## Optical OFDM - Generation

Before we go fur­ther into the sig­nal pro­cess­ing top­ics that start­ed with the cyclic pre­fix, we shall take a look at how we can gen­er­ate OFDM sig­nals ready for trans­mis­sion and then, next, how to get them into the fiber. By now you should have a firm grasp on what an OFDM sig­nal is. If not, take a look here and here.

There is no sin­gle, unique way of gen­er­at­ing and trans­mit­ting OFDM over fiber. Luck­i­ly so, since oth­er­wise read­ing these heaps on OFDM papers would have been real­ly bor­ing. To give an (incom­plete) overview of some of the meth­ods, Fig. 1 shows a pos­si­ble hier­ar­chy that would have made old Aris­to­tle proud$^1$.

Fig. 1: Hier­ar­chy of OFDM vari­ants.

At the root “node” OFDM can be sep­a­rat­ed into opti­cal­ly and elec­tri­cal­ly mul­ti­plexed OFDM. In the elec­tri­cal ver­sion (let’s call that E-OFDM) the mul­ti­plex­ing is usu­al­ly done by using an inverse FFT in a dig­i­tal sig­nal proces­sor (DSP). That the inverse FFT can be used for that pur­pose was shown in this post. In the opti­cal ver­sion (O-OFDM) the sub­chan­nels are mul­ti­plexed opti­cal­ly, gen­er­al­ly by com­bin­ing spec­tral­ly over­lap­ping sig­nals in an opti­cal cou­pler. There are var­i­ous meth­ods to achieve this, although some only gen­er­ate “OFDM-like” sig­nals whose sub­chan­nels are not all mod­u­lat­ed inde­pen­dent­ly.

#### Electrically multiplexed OFDM (E-OFDM)

The out­put of an inverse FFT is gen­er­al­ly com­plex-val­ued even if the inputs were real-val­ued. This is a con­se­quence of the asym­me­try of the sig­nal spec­trum which is defined by the inputs to the FFT$^2$. This is shown in Fig. 2. Since there is no such thing as a com­plex elec­tri­cal cur­rent, the DSP will out­put two real-val­ued sig­nals, one for the real part and one for the imag­i­nary part of the OFDM sig­nal. These two out­put sig­nals can be used to direct­ly dri­ve an opti­cal IQ-mod­u­la­tor which can imprint the com­plex OFDM sig­nal direct­ly onto a con­tin­u­ous light source. The dis­ad­van­tage of this approach is that the opti­cal com­po­nents need­ed are still quite cost­ly, and peo­ple start­ed look­ing for ways to work with only a sin­gle elec­tri­cal dri­ve sig­nal.

Fig. 2: Illus­tra­tion of the rela­tion between a real-val­ued sig­nal spec­trum (which is dic­tat­ed by the inverse FFT inputs) and the cor­re­spond­ing time sig­nal. The sig­nal has both real and imag­i­nary com­po­nents; the com­plex sig­nal is con­ju­gate sym­met­ric over the DFT time inter­val. Hence half of the time sig­nal is redun­dant.

##### real-valued baseband signals

The most sim­ple way is to adjust the inputs into the inverse FFT in such a way that cor­re­spond­ing pos­i­tive and neg­a­tive fre­quen­cy com­po­nents are con­ju­gate com­plex. The result­ing out­put of the inverse FFT must then be real-val­ued, as illus­trat­ed in Fig. 3 and as can be derived from the def­i­n­i­tion of the real part of a com­plex num­ber,

$$c_n(t) \exp\bigl(i \omega_n t\bigr) + c_n^*(t) \exp\bigl(-i \omega_n t\bigr) = 2 \Re \bigl\lbrace c_n(t) \exp\bigl(i \omega_n t\bigr) \bigr\rbrace\tag{1}$$

Hence, when $c_n = c_{-n}^* = c_{N-n}^*$ (the lat­ter equal­i­ty results from the cyclic nature of the N-point DFT, see also foot­note 2), the result­ing sig­nal will be real-val­ued. While this is one of the eas­i­est ways to gen­er­ate real-val­ued OFDM sig­nals and needs only a sin­gle dig­i­tal-to-ana­log con­vert­er (DAC), it wastes half of the DSP band­width. Wast­ing half the DSP capac­i­ty is usu­al­ly not a good way to set capac­i­ty records.

Fig. 3: Gen­er­a­tion of real-val­ued time sig­nals in the base­band: When the inverse FFT inputs (cor­re­spond­ing to the sig­nal spec­trum) are con­ju­gate com­plex, the result­ing time sig­nal is real. This is the same case as in Fig. 2, with time and fre­quen­cy domain reversed. In this case, the spec­trum is ful­ly redun­dant.

A more ele­gant, though slight­ly more expen­sive, way to turn com­plex-val­ued sig­nals into some­thing real (I stole that from some pop song), is to use an elec­tri­cal IQ-mod­u­la­tor which “mix­es” the OFDM sig­nal up to an inter­me­di­ate radio fre­quen­cy. This is basi­cal­ly the same as the opti­cal IQ-mod­u­la­tor, but whol­ly in the elec­tri­cal domain. But we didn’t talk about the opti­cal IQ-mod­u­la­tor yet, either, so here’s the prin­ci­ple of IQ-mod­u­la­tion:

To mod­u­late two elec­tri­cal dri­ve sig­nals, $b^{re}(t)$ and $b^{im}(t)$ onto an inter­me­di­ate fre­quen­cy, we use the set­up of Fig. 4. Math­e­mat­i­cal­ly,

$$I(t) = b^{re}(t) \cos\bigl(\omega t\bigr) - b^{im}(t) \sin\bigl(\omega t\bigr)\tag{2}$$

Now if you have paid atten­tion so far and always read the post foot­notes, this will look famil­iar. Look­ing at foot­note 2 of the first post, we can write the sig­nal $I(t)$ as

$$I(t) = \Re\Bigl\lbrace \Bigl[b^{re}(t) + i\, b^{im}(t)\Bigr] \exp\bigl( i \omega t \bigr) \Bigr\rbrace$$

which should explain the nam­ing of the vari­ables. Obvi­ous­ly, the sig­nal $I(t)$ must be real-val­ued and can be used to dri­ve an inten­si­ty mod­u­la­tor, such as an elec­troab­sorp­tion mod­u­la­tor or field mod­u­la­tor such as a Mach-Zehn­der mod­u­la­tor. How­ev­er, you could also use it to dri­ve a phase mod­u­la­tor or fre­quen­cy mod­u­la­tor, and this has all been done, I think. The spec­trum of $I(t)$ is shown in Fig. 5. While we have used the full capac­i­ty of the DSP, we need two DACs (one for $b^{re}$ and one for $b^{im}$) and a large mod­u­la­tor band­width due to the inter­me­di­ate fre­quen­cy. And of course the IQ-mix­er. The mod­u­la­tor band­width is used more effi­cient­ly by doing the IQ-upmix­ing direct­ly onto the opti­cal car­ri­er fre­quen­cy, with­out an inter­me­di­ate fre­quen­cy, but this requires an opti­cal IQ-mod­u­la­tor.

Fig. 4: Prin­ci­ple of elec­tri­cal upmix­ing: The real and imag­i­nary parts of com­plex vari­able b are mod­u­lat­ed onto both quad­ra­tures of an RF car­ri­er which is gen­er­at­ed using a volt­age-con­trolled oscil­la­tor or some­thing sim­i­lar. A π/2 phase shifter is used to gen­er­ate the sine quad­ra­ture from the cosine oscil­la­tor out­put. Both quad­ra­tures are then com­bined using a direc­tion­al cou­pler to obtain the out­put accord­ing to eq. (2). The ele­ments of the coupler’s scat­ter­ing matrix are S11 = S12 = S21 = 1, S22 = –1.

Fig. 5: Illus­tra­tion of real-val­ued inter­me­di­ate fre­quen­cy OFDM sig­nal gen­er­a­tion. The com­plex-val­ued base­band spec­trum is upmixed to an inter­me­di­ate fre­quen­cy, result­ing in a con­ju­gate sym­met­ric spec­trum. This con­ju­gate sym­me­try cor­re­sponds to a real-val­ued time sig­nal. Due to the inter­me­di­ate fre­quen­cy, the sig­nal has fre­quen­cy com­po­nents sig­nif­i­cant­ly high­er than in the base­band sig­nal and needs high­er sam­pling rates to be received cor­rect­ly. The IF spec­trum is again ful­ly redun­dant.

##### optical IQ-modulation

The opti­cal IQ-mod­u­la­tor, shown in Fig. 6, does basi­cal­ly the same as the elec­tri­cal IQ-mod­u­la­tor and as was described with (2), except that in this case the car­ri­er fre­quen­cy $\omega$ is not inter­me­di­ate but the opti­cal fre­quen­cy of the laser used to trans­mit the OFDM sig­nal. This is usu­al­ly around 190 THz or so, and the band­width of the OFDM sig­nal is just a tiny frac­tion of the full spec­trum. This makes IQ-mod­u­la­tion a bit eas­i­er from the tech­ni­cal view­point. Here’s why:

To upmix a com­plex input sig­nal (com­pris­ing two real-val­ued sig­nals) onto a fre­quen­cy $\omega$, we need a cosine and a sine func­tion of $\omega$, as giv­en in (2). Usu­al­ly, one would use the same source for both, which out­puts, say, the cosine oscil­la­tion at $\omega$ (the light ampli­tude, being a real-world elec­tric field, must always be real-val­ued - work­ing with the com­plex expo­nen­tial func­tion makes one for­get this some­times). To get the sine oscil­la­tion, we sim­ply need to shift the phase of the cosine sig­nal by $\pi/2$ (see Fig. 6). The opti­cal path length need­ed to achieve such a phase shift is

$$L_{\pi/2}(\omega) = \frac{\lambda(\omega)}{4 n} \approx \frac{0.4\,\mu\text{m}}{n}$$

where $\lambda(\omega) = 2\pi \, c / \omega$ is the free-space wave­length and $n$ is the refrac­tive index of the wave­guide used. Such small opti­cal path lengths are usu­al­ly imple­ment­ed by just heat­ing a wave­guide or using the elec­tro-opti­cal effect to change its refrac­tive index $n$ a bit until it fits. Any­way, the point here is that $\lambda_0$ will be very near­ly the same for all fre­quen­cy com­po­nents of the tiny OFDM band around the cen­tral $\omega$ up to a few dec­i­mals, so we need not wor­ry about the fre­quen­cy depen­dence of $L_{\pi/2}$. With elec­tri­cal upmix­ing, on the oth­er hand, the OFDM band may span an octave or more (a band from some $\omega_\text{min}$ to an $\omega_\text{max} \ge 2 \omega_\text{min}$), which means that $L_{\pi/2}$ should also span an octave or more. Even though my RF knowl­edge is some­what basic, I think this might be a chal­lenge. But that’s not part of this overview.

Fig. 6: Opti­cal IQ-mod­u­la­tor with laser source, direc­tion­al cou­plers, phase shifter, and Mach-Zehn­der mod­u­la­tors. The prin­ci­ple of oper­a­tion is the same as in Fig. 4.

Any­way, opti­cal IQ-mod­u­la­tors have been avail­able for a few years now and are, at least in the labs, com­mon equip­ment. Since there is no longer any redun­dan­cy in the (one-sided) opti­cal spec­trum – we only con­sid­er the right side of the spec­tra in Fig. 5 when the car­ri­er fre­quen­cy is opti­cal, as sort of explained around eq. (6) of this post – the spec­tral effi­cien­cy will be high­er with the opti­cal IQ-mod­u­la­tor by a fac­tor of 2 over the above approach­es, but they are much more expen­sive than “reg­u­lar” Mach-Zehn­der or elec­troab­sorp­tion mod­u­la­tors. Like almost every­where, one has to make a trade-off between per­for­mance and cost.

#### Optically multiplexed OFDM (O-OFDM)

The opti­cal mul­ti­plex­ing of OFDM sig­nals is actu­al­ly fair­ly sim­ple. The demul­ti­plex is some­what more involved, but we will dis­cuss this anoth­er time. The OFDM sym­bol $C(t)$ was defined in eq. (10) of this post as

$$C(t) = \sum_{k=0}^{N-1} C_k(t) = \sum_{k=0}^{N-1} c_k \cdot \exp\bigl(i\omega_k t\bigr)$$

Hence, if we have a bunch of opti­cal oscil­la­tors oper­at­ing at the var­i­ous required sub­car­ri­er fre­quen­cies $\omega_k$, we must “sim­ply” impart the sig­nals $c_k$ onto each of them, syn­chro­nous­ly. This last word here is real­ly impor­tant, since we need to have all sub­chan­nel synced at the receiv­er to per­form the DFT over the OFDM symbol.$^3$ What makes this a lit­tle bit dif­fi­cult is the need for all sub­car­ri­er sources to be locked in fre­quen­cy for a dura­tion of at least a sym­bol, but like­ly much longer, and also con­stant in phase over an OFDM sym­bol length.

There are sev­er­al meth­ods to gen­er­ate the required opti­cal sub­car­ri­ers. The eas­i­est, “brute-force” method is to use $N$ tun­able laser diodes and to lock them to the sub­car­ri­er fre­quen­cies. How­ev­er, with each laser hav­ing finite phase noise and fre­quen­cy drift, it is very dif­fi­cult to keep all phas­es con­stant for any­thing but the short­est OFDM sym­bols – and short sym­bols usu­al­ly means few sub­chan­nels, which makes OFDM rather use­less. Anoth­er approach is to use a sin­gle source and expand its out­put into a fre­quen­cy comb, sep­a­rate the comb into its spec­tral lines, mod­u­late these lines sep­a­rate­ly, and com­bine all these sub­chan­nel sig­nals to form the OFDM sig­nal. This is shown in Fig. 7. There are again sev­er­al meth­ods to gen­er­ate the comb, like mode-locked lasers pos­si­bly suc­ceed­ed by a spec­tral expan­sion using e.g. four-wave mix­ing, or the very nifty recir­cu­lat­ing fre­quen­cy shifter [1], which we shall not dis­cuss at the moment.

Fig. 7: Opti­cal mul­ti­plex­ing of OFDM sub­chan­nels. A spec­tral comb source is demul­ti­plexed into its trib­u­tary spec­tral lines (CW sig­nals) when are then mod­u­lat­ed sep­a­rate­ly at sym­bol rate T = 1/Δf and com­bined to form the OFDM out­put sig­nal.

To mod­u­late all sub­chan­nels inde­pen­dent­ly, an opti­cal trans­mit­ter is need­ed for each sub­chan­nel, includ­ing the nec­es­sary pro­cess­ing elec­tron­ics, dig­i­tal-to-ana­log con­vert­ers, and mod­u­la­tors. The exact require­ments on these parts depend on the mod­u­la­tion for­mat and sym­bol rate used. For the quad­ra­ture ampli­tude mod­u­la­tion for­mats, which offer very high spec­tral effi­cien­cy and are eas­i­ly gen­er­at­ed in the DSP in the E-OFDM meth­ods detailed above, the out­put sig­nals are com­plex-val­ued, both in each sub­chan­nel and in the com­bined OFDM chan­nel. To use these in O-OFDM, we need two DACs and an opti­cal IQ-mod­u­la­tor. Not the cheap­est solu­tion. But since we need to spend a lot of mon­ey on this equip­ment, we at least get to mod­u­late the sub­car­ri­ers at much high­er sym­bol rates. We are no longer lim­it­ed by elec­tron­ics to Mbaud, but eas­i­ly can do Gbaud. These high sym­bol rates and the asso­ci­at­ed large sub­chan­nel spac­ing, on the oth­er hand, make the OFDM sig­nal very sen­si­tive to group-veloc­i­ty dis­per­sion (GVD).

In the lab, we usu­al­ly mod­u­late the same sig­nals onto all even-num­bered and all odd-num­bered sub­chan­nels so that for the exper­i­ments one only needs a total of two trans­mit­ters, but this does not trans­fer into “real” com­mu­ni­ca­tion links.

Anoth­er draw­back of this approach is the pow­er loss in the $N:1$ sub­chan­nel com­bin­er. Such a com­bin­er inher­ent­ly must have very high loss­es (reci­procity is the mag­ic word) and only $1/N$ of the pow­er in each sub­chan­nel will make it into the OFDM sig­nal. Again in the demo lab, when the num­ber of trans­mit­ters is 2 no mat­ter how large $N$, these loss­es will be around 3 dB, which is accept­able. In a 32-sub­chan­nel sys­tem, this would be 15 dB at least. Sure, one can always ampli­fy the sig­nals again, but does so with degrad­ing sig­nal-to-noise ratio.

How­ev­er, O-OFDM is rel­a­tive­ly new and the require­ments shown here may not be the final word.

[[1019]]

1 Please note that this is only one way of struc­tur­ing the OFDM meth­ods that appeared very sen­si­ble to me. Every­one seems to have their own way of doing this and nam­ing things. Fig. 1 is just a ref­er­ence for the remain­der of these posts.

2 At this point, look­ing at the fig­ures, one may won­der where these neg­a­tive fre­quen­cy com­po­nents come from that make the spec­trum either sym­met­ric or not. By def­i­n­i­tion of the OFDM sym­bol,

$$C(t) = \sum_{k=0}^{N-1} c_k \cdot \exp\bigl(i\omega_k t\bigr)$$

there are only pos­i­tive fre­quen­cy com­po­nents in the sig­nal. The rea­son lies in the sam­pling. Since the $N$-point DFT has an out­put con­sist­ing of $N$ time sam­ples, the high­est con­tained fre­quen­cy com­po­nent is, accord­ing to Nyquist,

$$f_\text{Nyquist} = \frac{1}{2\,\Delta t} = \frac{N}{2} \Delta f = \frac{\omega_{N/2-1}}{2\pi}$$

where $1/\Delta t$ and $\Delta f$ are the sam­ple rate and sub­chan­nel spac­ing, respec­tive­ly. It is quite eas­i­ly imag­in­able by look­ing at the counter-clock­wise rotat­ing pha­sor in the com­plex plane. For all fre­quen­cies $\gt \omega_{N/2-1}$, the pha­sor rotates more than a half cir­cle from one sam­ple to the next and it rather looks like it (slow­ly) rotat­ing clock­wise instead, see Fig. 8. At $\omega_{N}$ it would rotate a full cir­cle between sam­ples and this could not be dis­tin­guished from it not rotat­ing at all. This is the under­ly­ing rea­son for the peri­od­ic­i­ty of the spec­trum of a sam­pled sig­nal and the whole alias­ing prob­lem. Any­way, it makes more sense to show any­thing over $\omega_{N/2-1}$ to have neg­a­tive fre­quen­cies, since it makes the whole sym­me­try thing and any esti­ma­tions of required band­width eas­i­er.

Fig. 8: Illus­tra­tion of neg­a­tive fre­quen­cies.

3 One advan­tage of O-OFDM is that (ide­al­ly) we can fix the tim­ing for each sub­chan­nel sep­a­rate­ly (since there is a sep­a­rate mod­u­la­tor for each sub­chan­nel). Hence, in the pres­ence of GVD we can avoid the need for an extra cyclic pre­fix by delay­ing the faster sub­chan­nels right at the trans­mit­ter so that they will arrive simul­ta­ne­ous­ly at the receiv­er where we can then per­form the DFT with­out the need for GVD com­pen­sa­tion or cyclic exten­sions. This does require, how­ev­er, that the sub­chan­nel sym­bols are not sig­nif­i­cant­ly dis­tort­ed by GVD which then lim­its the effec­tive­ness of this approach.
Update: I had a look at this in MATLAB and it seems that the pulse dis­tor­tions do lim­it the effec­tive­ness of this. How­ev­er, when simul­ta­ne­ous­ly pre-com­pen­sat­ing the pulse shapes (in each sub­chan­nel sep­a­rate­ly), this works excel­lent.

[1] S. Chan­drasekhar, X. Liu, B. Zhu, and D. W. Peck­ham, “Trans­mis­sion of a 1.2-Tb/s 24-car­ri­er no-guard-inter­val coher­ent OFDM super­chan­nel over 7200-km of ultra-large-area fiber,” in Euro­pean Con­fer­ence on Opti­cal Com­mu­ni­ca­tion (ECOC), Sep­tem­ber 2009, paper PD2.6.

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