Optical OFDM - Spectrum

We have seen in this post that the DFT spec­trum of a sin­gle OFDM sym­bol is a line spec­trum with one line for each sub­chan­nel $k$ which car­ries the mod­u­lat­ed data $c_{k}$. With­in a sin­gle sym­bol there is no spec­tral over­lap, which is the rea­son why OFDM works. What’s also inter­est­ing is the long-time (more than a sin­gle sym­bol) spec­trum as seen e.g. by an opti­cal spec­trum ana­lyz­er (OSA) and com­par­ing that to the spec­tra of “reg­u­lar” mod­u­la­tion for­mats as well as between var­i­ous imple­men­ta­tions of OFDM.

The pow­er spec­tral den­si­ty (PSD) of the OFDM chan­nel is shown below in log­a­rith­mic dB units.$^1$ The spec­trum is almost rec­tan­gu­lar, and the sig­nal pow­er is spread very even­ly over the used band­width (sim­i­lar to white noise), which is one of the advan­tages of OFDM. The first out-of-band side­lobes are always ~10.5 dB below the peak PSD value$^2$, which are an arti­fact of the sinc-shaped sub­chan­nel spec­trum. The width of these side­lobes decreas­es with the spec­tral width of each sub­chan­nel and thus with increas­ing sub­chan­nel count, as can be seen in the fig­ure.

Fig. 1: spec­tra of OFDM chan­nels with a dif­fer­ing num­ber of sub­chan­nels at the same data rate (T is inverse­ly pro­por­tion­al to N, as demand­ed by the orthog­o­nal­i­ty con­di­tion).

This near­ly rec­tan­gu­lar shape also allows mul­ti­ple WDM OFDM chan­nels to be locat­ed spec­tral­ly close to each oth­er. How­ev­er, if not fil­tered away (and thus affect­ing a num­ber of the out­side chan­nels) the con­sid­er­able width of the out-of-band OFDM spec­trum impos­es a min­i­mum spec­tral dis­tance – unless of course their sub­chan­nels are mutu­al­ly orthog­o­nal (i.e. all sub­chan­nels of all OFDM chan­nels ful­fill the orthog­o­nal­i­ty con­di­tion). This has been dubbed orthog­o­nal band mul­ti­plexed OFDM and was pub­lished e.g. in [2]. The advan­tage of this approach is that each OFDM band can be gen­er­at­ed by a trans­mit­ter that needs only a frac­tion of the band­width required to gen­er­ate the full-width OFDM band. At the receiv­er, opti­cal fil­ters can sep­a­rate the dif­fer­ent bands with each band being received in a dif­fer­ent low-band­width cir­cuit.

Fig. 2: mul­ti­plex­ing OFDM chan­nels; top: each OFDM chan­nel is fil­tered so as (almost) not to over­lap with its neigh­bor (dot­ted line shows a 5th-order Gauss­ian fil­ter curve); bot­tom: all sub­chan­nels are on an orthog­o­nal fre­quen­cy grid and OFDM bands are thus allowed to over­lap. In this case the OFDM bands may even be put clos­er togeth­er.

Fig. 3 com­pares the (unfil­tered) spec­tra of an OFDM chan­nel and a com­pa­ra­ble (i.e. same total bit rate) “reg­u­lar­ly” mod­u­lat­ed chan­nel. Both are assumed to car­ry the same mod­u­la­tion for­mat and have rec­tan­gu­lar puls­es. The sin­gle chan­nel can be seen as the lim­it of Fig. 1 for a small num­ber of sub­chan­nels. Thus, the side­lobes con­sume a con­sid­er­able part of the spec­trum. How­ev­er, these side­lobes can be fil­tered very gen­er­ous­ly, where­as such fil­ter­ing is detri­men­tal for the orthog­o­nal­i­ty required for the OFDM sub­chan­nels (the sub­chan­nel sym­bols must remain con­stant over the DFT win­dow length $T$).

Fig. 3: Pow­er spec­tral den­si­ty of an OFDM chan­nel and a sin­gle mod­u­lat­ed chan­nel at the same bit rate (each using rec­tan­gu­lar sym­bols).

Final­ly, here is a pic­ture of an “OFDM spec­trum” often seen in the lit­er­a­ture (albeit usu­al­ly more col­or­ful):

Fig. 4: ‘spec­tra’ of the OFDM sub­chan­nels, equal to the trans­fer func­tion of the Fouri­er trans­form. The trans­fer func­tion has zeros at the cen­ter fre­quen­cies of all chan­nels but one, thus is can ide­al­ly demul­ti­plex the line spec­trum of a sin­gle OFDM sym­bol.

This is a graph­i­cal super­po­si­tion of mul­ti­ple sub­chan­nel spec­tra. It is the cor­rect con­tin­u­ous Fouri­er trans­form for a sin­gle sym­bol of each sub­chan­nel by itself, but the com­pos­ite spec­trum of the OFDM sig­nal will not just be the sum of all these curves. The Fouri­er spec­trum of more sym­bols of a sin­gle sub­chan­nel would not even look like this, but would instead be a sin­gle delta peak in the lim­it of infi­nite sym­bols (see foot­note 1 for a deriva­tion). When deal­ing with ran­dom sequences you have to work with the PSD instead of the direct spec­trum. Fur­ther­more, the PSD is real-val­ued and has no neg­a­tive val­ues. The curves in Fig. 4 cor­re­spond also to the trans­fer func­tions of a DFT. With­out going into too much math­e­mat­i­cal detail, the sinc-shaped spec­trum is a result of the rec­tan­gu­lar time win­dow of the DFT in e.g. (12) which is then fre­quen­cy-shift­ed by the expo­nen­tial term. Since the DFT trans­fer func­tion is rough­ly the same as the square root of the PSD of each sub­chan­nel,$^3$ the DFT is the so-called matched fil­ter to ide­ally demul­ti­plex OFDM sig­nals.

1 Work­ing out the con­tin­u­ous spec­trum of a ran­dom­ly mod­u­lat­ed sub­chan­nel or chan­nel is actu­al­ly a bit tricky. Instead of look­ing at a sin­gle OFDM sym­bol, as in (10) from the basics post, we need to look at the sequence of all sym­bols in a sin­gle sub­chan­nel $k$. This can be writ­ten as

$$C_{k}(t) = \sum_{m = -\infty}^{\infty} c_{mk} \cdot \Pi_{T}(t-mT) \exp\bigl(i \omega_{k} [t-mT]\bigr)$$

where $c_{mk}$ is a ran­dom vari­able and $\Pi_{T}(t)$ is a rec­tan­gu­lar win­dow of width $T$, cen­tered on $t=0$. The expo­nen­tial func­tion describes the (unmod­u­lat­ed) OFDM sym­bol shape in sub­chan­nel $k$. Because of the lin­ear­i­ty of the Fouri­er trans­form, we can trans­form each term of the sum sep­a­rate­ly and have

$$\tilde C_{k}(\omega) = \sum_m \tilde C_{mk}(\omega)$$

with

$$\tilde C_{mk}(\omega) = c_{mk} \, \mathrm{sinc}\Bigl(\frac{\omega - \omega_k}{2}\Bigr) \exp\bigl(-i \omega m T\bigr)$$

where we used both the time-shift and fre­quen­cy-shift prop­er­ties of the Fouri­er trans­form. Now the expo­nen­tial term will add a rapid­ly vary­ing phase to the spec­trum for all $\omega \ne 2\pi n / T$ with inte­ger $n$, and the sinc-func­tion has zeros at all $\omega = 2\pi n / T$ except for $n=k$. The phas­es will gen­er­al­ly be dif­fer­ent for all $m$ so that when super­pos­ing infi­nite­ly many sin­gle-sym­bol con­tin­u­ous spec­tra, as required by the sum above, we will have some­thing like a ran­dom walk with very many steps in the com­plex plane. This means that $\tilde C_{mk}(\omega)$ will have a zero aver­age, but $\bigl|\tilde C_{mk}(\omega)\bigr|$ will on aver­age increase with the square root of the num­ber of steps. For each $\omega$, the ran­dom walk will be dif­fer­ent due to the $\omega$-dependent phase term above. And that’s with­out assum­ing any­thing about the sta­tis­tics of the $c_{mk}$. We could obtain some­thing mean­ing­ful out of this by ensem­ble-aver­ag­ing $\bigl|\tilde C_{mk}(\omega)\bigr|$, which would give us some­thing pro­por­tion­al to the sinc-func­tion after suf­fi­cient­ly long aver­ag­ing (actu­al­ly, its absolute val­ue), but hard­ly any­thing like Fig. 4.

We can make use of the known auto­cor­re­la­tion prop­er­ties of the ran­dom vari­able $c_{mk}$ to more eas­i­ly get an equal­ly use­ful quan­ti­ty, the pow­er spec­tral den­si­ty (PSD), which is defined for a sin­gle sub­chan­nel as

$$P_k(\omega) = \tilde C_{k}(\omega) \, \tilde C_{k}^*(\omega)$$

The PSD tells us how much pow­er (or ener­gy, depend­ing how you look at it) is con­tained in an infin­i­tes­i­mal spec­tral slice $d\omega$ and is thus quite use­ful. Insert­ing $\tilde C_{k}(\omega)$ from above, we have

$$P_k(\omega) = \mathrm{sinc}^2\Bigl(\frac{\omega - \omega_k}{2}\Bigr) \sum_m \sum_n c_{mk} c_{nk}^* \exp\bigl(-i \omega \bigl[m - n\bigr] T\bigr)$$

The term $\sum_n c_{mk} c_{nk}^*$ cor­re­sponds to the auto­cor­re­la­tion of the time-dis­crete vari­able $c_{mk}$ which is zero for all $n \ne m$ for a ran­dom vari­able. Hence we have

$$P_k(\omega) = \mathrm{sinc}^2\Bigl(\frac{\omega - \omega_k}{2}\Bigr) \sum_m c_{mk} c_{mk}^*$$

The PSD of a sin­gle sub­chan­nel is thus sinc$^2$-shaped and scales with the ener­gy $\sum_m c_{mk} c_{mk}^*$ in the sig­nal, as it should. If we now add more sub­chan­nels $l$ with inde­pen­dent ran­dom data, the PSD will be sim­ply the sum of the indi­vid­ual sub­chan­nel PSDs (the rea­son being that the cross-cor­re­la­tion between $c_{k}$ and $c_{k}$ is zero for $k\ne l$ and so are the mix­ing terms that would appear in the prod­uct term of the above equa­tion for mul­ti­ple sub­chan­nels). This is what is shown in Fig. 1.

2 Arm­strong in [1] claims that the side­lobes are 13 dB below the peak, which is only true for a sin­gle sub­chan­nel (a sinc func­tion), but not for the super­po­si­tion of mul­ti­ple sub­chan­nels, where all side­lobes add.

3 Actu­al­ly, the DFT trans­fer func­tion must be peri­od­ic as a result of the sam­pling in the time domain. The sinc-shaped trans­fer func­tion of the Fouri­er trans­form thus over­laps with its images in the neigh­bor­ing peri­ods, which is the same as “wrap­ping around” at the peri­od edges. This alters the func­tion shape at the edges of the fre­quen­cy win­dow a bit. This is shown in Fig. 5 for the cen­ter chan­nel, which is large­ly unaf­fect­ed, and an edge chan­nel, which dif­fers sig­nif­i­cant­ly on the oppo­site edge. How­ev­er, for a sin­gle OFDM sym­bol, the trans­fer func­tion zeros will still be at the right places, and every­thing is okay. We just need to take care that there is no sig­nal at the next trans­fer func­tion max­i­mum (shown dashed) because this will inter­fere with our sub­chan­nel of inter­est.

Fig. 5: Illus­tra­tion show­ing the dif­fer­ence between the OFDM sub­chan­nel spec­trum and the DFT trans­fer func­tion; top: for the cen­ter chan­nel of this 16-sub­chan­nel OFDM sig­nal the dif­fer­ence between the sub­chan­nel spec­trum (gray) and the DFT trans­fer func­tion (black) is neg­li­gi­ble; bot­tom: since the DFT trans­fer func­tion ‘wraps around’ at the edges, the dif­fer­ences become more pro­nounced. The peri­od­ic trans­fer func­tion may become prob­lem­at­ic when there are oth­er sig­nals out­side the OFDM main spec­tral lobe/rectangle.

[1] J. Arm­strong, “OFDM for opti­cal com­mu­ni­ca­tions,” Jour­nal of Light­wave Tech­nol­o­gy, vol. 27, no. 3, pp. 189-204, Feb­ru­ary 2009. http://​dx​.doi​.org/​1​0​.​1​1​0​9​/​J​L​T​.​2​0​0​8​.​2​0​1​0​061

[2] W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coher­ent opti­cal OFDM trans­mis­sion over 1000-km SSMF fiber using orthog­o­nal band mul­ti­plex­ing,” Optics Express, vol. 16, no. 9, pp. 6378-6386, April 2008. http://​dx​.doi​.org/​1​0​.​1​3​6​4​/​O​E​.​1​6​.​0​0​6​378

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