IERG 4100 Wireless Communications Part IIX: Multiple antenna systems

Motivation Current wireless systems Increasing demand Cellular mobile phone systems, WLAN, Bluetooth, Mobile LEO satellite systems, … Increasing demand Higher data rate ( > 100Mbps) IEEE802.11n Higher transmission reliability (comparable to wire lines) 4G Physical limitations in wireless systems Multipath fading Limited spectrum resources Limited battery life of mobile devices …

Motivation Time and frequency processing Coding Adaptive modulation Equalization Dynamic bandwidth and power allocation … Multiple antenna open a new signaling dimension: space Higher transmission rate (Multiplexing gain) Higher link reliability (Diversity gain) Wider coverage

Multiple antenna systems SU-MISO, TX diversity SU-SIMO, RX diversity SU-MIMO, Diversity vs. Multiplexing

Multiple antenna systems MISO Broadcast SIMO Multi-access MIMO Broadcast MIMO Multi-access

Multiplexing gain Multiple antennas at both Tx and Rx Can create multiple parallel channels Multiplexing order = min(M, N) Transmission rate increases linearly

Diversity gain Multiple Tx or multiple Rx or both Can create multiple independently faded branches Diversity order = MN Link reliability improved exponentially

Today’s Lecture Diversity schemes Beamforming Space time coding

Achieving diversity: Maximum ratio combining Recall Fading flattens BER curves Space-domain diveristy Improve BER from ~(SNR)-1 to (SNR)-n Assume N=1 or M=1 for the time being Diversity order n BER~exp(-SNR) BER~(SNR)-1

Maximum ratio combining (SIMO) Ways to combine the received signals Equal gain combining All paths co-phased and summed with equal weighting Maximum ratio combining All paths co-phased and summed with optimal weighting

Maximum ratio combining Variance of noise Maximal Ratio Combining (MRC) Optimal technique (maximizes output SNR) Combiner SNR is the sum of the branch SNRs. Achieve diversity order of N

Distribution of SNR in Rayleigh Fading Channel ： exponential distribution : chi-square distribution with degree of freedom 2N when hn are independent for different n Recall the BER Calculation: Through simple calculation, it can be seen that Diversity order Average SNR

Diversity

Diversity Gain

Maximum ratio transmission (MISO) The signal transmitted by M antennas Transmitter must know the channel! What if it does not know?

Achieving diversity without CSIT: Space time coding Core idea: complement traditional time with added space Without channel knowledge at the transmitter ST trellis codes (Tarokh’98), ST block codes (Alamouti’98) Coding techniques designed for multiple antenna transmission. Coding is performed by adding properly designed redundancy in both spatial and temporal domains which introduces correlation into the transmitted signal.

Space time coding The ST encoder maps a block of information symbols X to coded symbols S

An Introductory Example Two transmit antennas and 1 receive antenna If two time intervals for the transmission of 1 symbol is allowed Received signal Equal to 1 by 2 MIMO systems Diversity gain = 2 Data rate is reduced!!

Space time block code: Alamouti code Two transmit antennas and 1 receive antenna Assume channel does not change across two consecutive symbols Space A1 A2 t x1 x2 t+1 -x2* x1* time

Alamouti code The combining scheme The decision statistics Maximum-likelihood estimates of the transmitted symbols Choose xi if The combined signals are equivalent to that obtained from two-branch MRC! Diversity gain =2 Data rate is not reduced!

Alamouti code Full-rate complex code Optimality of capacity Is the only complex S-T block code with a code rate of unity. Optimality of capacity For 2 transmit antennas and a single receive antenna, the Alamouti code is the only optimal S-T block code in terms of capacity

Alamouti Code Performance From Alamouti, A simple transmit diversity technique for wireless communications

Alamouti Code The performance of Alamouti code with two transmitters and a single receiver is 3 dB worse than two-branch MRC. The 3-dB penalty is incurred because is assumed that each transmit antenna radiates half the energy in order to ensure the same total radiated power as with one transmit antenna. If each transmit antenna was to radiate the same energy as the single transmit antenna for MRC, the performance would be identical.

Space time block code Alamouti code can be generalized to an arbitrary number of antennas A S-T code is defined by an k x M transmission matrix M – number of TX antennas k – number of time periods for transmission of one block of coded symbols Fractional code rate Reduced Spectral efficiency Non-square transmission matrix Ref.: V. Tarokh, et al. “Space-time block codes from orthogonal designs.”

SVD SVD-Singular value decomposition Allows H to be decomposed into parallel channels as follows where S is a N-by-M diagonal matrix with elements only along the diagonal n=m that are real and non-negative U is a unitary N-by-N matrix and V is a unitary M-by-M matrix A Matrix is Unitary if AH=A-1 so that AHA= I For example

What are Singular Values? Note we can generate a square M-by-M matrix as HHH= (USVH)H(USVH)=V(SHS)VH Alternatively we can generate a square N-by-N matrix as HHH= (USVH)(USVH)H= UH (SSH)U We can see that the square of the singular values are the eigenvalues of HHH and HHH

SVD—What does it mean? Implies that UHHV=S is a diagonal matrix Therefore if we pre-process the signals by V at the transmitter and then post-process them with UH we will produce an equivalent diagonal matrix This is a channel without any interference and channel gains s11 and s22 for example

Water-Filling When we have parallel multiple channels each with different attenuations we can use water filling to optimize the capacity by modifying the transmit powers The capacity of multiple channels is given by The question is how to find the distribution of powers to maximize the capacity under the constraint that

Water-Filling Use Lagrangian multiplier to find the solution Write Take partial derivatives wrt to power allocations

Water-filling Know as water filling Good channels get more power than poor channels Channel index

Adaptive Modulation in Fading Channels 6 bps/HZ 4 bps/HZ 2 bps/HZ 0 bps/HZ

Adaptive Modulation in Fading Channels

Adaptive Modulation Data rate varies with channel fading amplitude Variable data-rate transmission can also be achieved by adapting the code rate Adaptive coding and modulation are often combined Coding and modulation schemes can be chosen according to several criteria Maximize average data rate given a fixed BER (bit error rate) Minimize average BER given a fixed average data rate In practice, need to consider the modulation types being discrete

Example An adaptive modulation system can choose to use QPSK or 8-PSK for a target BER of 10-3. The channel is Rayleigh fading with average SNR Adaptation rule The BER should always be smaller than 10-3 If the target BER cannot be met with either scheme, then no data is transmitted

Apply to MIMO with SVD Decompose MIMO channels with SVD Allocate the power according to water-filling principle and adaptive modulation Can transmit same (achieve diversity gain) or different (achieve multiplexing gain) data streams on the parallel channels

Achieving diversity using SVD Achieve diversity order of 4!

Achieving multiplexing gain using SVD Transmit different data streams on parallel channels Use water filling to distribute power on the channels Transmission rate on each channel is adapted to the effective channel gain

MIMO Detection SVD requires channel state information at both the transmitter and receiver When the transmitter doesn’t have knowledge of the channel, each antenna transmits independent data streams The received signal Our target is to detect original signals x from the received signal y

MIMO MLD Let’s first consider optimum receivers in the sense of maximum likelihood detection (MLD) In MLD we wish to maximize the probability of p(y|x) To calculate p(y|x) we observe that the distribution must be jointly Gaussian We need to find an optimal x from the set of all possible transmit vectors Complexity grows exponentially!

MIMO Zero-Forcing We still use the idea of Instead of minimizing only over the constellation points of x we minimize over all possible complex numbers (this is why it is sub-optimum) In other words, we want to force We then quantize the complex number to the nearest constellation point of x

MIMO Zero-Forcing The decision statistics is given by where is the pseudo-inverse of H Requirement: N>=M

MIMO Zero-Forcing Similar to passing the signal through a Gaussian channel, but with a different noise variance The variance of the noise added to xi is given by Problem: Noise enhancement The diversity order achieved by each stream is given by N-M+1

V-BLAST The performance of ZF is not good enough, while the complexity of MLD is too high Motivate different sub-optimal approaches V-BLAST (Vertical-Bell laboratories layered space time) Information stream is split into M sub-streams, each of them is modulated and transmitted from an antenna Only applicable when N>=M Based on interference cancellation Ref.: P. W. Wolniansky, G. J. Foschini, G. D. Golden, R. A. Valenzuela, “V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel”, 1998

VBLAST

V-BLAST: Key idea Successive interference cancellation Select the best bit stream and output its result using ZF Use this result to remove the interference of the detected bit stream from the other received signals Then detect the best of the remaining signals and continue until all signals are detected It is a non-linear process

V-BLAST receiver V-BLAST successive interference cancelling (SIC) The ith ZF-nulling vector wi is defined as the unique minimum-norm vector satisfying is the ith row of H+

V-BLAST optimal ordering Problem in SIC: error propagation If the first decode channel is in low SNR, may decode in error and propagate to subsequent decoding process Ordered Successive Interference Cancellation (OSIC) Idea: Detect the symbols in the order of decreasing SNR Provides a reasonable trade–off between complexity and performance (between MMSE and ML receivers) Achieves a diversity order which lies between N − M + 1 and N for each data stream

V-BLAST Performance M N