Patent Application: US-8902600-A

Abstract:
the invention concerns a method for transmitting a biorthogonal multicarrier signal bfdm / om , using a transmultiplexer structure providing : a modulating step , using a synthesis filter bank , having two 2m parallel branches , m ≧ 2 , each supplied by source data , and comprising an expander of order m and filtering means ; a demodulating step , using an analysis filter bank , having two 2m parallel branches , each comprising a decimation unit of order m and filtering means , and delivering received data representing said source data ; said filtering means being derived from a predetermined prototype modulating function .

Description:
as shown earlier , the technique of the invention is notably based on a special approach to discretization , aiming at directly obtaining a description of the modulated transmultiplexer type system . in addition to the advantage of a more general descriptive framework , this approach provides many possibilities for utilizing connections between the banks of filters and the transmultiplexers , for optimizing the realization structures and the computation of the associated coefficients . after having shown the general structure for representing bfdm / om systems , as a discrete model of the transmultiplexer type , four special embodiments of the invention are shown hereafter which respectively correspond to : two bfdm / om embodiments which , at the modulator and the demodulator , both use a fast reverse fourier transform algorithm ( ifft ) and differ by the type of implantation of the polyphase components of the prototype filter : mode 1 : ifft algorithm + transverse polyphase filtering ; mode 2 : ifft algorithm + ladder filtering . two embodiments adapted to ofdm / om , derived from bfdm / om : mode 3 : an alternative mode 1 verifying the discrete orthogonality of ofdm / om with transverse polyphase filtering and the possibility of implementing a symmetrical prototype filter or not ; mode 4 : and alternative mode 2 verifying the discrete orthogonality of ofdm / om with polyphase filtering achieved by a trellis structure . methods for designing prototype filters illustrating these methods for achieving bfdm / om and ofdm / om modulations , are also shown . the additional possibilities of bfdm / om , so that the transmission delay remains adjustable for a given prototype filter length . for example , performances in terms of time - frequency localization of the transformation associated to the modulator may be enhanced for an identical transmission delay . with this , high performances may also be maintained from the point of view of selectivity while reducing the transmission delay ; in the case of so - called back - to - back systems , the possibility with modes 2 and 4 of totally cancelling out the interference between symbols ( ies ) and the interference between channels ( iec ) and thereby obtaining what may also be called perfect reconstruction . other examples not reported here , also show that it is possible to obtain localization performances comparable to those of ofdm / om , biorthogonally , and this with much shorter prototype filters . to facilitate interpretation , the following notations are retained : the sets , for example r the real number field , as well as vectors and matrices , for example e ( z ) and r ( z ), the polyphase matrices , are marked in bold characters . otherwise , all the mathematical symbols used are marked in standard characters with generally the time functions in lower case , and the functions of transformed domains ( both z - and fourier - transformed ) are in upper case . starting with a causal prototype filter p [ k ], derived from h ( t ) by translation and discretization , we obtain a realization scheme which is the one of fig1 . the modulation part 13 comprises 2m branches 131 0 to 131 2m − 1 receiving source data a i , n . each source data is multiplied by ( 1311 ) to obtain x i ( n ), which feed an expander of order m 1312 , and then a synthesis filter f i [ z ] 11 . the outputs of the synthesis filters feed an adder 132 to form a signal s sent through a channel 14 . in the demodulation part 15 , the signal s feeds 2m branches 151 0 to 151 2m − 1 each comprising analysis filter h i [ z ] 12 , a decimator of order m 152 , a multiplication 153 by ⅇ j ⁢ π 2 ⁢ ( n - α ) in this scheme , filters f 1 [ z ] 11 and h i [ z ] 12 , with 0 ≦ i ≦ 2m − 1 , are derived from p [ k ] ( or p ( z )) by complex modulation α and β , 0 ≦ β ≦ m − 1 , are two integers which are related to a parameter d of the modulation d = αm − β . the calculations by means of which this scheme may be achieved , are reported in appendix b . it may also be noted , that the prototype filters may be different . subsequently , we will merely study the particular case when q [ k ]= p [ d − k ], without this limiting the scope of the patent application . the realization of a modulation and demodulation scheme directly according to this fig1 would be extremely costly , in terms of operative complexity . according to the approach of the invention , the prototype filters p ( z ), are therefore broken down into their polyphase components g 1 ( z ) as shown in appendix c . appendix c also specifies the input / output relationship , the conditions to be observed on the polyphase components and the construction delay . all the embodiments described later on are based on the implementation of a discrete fourier transform ( dft ). of course , this technique has the advantage that the dft is expressed by fast computation algorithms , designated by their acronyms fft , or ifft for the inverse transform . ( it shall be noted that the referenced equations ( 1 ) to ( 54 ) are found in appendices a to c ). the schemes of the modulator of fig5 and of the demodulator of fig6 are derived from this , both achieved by means of an inverse fourier transform ifft 51 , 61 . in these fig5 and 6 , s is an integer defined by d = 2 . s . m + d , d being an integer between 0 and 2m − 1 . data feeding each branch of the modulator of fig5 are multiplied ( 53 ) by 2 ⁢ m ⁢ 2 ⁢ c - j ⁢ 2 ⁢ π 2 ⁢ m ⁢ ⅈ ⁢ d - m 2 , and then transformed through ifft 51 . the outputs of the ifft feed polyphase components 52 ( see annex c ) and expander 1312 . the received signal s ( k ) is directed to 2m branches ( fig6 ), each comprising a decimator 152 and a polyphase component 62 , which feeds an ifft 61 . the 2m outputs of the ifft comprise a multiplier 63 of course , the notations and data appearing in fig5 and 6 , as well as in the other figures , are a full part of the present description . for the sake of simplification , but without any loss in generality , it is assumed hereafter that the prototype filter p ( z ) has a length of 2 mm so that all the polyphase components have the same length m . again using the notations of fig5 and 6 , the following modulation and demodulation algorithms already mentioned above are derived : ⁢ x m 0 ⁡ ( n ) = a m , n ⁢ ⅇ j ⁢ π 2 ⁢ n ⁢ ⁢ x l 1 ⁡ ( n ) = 2 ⁢ ∑ λ = 0 2 ⁢ m - 1 ⁢ ⁢ x λ 0 ⁡ ( n ) ⁢ ⅇ l ⁢ 2 ⁢ π 2 ⁢ m ⁢ λ ⁢ d - m 2 ⁢ ⅇ j ⁢ 2 ⁢ π 2 ⁢ m ⁢ kl ⁢ ( 61 ) = 2 ⁢ m ⁢ 2 ⁢ γϝϝτ ⁡ ( x 0 0 ⁡ ( n ) , … ⁢ , x 2 ⁢ m - 1 0 ⁡ ( n ) ⁢ ⅇ - j ⁢ 2 ⁢ π 2 ⁢ m ⁢ ( 2 ⁢ m - 1 ) ⁢ d - m 2 ) ⁢ ( 62 ) ( 60 ) ⁢ x l 2 ⁢ ⁢ ( n ) = ∑ λ = 0 m - 1 ⁢ ⁢ p ⁡ ( l + 2 ⁢ km ) ⁢ x λ 1 ⁡ ( n - 2 ⁢ k ) ( 63 ) ⁢ s ⁡ [ k ] = ∑ n = ⌊ λ m ⌋ - 1 ⌊ λ m ⌋ ⁢ ⁢ x λ - nm 2 ⁡ ( n ) ( 64 ) x ^ l ′ 2 ⁡ ( n - α ) = s ⁡ [ nm - β - l ] ( 65 ) x ^ l ′ 1 ⁡ ( n - α ) = ∑ λ = 0 m - 1 ⁢ ⁢ p ⁡ ( l + 2 ⁢ km ) ⁢ x ^ l ′ 2 ⁡ ( n - α - 2 ⁢ k ) ( 66 ) x ^ l ′ 0 ⁡ ( n - α ) = 2 ⁢ ⅇ - j ⁢ 2 ⁢ π 2 ⁢ m ⁢ l ⁢ d + m 2 ⁢ ∑ k = 0 2 ⁢ m - 1 ⁢ ⁢ x ^ l ′ 1 ⁡ ( n - α ) ⁢ ⅇ j ⁢ 2 ⁢ π 2 ⁢ m ⁢ kl ( 67 ) = 2 ⁢ m ⁢ 2 ⁢ ⅇ - j ⁢ 2 ⁢ π 2 ⁢ m ⁢ l ⁢ d + m 2 ⁢ γϝϝt ⁡ ( x ^ l ′ 1 ⁡ ( n - α ) , … ⁢ , x ^ 2 ⁢ m - 1 ′ 1 ⁡ ( n - α ) ) ( 68 ) a ^ m ⁢ ⁢ n - α = ⁢ { ⅇ - j ⁢ π 2 ⁢ ( n - α ) ⁢ x ^ l ′ 0 ⁡ ( n - α ) } ( 69 ) ladder schemes are a implementation means recently suggested for producing banks of filters . the inventors have mathematically validated their application to bfdm / om , as described hereafter . it is seen that a bfdm / om modulation may be written as a transmultiplexer using two inverse ffts ( fig5 and 6 ), wherein the polyphase components of the used prototype appear explicitly . each polyphase filter may then be written as a ladder . according to whether s is even or odd , filters g j ( z ) of fig5 and 6 may be replaced with the schemes given by fig7 and 8 . in order to achieve such schemes , a matrix breakdown of the polyphase components is implemented , which is based on 2 × 2 matrices , the number and nature of which are determined according to the desired prototype length and reconstruction delay . for example , in order to generate the pair of polyphase components [ g 1 ( z ), g m + 1 ( z )], we proceed in two steps : initialization is performed by a couple ( f 0 , f 1 ) f 0 corresponds to a product of three matrices which will match the first three items of the upper schemes in fig7 and 8 . the exact form of f 1 depends on the parity of parameter s . this is the identity matrix for even s ( cf . upper scheme in fig7 ) or this is a product of two matrices c 0 and b 0 for odd s , which will match the following two items in the upper scheme in fig8 . a prototype of length 2m ( s = 0 ) or 4m ( s = 1 ) is thereby obtained . in order to increase the length of p ( z ), with or without increasing the delay , a set of matrices are then applied , which will be either ( a , b ), or ( c , d ) respectively . the continuation of the embodiment scheme is thereby obtained . the same principle is applied to the polyphase components [ c d − 1 ( z ), g d − m − 1 ( z )], this time taking the inverse matrices of the previous matrices . an advantage of this structure is that it guarantees perfect reconstruction even in the presence of an error on the calculated coefficients , in particular in the presence of quantification errors . furthermore , this structure also facilitates optimization of the prototype filter , for example by considering a localization or frequency selectivity criterion : it is sufficient to optimize ([( m − 1 )/ 2 ]+ 1 )( 2 m + 1 )= mm coefficients instead of 2 mm , without introducing any perfect reconstruction constraint . in order to perform a comparison of the different embodiments provided , we shall place ourselves in the common case where n = 2 mm . in this case each polyphase component has a length equal to m . each polyphase component may be produced as a transverse component , as a ladder component or in the orthogonal case , as a trellis component . even if the ladders and the trellises have two outputs , only one may be exploited . premodulation ( a phase shift , i . e . a complex multiplication ); an inverse fourier transform ; polyphase filtering . at the demodulator , the same operations are performed in the reverse order . therefore , the complexity of the complete transmultiplexer with premodulation may be derived in terms of complex operations ( table 1 ) or real operations ( table 2 ). the achieved gain with respect to direct realization of the scheme of fig1 is therefore a net gain , as the latter would require 2 mm − 1 complex additions and 2 mm + 1 complex multiplications per sub - band and per sample , at both the modulator and the demodulator . in terms of memory cells , 4m complex values must be stored in order to perform the premodulation , as well as the coefficients of the various structures . when the same filters upon transmission and reception are used , the first column of table 3 is obtained . moreover , in all cases , 4 ( m + 1 ) m complex values must be stored in a buffer for the polyphase filtering both at the modulator and the demodulator . the different techniques provided are notably characterized by the fact that for a modulator - demodulator system put “ back - to - back ”, their iess and iecs are exactly zero . practically , because of the inaccuracy of the numerical calculation , they are generally of the order of 10 − 14 . in the case of modes 2 and 4 , this perfect reconstruction characteristic is provided structurally , i . e . it is maintained after quantification of the ladder coefficients for bfdm / om or trellis coefficients for ofdm / om . two criteria may be taken into account for designing the prototype filters : localization and selectivity . other aspects may also be taken into account , such as representative channel distortions of different transmission channels , for example of the mobile radio type . as purely indicative examples , tables 4 and 5 of the appendix d give particular embodiments of the invention , the results of which are illustrated by fig1 a , 11 b , 12 a and 12 b . fig1 a and 11b show the time response and the frequency response for a biorthogonal prototype with m = 4 , n = 32 , α = 8 , ξ = 0 . 9799 ( localization ), ξ mod = 0 . 9851 ( modified localization , according to doroslovacki &# 39 ; s criterion ). they match the first column of table 4 ( transverse coefficients ) and table 5 ( ladder coefficients ). fig1 a and 11b respectively show the time response and the frequency response for a biorthogonal prototype with m = 4 , n = 32 , α = 2 , ξ = 0 . 9634 ( localization ), ξ mod = 0 . 9776 ( modified localization , according to doroslovacki &# 39 ; s measurement ). they match the second column of table 4 . in this appendix , as an introduction to bfdm / oqam modulations , a few essential definitions on biorthogonality ([ 16 ], [ 17 ], [ 18 ]) will be given as a reminder let e be a vector space on a field k , the definitions and properties which we are going to use for generating a bfdm / oqam modulation , may be summarized in the following way : let ( x i ) iεi and ({ tilde over ( x )} i ) iεi be two families of e vectors . ( x i ) iεi and ({ tilde over ( x )} i ) iεi are biorthogonal if and only if ∀( i , j ) ε i 2 , x i ,{ tilde over ( x )} j = δ i , j let ( x i ) iεi and ({ tilde over ( x )} i ) iεi be two families of e vectors . ( x i ) iεi and ({ tilde over ( x )} i ) iεi form a pair of biorthogonal bases of e if and only if : ( x i ) iεi and ({ tilde over ( x )} i ) iεi form two bases of e ( x i ) iεi and ({ tilde over ( x )} i ) iεi are two biorthogonal families let (( x i ) iεi , ({ tilde over ( x )} i ) iεi ) be a pair of orthogonal bases of e , then ∀ xε e : χ = σ iεi α i χ i , alorsα i = { tilde over ( χ )} i , χ χ = σ iεi { tilde over ( α )} i { tilde over ( χ )} i , alors { tilde over ( α )} i = χ i , χ ∥ χ ∥ 2 = σ iεi χ i , χ * { tilde over ( χ )} i , χ a frequency - modulated complex signal on 2m sub - carriers may be written as α m , n εr ; h is a real prototype filter with a bandwidth v 0 and a finite support : h ( t ) ε [− t 1 , t 2 ] with t 1 and t 2 real f 0 = 2 : in order to obtain a biorthogonal modulation , we try to express s ( t ) with a couple of ( x m , n , { tilde over ( x )} m , n ) biorthogonal bases : derivation of the expression of the associated discrete bases is shown is appendix 2 . after translation by t 1 and discretization with a period t e = τ 0 / m = ½mv 0 , it is also possible , cf . appendix 2 , to define a pair of discrete biorthogonal bases such as let n be the length of the prototype filter p [ k ], such that : 2 t = t 1 + t 2 =( n − 1 ) t e ( 8 ) with d an arbitrarily set parameter and with which , as it will be seen , the reconstruction delay may be handled considering equation ( 10 ), let us now set : the factor (− 1 ) mn appears both in the modulator and the demodulator so it may be deleted without changing anything , and we are then led to the multiplexer scheme of fig1 . if q [ k ]= p [ d − k ], we then have q [ k ]= p [ n − 1 − k ]; the prototype is symmetrical . however , unlike what may often be read implicitly or explicitly ([ 4 ], [ 6 ], [ 7 ], [ 9 ]), the symmetry of the prototype is absolutely not required . to persuade ourselves that this is the case , we shall take one of the following prototypes and numerically check out ( a direct check is rather tedious ) that it provides perfect reconstruction for m = 4 in the orthogonal case ( we then have n − 1 = d = 7 , α = 2 and d = 1 ): p ( z )= σ n = 0 γ p ( n ) z − n which verifies ( 26 ) to ( 31 ) also provides perfect reconstruction for m = 4 in the orthogonal case : achieving modulation and demodulation schemes according to fig1 would be extremely costly in terms of operating complexity . by breaking down the prototype p ( z ) into its polyphase components g 1 ( z ), such as it is then possible to express the analysis and synthesis banks as : from this , the expression for the polyphase matrices r ( z ) and e ( z ) of the banks of filters of the modulator and demodulator is thereby derived : e ⁡ ( z 2 ) = w 1 ⁡ ( g 0 ⁡ ( z 2 ) 0 ⋱ 0 g 2 ⁢ m - 1 ⁡ ( z 2 ) ) ( 35 ) r ⁡ ( z 2 ) = j ⁡ [ w 2 ⁡ ( g 0 ⁡ ( z 2 ) 0 ⋱ 0 g 2 ⁢ m - 1 ⁡ ( z 2 ) ) ] t = ( 0 g 2 ⁢ m - 1 ⁡ ( z 2 ) ⋰ g 0 ⁡ ( z 2 ) 0 ) ⁢ w 2 t ⁢ ⁢ where ⁢ ⁢ j , w 1 ⁢ ⁢ and ⁢ ⁢ w 12 ( 36 ) in order to isolate the transmultiplexer function , we introduce the following notation , wherein for convenience in writing , we shall no longer take x m ( n )= j n a m , n , but x m ( n )= a m , n , in order to have : x m ( n ) represents the real symbols to be transmitted and { circumflex over ( x )}′ m ( n ) represents the complex symbols received after extraction of the real part . fig2 gives a global view of the chain . with the polyphase matrices e ( z 2 ) and r ( z 2 ), a polyphase form representation of the transmultiplexer ( fig3 ) may be obtained . then , all we have to do is to take the real part of the output samples in { circumflex over ( x )}′ m ( n − α ) order to reconstruct the input with a delay of α samples . let us note as x ( z ), the vector which represents the transmitted data in the z - transformed domain . upon reception , after demodulation , let us note as { circumflex over ( x )}′(− jz ) the z - transform vector associated with the received data . extraction of the real part then provides vector x ( z ). now , our goal is : to determine the input / output relationship , i . e . the relationship between x ( z ) and { circumflex over ( x )}( z ); to determine the conditions on the polyphase components g 1 ( z ) of p ( z ) for guaranteeing the equality : the 3 main items of this scheme for determining the input / output relationship are 2 polyphase matrices e ( z ) and r ( z ) as well as the transfer matrix δ β ( z ), connected to the expanders , delays and decimators . to determine the latter , the elementary case illustrated in fig4 may serve as a basis , for which the transfer function is given by z − α { circumflex over ( x )} 1 (− jz )= e ( z 2 ) δ β ( z ) r ( z 2 ) x (− jz ) ( jz ) − α { circumflex over ( x )} 1 ( z )= ē (− z 2 ) δ β ( jz ) r (− z 2 )) x ( z ) { circumflex over ( x )} 1 ( z )= j α z α w 1 g ( jz ) w 2 t x ( z ) ( 41 ) x ^ ⁡ ( z ) = q ⁡ ( z ) ⁢ x ⁡ ( z ) ( 43 ) q ( z )= {( jz ) α w 1 g ( jz ) w 2 t } ( 44 ) q ⁡ ( z ) = ( q 0 ⁡ ( z ) 0 q 1 ⁡ ( z ) ⋯ q m - 1 ⁡ ( z ) 0 0 q 0 ⁡ ( z ) 0 q 1 ⁡ ( z ) q m - 1 ⁡ ( z ) q 1 ⁡ ( z ) 0 q 0 ⁡ ( z ) ⋱ ⋱ ⋮ ⋮ q 1 ⁡ ( z ) ⋱ ⋱ 0 q 1 ⁡ ( z ) q m - 1 ⁡ ( z ) ⋱ 0 q 0 ⁡ ( z ) 0 0 q m - 1 ⁡ ( z ) ⋯ q 1 ⁡ ( z ) 0 q 0 ⁡ ( z ) ) ( 45 ) the meaning of − d is specified later on . the exact expression for u 1 (− z 2 ) further depends on this parameter d ( positive or zero integer ), it may then be proved that perfect reconstruction is obtained if and only if : with d and s , integers defined by d = 2sm + d , s & gt ; 0 and 0 ≦ d ≦ 2m − 1 . the reconstruction delay α is related to parameter s by the relationships : the special orthogonal case may be derived from this result , for which d = n − 1 , with n the length of the paraunitary prototype filter , i . e . a symmetrical filter here . p ( z ) is said to be paraunitary if : p ( z )= z −( n − 1 ) { tilde over ( p )} ( z ) with { tilde over ( p )}( z )= p *( z − 1 ) g d − l ( z )= z − s ĝ l ( z ) si 0 ≦ l ≦ d ( 52 ) g 2m + d − l = z −( s − 1 ) ĝ l ( z ) sid + 1 ≦ l ≦ 2 m − 1 ( 53 ) thus , in the special orthogonal case , we have perfect reconstruction with a delay h . boelcskei , p . duhamel , and r . hleiss . a design of pulse shaping ofdm / oqam systems for wireless communications with high spectral efficiency . submitted to ieee transactions on signal processing , november 1998 h . boelcskei , p . duhamel , and r . hleiss , “ design of pulse shaping ofdm / oqam systems for high data - rate transmission over wireless channels ”. in proc . international conference on communications ( icc ); 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