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

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

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

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

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

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

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

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

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

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

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

Basic Assumption The additive noise is Gaussian The average power constraint

Matrices A channel matrix Trace of a square matrix

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

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

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

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

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

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

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

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

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

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

Solution Consider a vector Y with the covariance as X

Solution Since X and Y have the same covariance Q then

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

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

Parallel Channels At most parallel channels Power distribution across parallel channels

Parallel Channels A few useful notes

Parallel Channels A note

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

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

Example Consider a 3x3 channel Mutual information is maximized by

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

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

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

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

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

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

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

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

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

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

Example Slow fading SIMO, Then and Scalar

Example Slow fading MISO, The optimum The outage

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

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

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

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