1. Wireless Communication Elec 534 Set IV October 23, 2007
Behnaam Aazhang

2. Reading for Set 4 Tse and Viswanath Goldsmith Chapters 7,8
Appendices B.6,B.7
Goldsmith
Chapters 10

3. Outline Channel model Basics of multiuser systems
Basics of information theory
Information capacity of single antenna single user channels
AWGN channels
Ergodic fast fading channels
Slow fading channels
Outage probability
Outage capacity

4. Outline Communication with additional dimensions
Multiple input multiple output (MIMO)
Achievable rates
Diversity multiplexing tradeoff
Transmission techniques
User cooperation

5. Dimension Signals for communication Achievable rate per real dimension
Time period T
Bandwidth W
2WT natural real dimensions
Achievable rate per real dimension

6. Communication with Additional Dimensions: An Example
Adding the Q channel
BPSK to QPSK
Modulated both real and
imaginary signal dimensions
Double the data rate
Same bit error probability

7. Communication with Additional Dimensions
Larger signal dimension--larger capacity
Linear relation
Other degrees of freedom (beyond signaling)
Spatial
Cooperation
Metric to measure impact on
Rate (multiplexing)
Reliability (diversity)
Same metric for
Feedback
Opportunistic access

8. Multiplexing Gain Additional dimension used to gain in rate
Unit benchmark: capacity of single link AWGN
Definition of multiplexing gain

9. Diversity Gain Dimension used to improve reliability
Unit benchmark: single link Rayleigh fading channel
Definition of diversity gain

10. Multiple Antennas Improve fading and increase data rate
Additional degrees of freedom
virtual/physical channels
tradeoff between diversity and multiplexing
Transmitter
Receiver

11. Multiple Antennas The model where Tc is the coherence time Transmitter
Receiver

12. Basic Assumption The additive noise is Gaussian
The average power constraint

13. Matrices
A channel matrix
Trace of a square matrix

14. Matrices The Frobenius norm
Rank of a matrix = number of linearly independent rows or column
Full rank if

15. Matrices A square matrix is invertible if there is a matrix
The determinant—a measure of how noninvertible a matrix is!
A square invertible matrix U is unitary if

16. Matrices Vector X is rotated and scaled by a matrix A
A vector X is called the eigenvector of the matrix and lambda is the eigenvalue if
Then
with unitary and diagonal matrices

17. Matrices The columns of unitary matrix U are eigenvectors of A
Determinant is the product of all eigenvalues
The diagonal matrix

18. Matrices If H is a non square matrix then
Unitary U with columns as the left singular vectors and unitary V matrix with columns as the right singular vectors
The diagonal matrix

19. Matrices
The singular values of H are square root of eigenvalues of square H

20. MIMO Channels There are channels
Independent if
Sufficient separation compared to carrier wavelength
Rich scattering
At transmitter
At receiver
The number of singular vectors of the channel
The singular vectors are the additional (spatial) degrees of freedom

21. Channel State Information
More critical than SISO
CSI at transmitter and received
CSI at receiver
No CSI
Forward training
Feedback or reverse training

22. Fixed MIMO Channel A vector/matrix extension of SISO results
Very large coherence time

23. Exercise
Show that if X is a complex random vector with covariance matrix Q its differential entropy is largest if it was Gaussian

24. Solution
Consider a vector Y with the covariance as X

25. Solution
Since X and Y have the same covariance Q then

26. Fixed Channel The achievable rate with
Differential entropy maximizer is a complex Gaussian random vector with some covariance matrix Q

27. Fixed Channel Finding optimum input covariance
Singular value decomposition of H
The equivalent channel

28. Parallel Channels At most parallel channels
Power distribution across parallel channels

29. Parallel Channels
A few useful notes

30. Parallel Channels
A note

31. Fixed Channel Diagonal entries found via water filling Achievable rate
with power

32. Example
Consider a 2x3 channel
The mutual information is maximized at

33. Example
Consider a 3x3 channel
Mutual information is maximized by

34. Ergodic MIMO Channels A new realization on each channel use No CSI
CSIR
CSITR?

35. Fast Fading MIMO with CSIR
Entries of H are independent and each complex Gaussian with zero mean
If V and U are unitary then distribution of H is the same as UHV*
The rate

36. MIMO with CSIR The achievable rate
since the differential entropy maximizer is a complex Gaussian random vector with some covariance matrix Q

37. Fast Fading and CSIR Finally, with The scalar power constraint
The capacity achieving signal is circularly symmetric complex Gaussian (0,Q)

38. MIMO CSIR Since Q is non-negative definite Q=UDU*
Focus on non-negative definite diagonal Q
Further, optimum

39. Rayleigh Fading MIMO CSIR achievable rate
Complex Gaussian distribution on H
The square matrix W=HH*
Wishart distribution
Non negative definite
Distribution of eigenvalues

40. Ergodic / Fast Fading The channel coherence time is
The channel known at the receiver
The capacity achieving signal b must be circularly symmetric complex Gaussian

41. Slow Fading MIMO
A channel realization is valid for the duration of the code (or transmission)
There is a non zero probability that the channel can not sustain any rate
Shannon capacity is zero

42. Slow Fading Channel If the coherence time Tc is the block length
The outage probability with CSIR only
with and

43. Slow Fading
Since
Diagonal Q is optimum
Conjecture: optimum Q is

44. Example
Slow fading SIMO,
Then and
Scalar

45. Example
Slow fading MISO,
The optimum
The outage

46. Diversity and Multiplexing for MIMO
The capacity increase with SNR
The multiplexing gain

47. Diversity versus Multiplexing
The error measure decreases with SNR increase
The diversity gain
Tradeoff between diversity and multiplexing
Simple in single link/antenna fading channels

48. Coding for Fading Channels
Coding provides temporal diversity or
Degrees of freedom
Redundancy
No increase in data rate

49. M versus D
(0,MRMT)
Diversity Gain
(min(MR,MT),0)
Multiplexing Gain