## Optical OFDM - Blessings and Curses

Due to the time-dis­trib­uted nature of the sub­chan­nel sym­bols and the unique algo­rithm used to demul­ti­plex the com­pound sig­nal, the influ­ence of var­i­ous trans­mis­sion impair­ments on the sub­chan­nel sig­nals is very dif­fer­ent from what it is in sin­gle-chan­nel sys­tems. As we will see, this makes OFDM much eas­i­er to ana­lyze and pre­dict its per­for­mance, but can also cause sig­nif­i­cant­ly worse impair­ments than in sin­gle-chan­nel trans­mis­sion in cer­tain cir­cum­stances. Curi­ous­ly, I have not seen this explored in more detail in the lit­er­a­ture on opti­cal OFDM.

Any­way, as so often, the secret lies in the sta­tis­tics. A prop­er­ty that is very par­tic­u­lar to OFDM is that each sub­chan­nel sym­bol is spread out over a large time and must be recon­struct­ed (via the FFT) from a large num­ber of sam­ples tak­en at the receiv­er. In this post, we had writ­ten this as

$$b_n = \frac{1}{N} \sum_{m=0}^{N-1} C\biggl(m\frac{T}{N}\biggr) \exp\biggl(-i 2 \pi \, \frac{n\cdot m}{N}\biggr)\tag{1}$$

where $b_n$ is sym­bol in the $n$th sub­chan­nel extract­ed from a par­tic­u­lar OFDM sym­bol of length $T$, $N$ is the num­ber of sub­chan­nels, and $C(t)$ is the received field. Hence, $b_n$ is the result of an aver­ag­ing process with com­plex weights of unit mod­u­lus (the expo­nen­tial func­tion). Now, gen­er­al­ly $C(t)$ will com­prise the actu­al sig­nal, which may or may not be dis­tort­ed in some way, plus some devi­a­tion from e.g. noise, so we may write

$$C(t) = C^{\,\prime}(t) + \mathcal{N}(t)\tag{2}$$

where $\mathcal{N}(t)$ describes the devi­a­tion, which in the case of opti­cal ampli­fi­er noise is white Gauss­ian noise. But we are not lim­it­ed to Gauss­ian devi­a­tions. A typ­i­cal non-Gauss­ian noise source for OFDM sig­nals is clip­ping noise which is cre­at­ed when the OFDM sig­nal is lim­it­ed to some max­i­mum ampli­tude and every­thing above that ampli­tude is “clipped off.” Now using (2) in (1) yields

\begin{aligned} b_n &= b’_n + \frac{1}{N} \sum_{m=0}^{N-1} \mathcal{N}\biggl(m\frac{T}{N}\biggr) \exp\biggl(-i 2 \pi \, \frac{n\cdot m}{N}\biggr)\\ &= b’_n + \bigl\langle \mathcal{N_n} \bigr\rangle\tag{3} \end{aligned}

where $b’_n$ is the orig­i­nal, dis­tor­tion-free sub­chan­nel sym­bol. We see that the dis­tor­tion con­tri­bu­tion to $b_n$ is the aver­age of the dis­tor­tion of all sam­ples belong­ing to the OFDM sym­bol, again with some unit-mag­ni­tude com­plex weights which are spe­cif­ic to each sub­chan­nel $n$.$^1$ Accord­ing to our favorite sta­tis­ti­cal law, the cen­tral lim­it the­o­rem, the dis­tri­b­u­tion of $\bigl\langle \mathcal{N_n} \bigr\rangle$ will be (two-dimen­sion­al) Gauss­ian for $N \gt 10$ (approx­i­mate­ly). Big deal, you may say, ASE noise is approx­i­mate­ly Gauss­ian any­way. Yes, but many dis­tor­tions are not. Clip­ping, men­tioned above, is one. Quan­ti­za­tion noise is anoth­er (it has a max­i­mum ampli­tude). Sig­nal dis­tor­tion by the non­lin­ear Mach-Zehn­der mod­u­la­tor char­ac­ter­is­tic is still anoth­er impair­ment that will be con­vert­ed to Gauss­ian noise in the OFDM sub­chan­nels. And that is just the trans­mit­ter.

To illus­trate the point, we can com­pare a cou­ple of con­stel­la­tion dia­grams of sin­gle-chan­nel (SC) and OFDM sys­tems. Start­ing with Fig. 1, we have two sig­nals with the same opti­cal pow­er and add the same amount of white Gauss­ian noise to both. This should give us (ide­al­ly) the same con­stel­la­tion dia­grams for SC and OFDM. How­ev­er, the SC sys­tem is RZ-mod­u­lat­ed which leads to a larg­er dis­tance between the points in the constellation.$^2$ Addi­tion­al­ly, the OFDM sig­nal has been clipped at the trans­mit­ter, which increas­es the vari­ance of the noise $\bigl\langle\mathcal{N_n}\bigr\rangle$ of each sym­bol, which does how­ev­er remain Gauss­ian.

Fig.1: Con­stel­la­tion dia­grams of sin­gle-chan­nel (SC) and OFDM sig­nals with equal opti­cal pow­er and equal added noise pow­er. SC is RZ-mod­u­lat­ed, increas­ing the dis­tance between con­stel­la­tion points. OFDM sig­nal was addi­tion­al­ly clipped, increas­ing its noise vari­ance.

Anoth­er impor­tant impair­ment in fiber-optic trans­mis­sion stems from non­lin­ear effects. To illus­trate this I used a very crude fiber mod­el that just iter­ates a loop in which the addi­tion of white Gauss­ian (ampli­fi­er) noise is alter­nat­ed with an inten­si­ty-depen­dent non­lin­ear phase shift – sort of a dis­per­sion-free fiber – and sent our two sig­nals through. Fig. 2 shows the results. On the left, the SC sig­nal clear­ly shows the inten­si­ty-depen­dent dis­tor­tion of the con­stel­la­tion dia­gram, where the out­er con­stel­la­tion points have been rotat­ed. Also, for the out­er points we can see some signs of (asym­met­ric) non­lin­ear phase noise. The OFDM sig­nal, on the oth­er hand, does see an com­mon non­lin­ear rota­tion of all con­stel­la­tion points – described by the mean of $\bigl\langle\mathcal{N_n}\bigr\rangle$ – but the dif­fer­ent phase shifts for high-lev­el and low-lev­el parts of the sig­nal have been con­vert­ed into (a con­sid­er­able amount of) Gauss­ian noise. Hence, the Gauss­ian aver­ag­ing works even for qua­si-deter­min­is­tic effects like non­lin­ear­i­ty. Both sig­nals would be dif­fi­cult to decode with­out errors unless some form of non­lin­ear­i­ty com­pen­sa­tion is per­formed (the OFDM sig­nal must be com­pen­sat­ed before demul­ti­plex­ing).

Fig.2: Con­stel­la­tion dia­grams of sin­gle-chan­nel (SC) and OFDM sig­nals with equal opti­cal pow­er and equal added noise pow­er, after (dis­trib­uted) non­lin­ear phase rota­tion. The vari­a­tion in the indi­vid­ual points of the SC sig­nal is con­vert­ed into uni­form­ly dis­trib­uted Gauss­ian noise in the OFDM sig­nal.

A dif­fer­ent pic­ture presents itself when we use QPSK mod­u­la­tion instead of 16-QAM. Since in the SC sig­nal, all sam­pling points now have approx­i­mate­ly the same ampli­tude, there is a sig­nif­i­cant non­lin­ear rota­tion of the con­stel­la­tion, but all con­stel­la­tion points are rotat­ed equal­ly. Sim­ple dif­fer­en­tial encod­ing can get rid of that impair­ment. The noise is near­ly unchanged from Fig. 2. The mul­ti­plexed (and trans­mit­ted) OFDM sig­nal, how­ev­er, still con­sists of a mul­ti­tude of dif­fer­ent ampli­tude sam­ples, even when the sub­chan­nel mod­u­la­tion is QPSK, which incurs the same lev­el of non­lin­ear noise and aver­age con­stel­la­tion rota­tion as for the 16-QAM sig­nal, mak­ing OFDM unat­trac­tive for non­lin­ear­ly impaired trans­mis­sion (at least for QPSK mod­u­la­tion).

Fig.3: Same as Fig. 2 for QPSK mod­u­la­tion, keep­ing aver­age opti­cal pow­er con­stant. The vari­a­tion of the SC con­stel­la­tion is gone due to all con­stel­la­tion points hav­ing equal pow­er, while the Gauss­ian noise vari­ance (and mean rota­tion) of the OFDM sig­nal has remained the same.

As a side note, Xiang Liu of Bell Labs USA recent­ly devot­ed a por­tion of his post-dead­line ECOC pre­sen­ta­tion to the demon­stra­tion of the noise in an OFDM sig­nal being Gauss­ian [1], antic­i­pat­ing (if not inspir­ing) this blog post. He not­ed that the Gauss­ian dis­tri­b­u­tion of post-mul­ti­plex­ing noise enables the use of soft-deci­sion FEC, which has a much bet­ter per­for­mance than reg­u­lar hard-deci­sion FEC and allowed for trans­mis­sion dis­tances of up to 1600 km.

And as a final side note, the above also hints at why clip­ping, while being a noise source at the trans­mit­ter, can actu­al­ly improve the received sig­nal. By reduc­ing the vari­a­tion of the sig­nal ampli­tudes and thus the vari­a­tion of the non­lin­ear­i­ty-relat­ed phase rota­tion, the relat­ed noise in the OFDM sig­nal is also reduced. This of course only works as long as dis­per­sion does not regen­er­ate the clipped peaks, which hap­pens very fast in broad­band OFDM sig­nals. Maybe I should do a post on clip­ping one day…

UPDATE: Some time after writ­ing this entry, I noticed that Jean Arm­strong did men­tion that the noise in OFDM sub­chan­nels is Gauss­ian even when the noise added to the com­pound OFDM sig­nal is not – in her very nice overview of opti­cal OFDM [1] which any­one work­ing on opti­cal OFDM real­ly should have read. She set the lim­it for that to hap­pen at $N \ge 64$ sub­car­ri­ers, which I think is a bit much if all the sub­car­ri­ers have approx­i­mate­ly equal pow­er. Since the OFDM sig­nals above use $N = 256$, we’re safe either way.

[1] J. Arm­strong, “OFDM for opti­cal com­mu­ni­ca­tions,” Jour­nal of Light­wave Tech­nol­o­gy, vol. 27, no. 3, pp. 189-204, Feb 2009.

1 One should not make the mis­take to believe that this noise aver­ag­ing improves the sig­nal-to-noise ratio in some way – the mean of $\mathcal{N}(t)$ may be zero or near-zero and one may intu­itive­ly think that we approach this mean as the aver­ag­ing win­dow length $N$ increas­es. The $1/N$ fac­tor in (3) might sug­gest some­thing like that. How­ev­er, the same fac­tor mul­ti­plies the usable part of the sig­nal $C^{\,\prime}$ so that the sig­nal-to-noise ratio does not ben­e­fit.

2 The nec­es­sary opti­cal band­width to trans­mit the RZ sig­nal is how­ev­er much high­er and the RZ sig­nal would nor­mal­ly incur more noise if both sys­tems were oper­at­ing with the same noise spec­tral den­si­ty. We shall dis­re­gard this for the pur­pose of the fig­ures here­in as this is not a post on the mer­its of SC vs. OFDM trans­mis­sion.

[1] X. Liu, S. Chan­drasekhar, P. J. Winz­er, S. Drav­ing, J. Evan­ge­lista, N. Hoff­man, B. Zhu, and D. W. Peck­ham, “Sin­gle coher­ent detec­tion of a 606-Gb/s CO-OFDM sig­nal with 32-QAM sub­car­ri­er mod­u­la­tion using 4x 80-Gsam­ples/s ADCs,” in Euro­pean Con­fer­ence on Opti­cal Com­mu­ni­ca­tion (ECOC), Sep 2010, paper PD2.6.

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#### One Comment

• Cool posts mate!
I real­ly liked your thoughts on the form of the spec­trum of OFDM sig­nals!
keep writ­ing!
Viele Grüße aus Oz