Grassmannian Packings for Efficient Quantization in MIMO Broadcast Transmission

1
Grassmannian Packings for Efficient Quantization
in MIMO Broadcast Transmission

• Alexei Ashikhmin and RaviKiran Gopalan
• Bell Labs Texas
  Instrument

• MIMO Broadcast Transmission
• Examples GRM(m) for MIMO Broadcast Systems
• transmission to mobiles with orthogonal channel
  vectors
• transmission to mobiles with almost orthogonal
• channel vectors
• Simulation Results
• Algebraic Construction of GRM(m)

2
MIMO Broadcast Transmission
is a quantization code
3
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 )

BS
If are pairwise orthogonal
then signals sent to do
not interfere with each other
4

• 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

5
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

6

• The number of mobiles is small, say
• Still if there are many sets of orthogonal
  codewords, there is a chance to find occupied
  orthogonal codewords
• For example, let
• be sets of orthogonal codewords. Then

7
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
8

• 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

9
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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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.
11

• 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
13
Simulation Results
All results for M8, i.e. the number of Base
Station antennas is 8
GRM(3)
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If K50 typically we can find 5 or 6 occupied
codewords
15
greedy alg.
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are orthogonal
and
are orthogonal
17
Construction of GRM(m)

• GRM(m) is a code in
• There are two methods for construction of GRM(m)
• Group theoretic approach a particular case of
  the Operator Reed-Muller codes (A.Ashikhmin and
  A.R.Calderbank, ISIT 2005)
• Coding theory approach

18
Group Theoretic Construction of GRM(m)
Pauli matrices
where
19

• Def. Vectors and are
  orthogonal (with respect
• to the symplectic inner product) if
• Construction
  of GRM(m)
• is a set of
  orthogonal independent vectors
• .
• Lemma 2 The operator is an orthogonal
  projector on a subspace ,

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Coding Theory approach for construction of GRM(m)

• GRM(m) is obtained by
  merging of
• Binary Reed-Muller codes RM(r,m)
• 2. Reed-Muller codes ZRM(2,m) codes over
• ZRM(2,m) is generated by the Boolean
  functions

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ZRM(2,m) is generated by
the Boolean functions Let us construct the
code from ZRM(2,m) by
mapping For example

22
Merging RM(r,2) and
CRM(2,2) into GRM(2)
r changes from m2 to 0

• rm2 take the all minimum weight codewords of
  RM(r,m)RM(2,2)
• rm-11 substitute codewords of
• into the minimum weight codewords of
  RM(r,m)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 GRM(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 GRM(m)).
For any we have and the number of
such that is
Example in GRM(2) there are 15 vectors
such that in GRM(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
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Decoding
Example M8
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Mobiles quantize
If some channel vector is quantized into
we say that is occupied
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Example The number of BS antennas M4, hence

(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
29
Example The number of BS antennas M4, hence

(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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Merging of RM(r,m) and
CRM(2,m)
changes from m to 0

• rm2 take the all minimum weight codewords of
  RM(r,2)RM(2,2)
• rm-11 substitute codewords of
• into minimum weight codewords of
  RM(r,2)RM(1,2)
• 3. rm-20 take the only minimum weight codeword
  of RM(r,m)RM(0,m)
• (1,1,1,1) and substitute into its nonzero
  positions codewords of

(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 G-ZRM(2)
(1,i) (1,-i) (1,1) (1,-1)
(1,i,0,0)
(0,1,i,0,0)
(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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• Lemma 1 The operator is an orthogonal
  projector,
• Def. Vectors and are
  orthogonal (with respect
• to the symplectic inner product) if
• is a set of
  orthogonal vectors
• .
• Lemma 2 The operator is an orthogonal
  projector on a subspace

32
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 full size MUB
set.