Notes Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Lecture 8: MIMO Architectures (II) Receiver Theoretical Foundations of Wireless Communications 1 Architectures D-BLAST Ragnar Thobaben CommTh/EES/KTH Wednesday, May 25, 2016 09:15-12:00, SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 20 Overview Notes Lecture 7: MIMO Architectures Lecture 8 • Generalized structure: V-BLAST. MIMO Architectures (II) • Fast fading channels: capacity with CSI-R. Ragnar Thobaben CommTh/EES/KTH • Slow fading channels: outage probability. Receiver Architectures Lecture 8: MIMO Architectures (Ch. 8.3+5) D-BLAST 1 Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC 2 D-BLAST Outage Probability Outage-Suboptimality of V-BLAST Coding Across the Antennas: D-BLAST 2 / 20
Receiver Architectures Notes – Linear Decorrelator Motivation: achieving the capacity. • With CSI-T: use SVD and transmit along the eigenmodes. Lecture 8 MIMO Architectures (II) • With CSI-R and rich scattering: use the angular representation and Ragnar Thobaben transmit along the angular windows. CommTh/EES/KTH • Goal: make sure that the receiver can separate the data streams Receiver efficiently. Architectures Linear Decorrelator Successive Cancellation Linear decorrelator Linear MMSE Receiver • Time-invariant channel model (with H = [ h 1 . . . h n t ]): MMSE-SIC D-BLAST n t � 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 ] + h i x i [ m ] + w [ m ] . i � = k → Interference from other streams. 3 / 20 Receiver Architectures Notes – Linear Decorrelator • 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 n t . Lecture 8 MIMO Architectures • Assuming V k is d k -dimensional, the projection (II) can be described by a matrix multiplication with a ( d k × n r ) matrix Q k : Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive ˜ y [ m ] = Q k y [ m ] Cancellation Linear MMSE = Q k h k x k [ m ] + ˜ w [ m ] , Receiver MMSE-SIC with D-BLAST w [ m ] = Q k w [ m ] . ˜ (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .) → 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 n t . → 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
Receiver Architectures Notes – Linear Decorrelator • Optimal demodulation • Matched filtering of ˜ y [ m ] with Q k h k , or equivalently filtering y [ m ] Lecture 8 MIMO Architectures with a filter (II) c k = ( Q ∗ k Q k ) h k . Ragnar Thobaben • SNR after matched filtering ( k -th stream with power P k ): CommTh/EES/KTH P k � Q k h k � 2 / N 0 . Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC • Decorrelating multiple streams D-BLAST simultaneously • Multiplying with the pseudoinverse: H † = ( H ∗ H ) − 1 H ∗ . • Bank of decorrelators. (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .) 5 / 20 Receiver Architectures Notes – Linear Decorrelator: Performance Lecture 8 MIMO Architectures Case 1: deterministic H . (II) Ragnar Thobaben • Maximum rate for the k − th stream and sum rate CommTh/EES/KTH n t 1 + P k � Q k h k � 2 � � Receiver � C k := log and R decorr = C k . Architectures N 0 Linear Decorrelator k =1 Successive Cancellation • No inter-stream interference: SNR = P k � h k � 2 / N 0 . Linear MMSE Receiver MMSE-SIC • Inter-stream interference reduces rate since � Q k h k � ≤ � h k � . D-BLAST • � Q k h k � = � h k � if h k is orthogonal to the other spatial signatures h i , with i � = k . 6 / 20
Receiver Architectures Notes – Linear Decorrelator: Performance Case 2: fading channels. Lecture 8 • Fast fading, average over realizations of the channel process: MIMO Architectures (II) n t 1 + P k � Q k h k � 2 � � �� Ragnar Thobaben ¯ � ¯ CommTh/EES/KTH C k := E log and R decorr = C k N 0 k =1 Receiver Architectures → Generally less or equal to the capacity with CSI-R. Linear Decorrelator Successive • High SNR , i.i.d. Rayleigh fading, n min = n t : Cancellation Linear MMSE � n t Receiver � n min log SNR � log( � Q k h k � 2 ) MMSE-SIC R decorr ≈ + E n t D-BLAST k =1 n min log SNR + n t E[log χ 2 = 2( n r − n t +1) ] n t → 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 Receiver Architectures Notes – Linear Decorrelator: Performance • Example Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .) 8 / 20
Receiver Architectures Notes – Successive Cancellation • Decorrelator: bank of separate filters for estimating the data Lecture 8 streams. MIMO Architectures (II) • But: the result from one of the filters can be used to improve the Ragnar Thobaben operation of the others; successive interference cancellation, SIC. CommTh/EES/KTH Receiver Architectures Linear Decorrelator • Modified detector structure: Successive Cancellation • For the k -th decorrelator, the Linear MMSE Receiver k − 1 previous streams have MMSE-SIC been removed. D-BLAST → Error propagation! (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .) • h k is projected by ˜ Q k on a higher dimensional subspace orthogonal to that spanned by h k +1 , . . . , h n t . • Improved SNR on the k -th stream: SNR k = P k � ˜ Q k h k � 2 / N 0 9 / 20 Receiver Architectures Notes – Successive Cancellation: Performance • A similar derivation as above yields � n t � n min log SNR � log( � ˜ Q k h k � 2 ) R dec-sic ≈ + E Lecture 8 n t MIMO Architectures k =1 (II) n t n min log SNR Ragnar Thobaben � E[log χ 2 = + 2( n r − n t + k ) ] CommTh/EES/KTH n t k =1 Receiver Architectures • SIC does not gain additional degrees of freedom. Linear Decorrelator • Constant term is equal to that in the capacity expansion Successive Cancellation (cf. (8.18)-(8.20) in the book) Linear MMSE Receiver ⇒ Power gain by decoding and subtracting! MMSE-SIC • Example D-BLAST 10 / 20 (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .)
Receiver Architectures Notes – Linear MMSE Receiver Comparison: decorrelator bank versus a bank of matched filters Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .) • 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 Receiver Architectures Notes – Linear MMSE Receiver Derivation of a generic MMSE receiver Lecture 8 • Generic model: y = h x + z , with MIMO Architectures (II) • Complex circular symmetric colored noise z ; Ragnar Thobaben • An invertible covariance matrix K z ; CommTh/EES/KTH • A deterministic vector h ; Receiver • A scalar data symbol x . Architectures Linear Decorrelator • Apply a linear transform 2 K − 1 / 2 z = K − 1 / 2 such that ˜ z is white, Successive z z Cancellation Linear MMSE K − 1 / 2 y = K − 1 / 2 Receiver h x + ˜ z . z z MMSE-SIC D-BLAST • Matched filtering with ( K − 1 / 2 h ) ∗ : z ( K − 1 / 2 h ) ∗ K − 1 / 2 y = h ∗ K − 1 z y = h ∗ K − 1 z h x + h ∗ K − 1 z z z z → The linear receiver v mmse = K − 1 z h maximizes the SNR. • Achieved SINR: σ 2 x h ∗ K − 1 z h . 2 Reminder: if K z is invertible, then K z = UΛU ∗ and K 1 / 2 = UΛ 1 / 2 U ∗ . z 12 / 20
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