MIMO Systems

An Efficient QR Decomposition Design for MIMO Systems

1 Jing-Shiun Lin, ² Yin-Tsung Hwang, ¹ Po-Han Chu,

1 Ming-Der Shieh, and ¹ Shih-Hao Fang

1 Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan

2 Department of Electrical Engineering, National Chung Hsing University,

Taichung, Taiwan

Contents

Introduction

Related Works

Experimental Results Conclusions

Proposed Method

Introduction (1/2)

 MIMO system

 N _t transmit antennas and N _r receive antennas

MIMO Detector Input data

Nt

x

x

1

n

¹

Nr

n y

1

Nr

y

Estimated Data

ˆx

1

Nt

xˆ h

11

h

12

Transmitter Receiver

Nt

h

₁

1

N

r

1 1 1

r r t t r

N   N  N N   N 

y H x n

ˆ _ML arg min



 

x

x y Hx

Ω is a possible signal set

Introduction (2/3)

 QR decomposition

Q is an unitary matrix

R is an upper-triangle matrix

Q H

Q ^{ 1} 

r t r t t t

N  N  N  N N  N

H Q R

11 12 1

21 22 2

1 2

t

t

r r r t

N N

N N N N

q q q

q q q

q q q

 

 

 

  

 

 

 

Q

11 12 1

22 2

0

0 0

t

t

t t

N N

N N

r r r

r r

r

 

 

 

  

 

 

 

R

Introduction (3/3)

 A MIMO detector solution

2 2

1 1 1

ˆ arg min arg min ' ,

t t t t

Nt Nt

N N N N

i ij j i ij j

x i j x i j i

H

x y h x y r x

where

     

   



   

y' Q y

5

Ω

Ω 1

1

1 Ω

Layer

Ω

1 1 Ω

N t

N t

Layer N _t  1

Layer 1

    _



 

 ^N

^t

i j

j ij i

i

i x y r x

e

    1   ^{  1}   ^{  2}

i i

i i

i

i x T x e x

T  _ ^ 

Partial Euclidean Distance (PED)

PED Accumulation

ISCAS 2012 - 05/22

Related Methods

 Gram-Schmidt method

 Use projection principle to construct a new basis

 Need extra operation to compute Q ^H y

 Low latency and low throughput

 Householder reflection

 Find a orthogonal matrix to reflect a column

vector onto a multiple of a standard basis vector

 Preserve vector norm

Related Methods

 Givens rotation

 2-dimension rotation method

 High computation parallelism

 CORDIC implementation

• Low complexity and high latency

• Pipeline & high throughput

   

    ^ _ ^

 



 









cos sin

sin G cos

   

   

   

    _ ^

 







 

 









 

 





 

 





 

 





 

 









2 44 2

43 2

42

2 34 2

33 2

32 2

31

1 24 1

23 1

22

1 14 1

13 1

12 1

11

44 43

42 41

34 33

32 31

24 23

22 21

14 13

12 11

2 2

2 2

1 1

1 1

0 0

cos sin

0 0

sin cos

0 0

0 0

cos sin

0 0

sin cos

h h

h

h h

h h

h h

h

h h

h h

h h

h h

h h

h h

h h

h h

h h

h h

















CORDIC

 CORDIC (COordinate Rotation DIgital Computer)

 Decompose the rotation angle  into predefined elementary angles.(iterative)

 

 



 



 





 

 

 









k k k

k

k k

k k

y x y

x









cos sin

sin cos

1 1

 

 



 



 









 





 



 









_







0 1 0

0 1

0

tan 1

tan cos 1

y x y

x

k k

k n k

k k n

k

 



 

Let tan(α _k )=2 ^-k , and we give n an iteration number and  _k = ±1, a direction index.

Rotation operation:

Multi-rotation

 

 



 



 









 



 



 







 



0 1 0

0

2 1

2 1

y K x

y x

k k

k k n

k n

n





K is a constant value.

(x⁽⁰⁾, y⁽⁰⁾) (x⁽¹⁾, y⁽¹⁾)

(x⁽²⁾, y⁽²⁾) (x⁽³⁾, y⁽³⁾)

(x^(m), y^(m))

45^◦ -26.6^◦ 14^◦

30°

Block-wise Symmetric

 Real-value decomposition

 Symmetric process

       

     

   

  ^ _ ^

 

 

 

 



 



 



 

 



 





n n x

x H

H

H H

y y

Im Re Im

Re Re

Im

Im Re

Im Re

Re(H) -Im(H)

Im(H) Re(H)

Re(y)

Im(y)

0

y 

R 

  ^y

Re

  ^y

Im

  R Re

  R

  R Re Im

  ^R

 Im 0

0

0

Symmetric process 0

Traditional process

?

Modified Real-valued GR Scheme

( ) ( )

[Re(

^k

); Im(

^k

)]



H H H

For c ₁ = 1 : N For i ₁ = k : N

1 1 1 1

1

, ,

tan ( h

_{i N c}

h

_{i c}

)

 

^ _

 ^{H y |}  ^{  ( , i i}

^{1 1}

^{ N , )  }  ^{H y |} 

End

2 1 2 1

1

1, ,

tan ( h

_{i c}

h

_{i c}

)

 

^ _

 ^{H y |}  ^{  ( , i i}

^{2 2}

^{ 1, )  }  ^{H y |} 

 H y |     ( i

2

N i ,

2

  N 1, )    H y | 

If(c ₁ < N)

For i ₂ = N-1 : c ₁

end end end

[Re( ), Im( );Im( ), Re( )]

 

R H H H H

For c ₂ = 1 : N

   



2 2



2 2

:,1 2 1 : 2 2 1

[ (:, ), (:, 0.5 )]

c c

c c N

    

 

R

R R

   



2 2



2 2

1 2 1 : 2 2 1 ,:

[ ( ,:), ( 0.5 ,:)]

c c

c c N

   

 

 

R

R R

   



2 2



2 2

ˆ 1 2 1 : 2 2 1 ,:

[ ( ,:), ( 0.5 ,:)]

c c

c c N

   

 

Y

y y

end

12 13 14 1

22 23 24 2

32 33 34 3

42 43 44 4

52 53 54 5

62 63

11

2

64 6

72 73 74 7

82 83 84 8

31

41

7 1

6

1

1 1

8 51

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0

0 0

0

0 0

0

0 0

0 0 0

0 0 0

0

0 0

0

h h h y

h h h y

h h h y

h h h y

h h h y

h h h

h h

h

y

h h h y

h h h y

h h h h

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11 12 13 14 1

21 22 23 24 2

31 32 33 34 3

41 42 43 44 4

52 53 54 5

62 63 64 6

72 73 74 7

82 83 84 8

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

0 0 0 0

h h h h y

h h h h y

h h h h y

h h h h y

h h h y

h h h y

h h h y

h h h y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G(1,5,1) G(2,6,1) G(3,7,1) G(4,8,1)

R(1,5,2) R(2,6,2) R(3,7,2) R(4,8,2)

R(1,5,3) R(2,6,3) R(3,7,3) R(4,8,3)

R(1,5,4) R(2,6,4) R(3,7,4) R(4,8,4)

R(1,5,y) R(2,6,y) R(3,7,y) R(4,8,y)

Example for N _t =N _r =4

Matrix form Computation schedule

G(i,j,k) is angle generation based on matrix element (i,k) and (j,k)

R(i,j,k) is vector rotation based on matrix elements (i,k) and (j,k)

G(1,2,1) G(3,4,1) R(1,2,2) R(3,4,2)

R(7,8,2) R(5,6,2) R(1,2,3) R(3,4,3)

R(7,8,3) R(5,6,3) R(1,2,4) R(3,4,4)

R(7,8,4) R(5,6,4) R(1,2,y) R(3,4,y)

R(7,8,y) R(5,6,y)

11

21

31

12 13 14 1

22 23 24 2

32 33 34 3

42 43 44 4

53 54 5

63 64

52

62 6

73 74 7

83 84 8

41

72

82

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1

1 1 1

1 1 1

1 1

1 1 1 1

1

1 1

1 1

0 0 0 0

h h h y

h h h y

h h h y

h h h y

h h y

h h y

h h y

h h y

h

h h h

h h

h

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11 12 13 14 1

22 23 24 2

31 32 33 34 3

42 43 44 4

52 53 54 5

82 63 64 6

72 73 74 7

82 83 84 8

2 2 2 2 2

2 2 2 2

2 2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

0

0 0 0 0 0

h h h h y

h h h y

h h h h y

h h h y

h h h y

h h h y

h h h y

h h h y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example for N _t =N _r =4

Matrix form Computation schedule

12 13 14 1

22 23 24 2

32 33 34 3

42 43 44 4

53 54 5

82 63 64 6

52

73 74 7

82 11

3

83 8

7 1

8 2

4

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2

2 2 2 2

2 2 2

2 2

2 2

2 2

2

2

0

0 0 0 0 0

h h h y

h h h y

h h h y

h h h y

h h y

h h h y

h h y

h h h

h

h

y h

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12 13 14 1

22 23 24 2

32 33 34 3

42 43 44 4

53 54 5

82 63 64 6

73 74 7

82 83 84 8

7 1

52

2 1

3 3 3 3

2 2 2 2

3 3 3 3

2 2 2 2

3 3 3

2 2 2 2

3 3 3

2 3

2 3

2 3

2

0 0

0 0 0 0 0

h h h y

h h h y

h h h y

h h h y

h h

h

h

y

h h h y

h h y

h h h

h

y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G(1,3,1) R(1,3,2)

R(5,7,2) R(1,3,3)

R(5,7,3) R(1,3,4)

R(5,7,4) R(1,3,y)

R(5,7,y)

Example for N _t =N _r =4

Matrix form Computation schedule

3

11 12 13 14 1

23 24 2

33 34 3

43 44 4

52 53 54 5

63 6

42

4 6

73 74 7

22

82 2

72

2 84 8

8 83

3 3 3 3 3

2 2 2

3 3 3

2 2 2

3 3 3

2

3

2 2 2

3 3 3

2 2

2 2

2 3 2

3

0 0 0 0 0 0 0

h h h h y

h h y

h h y

h h y

h h

h

h y

h h y

h h y

h h y

h

h h

h

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11 12 13 14 1

23 24 2

33 34 3

43 44 4

52 53 54 5

63 64 6

73 74 7

22

42

83 84 8

32

3 3 3 3 3

3 3 3

4 4 4

3 3 3

3 3 3

4

3

3 3 3

4 2

4 4

3 3 3

0 3

0 0 0

0 0 0 0 0 0

h h h h y

h h y

h h y

h h y

h h h y

h h y

h h

h y

h h y

h

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G(2,6,2) G(4,8,2) G(3,7,2)

R(4,8,3) R(3,7,3)

R(4,8,4) R(3,7,4)

R(4,8,y) R(3,7,y)

R(2,6,4) R(2,6,y) R(2,6,3)

Example for N _t =N _r =4

Matrix form Computation schedule

11 12 13 14 1

22 23 24 2

33 34 3

43 44 4

52 53 54 5

63 64 6

73 7

8

4 7

8 3

32

4 8

3 3 3 3 3

3 3 3 3

5 5 5

4 4 4

3 3 3 3

3 3 3

5 5

4 5

5

4 4

0 0 0 0

0 0 0 0 0 0 0

h h h h y

h h h y

h h y

h h y

h h h y

h h y

h h h

h

y y h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11 12 13 14 1

22 23 24

32

73

83

2

33 34 3

43 44 4

52 53 54 5

63 64 6

74 7

84 2

8 4

3 3 3 3 3

3 3 3 3

4 4 4

3 3 3

3 3 3 3

3 3 3

4 4

3 2

4 4

3 3

0 0 0 0

0 0 0 0 0 0

h h h h y

h h h y

h h y

h h y

h h h y

h h y

h h h

h y

y h

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G(3,4,2) R(3,4,3)

R(7,8,3) R(3,4,4)

R(7,8,4) R(3,4,y)

R(7,8,y)

Example for N _t =N _r =4

Matrix form Computation schedule

11 12 13 14 1

23 24 2

33 34 3

43 44 4

52 53 54 5

64 6

74 7

83 84 8

63

7 2

3 2

3 3 3 3 3

4 4 4

6 6 6

4 4 4

3 3 3 3

4 6

4 4

6 6

4 4 4

0 4

0

0 0 0

0 0 0 0

0

0

0

h h h h y

h h y

h h y

h h y

h h h y

h y

h y

h h

h

y h

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G(2,3,2) R(2,3,3)

R(6,7,3) R(2,3,4)

R(6,7,4) R(2,3,y)

R(6,7,y)

11 12 13 14 1

23 24 2

33 34 3

43 44 4

52 53 54 5

64 6

74 7

83 84 8

2

7 3

6

3 22

3

3 3 3 3 3

3 3 3

5 5 5

4 4 4

3 3 3 3

3 3

5 5

4 4

3 5

4 3

5

0 0

0 0 0

0 0 0 0 0 0

h h h h y

h h y

h h y

h h y

h h h y

h

h y

h y

h h

h y

h h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example for N _t =N _r =4

Matrix form Computation schedule

11 12 13 14 1

22 23 24 2

34 3

44 4

52 53 54 5

63 64 6

33

74 7

84 8

43

3 3 3 3 3

4 4 4 4

7 7

5 5

3 3 3 3

4 4

5

7 5 7

4 7 5

0

0 0 0 0 0

0 0 0 0

0 0 0

0

h h h h y

h h h y

h y

h y

h h h y

h h y

h h

h

y

h y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G(4,8,3) G(3,7,3)

R(4,8,4) R(3,7,4)

R(4,8,y) R(3,7,y)

11 12 13 14 1

22 23 24 2

34 3

44 4

52 53 54

33

73

5

63 64 6

74 7

8 43

84 83

3 3 3 3 3

4 4 4 4

6 6

4 4

3 3 3 3

4 4

4

4

6 6

4 4

6

6 4

0

0 0 0 0 0

0 0 0 0 0 0

h

h

h h h h y

h h h y

h y

h y

h h h y

h h y

h h

h h y

y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example for N _t =N _r =4

Matrix form Computation schedule

11 12 13 14 1

22 23 24 2

34 3

44 4

52 53 54 5

63 64 6

33

7

8 3

84 4

74

3 3 3 3 3

4 4 4 4

6 6

4 4

3 3 3 3

4 4 4

6 4

4

4 6

6

0

0 0 0 0 0

0 0

0 0 0

0 0 0

h h h h y

h h h y

h y

h

h y

h h h y

h h y

y y h

h

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11 12 13 14 1

22 23 24 2

34 3

44 4

52 53 54 5

63 64 6

74 7

8 33

4 8

3 3 3 3 3

4 4 4 4

7 7

5 5

3 3 3 3

4 4 4

7

7 7

5 5

0

0 0 0 0 0

0 0

0 0 0

0

0

0 0

h h h h y

h h h y

h y

h y

h h h y

h h y

h y

h y

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G(3,4,3) R(7,8,4) R(3,4,4)

R(7,8,y) R(3,4,y)

Example for N _t =N _r =4

Matrix form Computation schedule

11 12 13 14 1

22 23 24

44

8

2

33 34 3

4

52 53 54 5

63 64 6

74 7

8 4

3 3 3 3 3

4 4 4 4

7 7 7

5

3 3 3 3

4 4 4

5

7 7

5 5

0

0 0

0 0 0

0

0 0

0 0 0

0 0 0

h h h h y

h h h y

h h y

y

h h h y

h h y

h y

y h

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11 12 13 14 1

22 23 24 2

33 34 3

4

52 53 54 5

63 64 6

74 7

8 44

3 3 3 3 3

4 4 4 4

7 7 7

6

3 3 3 3

4 4 4

7 7

6

6

0

0 0

0 0 0

0

0 0

0 0 0

0 0 0 0

h h h h y

h h h y

h h y

y

h h h y

h h y

h y

y h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G(4,8,4)

R(4,8,y)

Example for N _t =N _r =4

Matrix form Computation schedule

Example for N _t =N _r =4

 Block-wise symmetric from

11 12 13 14 52 53 54 1

22 23 24 63 64 2

33 34 74 3

44 4

52 53 54 11 12 13 14 5

63 64 22 23 24 6

74 33 34 7

44 8

3 3 3 3 3 3 3 3

4 4 4 4 4 4

7 7 7 7

6 6

3 3 3 3 3 3 3 1

4 4 4 4 4 4

7 7 7 7

6 6

0

0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

0

0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

h h h h h h h y

h h h h h y

h h h y

h y

h h h h h h h y

h h h h h y

h h h y

h y

  

 



Re{H}

Im{H}

Re{H}

-Im{H}

  H

y Q y

Example for N _t =N _r =4

 Interleaved operation for column vectors

52 53 54

63 64

74

1

11 12 13 14 1

22 23 24 2

33 34 3

44 4

52 53 54 5

6

1 12 13 14

22 23 24

3

3 64 6

74 7

3 34

44 8

3 3 3 3 3

5 5 5 5

7 7 7

7 7

1

1 1 1

2 2

3

3 3 1 3 1 1

2 2 2

3 3

4 3

5 5 5

7 7

7

0

0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

0

0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

h h h h y

h h h y

h h y

h y

h h

h h h

h h

h

h h h h

h h

h y

h h y

h y

y h

h h

h

  



 

 





 

 

 

 

 

 

 

  

 

 

 



11 12 13 14 1

22 23 24 2

33 34 3

44 4

5

52 53 54

63 64

74

1

2 53 54 5

63 64 6

7

1 12 13 14

22 23 24

33 34

4

4 7

4 8

3 3 3

4 4

7

3 3 3 3

4 4 4

7

3 3 3 3 3

4 4 4 4

7 7 7

6 6

3 3 3 3

4 4 4

7

6

7 7

6

0

0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

0

0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

h h h

h h

h

h h h

h h h h y

h h h y

h h y

h y

h h h h y

h h h

h

h y

h

y h

h

y h

  

 



 

 

 

 





 

 

 

 

  

 

 

 



11 52 12 53 13 54 14 5

22 63 23 64 24 6

33 7

11 12 52 13 53 14 54 1

22 23 63 24 64 2

33 34 74 3

44 4

4 34 7

44 8

3 3 3 3 3 3 3 3

4 4 4 4 4 4

7 7

3 3 3 3 3 3 3 3

4 4 4 4 4 5

7 7

7 7 7

6 6

6 6

7

0

0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

0

0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

h h h h h h h y

h h h h

h h h h h h h y

h h h h h y

h y

h h h y

h y

h h h y

h y

   

  

  

 

 





 

 

 

 

  

 

 

 



Example for N _t =N _r =4

 Interleaved operation for row vectors

11 52 12 53 13 54 14 5

22 63 23 64 24 6

33 74 3

11 12 52 13 53 14 54 1

22 23 63 24 64 2

33 34

4

74 3

44 4

7

44 8

3 3 3 3 3 3 3 3

4 4 4 4 4 4

7 7 7 7

3 3 3 3 3 3 3 3

4 4 4 4 4 4

7 7 7 7

6 6

6 6

0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

h h h h h h h y

h h h h

h h h h h h h y

h h

h y

h h h y

h

h h h y

h h h y

h y

y

   

 

  

 

  





 

 

 

 

  

 

 

 



CORDIC Implementation

 Generation Processing Element (GPE)

 Rotation Processing Element (RPE)

2*adder, 2*barrel shifter, others

M u x R eg

A d d /S u b A d d /S u b

>>n

GPE

>>n

R eg R eg

M u x

MSB

MSB

Y

ⁱ⁺¹

X

ⁱ⁺¹

X

⁰

Y

⁰

X

^l

Y

^l

R

d

M u x R eg

A d d /S u b A d d /S u b

>>n

X

^l

Y

^l

R

_d

RPE

>>n

R eg R eg

X

⁰

Y

⁰

M u x

R

_d

Y

ⁱ⁺¹

X

ⁱ⁺¹

Low-complexity CORDIC Array (1/2)

 Mapping direction for 8 parallel CORDIC modules

 Pipeline structure

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

PE PE PE PE

PE PE PE PE

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

Add

Add

Add

Add

...

Add

Add Add

Add

Shift

Shift

Traditional

Proposed

Mapping direction

CORDIC Array

Low-complexity CORDIC Array (2/2)

 Synthesis results

 TSMC 0.18 m, slow library, 120 MHz

Components Gate Count CORDIC Array ¹ 8*2 adder+ others 4.3~4.6K 8*Traditional PE 8*(2 adder+2 barrel

shifter+ others)

8*1.16

=9.28K

1 Excluding constant multipliers

Experimental Results

Designs Salmela[7] Burg[8] Hwang[9] Tasi[10] Proposed Algorithm Gram-Schmit Givens Givens Givens Givens

CMOS Process 0.13 m 0.18 m 0.18 m 0.18 m 0.18 m

Processing Cycle 139 cycles 67 cycles 8 cycles 4 cycles 8 cycles

Logical Gates 23.2K 54K 134.6K 111K 103.7 K

Operating

Frequency 269 MHz 125 MHz 120 MHz 100 MHz 200 MHz

N.T. 1.398 2.591 15 25 25

A/N.T. 16.595 20.841 8.974 4.44 3.27

m 0.18 Technology cycle

Processing

frequency Operating

rate) QR d (Normalize

N.T. ^{ } 

Conclusion

 The proposed design employs both pipelined and folded CORDIC structures to reduce the hardware complexity.

 Implementation results show that the proposed design, with a gate count of 103.7K, can offer 25M CQRFs per second in 4×4 MIMO systems.

 Compared to other QR factorization works, the

proposed design excels in terms of the product

of area and time.