Outline

1. Outline • Introduction • Signal, random variable, random process and spectra • Analog modulation • Analog to digital conversion • Digital transmission through baseband channels • Signal space representation • Optimal receivers • Digital modulation techniques • Channel coding • Synchronization • Information theory

2. Signal, random variable, random process and spectra

3. Signal, random variable, random process and spectra • Signals • Review of probability and random variables • Random processes: basic concepts • Gaussian and White processes Selected from Chapter 2.1-2.6, 5.1-5.3

4. Signal • In communication systems, a signal is any function that carries information. Also called information bearing signal

5. Signal • Continuous-time signal vs. discrete-time signal • Continuous-valued signal vs. discrete-valued signal Continuous-time continuous-valued: analog signal Discrete-time and discrete-valued: digital signal Discrete-time and continuous-valued : sampled signal Continuous-time and discrete-valued: quantized signal

6. Signal

7. Signal • Energy vs. power signal • Energy • Power • A signal is an energy signal iff energy is limited • A signal is a power signal iff power is limited

8. Signal • Fourier Transform

9. Random variable • Review of probability and random variables • Two events A and B • Conditional probability P(A|B) • Joint probability P(AB)=P(A)P(B|A)=P(B)P(A|B) • A and B are independent iff P(AB)=P(A)P(B) • Let be mutually exclusive events with • . Then for any event , we have

10. Random variable • Review of probability and random variables • Bayes’ Rule: Let be mutually exclusive such that . For any nonzero probability event B, we have

11. Random variable • Review of probability and random variables Consider a binary communication system

12. Random variable • Review of probability and random variables • A random variable is a mapping from the sample space to the set of real numbers • Discrete r.v.: range is finite ({0,1}) or countable infinite ({0,1,…}) • Continuous r.v.: range is uncountable infinite (real number)

13. Random variable • Review of probability and random variables • The cumulative distribution function (CDF) of a r.v. X is • The key properties of CDF

14. Random variable • Review of probability and random variables • The probability density function (PDF) of a r.v. X is • The key properties of PDF

15. Random variable • Review of probability and random variables • Common random variables: Bernoulli, Binomial, Uniform, and Gaussian • Bernoulli distribution: • Binomial distribution: the sum of n independent Bernoulli r.v.

16. Random variable • Review of probability and random variables • Suppose that we transmit a 31-bit sequence with error correction capability up to 3 bit errors • If the probability of a bit error is p=0.001, what is the probability that the sequence received is in error? • On the other hand, if no error correction is used, the error probability is

17. Random variable • Review of probability and random variables • Uniform distribution: • The random phase of a sinusoid is often modeled as a uniform r.v. between 0 and • The mean or expected value of X is first moment of X

18. Random variable • Review of probability and random variables • The n-th moment of X • If n=2, we have the mean-squared value of X • The n-th central moment is • If n=2, we have the variance of X • is called the standard deviation

19. Random variable • Review of probability and random variables • Gaussian distribution: • A Gaussian r.v. is completely determined by its mean and variance, and hence usually denoted as The most important distribution in communications!

20. Random variable • Review of probability and random variables • Q-function is a standard form to express error probabilities without a closed-form • The Q-function is the area under the tail of a Gaussian pdf with mean 0 and variance 1 Extremely important in error probability analysis!

21. Random variable • Review of probability and random variables • Some key properties of Q-function Q-function is monotonically decreasing Craig’s alternative form of Q-function Upperbound For a Gaussian variable

22. Random variable • Review of probability and random variables • Joint distribution of two r.v.s X and Y is • And joint PDF is

23. Random variable • Review of probability and random variables • Marginal distribution • Marginal density • X and Y are said to be independent iff

24. Random variable • Review of probability and random variables • Correlation of the two r.v.s X and Y is • Correlation of the two centered r.v.s X-E[X] and Y-E[Y] is called the covariance of X and Y • If then X and Y are called uncorrelated.

25. Random variable • Review of probability and random variables • The covariance of X and Y normalized w.r.t. is referred to the correlation coefficient of X and Y • If X and Y are independent, then they are uncorrelated. • The converse is not true except for the Gaussian r.v.

26. Random variable • Review of probability and random variables • Functions of random variable. • How to obtain the PDF of r.v. ? • Two steps: • Calculate the CDF of Y through • Take the derivative of CDF • For r.v.s , generally

27. Random variable • Review of probability and random variables • are jointly Gaussian iff

28. Random variable • Review of probability and random variables • In case of n=2, we have

29. Random variable • Review of probability and random variables • And if X1 and X2 are uncorrelated, i.e., If X1 and X2 are Gaussian and uncorrelated, they are independent.

30. Random variable • Review of probability and random variables • If n random variables are jointly Gaussian, then any set of them is also jointly Gaussian. • Jointly Gaussian r.v.s are completely characterized by mean vector and the covariance matrix. • Any linear combination of the n r.v.s is Gaussian.

31. Random variable • Review of probability and random variables • Law of large numbers Consider a sequence of r.v. , let , if the r.v.s are uncorrelated with the same mean , and variance , then The average r.v. converges to the mean value

32. Random variable • Review of probability and random variables • Central limit theorem If are i.i.d. r.v.s with same mean , and variance , then Thermal noise results from the random movements of many electrons is well modeled by a Gaussian distribution.

33. Random variable • Review of probability and random variables • Consider for example a sequence of uniform distribution r.v.s with mean 0 and variance 1/12, then n=2 n=4 n=1 n=8 Dashed line: Solid line:

34. Random variable • Review of probability and random variables • Rayleigh distribution: Rayleigh distribution is usually used to model fading for non-line-of-sight (NLOS) signal transmissions. Very important distribution for analysis in mobile and wireless communications.

35. Random variable • Review of probability and random variables • Consider for example h=a+jb, where a and b are i.i.d. Gaussian r.v.s with mean 0 and variance , then the magnitude of h follows Rayleigh distribution • and the phase follows uniform distribution.

36. Random process and spectra • Random processes: basic concepts • A random process (stochastic process, or random signal) is the evolution of random variables over time. • A random process is a function defined over time and sample space with value at a specific time corresponding to a sample value . • Given , are the random variables. • Given , is the sample path function.

37. Random process and spectra • Random processes: basic concepts • Let , for fixed amplitude and frequency, the random process • have different sample path functions First trial Second trial Third trial

38. Random process and spectra • Random processes: basic concepts • The statistics of random variables at different times • CDF • Expectation function • Auto-correlation function • Covariance

39. Random process and spectra • Random processes: basic concepts • Consider for example the • with , we have • Expectation function: • Auto-correlation function (assume ):

40. Random process and spectra • Random processes: Stationary processes • A stochastic process is said to be stationary if for any n and The expectation function is independent of time The auto-correlation function only depends on the time difference

41. Random process and spectra • Random processes: Stationary processes • A random process is said to be Wide sense stationary (WSS) if • A strict stationary process satisfies for any n and

42. Random process and spectra • Random processes: Stationary processes • Some properties of WSS If , then

43. Random process and spectra • Random processes: Stationary processes • Statistical averaging • Time averaging • If time averaging = statistical averaging, then the process is said to be ergodic. (what does it mean?)

44. Random process and spectra • Random processes: Stationary processes

45. Random process and spectra • Random processes: Stationary processes • Frequency domain characteristics

46. Random process and spectra • Random processes: Stationary processes • Given a deterministic signal , the power is • Truncate to get an energy signal • Parseval’s theorem • Hence

47. Random process and spectra • Random processes: Stationary processes • Consider as a sample path function of a random process X(t) , the average power of X(t) is • Power spectral density of X(t)

48. Random process and spectra • Random processes: Stationary processes • Consider a binary semi-random signal • with • Find its power spectral • density?

49. Random process and spectra • Random processes: Stationary processes • Wiener-Khinchin theorem • Property

50. Random process and spectra • Random processes: Stationary processes • Consider for instance the random process • , we have • Then,