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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
Base
Station
is a quantization code
The Base Station (BS):
• chooses some mobiles, for example mobiles 1,2,3
• forms and using computes a precoding matrix
• transmits to mobiles 1,2,3 using the precoding matrix
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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 )
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
is the channel vector of
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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
Base
Station
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If a channel vector is quantized into we say that is occupied
and mark by
• If the number of mobiles (channel vectors) is large, e.g. , then
with a high probability all codewords will be occupied
• In this case even if we have only a few sets of orthogonal codewords, we easily find a set of occupied orthogonal codewords
Let us have a quantization code
orthogonal codewords
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• 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
Example
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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)
105 orthogonal bases
(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)
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The number of mobiles
• The bases form the constant weight code (n=60, |C|=105, w=4).
• 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 (n=60, |C|=105, w=4).
2.
The number of orthogonal bases is 1076625. Each codeword belongs to
7975 bases. The bases form the constant weight code
(n=1080, |C|=1076625, w=8)
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 M=8, i.e. the number of Base Station antennas is 8
K=1000
Q(3)
Yoo and Goldsmith greedy alg. with RVQ
RVQ with Reg. ZF
RVQ with ZF
Q(3)
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If K=50 typically
we can find 5 or 6
occupied codewords
Q(3),
Q(3),
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Q(3)
greedy alg.
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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.
Mutually Unbiased Bases (MUB)
Bases form a full size MUB set
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• MUB sets form a constant weight code C (n=15, |C|=6, w=5)
• 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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Transmission to
Transmission to
are orthogonal
are orthogonaland
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Decoding
Q(3), |Q(3)|=1080 Random Code C, |C|=1080
• Complex
multiplications 0 8*1080
• Complex
summations 1500 7*1080
Example M=8
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Q(m) is a code in
There are two equivalent methods for construction of Q(m):
1. Group theoretic approach
2. Coding theory approach
Construction of Q(m)
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• A subspace of can be defined by its orthogonal
projector , i.e.
• a
• is an orthogonal projector iff
Orthogonal Projectors
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Pauli matrices:
Group Theoretic Construction of Q(m)
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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Q(m) is obtained by merging of
1. 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
Coding Theory approach for construction of Q(m)
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1. r=m=2: take the all minimum weight codewords of RM(2,2):
2. r=m-1=1: substitute codewords of
into the minimum weight codewords of RM(1,2)
Codewords of Q(2):
Merging RM(r,m) and B(m) into Q(m)
(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)
r changes from m=2 to 0:
(1,i)
(1,-i)
(1,1)
(1,-1)
Minimum weight codeword of RM(1,2):
(1,1,0,0)
(1,i,0,0)
(1,-i,0,0)
(1,1,0,0)
(1,-1,0,0)
(0,1,i,0)
(0,1,-i,0)
(0,1,1,0)
(0,1, -1,0)
(0,1,1,0)
3. r=m-2=0: 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
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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)
r=0, minimum weights v codewords of RM(2,2)
r=1, minimum weights v codewords of RM(1,2) v +codewords of B(1)
r=2, minimum weights v codewords of RM(0,2) v +codewords of B(2)
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Theorem (Inner product distribution of Q(m)). For any
we have
and the number of such that is
Theorem
Example:
Example: in Q(2) there are 15 vectors such that
in Q(3) there are 315 vectors such that
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Theorem The maximum root-mean-square (RMS) inner product is
Theorem For any basis there exist bases
such that is an MUB set.