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| Visualizing MIMO Beamforming | |
| This nugget shows how beamforming between two mobile MIMO radios works. It gives an overview of the MIMO beamforming concept and the underlying algebra. Then it shows several example animations of mobile MIMO beamforming. Finally, the nugget shows how the idea of untangling random matrix processes applies to mobile MIMO beamforming. | |
| Background | |||||||||||||
 Imagine a group of people
talking all at once to a group of listeners. The objective of the
listeners
is to understand each of the talkers as clearly as if they
were taking turns talking. This problem is
analogous to radio communications between multi-antenna
radios. The solution is a beamforming technique which allows
multi-antenna radios
to
communicate multiple
streams of information across a multipath channel such
that all streams use the same radio spectrum but do not interfere. First, a let's look at the algebra behind MIMO beamforming (or just skip to the beamforming animations).
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| Physical Interpretation | |
 The algebra above has
an important physical interpretation. A given L x L MIMO channel matrix, H[n], is determined by the geometries of: (a) the propagation pathways and (b) the antenna arrays. The SVD of H[n] takes a matrix containing joint information about the propagation paths between TX and RX and separates it into two unitary matrices. V[n] describes the propagation perceived by the TX array and U[n] describes the propagation perceived by the RX array. Each triplet of left singular vector, right singular vector, and corresponding singular value forms a singular-mode (often described as an "eigenmode" - a name that is not strictly correct) . These singular-modes are orthogonal because U[n] and V[n] are unitary and thus allow parallel communication of symbols without interference. Now think of a singular vector of H[n] as a beamforming vector for a phased array. The difference being that while a phased array produces a single radiation pattern, the MIMO beamforming system produces multiple radiation patterns simultaneously - one for each pair of left and right singular vectors. When used as beamforming vectors, the singular vectors of H[n] cause the antenna arrays' radiation patterns to focus in directions that are aligned with certain propagation paths. To see this, let us consider a scenario involving two multi-antenna radios moving through an environment with 3 scatterers and no line-of-sight path. Each radio has a Uniform Circular Arrays (UCA) of 8 antennas with a 6cm radius (a half wavelength at 2.49 GHz). The transmitter moves 3m downwards while the receiver moves 3m upwards. At each time instant, the SVD of H[n] is computed and used to beamform as is done in (eq. 1). Each of the L scaled radiation patterns are computed at each time instant according to,  where A is a matrix whose columns are steering vectors and whose elements are given by,  for direction theta and the kth antenna element whose coordinates are (xl , yl). The resulting singular-mode radiation patterns evolve over time as shown below for L = 8. mouse
over image to see animation
There are
potentially 8 singular-modes for this 8x8 MIMO
system. However, there are only 3 degrees of freedom in this channel (3
scatterers) and therefore only 3
significant singular-modes are
seen to be involved in beamforming. Note that the individual paths and
antenna
elements are so close with respect to the dimensions of the scenario
that that they cannot be seen as separate. Now consider the the same scenario but with a line-of-sight path. mouse
over image to see animation
The singular-mode using the line-of-sight path is now seen to
dominate
while the other singular-modes are noticeably weakened. Furthermore, the
beam patterns have changed altogether from the non-line-of-sight
scenario. |
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| Untangling MIMO Beamforming | ||
| The
above examples showed MIMO beamforming in a channel that was
deficient of degrees of freedom. Now let us consider a channel where
there is an excess of degrees of freedom as this is a more typical
scenario in terrestrial communications with mobile MIMO radios. The scenario shown below has two multi-antenna radios moving through an environment with 10 scatterers and no line-of-sight path. Each radio has a UCA of 4 antennas with a 6cm radius. This 4x4 MIMO system is capable of communicating over at most 4 singular-modes but the channel offers 10 degrees of freedom. The surplus of available singular-modes results in beamforming that is more complex than the two previous scenarios. mouse
over image to see animation
The lower plot shows the temporal
evolution of the singular values of H[n]. These singular value sample
paths evolve as separate layers. While this layered evolution has
commonly been accepted as normal, it is actually a result of the SVD
algorithm imposing an artificial ordering on the singular values for each H[n] (and therefore also on the
columnwise ordering of the singular vectors). This ordering is
problematic because it destroys the natural evolution of the singular
vectors by tangling the singular-modes. This has very severe consequences for MIMO
beamforming systems because the channel state information has to be
updated far more frequently than necessary. The solution to this problem is to untangle the singular-modes. The scenario shown below is exactly the same as that above except that the untangled beamforming solution is used. mouse
over image to see animation
It is clear from the animation above that the untangled solution
restores the natural evolution of the singular-modes. Note that as part of
the untangling process, the singular values are complex-valued and
thier magnitude is plotted. The singular value sample
paths appear to cross each other in smooth transitions. This smooth
weaving of the singular values is a result of the change of dominant
paths as the radios move within the scattering
environment. To find out more about the untangling solution, see...
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