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MIMO

MIMO. Lecture 7. Main Story. So far we have mainly considered single-input multi output (SIMO ) and multi-input single-output ( MISO) channels . They provide diversity and power gains but no degree-of freedom ( DoF ) gain. DoF gain is most useful in the high SNR regime.

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MIMO

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  1. MIMO Lecture 7

  2. Main Story • So far we have mainly considered single-input multi output (SIMO) and multi-input single-output (MISO) channels. • They provide diversity and power gains but no degree-of freedom (DoF) gain. • DoFgain is most useful in the high SNR regime. • MIMO channels have a potential to provide DoF gain. • We start with deterministic models and then progress to statistical ones.

  3. MIMO Capacity via SVD(CSI at both Txand Rx)

  4. Spatial Parallel Channel • Capacity is achieved by waterfilling over the eigenmodes of H. (Analogy to frequency-selective channels).

  5. Rank and Condition Number At high SNR, equal power allocation is optimal: where k is the number of nonzero λi's, i.e. the rankof H. The closer the condition number: to 1, the higher the capacity. Rank of H is always less than min(nr, nt): Max Degrees of Freedom.

  6. Transmitter and Receiver CSI • Can decompose the MIMO channel into a bunch of orthogonal sub-channels. • Can allocate power and rate to each sub-channel according to waterfilling

  7. Receiver CSI Only • The channel matrix H and its singular values 's are random and unknown to the transmitter. • Has to fix a Q and a power allocation independent of H. • Q=I and uniform power allocation is a good choice in many cases. • It is not trivial to come up with capacity-achieving architectures.

  8. Capacity

  9. Fast Fading Capacity for i.i.d. Rayleigh Fading

  10. DoFmin(nt,nr) Determines the High SNR Slope

  11. Nature of Performance Gain • At high SNR (D.o.F. limited): min(nt,nr)-fold DoF gain. MIMO is crucial. • At low SNR (power limited): nr-fold power gain. Only need multiple receive antennas. • At all SNR, min(nt,nr)-fold gain due to a combination of both effects.

  12. Slow Fading MIMO Channel • So far we have emphasized the spatial multiplexingaspect of MIMO channels. • But we also learnt that in slow fading scenario, spatial diversity is an important thing. • How do the two aspects interact? • It turns out that you can get both in a slow fading channel but there is a fundamental tradeoff. • We characterize the optimal diversity multiplexing tradeoff.

  13. Diversity and Freedom (Review) • Two fundamental resources of a MIMO fading channel: • Diversity • Degrees of Freedom

  14. Diversity-Multiplexing Tradeoff Definition A space-time coding scheme achieves a diversity multiplexing tradeoff curve d(r) if for each multiplexing gain r, simultaneously and The largest multiplexing gain is rmax, the DoFutilized by the scheme. The largest diversity gain is dmax = d(0), the classical diversity gain.

  15. Optimal D-M Tradeoff for General nt x nrChannel • As long as block length • For integer r, it is as though r transmit and rreceive antennas were dedicated for multiplexing and the rest provide diversity.

  16. Diversity-Multiplexing Tradeoff of the Channel

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