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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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Lecture 7

main story
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.
spatial parallel channel
Spatial Parallel Channel
  • Capacity is achieved by waterfilling over the eigenmodes of H. (Analogy to frequency-selective channels).
rank and condition number
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.

transmitter and receiver csi
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
receiver csi only
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.
nature of performance gain
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.
slow fading mimo channel
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.
diversity and freedom review
Diversity and Freedom (Review)
  • Two fundamental resources of a MIMO fading channel:
  • Diversity
  • Degrees of Freedom
diversity multiplexing tradeoff
Diversity-Multiplexing Tradeoff


A space-time coding scheme achieves a diversity multiplexing tradeoff curve d(r) if for each multiplexing gain r, simultaneously


The largest multiplexing gain is rmax, the DoFutilized by the scheme. The largest diversity gain is dmax = d(0), the classical diversity gain.

optimal d m tradeoff for general n t x n r channel
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.