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 ,

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. Consider a binary communication system
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. The most important distribution in communications!
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. Extremely important in error probability analysis!
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. If X1 and X2 are Gaussian and uncorrelated, they are independent.
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. The average r.v. converges to the mean value
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,

51. Random process and spectra
Random processes: Stationary processes
Given a binary random signal
with

52. Random process and spectra
Random processes: Stationary processes
Then, we have

53. Random process and spectra
Random processes: Stationary processes
Random process transmission through linear systems
Mean of the output

54. Random process and spectra
Random processes: Stationary processes
Auto-correlation of the output
If the input is a WSS, then the output is also a WSS.
PSD of the output

55. Random process and spectra
Random processes: Stationary processes
Comparison between deterministic and random signal

56. Random process and spectra
Random processes: Stationary processes
Consider for example

57. Random process and spectra
Random processes: Gaussian and White processes
Gaussian process
Some properties
If it is WSS, it is strictly stationary.
If the input to a linear system is a Gaussian process, the output is also a Gaussian process.

58. Random process and spectra
Random processes: Gaussian and White processes
Consider for example the noise, which is often modeled as Gaussian and stationary with zero mean
White noise

59. Random process and spectra
Random processes: Gaussian and White processes
Band-limited noise
Think about

60. Random process and spectra
Random processes: Gaussian and White processes
Band-pass noise
Canonical form of a band-pass noise process

61. Random process and spectra
Random processes: Gaussian and White processes
Consider the band-pass noise.
If n(t) is a zero-mean, stationary and Gaussian noise, then nc(t) and ns(t) satisfy the following properties:
nc(t) and ns(t) are zero-mean, jointly stationary and Gaussian process.

62. Random process and spectra
Random processes: Gaussian and White processes
Consider the band-pass noise.
Angular representation of n(t)
with

63. Random process and spectra
Random processes: Gaussian and White processes
Find the statistics of the envelop and phase of the angular representation of n(t).

64. Random process and spectra
Random processes: Gaussian and White processes
Overview

65. Signal, random variable, random process and spectra
Suggested reading & Homework
Chapter , Chapter of the textbook