Linear Precoding in MIMO Wireless Systems

Transcription

1 Linear Precoding in MIMO Wireless Systems Bhaskar Rao Center for Wireless Communications University of California, San Diego Acknowledgement: Y. Isukapalli, L. Yu, J. Zheng, J. Roh 1 / 48

2 Outline 1 Promise of MIMO Systems 2 Point to Point MIMO 3 Limited Feedback MIMO Systems 4 MIMO-OFDM 5 Multi-User MIMO 2 / 48

3 Outline 1 Promise of MIMO Systems 2 Point to Point MIMO 3 Limited Feedback MIMO Systems 4 MIMO-OFDM 5 Multi-User MIMO 3 / 48

4 Multiple Input Multiple Output (MIMO) Systems A system with multiple antennas at the transmitter and multiple antennas at the receiver. Multi-Input Multi-Output System Enables Spatio-Temporal processing and the goal is to exploit the spatial dimension to increase system throughput 4 / 48

5 Textbooks Introduction to Space-Time Wireless Communications, A. Paulraj, R. Nabar and D. Gore, Cambridge University Press Fundamentals of Wireless Communications, D. Tse and P. Vishwanath Space-Time Coding, H. Jafarkhani MIMO Wireless Communications, Edited by Biglieri, Calderbank, et al 5 / 48

6 Benefits of MIMO Systems Increased Network Capacity Improved Signal Quality Increased Coverage Lower Power Consumption Higher Data Rates These requirements are often conflicting. Need balancing to maximize system performance 6 / 48

7 Technical Rationale Spatial Diversity to Combat Fading Spatial Signature for Interference Management Array Gain enables Lower Power Consumption Capacity Improvements using Spatial Multiplexing 7 / 48

8 Outage Capacity of MIMO Systems Capacity of MIMO systems 8 / 48

9 Outline 1 Promise of MIMO Systems 2 Point to Point MIMO 3 Limited Feedback MIMO Systems 4 MIMO-OFDM 5 Multi-User MIMO 9 / 48

10 MIMO Channel Model Input-Output relation for a discrete-time frequency-flat r t MIMO channel Es y = t Hs + n y = [y 1, y 2,, y r ] T r 1 receive signal vector s = [s 1, s 2,, s t ] T t 1 transmit signal vector n = [n 1, n 2,, n r ] T r 1 noise vector at the receiver H is the r t channel matrix E s average energy over a symbol period n i N C(0, N o ) with E[nn H ] = N o I r 10 / 48

11 MIMO Options Channel assumed known at Receiver Channel unknown at transmitter Diversity Gain: Orthogonal space-time block codes, Space time trellis codes Spatial Multiplexing: V-Blast, D-Blast Channel known at the transmitter- Transmit precoding 11 / 48

12 Transmitter With Channel Knowledge SVD of H can be expressed as H = UΣV H U H U = V H V = I r Σ = diag(σ m ) k m=1, σ m > 0 Further, HH H is Hermitian with eigendecomposition HH H = UΛU H Λ = diag(λ m ) k m=1, σ m σ m+1 with λ m = 0 for m > k and λ m = σ 2 m 12 / 48

13 Transmitter With Channel Knowledge Cont d Transmitted vector s = V s Input vector s is of dimension r 1 with E[ s s H ] = Γ t, Γ t diagonal Received signal transformed to ỹ = U H y Es ỹ = t Σ s + ñ 13 / 48

14 Transmitter With Channel Knowledge Cont d H is decomposed into k parallel sub-channels satisfying ỹ m = Es t σ m s m + ñ m, m = 1, 2,, k The channels are of different quality with the gain on each channel determined by σ m Number of channels depends on the rank of H. 14 / 48

15 Transmitter with Channel Knowledge Transmitter with Channel Knowledge ŝ Transmitter Channel Receiver n V s H H U y ~ y ~n 1 ~ s 1 λ 1 ~n 2 ~ s 2 λ 2 ~ y1 ~ y 2 s~k λ k n ~ k y~ k 15 / 48

16 Capacity of a deterministic MIMO Channels The channel capacity is given by C = max γ m k m=1 [ log E ] sλ m N o t γ m γ m = E[ s m 2 ] is the transmit energy in the m th sub-channel k m=1 γ m = t is the transmit energy constraint Optimum power allocation across the sub-channels is obtained as a solution to the lagrangian optimization problem 16 / 48

17 Optimal Power Allocation Optimal power allocation satisfies ( γm opt = µ N ) + ot, m = 1, 2,, k E s λ m k m=1 γ opt m = t where µ is a constant and (x) + implies { (x) + x if x 0 = 0 if x < 0 γ opt m is found iteratively by waterpouring algorithm 17 / 48

18 Waterpouring Solution Waterpouring Solution 18 / 48

19 High SNR At high SNR, equal power allocation is optimal C = k m=1 [ log E ] sλ m N o t k m=1 [ ] [ ] Es λ m Es log 2 = k log N o t 2 + N o k [ λm log 2 t m=1 Capacity grows linearly with k, the rank of the channel. Significant increase in Capacity. 19 / 48

20 Special Cases SIMO: H = h. Rank one and all power allocated to one mode C SIMO = log 2 (1 + E s h 2 N o ) MISO: H = h H. Rank one and all power allocated to one mode When Channel known at Tx C MISO = log 2 (1 + E s h 2 N o ) C SIMO = C MISO 20 / 48

21 Maximum Ratio Transmission (MRT) Input-Output relation for a r t MIMO channel Es y = t Hs + n When the channel is known at the transmitter, the information can be used to design an optimum precoder w The new Input-Output relation becomes Es y = t Hws + n 21 / 48

22 Maximum Ratio Transmission Cont d The receiver forms a weighted sum of the antenna outputs ỹ = g H y The objective is to maximize the received SNR Optimal scheme is given by η = gh Hw 2 F t g 2 F ρ w = v 1, g = u 1 Where, v 1 and u 1 are the left and right singular vectors of H corresponding to the maximum singular value The scheme achieves full diversity 22 / 48

23 MRT Transmission: 2 2 MIMO MRT Transmission: 2x2 MIMO 23 / 48

24 Outline 1 Promise of MIMO Systems 2 Point to Point MIMO 3 Limited Feedback MIMO Systems 4 MIMO-OFDM 5 Multi-User MIMO 24 / 48

25 Importance of CSI Feedback A. Improved system performance, in terms of capacity, SNR, BER, etc. Example: An MISO system with M transmit antennas and single receive antenna NO CSIT Perfect CSIT B. Reduced implementation complexity Example: An MIMO system with M transmit and receive antennas, No CSIT, capacity can be achieved by some 2-D (space-time) code Pre-coder with perfect CSIT, system is equivalent to M parallel SISO channels 2

26 Importance of CSI Feedback C. Enables exploitation of multi-user diversity With CSIT, effective selection of active users and route selection can be made. D. Greatly increase the system capacity region as well as the sum capacity Example: A multi-user MISO broadcasting channel with M transmit and single receive antenna users are not allowed to cooperate, and hence cause serious multi-user interference. CSI Feedback Proper pre-coding is possible, such as Zero-forcing, MMSE, etc E. Improve the robustness of the communication link (QoS requirements) Power and rate control is possible when CSIT is available and the network throughput is increased. 3

27 Block Diagram Sources of feedback imperfection Channel estimation Channel quantization Feedback delay () 6 / 34

28 Nature of CSI Feedback Channel state information (CSI) is a complex vector or matrix of continuous values For example: An MIMO system with M transmit antennas and N receive antennas,. It is not reasonable to feedback total 2MN real numbers of continuous values. Practical Feedback Schemes: Integer Index Channel Quantizer Adaptive Transmitter Each index represents a particular mode of the channel, which corresponds to a particular transmission strategy 4

29 Considerations in Feedback Systems A. Design of Optimal Quantizers (at the receiver) & Optimization of the Codebook? 1) The quantizer (or the encoder) should be simple as well as effective. 2) The quantizer and the codebook should be designed to match both the channel distribution and the system performance metrics, such as capacity, SNR, BER, etc. B. Performance Analysis of Finite Rate Feedback Multiple Antenna Systems 1) To understand the effects of the finite rate feedback on the system performance, to be specific, performance metric vs feedback rate. 2) Shed insights on the choice of the feedback schemes as well as the quantizer design. 5

30 MISO Channel Quantizer MISO Channel System Model: (scalar) (vector) If ideal CSIT available, the transmit beamforming scheme is chosen to be: capacity If only finite rate feedback is available, the beamforming vector is quantized to, capacity (codebook) 6

31 Codebook Design (Optimization) The capacity loss due to the finite rate quantization of the beamforming vectors is: Motivation: Minimize the capacity loss by optimizing the codebook vectors It is a difficult problem (non-convex optimization problem)! Simplifications: 1). The capacity loss can be approximated by the following form in high resolution regimes, 2). A New Design Criterion that can minimize the system capacity loss: (MSwIP) High SNR (MSIP) 7

32 Codebook design using the Lloyd Algorithm Nearest Neighborhood Condition (NNC): For given codebook vectors the optimum partitions are given by: partitioning the regions Centroid Condition (CC): For given partitions, the optimal code matrices are given by: Shifting new centers 8

33 Codebook Design Examples 9

34 MISO Capacity With Quantized Feedback 10

35 Extension to MIMO Channel Quantizer MIMO Channel System Model: Precoding Matrix Equal Power Allocation Channel Model With Quantized Feedback: 11

36 Sequential Vector Quantizer A simple approach to quantize the precoding matrix: How? Consider a unitary matrix whose first column is and the remainder columns are arbitarily chosen to satisfy. Then, has the form of where is a orthogonormal column matrix. 12

37 The Sequential Quantization Method Vector Parameterization: An orthonormal column matrix can be uniquely represented by by a set of unit-norm vectors with different dimensions,. Statistical Property: For random channel with entries,, for, and they are statistically independent. Quantization: For, unit-norm vector is quantized using a codebook that is designed for random unit-norm vectors In with the MSIP criterion. Practical applications: Under consideration by the Broadband Wireless Group (802.16e) 13

38 Joint Quantization for MIMO Systems Joint Quantization: by quantizing the entire precoding matrix at one shot The codebook is designed to minimize the system mutual information rate loss With ideal CSI Feedback With Quantized CSI Feedback Under the high resolution assumptions, it can be approximated as Generalized Weighted Matrix Inner Product between and. The first n eigen-values 14

39 Codebook design using the Lloyd Algorithm Nearest Neighborhood Condition (NNC): For given code matrices, the optimum partitions are given by: partitioning the regions Centroid Condition (CC): For given partitions, the optimal code matrices are given by: Shifting new centers 15

40 Multi-mode Spatial Multiplexing Multi-mode SP transmission strategy: 1) The number of data streams n is determined by the system SNR: 2) In each mode, the simple equal power allocation over n spatial channels is employed. Intuitive Explanation: Inverse Water-Filling Power Allocation (Optimal) water level water level power allocated power allocated Case I: Low SNR Case II: High SNR 16

41 Performance of Multi-mode S-M Ideal CSI Feedback Quantized CSI Feedback 17

42 Performance Analysis Some Interesting Questions: Finite Rate Effects: What is the performance (capacity, SNR, BER) versus the feedback rate? Mismatched Analysis: What happens if a codebook designed for one system is used in another system? Transform Codebooks: The codebook for a particular system is transformed from another system through a linear or non-linear operation. What is the performance? & How to design? Feedback With Error: What happens if the feedback information also suffers from error (delay)? Quantization of Imperfect CSI: What happens if CSI to be quantized suffers from estimation error? 18

43 Capacity Loss Analysis for MISO Channels Assume MISO channel with entries Instantaneous Capacity (mutual information rate) Loss: Capacity Loss: For a given codebook Analysis is quite involved 19

44 Publications 1 J. C. Roh and B. D. Rao, Transmit Beamforming in Multiple-Antenna Systems with Finite Rate Feedback: A VQ-Based Approach, IEEE Transactions Information Theory. vol. 52, no. 3, Pages: , Mar J. C. Roh and B. D. Rao, Design and Analysis of MIMO Spatial Multiplexing Systems with Quantized Feedback, IEEE Transactions on Signal Processing, Vol. 54, no. 8, Pages , Aug J. C. Roh and B. D. Rao, Efficient Feedback Methods for MIMO Channels Based on Parameterizations, IEEE Transactions on Wireless Communications, Pages: , Jan J. Zheng, E. Duni, and B. D. Rao, Analysis of Multiple Antenna Systems with Finite-Rate Feedback Using High Resolution Quantization Theory, IEEE Trans. on Signal Processing, vol. 55,Issue 4,Pages: , April / 48

45 Outline 1 Promise of MIMO Systems 2 Point to Point MIMO 3 Limited Feedback MIMO Systems 4 MIMO-OFDM 5 Multi-User MIMO 26 / 48

46 Frequency Selective Channels: MIMO-OFDM Next generation wireless communication system uses MIMO- OFDM MIMO-OFDM transfers a wideband frequency-selective channel into a number of parallel narrowband flat fading MIMO channels Benefits of OFDM Achieves high spectral efficiency Cyclic prefix is capable of mitigating multi-path fading Allows for efficient FFT-based implementations and simple frequency domain equalization Exploits frequency diversity, in addition to time and spatial diversity 27 / 48

47 MIMO-OFDM Block Diagram MIMO-OFDM Transceiver Binary Data Binary Data Modulation & Mapping OFDM Modulation OFDM Demodulation Demodulation & Demapping IFFT Add CP P/S S/P Remove CP FFT S/P Space-Time Processing Space-Time Decoder & Equalizer P/S IFFT Add CP P/S S/P Remove CP FFT 28 / 48

48 MIMO-OFDM Signaling The input-output relation of a broadband MIMO channel is y[k] = Es t L H[l]s[k l] + n[k] l=0 k - discrete time index L - number of channel taps t - number of transmit antennas 29 / 48

49 MIMO-OFDM Signaling Cont d OFDM with FFT/IFFT and CP insertion/removal operations decuples the frequency selective MIMO channel to a set of parallel MIMO channels as Es ỹ[l] = t H[l] s[l] + ñ[l], l = 0, 1,.., N 1. N - Number of subcarriers H[l] - DFT Coefficient of the channel s[l] - data on carrier l 30 / 48

50 Spatial Diversity in MIMO-OFDM Take Alamouti scheme as an example, there are two ways to realize spatial diversity 1 Coding in frequency domain, rather than in time domain It requires that the channel remains constant over at least two consecutive tones 2 Coding on a per-tone basis across OFDM symbols in time It requires that the channel remains constant during two consecutive OFDM symbols 31 / 48

51 Outline 1 Promise of MIMO Systems 2 Point to Point MIMO 3 Limited Feedback MIMO Systems 4 MIMO-OFDM 5 Multi-User MIMO 32 / 48

52 Multi-User MIMO Main Issue is the utilization of the spatial degree of freedom in a multi-user environment Resource Management Interference Management Capacity of Multi-User systems Multi-user Diversity 33 / 48

53 Multi-User SIMO Systems r(t) = P h l s l (t) + n(t) l=1 To receive user j, can use beamformer w j y j (t) = wj H r(t) = wj H h j s j (t) + P l=1,l j w H j h l s l (t) + wj H n(t) The beamforming vector can be optimized for each user separately. 34 / 48

54 Multi-User MISO Systems Transmitted signal Signal received by user j s(t) = P w l s l (t) l=1 r l (t) = h H j s(t) = h H j w j s j (t) + P l=1,l j h H j w l s l (t) + n j (t) The transmit beamformers for the other users do interfere with the desired user. Beamformers have to be jointly selected. A more challenging problem. 35 / 48

55 Problem Statement Problem Statement Consider a multiuser MIMO beamforming network Arbitrary Network configurations (cellular networks, multi-hop networks, etc.) Heterogeneous communication nodes with different power costs Minimize the network power cost while satisfying the minimum SINR requirements of all links SINR (signal to interference plus noise ratio) Joint optimization of beamforming weights and transmit powers University of California, San Diego 36 / 48

56 Problem Statement Problem Statement JOP: min p, V, U subject to T J ( p) = w p SINR V = { v,..., v U = { u,..., u l γ where p = [ p,..., p ] T L L L } } w = [ w,..., w ] 1 T L l for all 1 l L (network power vector, L: no. of links) (unit norm tx. beamforming vectors) (unit norm rx. beamforming vectors) (weight vector defining power costs) Solved for SIMO and MISO cases for w = 1 = [1,...,1] MISO problem is solved by using the virtual uplink concept T University of California, San Diego 37 / 48

57 SINR Expression for MIMO Beamforming SINR Expression for MIMO Beamforming SINR (signal to interference plus noise ratio) Γ SINR = l l i l Gll pl = G p + n li i l i l u H v u H H l H l 2 ll l 2 livi pl p + n i l u Φ u = u Φ H l H l s l l in l ul tl : Transmitter of link l (1 l L) rl : Receiver of link l Hli : complex channel gain matrix from ti to rl vl : transmit antenna weight vector of link l ul : receive antenna weight vector of link l G = u H v : effective link gain from t to r li H l 2 li i i l Problem isolation for optimal Rx. beamforming vectors U MMSE/MVDR beamforming at the receivers No straightforward problem isolation for V University of California, San Diego 38 / 48

58 SIMO problem : Cellular Uplink (Rashid-Farrokhi SIMO problem : Cellular Uplink et al. 98) (Rashid-Farrokhi et al. 98) Problem : min p, U subject to l Γ l p l γ l l Joint Beamforming & Power Control Algorithm p ( n+ 1) = I( p ( n) where I ( p l ) ( n) Glj ( ul ) p j + nl j l γ ) = minγ l = p ul G ( u ) SINR ( u ) ll l l ( n) l * l ( n) l Convergence to the global optima is established. Desirable features MVDR beamforming : implemented using adaptive filters power control : using a simple power control loop University of California, San Diego 39 / 48

59 MISO Problem & Virtual Uplink MISO Problem & Virtual Uplink Concept Concept(Rashid-Farrokhi et al. 98) (Rashid-Farrokhi et al. 98) Dual relation between cellular downlink and uplink Virtual uplink : uplink with reciprocal channels and noise vector 1. Optimal transmit beamforming vectors are identical to the optimal receive beamforming vectors in the virtual uplink H 11 H 33 H H 11 H H 33 H 22 H H 22 (a) Downlink (Primal) (b) Virtual Uplink (Dual) University of California, San Diego 40 / 48

60 Generalization Generalization We generalize this idea to arbitrary multiuser MIMO networks with generalized cost function (e.g., MIMO multihop networks, energy-aware networking environment, etc.) We derive the dual relation using the well-established duality concept in optimization theory We take advantage of the dual relation for solving the stated problem We developed an improved Decentralized Algorithm University of California, San Diego 41 / 48

61 Construction of a Dual Network Construction of a Dual Network For any multi-user MIMO network with linear beamformers, one can construct a dual network using the following three rules: Reverse the direction of all links Replace any MIMO channel matrix H by H H Use transmit beamforming vectors as receive beamforming vectors, and vice versa. H55 H33 H H 33 H H 55 H 11 H 22 H 44 H H 11 H H 22 H H 44 University of California, San Diego 42 / 48

62 Duality 43 / 48

63 Applications to JOP Applications to JOP Theorem 2 suggests an iterative algorithm (Algorithm E) Primal Network : Update p and U for fixed V, so that w T p is minimized Dual Network : Update q and V for fixed U, so that n T q is minimized (n) Γ out ~ ( n) ( n) in Γout Γ = Γ = Γ ( n+ 1) ~ ( n ) in out ~ ( n ) Γ out Lemma 3. In the proposed algorithm, once the solution becomes feasible, i.e., all SINR values meet or exceed the minimum requirements, it generates a sequence of feasible solutions with monotonic decreasing cost. University of California, San Diego 44 / 48

64 Cellular Network -Downlink Cellular Network - Downlink Multiple wrapped around cells (19 three-sectored cells) Same channel is reused in every cell but only in one sector Three co-channel users per sector Propagation exponent = 3.5, 8dB shadow fading University of California, San Diego 45 / 48

65 Performance Comparison Performance Comparison Algorithm A, B, E and F The proposed algorithm presents significant improvement in the complexity-performance tradeoff, thereby greatly improving practical value. University of California, San Diego 46 / 48

66 Current Trends Multi-user OFDM systems Coordinated Multi-Point Transmission (CoMP) Cooperative MIMO MIMO Ad-Hoc Networks 47 / 48

67 Summary MIMO Systems offer unique opportunities in wireless communication Provides an opportunity to use spatial dimension to provide diversity and hence reliability. Can be used to significantly increase capacity in a rich scattering environment 48 / 48