Antennas and Propagation d: Diversity Techniques and Spatial Multiplexing
Introduction: Diversity Diversity Use (or introduce) redundancy in the communications system Improve (short time) link reliability This lecture Introduce different types of diversity Receive diversity (switched, MRC) Transmit diversity (Alamouti) Antennas and Propagation Slide 2
Rayleigh Fading Antennas and Propagation Slide 3
Types of Diversity Space (Phase) Diversity Spatial separation of elements Phases of received multipath are different (i.e. different array factor) Resulting sum of multipath different Rich multipath Placing elements 0.4λ apart leads to uncorrelated signals Antennas and Propagation Slide 4
Types of Diversity (2) Pattern (Angle) Diversity Antennas have different patterns Multipath are weighted differently depending on direction they come from (i.e. we exploit differences in element factor) Can think of phase diversity as a special case of this (phases of patterns are different wrt angle) Antennas and Propagation Slide 5
Types of Diversity (3) Frequency Diversity Transmit same signal on different carriers Spaced greater than coherence bandwidth of channel Components will fade independently But, requires additional bandwidth/power Antennas and Propagation Slide 6
Types of Diversity (4) Time Diversity Wait for times when the channel quality is good Send information then Requires buffering Multiuser Diversity Like time diversity Only users with best channels can transmit Potential Problem Long latencies incurred Focus of this lecture: Space (phase) diversity Antennas and Propagation Slide 7
Switched Diversity Antennas and Propagation Slide 8
MRC Antennas and Propagation Slide 9
MRC Antennas and Propagation Slide 10
SNR vs. M Antennas and Propagation Slide 11
Introduction: Spatial Multiplexing Spatial Multiplexing Send different information on different antennas (paths) Increase transmission rate of system This lecture Nulling (zero-forcing) V-BLAST (Successive interference cancellation) Capacity for Rapidly-Fading Channels Differential Unitary Space-Time Coding Antennas and Propagation Slide 12
Problem Transmitter N T antennas Send different (independent) signals on antennas Receiver N R antennas (assume N R >=N T ) Needs to recover the N T transmitted signals Antennas and Propagation Slide 13
Nulling (Zero-Forcing) Consider the relationship Assume we have more receivers than transmitters Can estimate transmit signals with Think of H + (i,:) as a nulling vector (i.e. nulls effect of all transmitters but i) Problem: Noise amplification Variance in estimate of ith transmitted symbol Input stream with largest nulling vector most susceptible Antennas and Propagation Slide 14
MMSE Method Instead of using pseudo-inverse, consider minimization problem MSE given by Antennas and Propagation Slide 15
MMSE Method (2) Assume Simple Model (R x =P I, R η =σ 2 I) MSE: Minimize: Want MSE/ W = 0 (Matrix derivative) Useful to get a table of matrix derivatives and learn how to use them! Antennas and Propagation Slide 16
MMSE Method (3) MSE Consider the trace terms we have Antennas and Propagation Slide 17
MMSE Method (3) MSE Note: α=σ 2 /P. As SNR, α 0, and W=H + Antennas and Propagation Slide 18
V-BLAST Algorithm Problem with nulling / MMSE Noise amplification Weak channels get corrupted by error in strong channels Clever Idea Channels with small have lowest error Estimate best channel first Subtract its effect from received signal Then estimate next weaker channel, etc. VBLAST = Vertical Bell LAbs layered Space-Time Architecture Antennas and Propagation Slide 19
V-BLAST Algorithm 1. Begin by setting nulling matrix to Let set of transmit symbols estimated be empty 2. Decide on which transmit symbol to estimate next Nulling vector (row of W) with lowest norm gives best estimate Assuming specific transmit constellation: Antennas and Propagation Slide 20
V-BLAST Algorithm (2) 3. Compute effect of this symbol on received signal vector 4. Remove the effect of this symbol from the received signal 5. Go back to step 2 until all symbols estimated Antennas and Propagation Slide 21
V-BLAST Algorithm (4) Measured Performance 16 QAM M=8 N=12 25.9 bits/s/hz Nulling Optimal Ordering Significant performance improvement! V-BLAST Other Detection Methods V-BLAST MMSE ML Decoder Sphere Decoding Antennas and Propagation Slide 22
Rapidly Fading Channels Typical MIMO System Training Phase / Transmission Phase For N T transmit antennas need N T training symbol times Rapidly fading channels What if channel coherence time is on the order of N T? Channel changes before we can use it! What is the capacity of such a channel? Can we still send information through it? Antennas and Propagation Slide 23
Rapidly Fading Channels (2) Paper T. Marzetta, B. Hochwald, Capacity of a Mobile Multiple Antenna Link in Rayleigh Flat Fading, IEEE Transactions on Information Theory, January 1999. Considers exactly this problem Scenario Transmitter and receiver do not know channel M transmitters N receivers Block fading channel (i.i.d. Gaussian) Channel matrix H constant for time T Then channel changes to completely new channel Antennas and Propagation Slide 24
Rapidly Fading Channels (3) Graphical Representation of Block Channel Model Antennas and Propagation Slide 25
Input/Output Antennas and Propagation Slide 26
Input/Output (2) Antennas and Propagation Slide 27
Output Signal Distribution conditioned on input signal p(x S): Gaussian. Why? Completely determined by covariance matrix Antennas and Propagation Slide 28
Properties of Conditional Distribution Antennas and Propagation Slide 29
Properties of Conditional Distribution (2) Antennas and Propagation Slide 30
Mutual Information Antennas and Propagation Slide 31
Capacity-Achieving Signal Means that rotating all symbol matrices S by common unitary transformations (to rows or columns) doesn t change M.I. Show by change of variables S Φ H SΨ X Φ H SΨ and using Properties 3 and 4 Antennas and Propagation Slide 32
Capacity-Achieving Signal (2) Theorem 1: Capacity for M>T same as capacity M=T In other words, using more antennas than the channel coherence T does not increase capacity! Conditional PDF (and capacity) only depend on S through SS H Note: S is TxM, SS H is only TxT. Can create the same statistics on SS H with S matrices that are TxT for M>T Those additional antennas are not useful Antennas and Propagation Slide 33
Capacity-Achieving Signal (3) Outline: 1. Consider SVD of S= ΨVΦ H 2. Show that letting Φ=I does not reduce MI (Lemma 3) 3. Show that Ψ isotropic unitary does not reduce MI 4. Reordering of diagonal matrix V does not reduce MI Antennas and Propagation Slide 34
Capacity-Achieving Signal (4) Antennas and Propagation Slide 35
Capacity-Achieving Signal (5) Property 3 Doesn t depend on Φ anymore. Can integrate to get marginal. Mutual Information for signal S= ΨV Antennas and Propagation Slide 36
Capacity-Achieving Signal (6) From Lemma 3, transmitted signal is S=ΦV (Note: Φ and V are jointly distributed) Let Θ be isotropically distributed unitary matrix (ind. of Φ and V) Let S 1 =ΘS Define: MI[p(S)] = mutual inf. when dist. on S given by p(s) MI[p(S)] = I 0 (by definition) MI[p(S 1 Θ) = I 0 also (from Lemma 1) MI[p(S 1 )] = MI[E Θ p(s 1 Θ)] = I 1 (defined) E Θ {MI[p(S 1 Θ)]} (concavity of MI and Jensen s ineq) = I 0 Result: Premultiplying by a isotropically distributed unitary matrix cannot reduce MI Antennas and Propagation Slide 37
Capacity-Achieving Signal (7) Let S= ΘΦV ΘΦ has same statistics as Θ (isotropically dist. unitary) Therefore S= ΘV will also achieve capacity Note, though that Θ and V are independent Capacity not changed by rearranging V Proven by considering all permutations Forming a new signal that is a mixture density Due to concavity, this does not decrease MI Final result Optimal form of signal: S=ΘV (Θ and V independent) Θ is isotropically distributed unitary matrix Ordering of V does not matter Antennas and Propagation Slide 38
Examples Single antenna Capacity vs. T Antennas and Propagation Slide 39
Examples (2) Very rapid fading T=1 Vary # of receive antennas (N) Antennas and Propagation Slide 40
Examples (3) Slowly fading channel T=100 N=1 Vary # of transmit antennas (M) Capacity close to that for exact CSI Antennas and Propagation Slide 41
Capacity of Rapidly Fading Channels Basic Result Transmission still possible for rapidly-fading channels (i.e. when training not possible) But, take a very big hit in terms of capacity MIMO is most beneficial when channel known Example of Practical Coding Method Differential Unitary Space-Time Coding Natural Extension of DPSK Antennas and Propagation Slide 42
Single-Antenna Communication T=2 Consider Channel phase is random, but constant for T=2 symbol times How can use channel? Idea 1: Send 1 pilot to train, then 1 symbol Works, but inefficient Idea 2: Code information as phase difference between symbols Called differential phase-shift keying (DPSK) Antennas and Propagation Slide 43
Differential Phase-Shift Keying (DPSK) Symbols (phase differences) Transmit Receive samples x t Antennas and Propagation Slide 44
DPSK in an Extensible Form Symbols last 2 time slots (2x1 vectors) Indistinguishable at receiver Rotate symbols to overlap in time. Transmission: Antennas and Propagation Slide 45
DPSK in an Extensible Form Decoding Antennas and Propagation Slide 46
Extension to MIMO Scenario Block fading Rayleigh channel Constant for T=2M symbol times (half as many antennas as time slots) Differential Unitary Space-Time Coding Unitary TxM = 2MxM Note: and indistinguishable at receiver Antennas and Propagation Slide 47
Differential Unitary Space-Time Coding Transmission Decoding Antennas and Propagation Slide 48
Summary of Diff. Unitary STC Features Case where channel is difficult to estimate (rapidly fading) Allows efficient communications For block size of T, can acommodate M=T/2 antennas Bad Aspects? What about noise performance? 3dB penalty since channel is used twice for each symbol Also, how efficient are constellations? Think of PSK Difficult to find many well-separated unitary matrices Antennas and Propagation Slide 49
Summary: Spatial Multiplexing Spatial Multiplexing Use multipath to send different information from multiple antennas Increased transmission rate Methods Nulling (zero-forcing) V-BLAST (Successive interference cancellation) Differential Unitary Space-Time Coding Antennas and Propagation Slide 50