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Relation between Boolean functions and spreading sequences for MC-CDMA transmission

Ekaterina Pogossova, Karen Egiazarian, Jaakko Astola Tampere University of Technology, Tampere, Finland

Outline

Multicarrier CDMA (MC-CDMA) and a major

problem with it

Haar wavelet packets Binary-valued spreading transforms and

Boolean logic

Relation to the bent functions Chrestenson transform and complex-valued

counterparts of Haar wavelet packets

Multiple-valued spreading transforms and

multiple-valued logic

What is MC-CDMA(1)

MC-CDMA is a radio access technique,

which will potentially play role in 4G mobile telephony

MC-CDMA = OFDM + CDMA

OFDM - Orthogonal Frequency Division Multiplexing CDMA - Code Division Multiple Access

What is MC-CDMA(2)

What is MC-CDMA(2)

In MC-CDMA systems, the symbol is multiplied (spread) by

user specific spreading sequence, and converted into a parallel data stream, which is then transmitted over multiple carriers

Major problem – peak-to-average power ratio (PAPR)

signal at the transmitter

• utput

(k,l )-th element

• f the spreading

matrix element of the Fourier matrix period

• f signal s (t)

Problem: how to minimize PAPR

• f the resulting transmitted signal

Haar wavelet packets(1)

Walsh-Hadamard (WH) orthogonal matrix: We can represent the way of constructing this matrix in a binary tree: At the terminal nodes we have the rows of WH matrix

Haar wavelet packets(2)

We can take any pruned subtree of such a tree and combine the rows, corresponding to the terminal nodes of the resulting tree. As a result, we will have a new (N x N) orthogonal matrix,where Special case – Haar matrix – corresponds to Figure (c).

• n

N 2 =

New orthogonal spreading transforms

Consider the following recursive diagonal matrix: Let HN be any (N x N) Haar wavelet packet (HWP) decomposition orthogonal matrix, and let us define GN as Spreading with the new orthogonal matrix GN results in a much lower PAPR

• f the resulting transmitted signal, than spreading with any HWP matrix HN.

N N N

D H G ⋅ =

Boolean functions and spreading matrices(1)

) ( , ), ( ), (

2 1

x f x f x f

n

• )

( log

2 N

n =

. , , 2 , 1 , ) ( n i x x f

i i

• =

=

Let be all Boolean functions of variables of the form: The corresponding truth vectors are the first order Reed-Muller codes. Example: N=8. In {+1, -1} – encoding the truth vectors are

[ ]

1 1 1 1 1 1 1 1

1

− − − − = f

• [

]

1 1 1 1 1 1 1 1

2

− − − − = f

• [

]

1 1 1 1 1 1 1 1

3

− − − − = f

• !

we denote the following diagonal matrices: n k i k f k k D k k D

i i N i N

, , 2 , 1 , ), ( ) , ( ) , ( ~

) ( ) (

• =

⋅ =

n i D

i N

• ,

2 , 1 , ~

) (

=

Boolean functions and spreading matrices(2)

The set of n new spreading matrices defined by result in an equally low PAPR, as GN.

• Element-by-element multiplying themain diagonal of matrix DN with the truth vector of

a BF of the form in {+1,-1} – encoding, we will get a new diagonal matrix, which retains the feature of reducing PAPR.

• In this way, for each HWP matrix HN, we obtain n+1 new sets of orthogonal spreading

sequences, which differ from HN only by the signs of its columns, but have significantly better PAPR properties.

• Note: In the particular case, when HN is the Walsh-Hadamard matrix (each symbol

goes to each subcarrier), are identical to the GN up to the row permutations.

) ( ) (

~ ~

i N N i N

D H G ⋅ =

) (

~

i N

G

n i G

i N

, , 2 , 1 , ~

) (

• =

Relation to the bent functions

• A Boolean function of n variables is called bent, if its Walsh transform

coefficients are all of equal magnitude ( )

• The corresponding truth vector is a bent sequence.
• If (n ie even), then as well as

have bent sequences on the main diagonal.

• As a matter of fact, for any bent sequence ,

will give an orthogonal spreading matrix, providing lower PAPR, than HN for the resulting signal.

2

2

n k

N 4 =

N

D n i D

i N

• ,

2 , 1 , ~

) (

=

f

• }

{ f diag H N

• ⋅

2 4 6 8 10 12 14 16 4 6 8 10 12 14 16

Sequence number PAPR[dB]

HN GN HN × diag{f}

Experimental results

2 4 6 8 10 12 14 16 5 6 7 8 9 10 11 12

Sequence number PAPR[dB]

HN GN HN × diag{f}

HN is Walsh-Hadamard matrix HN is some other HWP matrix

If HN is the Walsh matrix, GN just slightly outperforms

} { f diag H N

• ⋅

If HN is other HWP matrix, GN performs significantly better

Advantages of our approach

We provide a constructive way of spreading matrix

generation, which will necessarily provide low PAPR

It is valid for both odd and even n, while bent functions

are defined for even n only

It gives good results for any HWP matrix HN Our approach is naturally generalized to complex-valued

spreading, and multiple-valued logic.

Complex-valued generalizations(1)

• Complex-valued counterpart of the Walsh transform –

Chrestenson transform. The entries of the transform matrix are complex numbers, taking p - th roots of unity

∑

− = − −

= = + +

1 1 ) , ( 2 ,

) , ( , ) 1 , 1 (

n nn k j C p i p N

k j k j C e k j H

η η η π

; 1 }; 1 , , 1 , { , ; ≥ − ∈ = n p k j p N

n n

• )

, , , (

2 1 n

j j j

• )

, , , (

2 1 n

k k k

• where

and are p-ary expansions of j and k respectively

Complex-valued generalizations(2)

• The decomposition tree is p – ary in this case
• Wavelet packet approach allows to take any of its pruned subtrees

to form the orthogonal basis (a) - complete decomposition (b) - some other decomposition (c) - complex-valued Haar (generalized Haar) decomposition

Complex-valued spreading and multiple-valued logic(1)

Complex-valued spreading matrix:

p N p N p N

D H G

, , ,

⋅ =

p N

H

,

• matrix,resulting from any wavelet-packet type decomposition

between Chrestenson and complex-valued Haar

p p

D ,

• (p x p) identity matrix

p

γ

• p-th root of unity,

} 1 , , 2 , 1 { , 2 − ∈ = p k p ik e

p

• π

γ

k, p are relatively prime

Complex-valued spreading and multiple-valued logic(2)

• By taking the rows of Chrestenson matrix, corresponding to the

multiple-valued (p -valued) logic functions of the form

i

f

• },

) 1 ( , , 2 , 1 { }, 1 , , 2 , 1 { , ) ( n p i p c cx x f

k i

− ∈ − ∈ =

• we define new diagonal matrices

), ( log }, , , 2 , 1 { N n n k

p

= ∈

• ),

( ) , ( ) , ( ~

, ) ( ,

k f k k D k k D

i p N i p N

• ⋅

=

} , , 2 , 1 { }, ) 1 ( , , 2 , 1 { N k n p i

• ∈

− ∈ The new set of spreading matrices will be:

} ) 1 ( , , 2 , 1 { , ~ ~

) ( , , ) ( ,

n p i D H G

i p N p N i p N

− ∈ ⋅ =

• All described matrices

) ( ,

~ i

p N

G

p N

G

,

as well as , constructed earlier, will result in a low PAPR of the resulting transmitted signal

Relation to the complex-valued bent functions

• By complex-valued bent function we call the function, whose

Chrestenson transform coefficients are all of equal magnitude

Only a sequence of length can be complex-valued bent If , then the diagonals of and are complex-

valued bent sequences.

As in the binary case, for any complex-valued bent sequence ,

spreading the matrix will result in much lower PAPR of the transmitted signal, than spreading with

The PAPR reduction is not always that good as in the case of

spreading with or

f

• }

{

,

f diag H

p N

• ⋅

p N

H

,

p N

G

, ) ( ,

~ i

p N

G

k

p N

2

=

k

p N

2

=

p N

D

, ) ( ,

~ i

p N

D

1 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 10 11 12 13

Sequence number PAPR[dB] HN,p GN,p HN,p× diag{f} p=3, n=2

Experimental results

1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14

Sequence number PAPR[dB] HN,p GN,p HN,p × diag{f} p=3, n=2

HN,p is Chrestenson matrix

HN,p is complex-valued Haar matrix

If HN,p is the Chrestenson matrix, GN,p just slightly outperforms

} {

,

f diag H

p N

• ⋅

If HN ,p is complex-valued Haar matrix, GN,p performs significantly better

Conclusion

The problem of spreading code design for MC-CDMA

transmission has been explored from the perspective of Boolean and multiple-valued logic

As a result, an interesting relationship between well

known orthogonal transforms, Rademacher matrices,bent functions has been established

• A family of previously unknown orthogonal transforms

has been derived, which can be attractive for MC-CDMA system