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Wireless Communication Elec 534 Set IV October 23, 2007 Behnaam Aazhang Reading for Set 4 • Tse and Viswanath – 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 • Achievable rates • Transmission techniques Dimension • Signals for communication – Time period T – Bandwidth W – 2WT natural real dimensions • Achievable rate per real dimension Pav 1 log( 1 2 ) 2 n 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 2 Eb Pe Q( ) N0 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 C ( SNR) log( 1 SNR) bit per second per Hertz • Definition of multiplexing gain C ( SNR) r lim SNR log( SNR) Diversity Gain • Dimension used to improve reliability • Unit benchmark: single link Rayleigh fading channel out 1 SNR • Definition of diversity gain log( out ( SNR)) d lim SNR log( SNR) 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 rM R Tc H M R M T bM T Tc nM R Tc where Tc is the coherence time Transmitter Receiver Basic Assumption • The additive noise is Gaussian nM R N0 ~ Gaussian (0, I M R M R ) 2 • The average power constraint Trace{E[ bMT b ]} Pav H MT Matrices • A channel matrix H M R M T h11* h11 h1M R H , H M T M R h* hM 1 hM M T R T 1M R • Trace of a square matrix M Trace[ H M M ] hii i 1 hM* T 1 * hM M T R Matrices • The Frobenius norm H Trace[ HH ] Trace[ H H ] H F H • Rank of a matrix = number of linearly independent rows or column Rank [ H ] min{ M R , MT } • Full rank if Rank [ H ] min{ M R , MT } Matrices • A square matrix is invertible if there is a matrix 1 AA I • The determinant—a measure of how noninvertible a matrix is! • A square invertible matrix U is unitary if UU H I Matrices • Vector X is rotated and scaled by a matrix A y Ax • A vector X is called the eigenvector of the matrix and lambda is the eigenvalue if Ax x • Then A U U H 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 1 0 0 N Matrices • If H is a non square matrix then H M R M T U M R M R M RMT V H M T M T • Unitary U with columns as the left singular vectors and unitary V matrix with columns as the right singular vectors • The diagonal matrix 1 0 M R M T 0 0 1 0 M T or 0 0 M R 0 0 Matrices • The singular values of H are square root of eigenvalues of square H i singular( H M R M T ) i eigenvalue ( H M R M T H MHT M R ) MIMO Channels • There are M T M R channels – Independent if • Sufficient separation compared to carrier wavelength • Rich scattering – At transmitter – At receiver • The number of singular vectors of the channel min{ MT , M R } • 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 I (rM R ; bM T | H M R M T ) h(rM R | H M R M T ) h(rM R | bM T , H M R M T ) h(rM R | H M R M T ) h(nM R ) h(rM R | H M R M T ) M R log( eN 0 ) log[( e) M R det( N 0 I M R M R HQH * )] M R log( eN 0 ) 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 h(Y ) h( X ) fY log fY dY f Gaussian log f GaussiandX fY log fY dY fY log f GaussiandY f Gaussian fY log dY 0 fY Solution • Since X and Y have the same covariance Q then f log f GaussiandX f Gaussian[ X QX ]dX * Gaussian fY [Y QY ]dY * fY log f GaussiandY Fixed Channel • The achievable rate max I (b; r ) log[( e) M R det( N 0 I M R M R HQH * )] M R log( eN 0 ) pb log det( I M R M R HQH * ) N0 * Q E [ bb ] and E [ b b] Pav with MT MT • 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 H UV * min{ M R , M T } * u v mmm m 1 • The equivalent channel ~ ~ * ~ ~ ~ rM R M R M T bM T nM R with r U r and b V *b Parallel Channels • At most min{ MT , M R } parallel channels ~ ~ ~ rm mbm nm ; m 1,2,, min {M R ,M T } • Power distribution across parallel channels QMT MT E[bb ] and E[b*b] tr (Q) tr (VQV * ) Pav Parallel Channels • A few useful notes * log det( I M R M R HQH ) log det( I M T M T N0 log det( I M R M R log det( I M R M R QM T M T H M* T M R H M R M T N0 M R M T VM T M T QV **M T M R ~ * Q ) N0 ~ 2 log (1 Qmmm ) m ) N0 ) Parallel Channels • A note ~ * Q ~ 2 det( I M R M R ) (1 Qmmm ) m N0 ~ with equality w hen Q is diagonal Fixed Channel • Diagonal entries found via water filling • Achievable rate I ( r ; b) min{ M R , M T } m 1 P log( 1 ) N0 * 2 m m with power P ( * m N0 2 m ) with P * m m Pav Example • Consider a 2x3 channel H 32 1 1 1 / 3 1 1 1 / 3 6 1 1 1 / 3 1 / 2 1/ 2 • The mutual information is maximized at 6P * I (r; b) log( 1 ) with E[bi b j ] P / 2 N0 Example • Consider a 3x3 channel 1 0 0 H 0 1 0 0 0 1 • Mutual information is maximized by P P I (r ; b) 3 log( 1 ) with Q I 33 3N 0 3 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 I ( H M R M T , rM R ; bM T ) I ( H ; b) I (r; b | H ) I (r; b | H ) E[ I (r; b | H h] MIMO with CSIR • The achievable rate max I (r; b) log[( e) M R det( N 0 I M R M R HQH * )] M R log( eN 0 ) pb since the differential entropy maximizer is a complex Gaussian random vector with some covariance matrix Q Fast Fading and CSIR • Finally, I (b; r ) E[log det( I M R M R HQH )] N0 with Q E [ bb ] M T M T E [ b b] Pav • 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* HQH ( HU ) D( HU )* I (b; r ) E[log det( I )] E[log det( I )] N0 N0 • Focus on non-negative definite diagonal Q • Further, optimum Q I M T M T Pav HH I (b; r ) E[log det( I )] M T N0 Rayleigh Fading MIMO • CSIR achievable rate Pav HH I (b; r ) E[log det( I )] M T N0 • 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 Tc 1 • The channel known at the receiver C E{log det( I M R M R Pav HH )} M T N0 • The capacity achieving signal b must be circularly symmetric complex Gaussian (0, ( Pav / M T ) I M T M T ) 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 I (r ; b) log det( I M R M R H M R M T QH M* T M R N0 ) • The outage probability with CSIR only out ( R, Pav ) inf Pr[log det( I M R M R Q with E[bb] Pav and Q E[bb ] HQH ) R] N0 Slow Fading • Since Pr[log det( I M R M R HQH HUQU * H ) R] Pr[log det( I M R M R ) R] N0 N0 • Diagonal Q is optimum • Conjecture: optimum Q is Qopt 1 1 1 Pav m 0 0 Example • Slow fading SIMO, M T 1 • Then Qopt Pav and Pr[log det( I M R M R Pav H * H HQH ) R] Pr[log( 1 ) R] N0 N0 • Scalar H * H is 2 distribute d N 0 ( e R 1) Pav M R 1 u out ( R, Pav ) u e du 0 ( M R ) Example • Slow fading MISO, M R 1 • The optimum Qopt Pav I mm for some m M T m • The outage N 0 m ( e R 1) Pa v m 1 u * Pav HH Pr[log( 1 ) R] mN0 u e du 0 ( m) Diversity and Multiplexing for MIMO • The capacity increase with SNR SNR C k log( 1 ) k • The multiplexing gain C ( SNR) r lim SNR log( SNR) Diversity versus Multiplexing • The error measure decreases with SNR increase SNR d • The diversity gain log( out ( SNR)) d lim SNR log( SNR) • Tradeoff between diversity and multiplexing – Simple in single link/antenna fading channels Coding for Fading Channels • Coding provides temporal diversity FER g c SNR d or P(C E ) g c SNR d • Degrees of freedom – Redundancy – No increase in data rate M versus D Diversity Gain (0,MRMT) (min(MR,MT),0) Multiplexing Gain

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