q(2) = 0.0228, Q(3) = 0.00135 2

Functions of Random Variables - 4 Example Y = X + N X: P(X = -1) = P(X = 1) = 1/2 N: Gaussian RV with mean 0 and variance 1 X, N: independent P(error) = P(X = -1  Y > 0) + P(X = 1  Y < 0) = P(X = -1) P(Y > 0 | X = -1) + P(X = 1) P(Y < 0 | X = 1) = ? Xb Yb 1 1 0  -1 1  1 X Y Xb < 0  0 > 0  1 Yb N = P(Xb = 0  Yb = 1) + P(Xb = 1  Yb = 0) 0.5 0.5 P(Y < 0 | X = 1) = P(X + N < 0 | X = 1) = P(1 + N < 0) = P(N < -1) = Q(1) P(Y > 0 | X = -1) = P(X + N > 0 | X = -1) = P(-1 + N > 0) = P(N > 1) = Q(1) P(X = -1) f(Y | X = -1) P(X = 1) f(Y | X = 1) P(X = -1) P(Y > 0 | X = -1) P(X = 1) P(Y < 0 | X = 1) = 0.5 Q(1) = 0.5 Q(1) 0.5 0.5 -1 1

Correlation Correlation of X and Y Covariance of X and Y Correlation coefficient of X and Y How much X and Y are correlated Correlation can be also affected by the mean and variance of X and Y zero mean unit variance Uncorrelated

Power Spectral Density - 1 Random Process the outcome of a random experiment is mapped into a waveform that is a function of time Autocorrelation Power Spectral Density A measure of the frequency distribution of a single random process Cross-correlation Cross Spectral Density A measure of the frequency inter-relationship between two random processes Deterministic Signal (Fourier Transform) Frequency-domain Representation Random Signal  Autocorrelation (Fourier Transform) Power Spectral Density

Power Spectral Density - 2 Random process Autocorrelation Power spectral density slowly fluctuating random process Rapidly fluctuating random process narrow bandwidth wide bandwidth

Example: Power Spectral Density - 1 Signal Autocorrelation Power Spectral Density

Example: Power Spectral Density - 2 Signal Autocorrelation Power Spectral Density