Negaive frequency communicaion Fanping DU Email: dufanping@homail.com Qing Huo Liu arxiv:2.43v5 [cs.it] 26 Sep 2 Deparmen of Elecrical and Compuer Engineering Duke Universiy Email: Qing.Liu@duke.edu Absrac Specrum is he mos valuable resource in communicaion sysem, bu unforunaely, so far, a half of he specrum has been wased. In his paper, we will see ha he negaive frequency no only has a physical meaning bu also can be used in communicaion. In fac, he complee descripion of a frequency signal is a roaing complex-frequency signal, in a complee descripion, posiive and negaive frequency signals are wo disinguishable and independen frequency signals, hey can carry differen informaion. Bu he curren carrier modulaion and demodulaion do no disinguish posiive and negaive frequencies, so half of he specrum resources and signal energy are wased. The complex-carrier modulaion and demodulaion, proposed by his paper, use he complex-frequency signal as a carrier signal, he negaive and posiive frequency can carry differen informaion, so he specrum resources are fully used, he signal energy carried by complex-carrier modulaion is focused on a cerain band, so he signal energy will no be los by he complex-carrier demodulaion. Inroducion According o Shannon formula: C = W log (+S/N) 2 ()
where C is he channel capaciy, W is he channel bandwidh, S is he signal power, and N is he noise power. We can see ha he mos effecive way o increase he channel capaciy is increasing bandwidh, and enhance he signal-o-noise raio can also increase he channel capaciy. In curren communicaion sysems, o ake full advanage of he specrum, carrier modulaion echnology is used. The principle of curren carrier modulaion is shown in Fig., he real par of he baseband complex signal muliplied by cos() and he imaginary par muliplied by sin(), hen add, as he following formula: s CB () = Re{s BB () e i } (2) where s CB () is he modulaion signal, e i is he carrier signal, s BB () is he baseband complex signal, Re is o ake he real par. Obviously in he curren carrier modulaion, alhough he baseband signal is complex, he carrier signal is complex, bu he modulaion signal is aken he real par, so a real signal is sen. In his paper i is called real-carrier modulaion. In fac, he real-carrier modulaion echnology wases a half of he specrum resources and he signal energy, ha is because of he incorrec undersanding and using of negaive frequency. So far, in all curren communicaion sysems, including wireless, cable and fiber he available bandwidh is defined in he posiive specrum. For example, he bandwidh defined in he laes version of LTE proocol, see [2], negaive specrum is no used. So o make full use of specrum resources, we mus firs recify he undersanding of he negaive frequency. 2 Complex-frequency signal The negaive frequency has a physical meaning, see []. As shown in Fig. 2, we define he angle of counerclockwise roaion as +θ, he angle of clockwise roaion as θ, so he definiion of angular frequency is: = dθ() d 2 (3)
= d( θ()) d By (3), (4) we can see ha he posiive angular frequency corresponds o he speed of counerclockwise roaion, he negaive angular frequency corresponds o he speed of clockwise roaion, so negaive angular frequency does no due o he negaive ime, bu due o negaive angle, posiive and negaive frequency represen wo differen direcions of roaion in a plane, here are wo kinds of frequencies is essenially because of he angle is defined in a plane, a plane has wo sides. Wih he undersanding of he physical meaning of negaive frequency, hen how o disinguish beween posiive and negaive frequencies, or how o describe hese wo direcions of roaions in a plane? Tha is he Euler formula: (4) e ±i = cos()±i sin() (5) As shown in Fig. 3, e i and e i are wo kinds of complex-frequency signals, corresponding o he negaive and posiive frequency signals. Alhough in he ime-complex direc produc space, i is easy o disinguish posiive and negaive frequency signals, bu in ime-real direc produc space, he projecion of posiive and negaive frequency signals are he same real-frequency signal cos(), ha is: Re{e i } = Re{e i } = cos() (6) So when we see a real-frequency signal, we can no disinguish i beween posiive or negaive frequency signals from is projecion, he probabiliy of he frequency signal being negaive or posiive is equal, i is /2, ha is: cos() = (e i +e i )/2 (7) Similarly: sin() = i (e i e i )/2 (8) Therefore, a real-frequency signal cos() or sin() is no complee, he complee descripion of a frequency signal mus be a roaing complex-frequency signal e ±i, in complee 3
descripion, he negaive frequency signal e i and posiive frequency signal e ı are wo disinguishable and independen frequency signals, hey can carry differen informaion. In his paper, we define he posiive frequency, which direcion of roaion mee he righ hand rule as he R-frequency. We define he negaive frequency, which direcion of roaion mee he lef hand rule as he L-frequency. Unless oherwise cied, his paper will use he erms L-frequency and R-frequency, L-band and R-band insead of he posiive frequency and negaive frequency, posiive band and negaive band. 3 Real-carrier modulaion and demodulaion In his secion, we will see ha he curren real-carrier modulaion echnology occupies all he L-band and R-band in he specrum, so a half of he specrum resources are wased, and he curren real-carrier demodulaion echnology only receives one of he L-band or R-band, so a half of he signal energy is los. 3. Real-carrier modulaion The principle of curren real-carrier modulaion is as formula (2). I is modulaed by he R-frequency signal e ı, according o he formula (7) and (8): s CB () = Re{s BB () e i } = Re{s BB ()} cos() Im{s BB ()} sin() = Re{s BB ()} (e i +e i )/2 Im{s BB ()} i (e i e i )/2 (9) = (Re{s BB ()} Im{s BB ()} i) e i /2+(Re{s BB ()}+Im{s BB ()} i) e i /2 = s BB () e i /2+s BB () e i /2 where s CB () is he modulaion signal, s BB () is he baseband signal, s BB () is he conjugaed baseband signal, e i is he R-frequency signal. According o he muliplicaion of wo signals in ime domain is equivalen o heir convoluion in frequency domain, i can be seen from he above equaion ha he real-carrier modulaion will move he baseband o he 4
R-band and L-band, and he signal energy is divided equally. Therefore, afer being modulaed by he R-frequency signal e i and aking he real par, he R-bandsignal is he same as he baseband signal, and he ampliude is a half, he L-band signal is conjugaed symmery o he baseband signal, and he ampliude is also a half. Similarly, if i is modulaed by he L-frequency signal e i : s CB () = Re{s BB () e i } = s BB () e i /2+s BB () e i /2 () where s CB () is he modulaion signal, s BB () is he baseband signal, s BB () is he conjugaed baseband signal, e i is he L-frequency signal. Therefore, afer modulaed by he L-frequency signal e i and ake he real par, he L-band signal is he same as he baseband signal, and he ampliude is a half, he R-band signal is conjugaed symmery o he baseband signal, and he ampliude is also a half. In summary, he real-carrier modulaion occupies all he L-band and R-band, and he informaion on he L-band and R-band are conjugaed symmeric, no independen, and he signal energy is a half on each side. Furhermore, currenly people regard one of he bands as a image frequency componen, as if i is no realiy, in fac, he image frequency is caused by using he incomplee realfrequency signal as he carrier signal. A demo of band move of he real-carrier modulaion is shown in Fig. 4, modulaed by he R-frequency signal, ampliude specrum. 3.2 Real-carrier demodulaion The curren real-carrier demodulaion also assumed o receive a real signal, so i is deal as a real signal. As he real-carrier modulaion does no disinguish L-frequency and R- frequency, so he modulaion signal maybe modulaed by he L-frequency signal or he R-frequency signal, here assume he modulaion signal is modulaed by he R-frequency signal, as formula (9). 5
If demodulaed by he L-frequency signal e i, according o he formula (9): s RBB () = s CB () e i = (s BB () e i /2+s BB () e i /2) e i () = s BB () e 2 i /2+s BB ()/2 where s RBB () is he demodulaion signal, s CB () is he modulaion signal, s BB () is he baseband signal, s BB () is he conjugaed baseband signal, e i is he L-frequency signal. Afer demodulaion, he L-band s BB () /2 is moved o wo imes far away and he R-band s BB ()/2 is moved o he baseband, afer a low-pass filer, he L-band signal energy is discarded. Similarly, if demodulaed by he R-frequency signal e i, according o he formula (9): s RBB () = s CB () e i = (s BB () e i /2+s BB () e i /2) e i (2) = s BB () e 2 i /2+s BB () /2 where s RBB () is he demodulaion signal, s CB () is he modulaion signal, s BB () is he baseband signal, s BB () is he conjugaed baseband signal, e i is he R-frequency signal. Afer demodulaion, he R-band s BB ()/2 is moved o wo imes far away and he L-band s BB () /2 is moved o he baseband, afer a low-pass filer, he R-band signal energy is discarded. Bu he reserved L-band signal is conjugaed o original signal. Alhough informaion on he L-band and R-band is conjugaed, bu filering ou one of hem will los a half of signal energy. A demo of band move of he real-carrier demodulaion is shown in Fig. 5, demodulaed by he L-frequency signal, ampliude specrum. 4 Complex-carrier modulaion and demodulaion As menioned earlier, he complee descripion of a frequency signal is a roaing complexfrequency signal, in a complee descripions, he L-frequency signal e i and R-frequency 6
signal e i are wo disinguishable and independen frequency signals, so hey can carry differen informaion. Therefore, we can modulae he baseband signals by he L-frequency or R-frequency signal, in order o disinguish from he real-carrier modulaion, i is called complex-carrier modulaion in his paper. Because here are wo kinds of frequencies, so here are wo kinds of complex-carrier modulaions, in his paper he modulaion using he L-frequency signal is called L-complex modulaion, he modulaion using he R-frequency signal is called R-complex modulaion. In his secion, we will see ha compared wih he real-carrier modulaion, he complexcarrier modulaion uses he disinguishable and independen L-frequency signal e i and R-frequency signal e i as he carrier signal, hey can carry differen informaion, so he specrum resources are full used, he signal energy carried by complex-carrier modulaion is focused on a cerain band, so he signal energy will no be los by he complex-carrier demodulaion. 4. Complex-carrier modulaion There are wo kinds of complex-carrier modulaion, L-complex modulaion and R-complex modulaion. The principle of L-complex modulaion is as he following formula: s CB () = s BB () e i = (Re{s BB ()} +i Im{s BB ()}) (cos() i sin()) = (Re{s BB ()} cos()+im{s BB ()} sin())+i (Im{s BB ()} cos() Re{s BB ()} sin()) (3) where s CB () is he complex-carrier modulaion signal, s BB () is baseband signal, e i is he L-frequency signal. Similarly, he principle of R-complex modulaion is as he following formula: s CB () = s BB () e i = (Re{s BB ()} +i Im{s BB ()}) (cos() +i sin()) = (Re{s BB ()} cos() Im{s BB ()} sin())+i (Im{s BB ()} cos() +Re{s BB ()} sin()) (4) where s CB () is he complex-carrier modulaion signal, s BB () is baseband signal, e i is he R-frequency signal. 7
Bandmove ofhel-complexmodulaionisshown infig. 6. Bandmove ofher-complex modulaion is similar and omied. The L-frequency and R-frequency are wo disinguishable and independen frequency signals, so hey can carry differen informaion. The band move of carrying wo differen informaion is shown in Fig. 7, he baseband A and B are moved o L-band and R-band separaely. According o he formula (3) he principle of L-complex modulaion is shown in Fig. 8: I can be seen from he figure, he real par and he imaginary par of he L-complex modulaion signal are modulaed separaely, hey are wo orhogonal signals in space, so he L-complex modulaion signals in he ransmission medium are he lef roaing complex signals. The principle of R-complex modulaion is similar and omied. 4.2 Complex-carrier demodulaion In essence, he principle of complex-carrier modulaion and demodulaion are he same, hey are simply band moves, bu in opposie direcions. So he L-complex modulaion signal is demodulaed by he R-frequency signal, and he R-complex modulaion signal is demodulaed by he L-frequency signal. The L-complex modulaion signal is demodulaed by he R-frequency signal as he following formula: s RBB () = s CB () e i = (s BB () e i ) e i = s BB () (5) where s RBB () is he demodulaion signal, e i is he R-frequency signal, s CB () is he modulaion signal, s BB () is baseband signal. The band move of he demodulaion for he L-complex modulaion signal is shown in Fig. 9. The R-complex modulaion signal is demodulaed by he L-frequency signal as he following formula: s RBB () = s CB () e i = (s BB () e i ) e i = s BB () (6) 8
where s RBB () is he demodulaion signal, e i is he L-frequency signal, s CB () is he modulaion signal, s BB () is baseband signal. The band move of he demodulaion for he R-complex modulaion signal is similar and omied. The differen informaion on he L-band and R-band can be demodulaed separaely, as shown in Fig.. 4.3 Essence of complex-carrier modulaion and demodulaion As menioned earlier, in essence, he principle of complex-carrier modulaion and demodulaion are he same, hey are band moves. So in his paper, we use a single word band-move insead of he modulaion and demodulaion. Furhermore, band-move is a ransformaion, i is he roaional speed ransform in ime domain, he move ransform in frequency domain. Like he reference sysem ransform, we can see, he frequency is a relaive value, is value is relaed o he reference frequency, as he following formula: e i = e i e ic = e i(+c) (7) where e i is he frequency afer ransform, e i is he frequency before ransform, e ic is he reference frequency. Thus, a negaive frequency can become a posiive frequency afer he band-move ransform, which also confirms, on he oher hand, ha he negaive frequency has a physical meaning, and he sign of a frequency is relaed o he reference frequency. The ransformaion has he following properies: : The addiive propery, one band-move is equivalen o he sum of several band-moves, as he following formula: s CB () = s BB () e i e i 2 = s BB () e i( + 2 ) (8) where s CB () is he band-move signal, s BB () is he original signal. 2: The commuaive propery, he resul of several band-moves has nohing o do wih 9
he orders, as he following formula: s CB () = s BB () e i e i 2 = s BB () e i 2 e i (9) where s CB () is he band-move signal, s BB () is he original signal. So he ransformaions of band-move make up a coninuous Abel group. 5 Circularly polarized elecromagneic signal From his secion, we will see ha he circularly polarized elecromagneic signals are exacly he complex-frequency signals, he righ-hand and lef-hand circularly polarized elecromagneic signals correspond o he R-frequency and L-frequency signals. As we all know, ligh is elecromagneic wave, and he righ-hand circularly polarized ligh wavefron is described as he following formula: E = A e i( 2π x/λ+φ) (2) where E is elecric vecor, A is he elecric inensiy, is he angular frequency, x is he propagae disance, λ is he wavelengh, φ is he iniial phase. Assume we observed his ligh a a cerain place, ha means x is a consan, so 2π x/λ+φ is a consan, i can be seen as a iniial phase φ, so he (2) can be change o he following formula: E = A e i( 2π x/λ+φ) = A e i+φ = A cos(+φ )+i A sin(+φ ) = A e iφ e i (2) I is a R-frequency e ı which has a iniial phase φ and a non-normalized ampliude A. Also we can see i conains a real par A cos(+φ ) and a imaginary par A sin(+φ ), so he righ-hand circularly polarized ligh is a complex-frequency signal, and he real par and imaginary par are orhogonal in space, ha means we can send a complex-frequency signal by a circularly polarized signal. The righ-hand circularly polarized ligh is shown in Fig., i is he same as he Fig. 3.
Similarly, we can see he lef-hand circularly polarized ligh is he R-frequency signal. Furhermore, he linearly polarized elecromagneic signal is he real-frequency signal. So we can send ou he R-band and L-band signals by he righ-hand and lef-hand circularly polarized elecromagneic signals. By he way, formula (2) is a soluion of Schrodinger equaion, as shown in Fig., we can see, here i has a clearly physical meaning, i means orhogonal in space, no imaginary. 6 Conclusions In summary, we draw he following conclusions: : Negaive specrum exiss, bu is wased. 2: A complee descripion of a frequency signal is a roaing complex-frequency signal, in a complee descripion, R-frequency signal and L-frequency signal are wo disinguishable and independen frequency signals. 3: Curren real-carrier modulaion don disinguish beween R-frequency signal and L-frequency signal, so R-band and L-band are boh occupied. 4: Curren real-carrier demodulaion only receive one band, so half of he energy will be los. 5: Image frequency is caused by he real-carrier modulaion and demodulaion. 6: Complex-carrier modulaion use he complex-frequency signal as a carrier signal, so i can carry differen informaion on L-band and R-band, he specrum resources are fully used. 7: Complex-carrier demodulaion can receive he whole signal energy, so he channel capaciy is higher. 8: Frequency is a relaive concep, i is relaive o he selecion of reference frequency. 9: Complex-carrier modulaion and demodulaion is essenially a ransformaion, make up a coninuous Abel group. : Circularly polarized elecromagneic signal is he complex signal, while linearly po-
larized signal is he real signal. In fac, he concepion of complex-carrier is jus like he subcarrier in OFDM, bu expanded o he RF range. Due o he complex-carrier modulaion and demodulaion echnology can make full use of he specrum, and do no discard he signal energy, herefore, facing wih he scarce of he specrum resources, we sure ha he complex-carrier modulaion and demodulaion echnology will become he mainsream of he nex-generaion communicaion echnology. For example, in he curren LTE echnology, if he wo code-words are modulaed by he L-band and R-band separaely, hen clearly he channel capaciy will be grealy improved. For anoher example, if up-link and down-link are modulaed by he L-band and R-band separaely, hen up-link and down-link can communicae using he same frequency a he same ime. References [] Chen, Huaichen, Fang Haiyan, On The Physical Meaning of Negaive Frequency in Specrum, Journal of Elecrical & Elecronic Educaion, Jan. 28 [2] 3GPP TS 36. V9.5., Oc. 2 2
cos( ) Re{ sbb()} s () BB Spli + s () CB Im{ sbb()} sin( ) Figure : Principle of real-carrier modulaion. 3
+θ θ Figure 2: Definiion of he angle. 4
exp(iw) exp( iw) 3 2 3 2 cos(w) cos( w) 3 2 3 2 Figure 3: Posiive and negaive frequency signals and heir projecions. 5
F( ) Baseband F( ) F( ) F( ) C C Carrier frequency Modulaion band (before ake real par) C /2 C Modulaion band (afer ake real par) Figure 4: Band move of he real-carrier modulaion. F( ) C /2 F( ) C Modulaion band C F( ) Carrier frequency 2 C /2 Baseband Figure 5: Band move of he real-carrier demodulaion. 6
F( ) F( ) baseband C F( ) L-frequency C L-band modulaion Figure 6: Band move of he L-complex modulaion. 7
F( ) F( ) Baseband A F( ) Baseband B F( ) C R-frequency C F( ) L-frequency C C R-band and L-band Figure 7: Band move of modulaing wo differen informaion. 8
cos( ) Re{ sbb()} s () BB Spli + Re{ scb()} Im{ sbb()} sin( ) sin( ) Re{ sbb()} Spli + Im{ scb()} Im{ sbb()} cos( ) Figure 8: Principle of L-complex modulaion. 9
F( ) C F( ) L-band modulaion F( ) C R-frequency baseband Figure 9: Band move of he demodulaion for he L-complex modulaion signal. F( ) C F( ) C R-band and L-band F( ) C R-frequency C F( ) L-frequency F( ) Baseband A Baseband B Figure : Band move of demodulaing wo differen informaion. 2
Figure : Circularly polarized elecromagneic signal. 2