1. 디지털통신
Random Process
임 민 중
동국대학교 정보통신공학과 1

2. Random Processes

3. 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)

4. 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
= = 0.14

5. 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

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

7. 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 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) =

8. 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

9. Random Variables - 5 Average Symbol Energy 1 1 1 1 A t -A1 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

10. 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

11. 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

12. 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 + x x100) / sqrt(100)

13. Gaussian Random Variables - 3x
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 + x x100) / sqrt(100)

14. 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

15. 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) = , Q(2) = , Q(3) =
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

16. 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) = ,
Q(2) = , Q(3) = 2

17. 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

18. 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

19. 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

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

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

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