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Wireless Communication Elec 534 Set IV October 23, 2007

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Presentation on theme: "Wireless Communication Elec 534 Set IV October 23, 2007"— Presentation transcript:

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

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

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

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