Quantization%20Codes%20Comprising%20Multiple%20Orthonormal%20Bases

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Quantization Codes Comprising Multiple
Orthonormal Bases

• Alexei Ashikhmin
• Bell Labs

• MIMO Broadcast Transmission
• Quantizers Q(m) for MIMO Broadcast Systems
• transmission to mobiles with orthogonal channel
  vectors
• transmission to mobiles with almost orthogonal
• channel vectors
• Simulation Results
• Algebraic Constructions of Q(m)

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MIMO Broadcast Transmission
is a quantization code
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Requirements for a quantization code

• should provide good quantization (for given
  size )
• should afford a simple decoding
• should have many sets of M orthogonal
  codewords (bases of )

is the channel vector of
BS
is the channel vector of
is the channel vector of
If are pairwise orthogonal
then signals sent to do
not interfere with each other
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• Mobiles quantize
• Base Station strategy among
  find orthogonal codewords, say
  , and transmit to the corresponding mobiles
  1,3,5
• The channel vectors of these mobiles
  will be almost orthogonal

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Let us have a quantization code
If a channel vector is quantized into
we say that is occupied and mark by

• In this case even if we have only a few sets of
  orthogonal codewords, we
  easily find a set of occupied orthogonal
  codewords

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Example

• Let and the number of mobiles is
  small, say
• Let
• If are many sets of orthogonal code vectors there
  is a chance to find occupied orthogonal codewords
• For example, if
• are sets of orthogonal codewords. Then

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Example The number of antennas The first code
in the family (for practical applications
we add four vectors to the code to make the code
size 64)
(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0,
0, 0, 1) (1, 0, 1, 0), (0, 1, 0, 1), (1, 0,
-1, 0), (0, -1, 0, 1) (1, 0, -i, 0), (0, 1, 0,
-i), (1, 0, i, 0), (0, 1, 0, i) (1, 1, 0,
0), (0, 0, 1, 1), (1, -1, 0, 0), (0, 0, -1, 1)
(1, -i, 0, 0), (0, 0, 1, -i), (1, i, 0, 0), (0,
0, 1, i) (1, 0, 0, 1), (0, 1, 1, 0), (1, 0,
0, -1), (0, 1, -1, 0) (1, 0, 0, -i), (0, 1, i,
0), (1, 0, 0, i), (0, 1, -i, 0) (1, 1,
1, 1), (1, -1, 1, -1), (1, 1, -1, -1), (1, -1,
-1, 1) (1, 1, -i, -i), (1, -1, -i, i), (1, 1, i,
i), (1, -1, i, -i) (1, -i, 1, -i), (1, i, 1,
i), (1, -i, -1, i), (1, i, -1, -i) (1, -i, -i,
-1), (1, i, -i, 1), (1, -i, i, 1), (1, i, i, -1)
(1, -i, -i, 1), (1, i, -i, -1), (1, -i, i, -1),
(1, i, i, 1) (1, -i, 1, i), (1, i, 1, -i), (1,
-i, -1, -i), (1, i, -1, i) (1, 1, 1, -1),
(1, -1, 1, 1), (1, 1, -1, 1), (1,-1,-1,-1) (1,
1,-i, i), (1, -1, -i, -i), (1, 1, i, -i), (1, -1,
i, i)
105 orthogonal bases
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• The bases form the constant weight code (n60,
  C105, w4).
• With probability 0.65 will find four orthogonal
  occupied codewords
• With probability 0.349 will find three orthogonal
  occupied codewords

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Examples (continued) 1. The number of
orthogonal bases is 105. Each codeword belongs
to 7 bases. The bases form the constant weight
code (n60, C105, w4). 2. The number of
orthogonal bases is 1076625. Each codeword
belongs to 7975 bases. The bases form the
constant weight code
(n1080, C1076625, w8) If K is small
that the probability to find M occupied
orthogonal codewords is also small What to
do? - Use almost orthogonal codewords
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Simulation Results
All results for M8, i.e. the number of Base
Station antennas is 8
Q(3)
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If K50 typically we can find 5 or 6 occupied
codewords
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greedy alg.
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Mutually Unbiased Bases (MUB)
Def. Orthonormal bases of
are mutually unbiased if for any
we have Theorem The
number of MUBs Def.
(i.e. ) is a full size
MUB set.
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• MUB sets form a constant weight code C (n15,
  C6, w5)
• If K is small the chance that M occupied
  codewords are covered by
• an MUB set is significantly higher than that
  they are covered by a basis

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There are 840 full size MUB sets
, each belongs to 56 full size MUB
sets

• Let are orthogonal
• Let are orthogonal
• Let
• To transmit efficiently to mobiles with
• we design a special precoding matrix

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are orthogonal
and
are orthogonal
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Decoding
Example M8
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Construction of Q(m)

• Q(m) is a code in
• There are two equivalent methods for construction
  of Q(m)
• Group theoretic approach
• Coding theory approach

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Orthogonal Projectors

• A subspace of can be defined by its
  orthogonal
• projector , i.e.
• a
• is an orthogonal projector iff

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Group Theoretic Construction of Q(m)
Pauli matrices
where
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It is easy to check that Theorem is an
orthogonal projector and
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• Def. Vectors and are
  orthogonal (with respect
• to the symplectic inner product) if
• is a set of
  orthogonal independent vectors
• .
• Lemma 2 The operator is an orthogonal
  projector on a subspace ,
• and

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It is easy to check that
and Thus defines a
subspace . So
is a line.
therefore
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• Construction
  of Q(m)
• Take all sets
  of orthogonal independent vectors
• Take all choices of
• For each set and set
  compute
• defines a line, in other words
  defines a code vector of Q(m).

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Coding Theory approach for construction of Q(m)

• Q(m) is obtained by merging
  of
• Binary Reed-Muller codes RM(r,m)
• is the order or RM(r,m),
• the code length is
• 2. Codes B(m) over the alphabet 1,-1,i,-i
• the code length is

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Merging RM(r,m) and
B(m) into Q(m)
r changes from m2 to 0

• rm2 take the all minimum weight codewords of
  RM(2,2)
• rm-11 substitute codewords of
• into the minimum weight codewords of
  RM(1,2)

(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0,
0, 1)
Minimum weight codeword of RM(1,2)
Codewords of Q(2)
(1,i) (1,-i) (1,1) (1,-1)
(1,i,0,0)
(0,1,i,0)
(0,1,-i,0)
(1,-i,0,0)
(1,1,0,0)
(0,1,1,0)
(1,1,0,0)
(0,1,1,0)
(1,-1,0,0)
(0,1, -1,0)
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(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1
) (1,1,0,0),(1,i,0,0),(1,-1,0,0),(1,
-i,0,0) (1,0,1,0),(1,0,i,0),(1,0,-1,0
),(0,1,0,-i) (1,0,0,1),(1,0,0,i),(1,0
,0,-1),(1,0,0,-i) (0,1,1,0),(0,1,i,0)
,(0,1,-1,0),(0,1,-i,0)
(0,1,0,1),(0,1,0,i),(0,1,0,-1),(1,0,-i,0)
(0,0,1,1),(0,0,1,i),(0,0,-1,1),(0,0,1,-i)
(1,1,1,1), (1,-1,1,-1), (1,1,-1,-1),
(1,-1,-1,1), (1,1,-i,-i),
(1,-1,-i,i), (1,1,i,i), (1,-1,i,-i),
(1,-i,1,-i), (1,i,1,i), (1,-i,-1,i),
(1,i,-1,-i), (1,-i,-i,-1),
(1,i,-i,1), (1,-i,i,1), (1,i,i,-1),
(1,-i,-i,1), (1,i,-i,-1), (1,-i,i,-1),(1,i,i,1),
(1,-i,1,i), (1,i,1,-i), (1,-i,-1,-i),
(1,i,-1,i), (1,1,1,-1), (1,-1,1,1),
(1,1,-1,1), (1,-1,-1,-1),
(1,1,-i,i), (1,-1,-i,-i), (1,1,i,-i), (1,-1,i,i)
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Theorem
Example
Theorem (Inner product distribution of Q(m)).
For any we have and the number of
such that is
Example in Q(2) there are 15 vectors such
that in Q(3) there are 315
vectors such that
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Theorem For any basis
there exist bases such that
is an MUB set.
Theorem The maximum root-mean-square (RMS) inner
product is