CHAPTER 8 MIMO Xijun Wang
WEEKLY READING 1. Goldsmith, Wireless Communications, Chapters 10 2. Tse, Fundamentals of Wireless Communication, Chapter 7-10 2
MIMO 3
BENEFITS OF MIMO n Array gain The increase in receive SNR that results from a coherent combining effect of the wireless signals at a receiver. Can be realized through spatial processing at the receive antenna array and/or spatial pre-processing at the transmit antenna array. n Diversity gain Can be realized by providing the receiver with multiple (ideally independent) copies of the transmitted signal in space, frequency or time. The number of copies is referred to as the diversity order. 4
BENEFITS OF MIMO n Multiplexing gain Transmitting multiple, independent data streams within the bandwidth of operation. This increased data rate is called the multiplexing gain. n Interference reduction and avoidance Directing signal energy towards the intended user and minimizing interference to other users. 5
MIMO CHANNEL MODEL n The degree of correlation is determined by the scattering in the environment and antenna spacing at the transmitter and the receiver. n With rich (omni-directional and isotropic) scattering, the typical antenna spacing required for decorrelation is approximately λ/2. n The i.i.d. Rayleigh fading model is reasonable in a richly scattering environment, the channel elements are perfectly decorrelated. 6
MIMO SIGNAL MODEL n For a frequency-flat fading MIMO channel power constraint the average SNR per receive antenna under unity channel gain 7
MULTIPLEXING GAIN n A MIMO channel can be decomposed into a number R of parallel independent channels. n By multiplexing independent data onto these independent channels, we get an R-fold increase in data rate in comparison to a system with just one antenna at the transmitter and receiver. n Consider a MIMO channel with Mr Mt channel gain matrix H known to both the transmitter and the receiver. n Let R H denote the rank of H. 8
MULTIPLEXING GAIN n For any matrix H we can obtain its singular value decomposition (SVD) as the Mr Mr matrix U and the Mt Mt matrix V are unitary matrices Σ is an Mr Mt diagonal matrix of singular values {σi} of H. σi = λi for λi the ith eigenvalue of HH H R H of these singular values are nonzero If H is full rank, which is sometimes referred to as a rich scattering environment, then 9
MULTIPLEXING GAIN n The parallel decomposition of the channel is obtained by defining a transformation on the channel input and output x and y through transmit precoding and receiver shaping. 10
MULTIPLEXING GAIN n The transmit precoding and receiver shaping transform the MIMO channel into R H parallel singleinput single-output (SISO) channels with input x% and output y%. n Note that multiplication by a unitary matrix does not change the distribution of the noise. 11
MULTIPLEXING GAIN n Parallel Decomposition of the MIMO Channel. 12
DIVERSITY GAIN n MIMO beamforming the same symbol, weighted by a complex scale factor, is sent over each transmit antenna 13
DIVERSITY GAIN n Both transmit and receive weight vectors are normalized so that u = v = 1. n Beamforming provides diversity gain by coherent combining of the multiple signal paths. n Channel knowledge at the receiver is typically assumed since this is required for coherent combining. n The diversity gain then depends on whether or not the channel is known at the transmitter. 14
DIVERSITY GAIN n When the channel matrix H is known the received SNR is optimized by choosing u and v as the principal left and right singular vectors of the channel matrix H. The received SNR is λmax is the largest eigenvalue of n When the channel is not known at the transmitter the transmit antenna weights are all equal the received SNR equals 15
DIVERSITY/MULTIPLEXING TRADEOFFS n Two mechanisms for utilizing multiple antennas Obtain capacity gain by decomposing the MIMO channel into parallel channels and multiplexing different data streams onto these channels. The channel gains are coherently combined to obtain a very robust channel with high diversity gain. Some of the space-time dimensions can be used for diversity gain, and the remaining dimensions used for multiplexing gain. Should the antennas be used for diversity gain, multiplexing gain, or both? 16
DIVERSITY/MULTIPLEXING TRADEOFFS n multiplexing gain r n diversity gain d n For each r the optimal diversity gain d opt (r) is the maximum the diversity gain that can be achieved by any scheme. 17
SCALAR RAYLEIGH CHANNEL n Consider the scalar slow fading Rayleigh channel, n PAM The average error probability is governed by the minimum distance between the PAM points. The constellation ranges from approximately SNR to + SNR. for general R, there are 2R constellation points, the minimum distance is approximately the error probability at high SNR is approximately 18
SCALAR RAYLEIGH CHANNEL n PAM By setting the data rate R = r log SNR, we get diversity-multiplexing tradeoff n QAM with data rate R diversity-multiplexing tradeoff 19
SCALAR RAYLEIGH CHANNEL 20
FIXED RATE Increasing the SNR by 6dB decreases the error probability by 1/4 for both PAM and QAM due to a doubling of the minimum distance. 21
FIXED RELIABILITY Increasing the SNR by 6dB increases the data rate for QAM by 2 bits/s/hz but only increases the data rate of PAM by 1 bits/s/hz. 22
MISO RAYLEIGH CHANNEL n Consider the n t transmit and single receive antenna MISO channel with i.i.d. Rayleigh coefficients n The outage probability at target data rate R = r log SNR bits/s/hz n The optimal diversity-multiplexing tradeoff for the i.i.d. Rayleigh fading MISO channel is 23
MISO RAYLEIGH CHANNEL If we use QAM symbols in conjunction with the Alamouti scheme, we achieve the diversitymultiplexing tradeoff of the MISO channel. 24
2 2 MIMO RAYLEIGH CHANNEL 25
N T N R MIMO I.I.D. RAYLEIGH CHANNEL 26
ADDING ONE TRANSMIT AND ONE RECEIVE ANTENNA The entire tradeoff curve is shifted by 1 to the right increasing the maximum possible multiplexing gain by 1. 27
MIMO RAYLEIGH CHANNEL n The optimal tradeoff curve is based on the outage probability, so in principle arbitrarily large block lengths are required to achieve the optimal tradeoff curve. n However, it has been shown that in fact space-time codes of block length l = n t + n r 1 is sufficient to achieve the curve. 28