quaternion orthogonal designssali.seecs.nust.edu.pk/wp-content/uploads/2014/06/talk.pdf ·...

Novel Construction Methods of

Quaternion Orthogonal Designs

Based on Complex Orthogonal Designs

Sajid Ali Joint work with S. Ali Hassan and E. Mushtaq

NUST University, Pakistan.

ISIT-2017Aachen

Motivation

§ Orthogonal spacetime block codes provide multiple gains, however, maximal rate designs in MIMOs are difficult to construct

§ Dual-polarized antennas offer a good quality of service through reliable communication by mitigating multipath effects

§ Efficient codes for multiple dual-polarized antennas § Polarization diversity gain along with space & time diversities

§ Low-complexity decoding (de-coupled decoding)

By default Orthogonal

Tx RxOne Dual polarized One Dual polarized

Reflections, Scattering, Diffraction

Channel

By default Orthogonal

Polarization Diversity Gain

Low intensity indicates that the received signal is generally different from what is transmitted.

[Oestges C., Clerckx B., Guillaud M. and Debbah M., IEEE Tran. of Wireless Comm., 2008]

§ Space & Cost Effective § Optimal Channel Separation

Quick ReviewQuaternions

0 1 2 3q q q i q j q k 2 2 2 1i j k

i j k j i j k i k j

k i j i k

A quaternion is a generalization of the concept of complex numbers defined over a basis of non-commuting elements {1, , j, k}i


q

0 1 2 3q q q i q j q k

1z 2z j2 2 2 1i j k

i j k j i j k i k j

k i j i k



q

0 1 2 3q q q i q j q k

1z 2z j2 2 2 1i j k

i j k j i j k i k j

k i j i k

Quaternion conjugate * *1 2

Qq z j z 2| |Q Qqq q q q

| | 1q


For normalized signals

RealizationQuaternions

q 1z 2z j2 2 2 1i j k

i j k j i j k i k j

k i j i k

[Isaeva O. M. and Sarytchev V. A. , in Proc. 2nd IEEE Topical Symposium of Combined Optical-Microwave Earth and Atmosphere Sensing, Atlanta, US, April 1995, pp. 195–196.]


0 1 2 3q q q i q j q k

Quaternion Orthogonal Codes (QODs)A quaternion orthogonal code is an n x m matrix of quaternion elements which satisfy

[Seberry J., et. al., “The theory of quaternion orthogonal designs,” IEEE Trans. Signal Process., 2008.]

Quaternion Orthogonal Codes (QODs)

Such codes can be obtained easily by using complex orthogonal codes.


A quaternion orthogonal code is an n x m matrix of quaternion elements which satisfy


Symmetric-Pair Design:


Two complex orthogonal codes A and B form a symmetric-pair design if is symmetric.




and/or

Two complex orthogonal codes form a complex amicable design if Complex Amicable Design:







and/or

Two complex orthogonal codes form a complex amicable design if Complex Amicable Design:

QOD






System Model (Dual Polarized Antennas)

Brief Example

Almouti Scheme

Antenna 1 Antenna 2

Time Slot 1

Time Slot 2

1 2* *

2 1

z zz z


Brief Example

Almouti Scheme

Antenna 1 Antenna 2

Time Slot 1

Time Slot 2

1 2* *

2 1

z zz z

1 2 2 1* * * *

2 1 1 2

z z j z z jz z j z z j

QODsTwo dual-polarized Antennas


Brief Example

Almouti Scheme

Antenna 1 Antenna 2

Time Slot 1

Time Slot 2

1 2* *

2 1

z zz z

1 2 2 1* * * *

2 1 1 2



1 2 2 1* * * *

2 1 1 2

z z z zz z z z


Brief Example

Almouti Scheme

Antenna 1 Antenna 2

Time Slot 1

Time Slot 2

1 2* *

2 1

z zz z

1 2 2 1* * * *

2 1 1 2



1 2 2 1* * * *

2 1 1 2

z z z zz z z z

quasi-Orthogonal Code:Four single polarized antennas are transmitting two complex symbols in two time slots.


(1)1 1

1 1

1 12 2*

1 12 2

( ,..., z )( ,..., z )

l l

l l

l lH

l l

G z z Iz I G z

An efficient way to generate square CODs of order is

(Liang X. B., Tran. of Inf. Theo., 2003)

ll 22


Theorem 1:

(1)

Two CODs A and B of the form (1), where B is obtained by permuting the columns of A, satisfy both symmetry and complex amicable properties. Consequently, Q = A + B j is a QOD of rate . ( 1) / 2ll

1 1

1 1

1 12 2*

1 12 2

( ,..., z )( ,..., z )

l l

l l

l lH

l l

G z z Iz I G z



ll 22

For a given square COD , the matrix

is a QOD of rate .


Theorem 1:

(1)

Lemma 1:

Q =

Non-uniqueness

Two CODs A and B of the form (1), where B is obtained by permuting the columns of A, satisfy both symmetry and complex amicable properties. Consequently, Q = A + B j is a QOD of rate . ( 1) / 2ll

1 1

1 1

1 12 2*

1 12 2

( ,..., z )( ,..., z )

l l

l l

l lH

l l

G z z Iz I G z

( 1) / 2ll

1 12( ,..., )l lG z z

1 1

1 1

1 12 2*

1 12 2

( ,..., z )( ,..., z )

l l

l l

l lH

l l

G z z I jz I G z j



ll 22


Example: 1 2* *

2 1

z zz z


Example: 1 2* *

2 1

z zz z

1 2 3* *

2 1 3* *

3 1 2* *

3 2 1

00

00

z z zz z zz z z

z z z


Example: 1 2* *

2 1

z zz z

1 2 3* *

2 1 3* *

3 1 2* *

3 2 1

00

00

z z zz z zz z z

z z z

3 1 2* *

3 2 1* *

1 2 3* *

2 1 3

00

00

z z zz z z

z z zz z z


Example: 1 2* *

2 1

z zz z

1 3 2 3 1 2* * * *

2 1 3 2 3 1* * * *

3 1 2 1 3 2* * * *

2 3 1 2 1 3

z z j z z z j z jz z z j z j z z j

z z j z j z z j zz z z j z z z j

1 2 3* *

2 1 3* *

3 1 2* *

3 2 1

00

00

z z zz z zz z z

z z z

3 1 2* *

3 2 1* *

1 2 3* *

2 1 3

00

00

z z zz z z

z z zz z z


Example: 1 2* *

2 1

z zz z

1 2 3* *

2 1 3* *

3 1 2* *

3 2 1

00

00

z z zz z zz z z

z z z

1 3 2 3 1 2* * * *

2 1 3 2 3 1* * * *

3 1 2 1 3 2* * * *

2 3 1 2 1 3



3 1 2* *

3 2 1* *

1 2 3* *

2 1 3

00

00

z z zz z z

z z zz z z

Code rate = 3/4


Consider a MISO tranmission dual-polarized system ( ). The system model is 1tN


(1)

( )tN

H

H

( ) ( )

( ) 11 12( ) ( )

21 22

m mm

m m

h hH

h h

where

Rayleigh Fading Channel

(zero-mean and unit variance)



(1)

( )tN

H

H

( ) ( )

( ) 11 12( ) ( )

21 22

m mm

m m

h hH

h h

11 12

1 2t tN N

n n

n n

where



White noise (Gaussian RVs iid zero mean

and identical variance)



(1)

( )tN

H

H

( ) ( )

( ) 11 12( ) ( )

21 22

m mm

m m

h hH

h h

11 12

1 2t tN N

n n

n n

where



White noise (Gaussian RVs iid zero mean

and identical variance)


The complex matrix is obtained by decomposing a QOD with odd columns representing symbols transmitted through one polarization while even columns contain symbols transmitted through orthogonal polarization.

qC


Low Complexity Decoder:

21 (1) (2) (3)min ( )u

qzR C C H

(1) ,Qtr R R (2) 12Re ( )Qqtr R C C H

(2) 1 1( ( )) ( )Qq qtr C C H C C H

The ML-decoding rule for both constructions given in Theorem 1 and Lemma 1, simplifies to


Low Complexity Decoder:

Lemma 2:

21 (1) (2) (3)min ( )u

qzR C C H

(1) ,Qtr R R (2) 12Re ( )Qqtr R C C H

(2) 1 1( ( )) ( )Qq qtr C C H C C H

21 (2) 1min ( ) 2Re ( )u

Qq qz

R C C H tr R C C H


Example: 1 3 2 3 1 2* * * *

2 1 3 2 3 1* * * *

3 1 2 1 3 2* * * *

2 3 1 2 1 3



Code rate = 3/4


Example: 1 3 2 3 1 2* * * *

2 1 3 2 3 1* * * *

3 1 2 1 3 2* * * *

2 3 1 2 1 3



Decoupled Decoder: Code rate = 3/4

* *1 1 12 2 1 34 3 1 56 4 1 78

* *1 2 34 2 2 12 3 1 78 4 2 56

* *1 3 56 2 3 78 3 3 12 4 3 34

min ( 2Re{ }),

min ( 2Re{ }),

min ( 2Re{ }),

u

u

u

Q Q Q Qz

Q Q Q Qz

Q Q Q Qz

r z g r z g r z g r z g




Example: 1 3 2 3 1 2* * * *

2 1 3 2 3 1* * * *

3 1 2 1 3 2* * * *

2 3 1 2 1 3



Decoupled Decoder:

where such that mn m ng g g j (1) (3) (1) (3) (2) (4) (2) (4)

1 11 21 2 12 22 3 11 21 4 12 22, , , ,g h h g h h g h h g h h (1) (3) (1) (3) (2) (4) (2) (4)

5 21 11 6 22 12 7 21 11 6 22 12, , , .g h h g h h g h h g h h

Code rate = 3/4

* *1 1 12 2 1 34 3 1 56 4 1 78

* *1 2 34 2 2 12 3 1 78 4 2 56

* *1 3 56 2 3 78 3 3 12 4 3 34

min ( 2Re{ }),

min ( 2Re{ }),

min ( 2Re{ }),

u

u

u

Q Q Q Qz

Q Q Q Qz

Q Q Q Qz





Performance Analysis: Design 1 Theorem 1Design 2 Lemma 1

Thought To Take Away !!!

Thought To Take Away !!!

Have coeffee first and questions later ....

Conclusion

• Designs based on quaternions provide a feasible solution for dual-polarized antennas and easy to generate

• QODs exploit polarization diversity along with other diversities

• Decoupled decoding becomes an inherited characteristic of the approach

• Simulation results also confirm a performance up gradation in MIMOs against standard complex orthogonal or quasi-orthogonal codes which have other shortcomings

quaternion orthogonal designssali.seecs.nust.edu.pk/wp-content/uploads/2014/06/talk.pdf ·...

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