11 S ingle I nput- S ingle O utput systems (SISO) y(t) = g x(t) + n(t) x(t): transmitted signal y(t): received signal g(t): channel transfer function n(t): noise (AWGN, 2 ) Signal to noise ratio : Capacity : C = log 2 (1+ ) x(t) y(t) g
12 S ingle I nput- M ultiple O utput (SIMO) M ultiple I nput- S ingle O utput (MISO) Principle of diversity systems (transmitter/ receiver) +: Higher average signal to noise ratio Robustness - : Process of diminishing return Benefit reduces in the presence of correlation Maximal ratio combining > Equal gain combining > Selection combining
13 Idea behind diversity systems Use more than one copy of the same signal If one copy is in a fade, it is unlikely that all the others will be too. C 1xN >C 1x1 C 1xN more robust than C 1x1 1N1N
14 Background of Diversity Techniques Variety of Diversity techniques are proposed to combat Time-Varying Multipath fading channel in wireless communication –Time Diversity –Frequency Diversity –Space Diversity (mostly multiple receive antennas) Main intuitions of Diversity: –Probability of all the signals suffer fading is less then probability of single signal suffer fading –Provide the receiver a multiple versions of the same Tx signals over independent channels Time Diversity –Use different time slots separated by an interval longer than the coherence time of the channel. –Example: Channel coding + interleaving –Short Coming: Introduce large delays when the channel is in slow fading
15 Diversity Techniques Improve the performance in a fading environment –Space Diversity Spacing is important! (coherent distance) –Polarization Diversity Using antennas with different polarizations for reception/transmission. –Frequency Diversity RAKE receiver, OFDM, equalization, and etc. Not effective over frequency-flat channel. –Time Diversity Using channel coding and interleaving. Not effective over slow fading channels.
28 MIMO Wireless Communications: Combining TX and RX Diversity Transmission over Multiple Input Multiple Output (MIMO) radio channels Advantages: Improved Space Diversity and Channel Capacity Disadvantages: More complex, more radio stations and required channel estimation
29 MIMO Model Matrix Representation –For a fixed T T: Time index W: Noise
31 M ultiple I nput- M ultiple O utput systems (MIMO) H 1M1M 1N1N Average gain Average signal to noise ratio H 11 H N1 H 1M H NM
32 Shannon capacity K= rank(H): what is its range of values? Parameters that affect the system capacity Signal to noise ratio Distribution of eigenvalues (u) of H
33 Interpretation I: The parallel channels approach “Proof” of capacity formula Singular value decomposition of H: H = S·U·V H S, V: unitary matrices (V H V=I, SS H =I) U : = diag(u k ), u k singular values of H V/ S: input/output eigenvectors of H Any input along v i will be multiplied by u i and will appear as an output along s i
34 Vector analysis of the signals 1. The input vector x gets projected onto the v i ’s 2. Each projection gets multiplied by a different gain u i. 3. Each appears along a different s i. u1u1 u2u2 uKuK · v 1 · v 2 · v K u K s K u 1 s 1 u 2 s 2
35 Capacity = sum of capacities The channel has been decomposed into K parallel subchannels Total capacity = sum of the subchannel capacities All transmitters send the same power: E x =E k
36 Interpretation II: The directional approach Singular value decomposition of H: H = S·U·V H Eigenvectors correspond to spatial directions (beamforming) 1M1M 1N1N (si)1(si)1 (si)N(si)N
39 Space-Time Coding What is Space-Time Coding? –Space diversity at antenna –Time diversity to introduce redundant data Alamouti-Scheme –Simple yet very effective –Space diversity at transmitter end –Orthogonal block code design
63 Block and Data Model 1X(N+P) block of information symbols broadcast from transmit antenna: i S i (d, t) 1X(N+P) block of received information symbols taken from antenna: j R j = h ji S i (d, t) + n j Matrix representation:
64 Related Issues How to define Space-Time mapping S i (d,t) for diversity/channel capacity trade-off? What is the optimum sequence for pilot symbols? How to get “best estimated” Channel State Information (CSI) from the pilot symbols P? How to design frame structure for Data symbols (Payload) and Pilot symbols such that most optimum for FER and BER?
65 Specific Example of STBC: Alamouti’s Orthogonal Code Let’s consider two antenna i and i+1 at the transmitter side, at two consecutive time instants t and t+T: The above Space-Time mapping defines Alamouti’s Code. A general frame design requires concatenation of blocks (each 2X2) of Alamouti code,
66 Estimated Channel State Information (CSI) Pilot Symbol Assisted Modulation (PSAM)  is used to obtain estimated Channel State Information (CSI) PSAM simply samples the channel at a rate greater than Nyquist rate,so that reconstruction is possible Here is how it works…
68 Estimated CSI (cont.d) Block diagram of the receiver
69 Channel State Estimation (cont.d) Pilot symbol insertion length, P ins =6. The receiver uses N=12, nearest pilots to obtain estimated CSI
70 Channel State Estimation Cont.d Pilot Symbols could be think of as redundant data symbols Pilot symbol insertion length will not change the performance much, as long as we sample faster than fading rate of the channel If the channel is in higher fading rate, more pilots are expected to be inserted
71 Estimated CSI, Space-time PSAM frame design The orthogonal pilot symbol (pilots chosen from QPSK constellation) matrix is,  Pilot symbol insertion length, P ins =6. The receiver uses N=12, nearest pilots to obtain estimated CSI Data = 228, Pilots = 72
72 Channel State Estimation (cont.d) MMSE estimation Use Wiener filtering, since it is a Minimum Mean Square Error (MMSE) estimator All random variables involved are jointly Gaussian, MMSE estimator becomes a linear minimum mean square estimator : Wiener filter is defined as,. Note, and
73 Block diagram for MRRC scheme with two Tx and one Rx
74 Block diagram for MRRC scheme with two Tx and one Rx The received signals can then be expressed as, The combiner shown in the above graph builds the following two estimated signal
75 Maximum Likelihood Decoding Under QPSK Constellation Output of the combiner could be further simplified and could be expressed as follows: For example, under QPSK constellation decision are made according to the axis.
76 Space-Time Alamouti Codes with Perfect CSI,BPSK Constellation
77 Space-Time Alamouti Codes with PSAM under QPSK Constellation
78 Space-Time Alamouti Codes with PSAM under QPSK Constellation
79 Performance metrics Eigenvalue distribution Shannon capacity –for constant SNR or –for constant transmitted power Effective degrees of freedom(EDOF) Condition number Measures of comparison –Gaussian iid channel –“ideal” channel
80 Measures of comparison Gaussian Channel H ij =x ij +jy ij : x,y i.i.d. Gaussian random variables Problem: p outage “Ideal” channel (max C) rank(H) = min(M, N) |u 1 | = |u 2 | = … = |u K |
81 Eigenvalue distribution Ideally: As high gain as possible As many eigenvectors as possible As orthogonal as possible H 1M1M 1N1N Limits Power constraints System size Correlation
91 Equivalent treatment in the frequency domain Wideband channel = Many narrowband channels H(t) H(f) f Noise level
92 Extensions Optimal power allocation Optimal rate allocation Space-time codes Distributed antenna systems Many, many, many more!
93 Optimal power allocation IF the transmitter knows the channel, it can allocate power so as to maximize capacity Solution: Waterfilling
94 Illustration of waterfilling algorithm Stronger subchannels get the most power
95 Discussion on waterfilling Criterion: Shannon capacity maximization (All the SISO discussion on coding, constellation limitations etc is pertinent) Benefit depends on the channel, available power etc. Correlation, available power Benefit Limitations: –Waterfilling requires feedback link –FDD/ TDD –Channel state changes
96 Optimal rate allocation Similar to optimal power allocation Criterion: throughput (T) maximization B k : bits per symbol (depends on constellation size) Idea: for a given k, find maximum B k for a target probability of error P e
97 Discussion on optimal rate allocation Possible limits on constellation sizes! Constellation sizes are quantized!!! The answer is different for different target probabilities of error Optimal power AND rate allocation schemes possible, but complex
98 Distributed antenna systems Idea: put your antennas in different places +: lower correlation - : power imbalance, synchronization, coordination
100 Detection algorithms Maximum likelihood linear detector y = H x + n x est = H + y H + = (H H H) -1 H H : Pseudo inverse of H Problem: find nearest neighbor among Q M points (Q: constellation size, M: number of transmitters) VERY high complexity!!!
101 Solution: BLAST algorithm BLAST: B ell L abs l A yered S pace T ime Idea: NON-LINEAR DETECTOR –Step 1: H + = (H H H) -1 H H –Step 2: Find the strongest signal (Strongest = the one with the highest post detection SNR) –Step 3: Detect it (Nearest neighbor among Q) –Step 4: Subtract it –Step 5: if not all yet detected, go to step 2
102 Discussion on the BLAST algorithm It’s a non-linear detector!!! Two flavors –V-BLAST (easier) –D-BLAST (introduces space-time coding) Achieves 50-60% of Shannon capacity Error propagation possible Very complicated for wideband case
103 Coding limitations Capacity = Maximum achievable data rate that can be achieved over the channel with arbitrarily low probability of error SISO case: –Constellation limitations –Turbo- coding can get you close to Shannon!!! MIMO case: –Constellation limitations as well –Higher complexity –Space-time codes: very few!!!!
104 Channel estimation The channel is not perfectly estimated because –it is changing (environment, user movement) –there is noise DURING the estimation An error in the channel transfer characteristics can hurt you –in the decoding –in the water-filling Trade-off: Throughput vs. Estimation accuracy What if interference (as noise) is not white????
105 Interference Generalization of other/ same cell interference for SISO case Example: cellular deployment of MIMO systems Interference level depends on –frequency/ code re-use scheme –cell size –uplink/ downlink perspective –deployment geometry –propagation conditions –antenna types
106 Summary and conclusions MIMO systems are a promising technique for high data rates Their efficiency depends on the channel between the transmitters and the receivers (power and correlation) Practical issues need to be resolved Open research questions need to be answered