Grassmannian Packings for Efficient Quantization in MIMO Broadcast

Transmission

Alexei Ashikhmin and RaviKiran GopalanBell 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)

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

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

is the channel vector of

If are pairwise orthogonal then signals sent to do

not interfere with each other

• 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

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

• 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

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)

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

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

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

• 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

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

full size MUB sets

Simulation Results

All results for M=8, i.e. the number of Base Station antennas is 8

K=1000

GRM(3)

Yoo and Goldsmith greedy alg. with RVQ

RVQ with Reg. ZF

RVQ with ZF

GRM(3)

If K=50 typically

we can find 5 or 6

occupied codewords

GRM(3),

GRM(3),

GRM(3)

greedy alg.

Transmission to

Transmission to

are orthogonal

are orthogonaland

GRM(m) is a code in

There are two methods for construction of GRM(m):

1. Group theoretic approach – a particular case of the Operator Reed-Muller codes (A.Ashikhmin and A.R.Calderbank, ISIT 2005)

2. Coding theory approach

Construction of GRM(m)

Pauli matrices:

Group Theoretic Construction of GRM(m)

where

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 ,

GRM(m) is obtained by merging of

1. 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:

Coding Theory approach for construction of GRM(m)

ZRM(2,m) is generated by the Boolean functions

Let us construct the code from ZRM(2,m) by mapping

For example

Codewords of GRM(2):

Merging RM(r,2) and CRM(2,2) into GRM(2)

(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

1. r=m=2: take the all minimum weight codewords of RM(r,m)=RM(2,2):

2. r=m-1=1: 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)

(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 CRM(2,1)

r=2, minimum weights v codewords of RM(0,2) v +codewords of CRM(2,2)

Theorem (Inner product distribution of GRM(m)). For any

we have

and the number of such that is

Theorem

Example:

Example: in GRM(2) there are 15 vectors such that

in GRM(3) there are 315 vectors such that

Theorem The maximum root-mean-square (RMS) inner product is

Theorem For any basis there exist bases

such that is an MUB set.

Decoding

GRM(3), |GRM(3)|=1080 Random Code C, |C|=1080

• Complex

multiplications 0 8*1080

• Complex

summations 1500 7*1080

Example M=8

Mobiles quantize:

If some channel vector is quantized into we say that is occupied

The number of mobiles

is large, say

In this case, even if we

have only one set of

orthogonal vectors, say

we are doing

fine

Example: The number of BS antennas M=4, 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

Example: The number of BS antennas M=4, 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

Merging of RM(r,m) and CRM(2,m)

1. r=m=2: take the all minimum weight codewords of RM(r,2)=RM(2,2):

2. r=m-1=1: substitute codewords of

into minimum weight codewords of RM(r,2)=RM(1,2)

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

(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)

changes from m to 0:

(1,i)

(1,-i)

(1,1)

(1,-1)

Minimum weight codeword of RM(1,2): Codewords of G-ZRM(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)

(0,1,-i,0)

(0,1,1,0)

(0,1, -1,0)

(0,1,1,0)

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

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.

Mutually Unbiased Bases (MUB)

Bases form a full size MUB set