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) period of signal s ( t ) ( k,l ) - th element element of the signal at the of the spreading Fourier matrix transmitter matrix output Problem: how to minimize PAPR of 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. ��� ��� ��� n N 2 As a result, we will have a new ( N x N ) orthogonal matrix,where = Special case – Haar matrix – corresponds to Figure (c).
New orthogonal spreading transforms Consider the following recursive diagonal matrix: Let H N be any ( N x N ) Haar wavelet packet (HWP) decomposition orthogonal matrix, and let us define G N as G = H ⋅ D N N N Spreading with the new orthogonal matrix G N results in a much lower PAPR of the resulting transmitted signal, than spreading with any HWP matrix H N .
Boolean functions and spreading matrices(1) � f ( x ), f ( x ), , f ( x ) n = log 2 N ( ) be all Boolean functions of Let 1 2 n � f ( x ) x , i 1 , 2 , , n . = = variables of the form: i i are the first order Reed-Muller codes. The corresponding truth vectors Example: N =8 . In {+1, -1} – encoding the truth vectors are � [ ] f 1 1 1 1 1 1 1 1 = − − − − 1 � � [ ] [ ] f 1 1 1 1 1 1 1 1 f = 1 1 − 1 − 1 1 1 − 1 − 1 = − − − − 3 2 ~ !� ( i ) � we denote the following diagonal matrices: D , i = 1 , 2 , n N � ~ ( i ) ( i ) � D ( k , k ) = D ( k , k ) ⋅ f ( k ), i , k = 1 , 2 , , n N N i
Boolean functions and spreading matrices(2) ~ ( i ) � The set of n new spreading matrices defined by G , i = 1 , 2 , , n N ~ ~ ( i ) ( i ) G H D = ⋅ N N N result in an equally low PAPR, as G N . Element-by-element multiplying themain diagonal of matrix D N 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 H N , we obtain n +1 new sets of orthogonal spreading � sequences, which differ from H N only by the signs of its columns, but have significantly better PAPR properties. Note: In the particular case, when H N is the Walsh-Hadamard matrix (each symbol � goes to each subcarrier), ~ ��� ( i ) are identical to the G N up to the row permutations. G N
Relation to the bent functions A Boolean function of n variables is called bent , if its Walsh transform � n coefficients are all of equal magnitude ( ) 2 2 The corresponding truth vector is a bent sequence . � ~ ( i ) k D � If ( n ie even), then as well as N = 4 have bent D , i = 1 , 2 , n � N N sequences on the main diagonal. � � f As a matter of fact, for any bent sequence , H N ⋅ diag { f } � will give an orthogonal spreading matrix, providing lower PAPR, than H N for the resulting signal.
Experimental results 16 12 H N H N G N G N 11 H N × diag{f} 14 H N × diag{f} 10 12 H N is some other HWP matrix PAPR[dB] PAPR[dB] 9 10 8 H N is Walsh-Hadamard matrix 8 7 6 6 4 5 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 Sequence number Sequence number � H N ⋅ diag { f } If H N is the Walsh matrix, G N just slightly outperforms If H N is other HWP matrix, G N 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 H N � 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 2 i π n − 1 C ( j , k ) ∑ p H ( j + 1 , k + 1 ) = e , C ( j , k ) = j k N , p η nn − 1 − η 0 η = n n � N = p ; j , k ∈ { 0 , 1 , , p − 1 }; n ≥ 1 ; where � � ( j , j , , j ) ( k , k , , k ) and are p- ary expansions of j and k respectively 1 2 n 1 2 n
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: G = H ⋅ D N , p N , p N , p H - matrix,resulting from any wavelet-packet type decomposition N , p between Chrestenson and complex-valued Haar D , - ( p x p ) identity matrix p p 2 π ik γ - p -th root of unity, � k, p are relatively prime γ = e , k ∈ { 1 , 2 , , p − 1 } p p p
Complex-valued spreading and multiple-valued logic(2) � f By taking the rows of Chrestenson matrix, corresponding to the � i multiple-valued ( p -valued) logic functions of the form � � f ( x ) = cx , c ∈ { 1 , 2 , , p − 1 }, i ∈ { 1 , 2 , , ( p − 1 ) n }, i k � k ∈ { 1 , 2 , , n }, n = log ( N ), p we define new diagonal matrices � ~ ( i ) � � D ( k , k ) = D ( k , k ) ⋅ f ( k ), i ∈ { 1 , 2 , , ( p − 1 ) n }, k ∈ { 1 , 2 , , N } N , p N , p i The new set of spreading matrices will be: ~ ~ ( i ) ( i ) � G = H ⋅ D , i ∈ { 1 , 2 , , ( p − 1 ) n } N , p N , p N , p ~ i ( ) as well as G All described matrices , constructed earlier, will result G N , p N , p 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 2 k � Only a sequence of length can be complex-valued bent N = p ~ i ( ) D 2 k � If , then the diagonals of and are complex- D N = p N , p N , p valued bent sequences. � f � As in the binary case, for any complex-valued bent sequence , � spreading the matrix will result in much lower H ⋅ diag { f } N , p PAPR of the transmitted signal, than spreading with H N , p � The PAPR reduction is not always that good as in the case of ~ i spreading with or ( ) G G N , p N , p
Experimental results 14 13 H N,p H N,p G N,p 12 G N,p H N,p × diag{f} H N,p × diag{f} 12 11 10 10 9 PAPR[dB] PAPR[dB] p=3, n=2 p=3, n=2 8 8 7 6 6 H N,p is Chrestenson matrix H N,p is complex-valued Haar matrix 5 4 4 2 3 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Sequence number Sequence number � If H N,p is the Chrestenson matrix, G N,p just slightly outperforms H ⋅ diag { f } N , p If H N ,p is complex-valued Haar matrix, G N,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
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