Lecture 7: MIMO Capacity and Multiplexing Architectures - PowerPoint PPT Presentation

Lecture ¡7: ¡MIMO ¡Capacity ¡and ¡ Multiplexing ¡Architectures I-Hsiang Wang ihwang@ntu.edu.tw 5/15, 2014

Design ¡of ¡MIMO ¡Systems • Regarding MIMO, what we have done so far: - Established solid foundation on the statistical channel modeling - Analyzed AWGN (no fading) MIMO capacity • Indeed, MIMO is capable of the following: - Multiplex multiple data streams simultaneously - Provide spatial diversity - Increase power gain • What’s next: - Derive MIMO capacity under fading - Design transceiver architectures to extract multiplexing gain, diversity gain, and power gain 2

Plot • Derive capacity of fading MIMO channel - Fast fading: CSIR only and full CSI - Slow fading: outage probability • Discuss the nature of performance gains • Introduce transceiver architectures for fast fading - The V-BLAST family • Introduce a transceiver architecture for slow fading - D-BLAST 3

Outline • Capacity of fast fading MIMO • V-BLAST • Receiver architectures: - Linear filters: decorrelator, matched filter, MMSE - Successive interference cancellation (SIC) • Outage probability of slow fading MIMO • D-BLAST 4

Fast ¡Fading ¡MIMO ¡Channel 5

Fast ¡Fading ¡MIMO ¡with ¡Full ¡CSI • Channel model: y [ m ] = H [ m ] x [ m ] + w [ m ] - { H [ m ]} : random fading process which is stationary and ergodic • With full CSI, Tx and Rx can perform pre- and post- processing based on the SVD of H [ m ] at each time: • H [ m ] = U [ m ] Λ [ m ] V [ m ] * • Convert the fading MIMO into a fading parallel channel: ( n min := min{ n t , n r } ) y i [ m ] = λ i [ m ] e x i [ m ] + e w i [ m ] , i = 1 , 2 , . . . , n min e • Water-filling to find the optimal power allocation 6

Ergidic ¡Capacity ¡with ¡Full ¡CSI λ 1 [ m ] w 1 [ m ] e x [ m ] y [ m ] ... V [ m ] V * [ m ] U [ m ] U * [ m ] e x [ m ] y [ m ] e λ n min [ m ] e w n min [ m ] • Capacity via water-filling: n min 1 + λ 2  ✓ i P ∗ ( λ i ) ◆� X C MIMO = log E σ 2 i =1 n min "✓ ◆ + # ◆ + ν − σ 2 ν − σ 2 ✓ X P ∗ ( λ ) = ν satisfies = P , E λ 2 λ 2 i i =1 7

Transceiver ¡Architecture ¡with ¡Full ¡CSI ~ ~ { x 1 [ m ]} { y 1 [ m ]} AWGN Decoder coder . . n min . { w [ m ]} . information . . streams ~ ~ { x n min [ m ]} { y n min [ m ]} AWGN U * H V + Decoder coder {0} . . . {0} pre-processing post-processing 8

Receiver ¡CSI ¡Only: ¡V-­‑BLAST • Tx cannot apply the pre-processing matrix V • V-BLAST architecture: P 1 AWGN coder w [ m ] rate R 1 · y [ m ] · x [ m ] · Joint · · · + Q H [ m ] · · · decoder · · · AWGN coder P n t rate R n t - Tx prepares n t data streams, each encoded with a rate R i coder - Generate x by multiplying them with a unitary matrix Q - Rx carries out joint decoding of the streams (eg., ML) • We will discuss Rx architecture later 9

Capacity ¡of ¡Fast ¡Fading ¡MIMO ¡with ¡CSIR • Using information theoretic arguments (or a sphere packing argument), one can show that V-BLAST can achieve the capacity, which is given by the following: I n r + HK x H ∗  ✓ ◆� C = max log det K x :Tr( K x ) ≤ P E σ 2 • K x := the covariance matrix of transmit signal vector x • With V-BLAST, K x = Q diag ( P 1 , . . . , P n t ) Q ∗ • The issue boils down to finding the optimizing K x for a given stationary distribution of H 10

Multiplexing ¡in ¡the ¡Angular ¡Domain • In V-BLAST, Q can be thought of as the coordinate system onto which Tx modulates its data streams - The question is, which coordinate system Q should be used? • The choice of Q (and the power allocation { P 1 , … , P n } ) depends on the statistical property of H , so let’s focus on the angular domain representation: H a = U r * HU t • Under rich scattering, entries of H a are statistically independent and zero-mean ⟹ it is reasonable to multiplex data on the coordinate system (indeed, optimal. HW.) • Choose Q = U t and hence K x = U t Λ p U t * . • Still need to determine the power allocation Λ p 11

Uniform ¡Power ¡Allocation • If further symmetry is present the random H a , uniform Λ p = P power allocation ( � � � � ) turns out to be optimal n t I n t • Sufficient condition: the n t column vectors of H a are i.i.d. - For example, i.i.d. Rayleigh faded H a . • Hence the covariance matrix K x = U t Λ p U t * = P n t I n t • This gives us the capacity formula: ( SNR := P / σ 2 )  ✓ ◆� I n r + SNR HH ∗ C = E log det n t • In this case, Q can be any unitary matrix; in particular, it suffices to choose Q = I n t 12

V-­‑BLAST ¡under ¡i.i.d. ¡Rayleigh P 1 AWGN coder w [ m ] rate R 1 · y [ m ] · x [ m ] · Joint · I · · + Q H [ m ] · · · decoder · · · AWGN coder P n t rate R n t P 1 = P 2 = · · · = P n t = P n t • Effectively, each of the Tx antennas, say, antenna i , transmits an independent data stream of rate R i • How to determine R i ? - For joint ML, it does not matter as long as Σ R i = the total capacity - For other Rx architectures (later), the individual rate depends on the effective channel it faces with, after the MIMO detector 13

Receiver ¡CSI ¡vs. ¡Full ¡CSI • Capacity formula can be rewritten using singular values of the random matrix H : λ 1 ≥ λ 2 ≥ · · · ≥ λ n min ≥ 0 n min h ⇣ ⌘i 1 + SNR n t λ 2 P C CSIR = E log i i =1 - No CSIT ⟹ water-filling is not possible • Recall the capacity with full CSI: n min ⇥ � 1 + SNR ∗ ( λ i ) λ 2 �⇤ P C Full CSI = log E i i =1 ⇣ ⌘ + � n min � + , ν satisfies � 1 1 SNR ∗ ( λ ) = P = SNR ν − ν − E λ 2 λ 2 i i =1 14

DoF ¡Gain ¡at ¡High ¡SNR C (bits /s / Hz) 70 C (bits /s / Hz) 35 n t = n r = 1 60 30 n t = n r = 1 n t = 1 n r = 4 n t = 1 n r = 8 n t = n r = 4 50 25 n t = n r = 8 40 20 30 15 20 10 5 10 –10 30 –10 30 10 20 10 20 SNR (dB) SNR (dB) • High SNR regime: n min -fold DoF gain n min ⇣ ⌘ SNR ⇥ � λ 2 �⇤ P C CSIR ≈ n min log + E log i n t i =1 n min ⇣ ⌘ SNR ⇥ � λ 2 �⇤ P C Full CSI ≈ n min log + E log i n min i =1 • n min := min{ n t , n r } determines the high-SNR slope 15

Power ¡Gain ¡at ¡Low ¡SNR C (bits / s / Hz) C 1,1 8 C (bits / s / Hz) C 1,1 4 7 6 3.5 n t = 1 n r = 4 n t = 1 n r = 8 5 n t = n r = 4 n t = n r = 8 3 4 2.5 3 –30 –20 –10 10 –30 –20 –10 10 SNR (dB) SNR (dB) • n r determines the power gain under CSIR n min SNR log 2 e = SNR ⇥ λ 2 ⇤ P n t E [Tr ( HH ∗ )] log 2 e C CSIR ≈ n t E i i =1 = n r SNR log 2 e • If CSIT is available, power gain is larger due to both beamforming and dynamic power allocation 16

Nature ¡of ¡Peformance ¡Gain ¡(CSIR ¡only) • High SNR (DoF-limited): - min{ n t , n r } -fold DoF gain - capacity scaling linearly with n min := min{ n t , n r } - MIMO is crucial • Low SNR (Power-limited): - n r -fold power gain - capacity scaling linearly with n r - Only need multiple Rx antennas • At moderate SNR - min{ n t , n r } -fold gain - Due to a combination of both effects 17

Receiver ¡Architectures 18

Decoding ¡at ¡the ¡Receiver • In the previous V-BLAST architecture, Rx uses ML: - ML is optimal - But the complexity grows exponentially with the # of data streams • A natural approach: - First separate the signal of each data stream from others with certain linear operations - Then decode each data stream using single-user decoder • In the following we focus on Rx architectures that use linear operations in the first step - Assuming V-BLAST with Q = I n t , that is, each Tx antenna sends an independent data stream - If not, we can just lump Q into the channel matrix H 19

Decorrelator: ¡Interference ¡Nulling • Rewrite the received signal vector y as follows: X y = h i x i + h j x j + w j 6 = i - x i denotes the signal sent from the i -th Tx antenna - h i denotes the i -th column of channel matrix H , representing the signal direction of x i . • To decode x i , a simple idea is to use a decorrelator: - First null out interference by projecting y onto the null space of the directions of all interfering vectors { h j | j ≠ i } - Then apply matched filter to the projected signal • The Rx architecture consists of a bank of decorrelators. • Also called interference nulling, zero forcing, etc. 20

Bank ¡of ¡Decorrelators y 1 := ( Q 1 h 1 ) ∗ ( Q 1 y ) Decorrelator e Rows of Q k form a for stream 1 orthonormal basis of the null space y 2 := ( Q 2 h 2 ) ∗ ( Q 2 y ) Decorrelator e of { h j | j ≠ k } . for stream 2 nulling matched filter y [ m ] after projection 1 + P k ⇥ � σ 2 || Q k h k || 2 �⇤ R k = E log Note: for successful decorrelation, Decorrelator n t ≤ n r and hence n min = n t for stream n t 21

Performance ¡in ¡i.i.d. ¡Rayleigh • Uniform power allocation ⟹ the overall achievable rate: n min h ⇣ n t || Q k h k || 2 ⌘i 1 + SNR P R decorr = E log k =1 • At high SNR: n min ⇣ ⌘ SNR ⇥ � || Q k h k || 2 �⇤ P R decorr ≈ n min log + E log n t k =1 n min ⇣ ⌘ SNR ⇥ � λ 2 �⇤ P C CSIR ≈ n min log + E log i n t i =1 • At low SNR: lose the power gain (homework) • The decorrelator fully extracts the spatial multiplexing gain of V-BLAST, but not the power gain 22