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the main title of this paper is , apart from the term multiplex , literally identical to a forney , gallager and co - workers paper issued more than one decade ago [ for et al .
84 ] , which analyzed the benefits of coded - modulation techniques .
the large success achieved by ungerboeck s coded - modulation came from the way of introducing redundancy in the encoder [ ung 82 ] . in classical channel coding ,
redundant signals are appended to information symbols in a way somewhat analogous to time division multiplex tdm ( envelope interleaving ) .
it was believed that introducing error - control ability would increase bandwidth .
an efficient way of introducing such an ability without sacrificing rate nor requiring more bandwidth consists in adding redundancy by an alphabet expansion .
this technique is particularly suitable for channel in the narrow - band region .
a similar reasoning occurs in the multiplex framework where many people nowadays believe that mux must increase bandwidth requirements .
+   + one of the most powerful tools in communications is the fourier transform , specially its discrete version dft . on the other side ,
applications of finite field have a renewed explosion of interest in the last decades .
a discrete fourier transform for finite fields ( ffft ) was introduced by pollard [ pol 71 ] .
it has successfully applied to perform discrete convolution and as a tool of image processing [ ree et al .
77 , ree - tru 79 ] among many other applications .
besides classical ffft we are concerned with a new finite field discrete version [ cam et al .
98 ] of the integral transform introduced by r.v.l .
hartley [ har 42 , bri 92 ] .
alike classical galois - fourier transforms [ bla 79 ] , finite field hartley transform ght [ cam et al .
98 ] defined on a gaussian integer set gi(@xmath0 ) contains some redundancy and only the cyclotomic coset leaders of transform coefficients are needed to be transmitted .
this yields a new * efficient - bandwidth code division multiplex for band - limited channels*. these mux may present lower bandwidth requirements than tdm by using an argument somewhat analogous to coded - modulation .
tradeoffs between extension of the alphabet and bandwidth are exploited in the sequel .
+   + in the present work , the coded - modulation idea is adapted to multiplex : information streaming from users are not combined by interleaving ( like tdm ) but rather by signal alphabet expansion .
the mux of users sources over a galois field gf(@xmath1 ) deals with an expanded signal set having symbols from an extension field gf(@xmath0 ) , @xmath2 . as a consequence ,
the multiplex of @xmath3 band limited channels of identical maximal frequency @xmath4 leads to _ bandwidth requirements less than @xmath5 _ , in contrast with tdmed or fdmed signals .
+   + the design of such an _ efficient bandwidth mux _ is based upon the finite field structure , specifically , it consists of applying finite - field fourier transforms ( ffft ) .
this paper shows that the bandwidth compactness factor relatively to tdm depends on the length @xmath6 , the alphabet extension .
+   + another point to mention is that the superiority of digital mux regarding analogic mux is essentially due to the low complexity of tdm .
the majority of today multiplex systems follows plesiocrhonous ( pdh ) or synchronous ( sdh ) hierarchy . besides presenting higher spectral efficiency ( bits / s / hz ) than classical mux
, the new cdm schemes here introduced are based on fast transforms so they also seem to be attractive from the implementation viewpoint .
although most mux systems today be intended to optical fiber which are not yet bandlimited channels , multiplex has also been adopted on satellite channels .
probably applications of such multiplex will be on transponders and specially on cellular communications .
cable television ( catv ) can also have benefit by using such a technique .
digital multiplex normally alludes time division multiplex ( tdm ) .
however , it also can be achieved by coding division multiplex ( cdm ) .
the cdm has recently been focus of interest , specially after the is-154 standardization of the cdma system for cellular telephone . in this section
we introduce a new class of mux schemes based upon finite - field transforms which can be implemented by fast transform algorithms .
_ classical multiplex increases simultaneously the transmission rate and the bandwidth by the same factor _ , keeping thus the spectral efficiency unchanged . in order to achieve ( slight ) better spectral efficiencies ,
classical cdma uses waveforms presenting a non zero but residual correlation .
+   + given a signal @xmath7 over a finite field gf(@xmath1 ) we deal with the galois domain considering the spectrum @xmath8 over an extension field gf(@xmath0 ) which corresponds to the finite field transform ( galois transform ) [ bla 79 , cam et al .
+   + as an alternative and attractive implementation , the multiplex is carried out by a finite field transform ( ffft / ffht ) and the demux corresponds exactly to a inverse finite - field transform of length @xmath9 .
each symbol on the ground field gf(@xmath1 ) have duration @xmath10 seconds .
an @xmath3-user mux can be designed on the extension field gf(@xmath0 ) where @xmath9 . for the sake of simplicity , we begin with @xmath6=1 and consider a ( @xmath11)-channel mux as follows .
typically , we can consider gf(3 ) corresponding to alternate mark inversion ami signaling .        a ( @xmath11)-cd consider digital carrier sequences per channel as versions of cas function over the galois ( complex ) field gi(@xmath1 ) . the cas ( cos and sin ) function is defined in terms of finite field trigonometric functions [ cam et al .
98 ] , @xmath14 .
+   + carrier 0 : + @xmath15 + carrier 1 : + @xmath16 + ... + carrier @xmath17 : + @xmath18 + ... + carrier @xmath19 : + @xmath20 .
+   + the cyclic digital carrier has the same duration @xmath10 of a input modulation symbol , so that it carries @xmath3 slots per data symbol .
the interval of each cas - symbol is @xmath21 and therefore the bandwidth expansion factor by multiplexing @xmath3 channels may be roughly @xmath3 , the same result as fdm and tdm / pam . + a first scheme of the multiplex is showed : the output corresponds exactly to the galois - hartley transform of the `` users''-vector @xmath22 .
+ therefore , it contains all the information of the channels .
each coefficient @xmath23 of the spectrum has duration @xmath21 .
these carriers can also be viewed as spreading waveforms .
an @xmath3-user mux has @xmath3 spreading sequences , one per channel .
the requirements to achieve welch s lower bound according to massey and mittelholzer [ mas - mit 91 ] are hold by @xmath24 sequences .
the matrix @xmath25 $ ] presents both orthogonal rows and columns and have the same energy. +   + a naive example is present in order to illustrate such an approach .
a 4-channel mux over gf(5 ) can be easily implemented @xmath26 .
it is straightforward to see that such signal are not fdmed nor tdmed .
hlio magalhes de oliveira was born in arcoverde , pernambuco , brazil , in 1959 .
he received both the b.sc . and
degrees in electrical engineering from universidade federal de pernambuco ( ufpe ) , recife , pernambuco , in 1989 and 1983 , respectively .
then he joined the staff of the same university as a lecturer . in 1992
he earned the docteur de l__cole nationale suprieure des tlcommunications _ _ degree , in paris . he is currently with the statistics departement of ufpe
de oliveira was appointed as honored professor by fivety electrical engineering undergratuate classes and chosen as the godfather of sixteen engineering graduation .
his publications are available at arxiv .
research current interests include : applied statistics , communications theory , applied information theory , signal analysis , and wavelets .
a new efficient - bandwidth code - division - multiple - access ( cdma ) for band - limited channels is introduced which is based on finite field transforms . a multilevel code division multiplex exploits orthogonality properties of nonbinary sequences defined over a complex finite field . 
 galois - fourier transforms contain some redundancy and just cyclotomic coefficients are needed to be transmitted yielding compact spectrum requirements . 
 the primary advantage of such schemes regarding classical multiplex is their better spectral efficiency . 
 this paper estimates the _ bandwidth compactness factor _ relatively to time division multiple access tdma showing that it strongly depends on the alphabet extension . 
 these multiplex schemes termed galois division multiplex ( gdm ) are based on transforms for which there exists fast algorithms . 
 they are also convenient from the implementation viewpoint since they can be implemented by a digital signal processor .    
 digital multiplex , code - division multiple access , hartley - galois transform , finite field transforms , spread sequence design .