MIMO : MIMO Theoretical Foundations of Wireless Communications 1 Wednesday, May 25, 2016 09:15-12:00, SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 20 Overview MIMO Lecture 7: MIMO Generalized structure: V-BLAST. Fast fading channels: capacity with CSI-R. Slow fading channels: outage probability. : MIMO (Ch. 8.3+5) 1 2 Outage Probability Outage-Suboptimality of V-BLAST Coding Across the Antennas: 2 / 20
MIMO Motivation: achieving the capacity. With CSI-T: use SVD and transmit along the eigenmodes. With CSI-R and rich scattering: use the angular representation and transmit along the angular windows. Goal: make sure that the receiver can separate the data streams efficiently. Linear decorrelator Time-invariant channel model (with H = [h 1... h nt ]): y[m] = h i x i [m] + w[m]. i=1 Focusing on the k-th data stream (i.e., the k-th transmit antenna): y[m] = h k x k [m] + i k h i x i [m] + w[m]. Interference from other streams. 3 / 20 MIMO Goal: project y onto the subspace V k which is orthogonal to the space spanned by h 1,..., h k 1, h k+1,... h nt. Assuming V k is d k -dimensional, the projection can be described by a matrix multiplication with a (d k n r ) matrix Q k : ỹ[m] = Q k y[m] = Q k h k x k [m] + w[m], with w[m] = Q k w[m]. Rows of Q k form an orthonormal basis of V k. h k has to be linearly independent of h 1,..., h k 1, h k+1,... h nt. The maximum number of data streams is n t n r ; i.e., only subsets of antennas are used if n t > n r. 4 / 20
MIMO Optimal demodulation Matched filtering of ỹ[m] with Q k h k, or equivalently filtering y[m] with a filter c k = (Q k Q k)h k. SNR after matched filtering (k-th stream with power P k ): Decorrelating multiple streams simultaneously P k Q k h k 2 /N 0. Multiplying with the pseudoinverse: H = (H H) 1 H. Bank of decorrelators. 5 / 20 : Performance MIMO Case 1: deterministic H. Maximum rate for the k th stream and sum rate C k := log (1 + P ) k Q k h k 2 and R decorr = C k. No inter-stream interference: SNR = P k h k 2 /N 0. N 0 Inter-stream interference reduces rate since Q k h k h k. Q k h k = h k if h k is orthogonal to the other spatial signatures h i, with i k. 6 / 20
: Performance MIMO Case 2: fading channels. Fast fading, average over realizations of the channel process: [ C k := E log (1 + P )] k Q k h k 2 N 0 and R decorr = Generally less or equal to the capacity with CSI-R. High SNR, i.i.d. Rayleigh fading, n min = n t: [ R decorr n min log SNR nt ] + E log( Q k h k 2 ) n t = n min log SNR + n te[log χ 2 2(n n r n t +1) ] t C k Decorrelator is able to fully exploit the degrees of freedom of the MIMO channel. Second term shows the degradation in rate compared to the capacity. 7 / 20 : Performance Example MIMO 8 / 20
MIMO Decorrelator: bank of separate filters for estimating the data streams. But: the result from one of the filters can be used to improve the operation of the others; successive interference cancellation, SIC. Modified detector structure: For the k-th decorrelator, the k 1 previous streams have been removed. Error propagation! h k is projected by Q k on a higher dimensional subspace orthogonal to that spanned by h k+1,..., h nt. Improved SNR on the k-th stream: SNR k = P k Q k h k 2 /N 0 9 / 20 MIMO : Performance A similar derivation as above yields R dec-sic n min log SNR n t + E [ nt ] log( Q k h k 2 ) = n min log SNR + E[log χ 2 2(n n r n t +k) ] t SIC does not gain additional degrees of freedom. Constant term is equal to that in the capacity expansion (cf. (8.18)-(8.20) in the book) Power gain by decoding and subtracting! Example 10 / 20
Comparison: decorrelator bank versus a bank of matched filters MIMO Matched filter: good at low SNR, bad at high SNR. (Preserving the signal energy at the cost of interference.) Decorrelator: bad at low SNR, good at high SNR. (Eliminating all interference at the cost of a low SNR.) Desirable receiver: maximize the signal-to-interference-plus-noise ratio (SINR). 11 / 20 MIMO Derivation of a generic MMSE receiver Generic model: y = hx + z, with Complex circular symmetric colored noise z; An invertible covariance matrix K z ; A deterministic vector h; A scalar data symbol x. Apply a linear transform 2 K 1/2 z such that z = K 1/2 z z is white, K 1/2 Matched filtering with (K 1/2 z h) : z y = Kz 1/2 hx + z. (K 1/2 z h) Kz 1/2 y = h K 1 z y = h K 1 z hx + h K 1 z z The linear receiver v mmse = K 1 z h maximizes the SNR. Achieved SINR: σ 2 xh K 1 z h. 2 Reminder: if K z is invertible, then K z = UΛU and K 1/2 z = UΛ 1/2 U. 12 / 20
MIMO Specialization for the MIMO case Channel model for the k-th stream, y[m] = h k x k [m] + z k [m], with the noise plus interference term and its covariance matrix 3 z k [m] = i k h i x i [m] + w[m] and K zk = N 0I nr + P i h i h i. receiver: v mmse = N 0I nr + P i h i h i achieving the output SINR P k h k K 1 z k h k = P k h k 3 Note that K zk is invertible. i k 1 h k, N 0I nr + P i h i h i i k 1 i k h k. 13 / 20 MIMO Performance Low SNR: K zk N 0I nr, i.e., MMSE solution becomes matched filter. High SNR: MMSE receiver reduces to the decorrelator. Capacities for the k-th stream ( ) C k = log 1 + P k h k K 1 z k h k and [ ( )] Ck = E log 1 + P k h k K 1 z k h k. Example: i.i.d. Rayleigh fading and equal power allocation. 14 / 20
MIMO interference cancellation using MMSE filters. achieves the highest possible rate for CSI-R ) log det (I nr + 1N0 HK xh 15 / 20 MIMO : Information Theoretic Optimality Consider again the generic model y = hx + z, with additive colored noise and Gaussian x and z. MMSE receiver is information lossless; i.e., it provides a sufficient statistic to detect x such that I (x; y) = I (x; v mmsey). Consider now the MIMO model y[m] = Hx[m] + w[m], with x CN (0, diag{p 1,..., P nt }). Using the chain rule of mutual information, we get I (x; y) = I (x 1,..., x nt ; y) = I (x 1; y) + I (x 2; y x 1) +... + I (x nt ; y x 1,..., x nt 1). Considering the k-th term in the chain rule, we get using I (x k ; y x 1,..., x k 1 ) = I (x k ; y ) = I (x k ; v mmseky ) k 1 y = y h i x i = h k x k + i=1 i>k h i x i + w The rate achieved in the k-th stage is precisely I (x k ; y x 1,..., x k 1 ). 16 / 20
Outage Probability MIMO Outage Probability Suboptimality of V-BLAST Coding Across the Antennas Reliable communication at rate R is possible as long as ) log det (I nr + 1N0 HK xh > R subject to Tr[K x] P. Information theory guarantees the existence of a channel-state independent coding scheme that achieves reliable communications whenever this condition is met. Universal code: works for all channels that satisfy the above condition. If the condition is not satisfied, we are in outage. Outage probability [ ) ] pout MIMO (R) = min Pr log det (I nr + 1N0 HK xh < R. K x :Tr[K x ] P Choose the transmit strategy (i.e., K x ) to minimize the outage probability. Solution depends on statistics of H. 17 / 20 Outage-Suboptimality of V-BLAST (with ) MIMO Outage Probability Suboptimality of V-BLAST Coding Across the Antennas V-BLAST: capacity achieving for the fast fading MIMO channel Independent data streams are multiplexed and transmitted over the antenna; stream k with power P k and rate R k. receiver. V-BLAST: diversity Each stream has at most diversity order n r while the MIMO channel provides diversity gain n r n t. V-BLAST does not exploit full diversity and cannot be outage optimal. Due to interference, the diversity can be lower than n r. (Example: for SIC with decorrelator the diversity loss equals the number of uncanceled interferers.) 18 / 20
MIMO Outage Probability Suboptimality of V-BLAST Coding Across the Antennas Outage-Suboptimality of V-BLAST (with ) For a given H, V-BLAST achieves ) log det (I nr + 1N0 HK xh = log(1 + SINR k ). (SINR k is random since it is a function of H.) Assume that the outage-optimal transmit strategy K x is employed and that the target rate R is split into rates R 1,..., R nt. The channel is in outage if ) log(1 + SINR k ) = log det (I nr + 1N0 HK xh < R = V-BLAST is in outage whenever any stream is in outage (i.e., log(1 + SINR k ) < R k for some k), and this can occur even though the channel is not in outage. Summary: Problem with V-BLAST Each stream sees only one efficient channel with SINR k ; there is no coding across the channels. But each stream should see all channels. R k 19 / 20 MIMO Outage Probability Suboptimality of V-BLAST Coding Across the Antennas Coding Across the Antennas: Two-antenna example The i-th codeword x (i) is made up of two codewords x (i) A and x(i) B. First time slot: antenna-1 is silent; antenna-2 transmits x (1) A ; receiver performs MRC of the received antennas to estimate x (1) A. x (1) A sees effectively SINR 2. Second time slot: antenna-1 transmits x (1) B ; antenna-2 transmits x (2) A ; receiver performs linear MMSE treating x (2) A as noise and estimates x(1) B. x (1) B sees effectively SINR 1. The codeword x (i) can be decoded if log(1 + SINR 1 ) + log(1 + SINR 2 ) > R and x (1) B can be subtracted such that x(2) A sees again an interference-free channel. Drawbacks: Rate-loss due to initialization phase and error propagation. 20 / 20