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디지털통신 Random Process 임민중 동국대학교 정보통신공학과 Minjoong Rim, Dongguk University 1 Random Process Random Processes Probability • Definitions Sample space S - Event - a single sample point or a set of sample points in the space S Elementary event - the set of all possible outcomes of the experiment 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(AB) = P(A) + P(B) Minjoong Rim, Dongguk University 3 Random Process Conditional Probability • P(B|A) The probability of event B, given that A has occurred P( A B) P( B | A) P( A) P(A B) = P(A) P(B|A) • Bayes’ rule P( A B) P( B | A) P( A) P( A | B) P( B) P( A | B) P( B) P( B | A) P( A) • Example: binary data transmission Transmitted Data 0 X Y 1 P(X = 0) = 0.6 P(X = 1) = 0.4 P(Y = 0 | X = 0) = 0.9 P(Y = 1 | X = 0) = 0.1 P(Y = 1 | X = 1) = 0.8 P(Y = 0 | X = 1) = 0.2 0 1 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) Minjoong Rim, Dongguk University 4 Received Data = 0.6 0.1 + 0.4 0.2 = 0.14 Random Process 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 FX ( x ) P( X x ) 0 FX ( x ) 1 FX ( x1 ) FX ( x2 ) if x1 x2 FX ( ) 0 FX () 1 • Probability density function (PDF) p X ( x) dFX ( x ) dx P( x1 X x2 ) p X ( x )dx x2 Properties of PDF x1 p X ( x) 0 Minjoong Rim, Dongguk University 5 p X ( x )dx 1 Random Process Random Variables - 2 • Mean (Ensemble average) X mX E{ X } xpX ( x)dx E(X) • Variance V ( X ) E{( X mx ) } ( x mx ) 2 p X ( x)dx 2 X 2 X X Relationship between variance and mean-square value X2 E { X 2 2mX X m2X } E { X 2 } 2mX E { X } m2X E { X 2 } m2X Minjoong Rim, Dongguk University 6 Random Process Random Variables - 3 • Example: Rolling a dice pdf, pmf (probability mass function) The sum should be one 1/6 1 2 3 4 5 6 time average cdf Rolling a dice ... repeatedly increasing from 0 to 1 1 1/6 1 2 3 4 5 6 3 1 5 4 5 2 6 3 1 1 4 6 2 ... 6 2 ensemble average 3 1 E(X) = 1 (1/6) + 2 (1/6) + 3 (1/6) + 4 (1/6) + 5 (1/6) + 6 (1/6) = 3.5 V(X) = 12 (1/6) + 22 (1/6) + 32 (1/6) + 42 (1/6) + 52 (1/6) + 62 (1/6) - 3.52 = 2.9167 Minjoong Rim, Dongguk University 7 Random Process Random Variables - 4 • Example: Uniform random variable pdf 1/(b-a) f(x) area = 1 a cdf b 1 F(x) a E(X) = V(X) = b xf ( x)dx x b x m Minjoong Rim, Dongguk University a 1 ba dx ba 2 b a 1 b ab f ( x)dx x dx b a a 2 12 2 2 8 2 Random Process Random Variables - 5 • Average Symbol Energy E T s(t ) 2 dt 0 1 0 0 1 0 0 1 1 0 E(0) = E(1) = 1/2 A t A 2T ( A) 2 T P A 2T 2 -A T = symbol duration 10 11 01 10 00 01 11 3A 00 10 E(00) = E(01) = E(10) = E(11) = 1/4 A t -A -3A A 2T (3 A) 2 T ( A) 2 T (3 A) 2 T P 5 A 2T 4 T = symbol duration Minjoong Rim, Dongguk University 9 Random Process Random Variables - 6 • Average Symbol Energy 0 0 1 A 0 1 0 A assuming E(0) = E(1) = 1/2 E | s |2 E(00) = E(01) = E(10) = E(11) = 1/4 0 B 1 -B Symbol Energy = (0 + A2) / 2 = A2 / 2 00 01 10 11 C 3C -C -3C 11 10 00 01 -C C 0 -B B Symbol Energy = ((-B)2 + B2) / 2 = B2 00 01 10 11 D D D D j 2 1 -3C 1 (1 + j) (-1 + j) (1 - j) (-1 - j) 01 D -D 00 D 3C 11 Symbol Energy = ((-3C)2 + (-C)2 + C2 + (3C)2) / 4 = 5C2 -D 10 Symbol Energy = 2D2 Minjoong Rim, Dongguk University 10 Random Process Gaussian Random Variables - 1 • Gaussian Random Variable 1 n2 1 p( n) exp 2 2 • 2-dimensional Gaussian Random Variable 2-dimensional amplitude Rayleigh = Amplitude of Zero-Mean Complex Gaussian Complex Gaussian Minjoong Rim, Dongguk University 11 Rayleigh distribution in amplitude Random Process 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) Minjoong Rim, Dongguk University y = (x1 + x2 + x3) / sqrt(3) 12 y = (x1 + x2 + ... + x100) / sqrt(100) Random Process Gaussian Random Variables - 3 x y = (x1 + x2 + x3 + x4) / sqrt(4) Minjoong Rim, Dongguk University y = (x1 + x2) / sqrt(2) y = (x1 + x2 + x3 + x4 + x5) / sqrt(5) 13 y = (x1 + x2 + x3) / sqrt(3) y = (x1 + x2 + ... + x100) / sqrt(100) Random Process Functions of Random Variables - 1 • Functions of Random Variables Y g( X ) Expected Value E (Y ) g ( x) f X ( x)dx X • Properties Y aX b 0 X 1 X: random variable a, b: constants E (aX b) aE ( X ) b 1 2X V (aX b) a V ( X ) 2 0 • Example Y = 2X + 1, E(X) = 1, V(X) = 1 E(Y) = ? V(Y) = ? Minjoong Rim, Dongguk University 2X 1 1 14 Random Process Functions of Random Variables - 2 • Q function Random Variable mean = 0 X: Gaussian RV with zero mean and unit variance Q(x) P(X > x) Table is given - X variance = 1 Q x x 0 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) = ? -1 0 • Example Y: Gaussian random variable with mean = b and variance = a2 P(Y > T) = Q((T – b) / a) Y aX b b Minjoong Rim, Dongguk University T X Y b a T b Q a 0 (T-b)/a 15 Random Process Functions of Random Variables - 3 • Example • Example P(Y > 0) where Y is a Gaussian RV with mean = -1 and variance = 22 ? P(Y < 0) where Y is a Gaussian RV with mean = 1 and variance = (1/2)2 ? 1 2 2 X 1 -1 0 0 1 1 2 2X 0 X 1 -1 0 X 0 X 1 X -2 0.5 0 X Q(0.5) = 0.309, Q(1) = 0.159, Q(1.5) = 0.0668, Q(2) = 0.0228, Q(3) = 0.00135 Minjoong Rim, Dongguk University 0 16 2 Random Process Functions of Random Variables - 4 0 • Example 0 Xb Y=X+N X: P(X = -1) = P(X = 1) = 1/2 Yb 1 N: Gaussian RV with mean 0 and variance 1 X, N: independent 1 0 -1 1 1 Xb X Y <00 >01 Yb N P(error) = P(Xb = 0 Yb = 1) + P(Xb = 1 Yb = 0) P( A B) P( B | A) P( A) P( A | B) P( B) = 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) = ? P(Y > 0 | X = -1) = P(X + N > 0 | X = -1) = P(-1 + N > 0) = P(N > 1) = Q(1) 0.5 0.5 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 -1 Minjoong Rim, Dongguk University P(Y < 0 | X = 1) = P(X + N < 0 | X = 1) = P(1 + N < 0) = P(N < -1) = Q(1) 0.5 0 0 17 = 0.5 Q(1) 1 Random Process Correlation • Correlation of X and Y How much X and Y are correlated E[XY] Correlation can be also affected by the mean and variance of X and Y • Covariance of X and Y zero mean cov[XY ] E[( X E[ X ])(Y E[Y ])] E[ XY ] E[ X ]E[Y ] • Correlation coefficient of X and Y unit variance X ,Y cov( X , Y ) XY Minjoong Rim, Dongguk University 18 Uncorrelated Random Process Power Spectral Density - 1 • Random Process Deterministic Signal (Fourier Transform) Frequency-domain Representation Random Signal Autocorrelation (Fourier Transform) Power Spectral Density the outcome of a random experiment is mapped into a waveform that is a function of time • Autocorrelation X (t ) RX ( ) E[ X (t ) X (t )] • Power Spectral Density A measure of the frequency distribution of a single random process S X ( f ) RX ( )e j 2 f d • Cross-correlation X (t ) RXY ( ) E[ X (t )Y (t )] • Cross Spectral Density A measure of the frequency inter-relationship between two random processes S XY ( f ) RXY ( )e j 2f d Minjoong Rim, Dongguk University 19 Random Process Y (t ) Power Spectral Density - 2 • Random process slowly fluctuating random process • Autocorrelation Rapidly fluctuating random process • Power spectral density narrow bandwidth wide bandwidth f f Minjoong Rim, Dongguk University 20 Random Process Example: Power Spectral Density - 1 Signal Minjoong Rim, Dongguk University Autocorrelation 21 Power Spectral Density Random Process Example: Power Spectral Density - 2 Signal Minjoong Rim, Dongguk University Autocorrelation 22 Power Spectral Density Random Process