Optical OFDM - Cyclic Prefix

One of the most com­mon mis­un­der­stand­ings about OFDM that keeps com­ing in most dis­cus­sions on the top­ic is if sub­chan­nels are still orthog­o­nal when there is a cyclic pre­fix involved. But let’s start at the begin­ning…

Dispersion

We have seen in this post that the DFT can be used at the receiv­er to demul­ti­plex the OFDM sig­nal. For an OFDM chan­nel with $N$ sub­chan­nels, we sim­ply take $N$ sam­ples spread even­ly over the sym­bol length $T$ and per­form the DFT on these sam­ples. The $N$ out­puts of the DFT then cor­re­spond to the sig­nals in each sub­chan­nel. This works per­fect­ly in a back-to-back con­fig­u­ra­tion (where we pop the receiv­er input right onto the trans­mit­ter out­put). Here, the sym­bols in all sub­chan­nels are aligned in time and to the DFT win­dow, just as they are out­put by the inverse DFT at the trans­mit­ter. This is shown in Fig. 1a. Now let’s assume that we send the sig­nal over a dis­per­sive chan­nel – i.e. a chan­nel that intro­duces dif­fer­ent group delays in some form or anoth­er. In wire­less appli­ca­tions, there is mul­ti-path dis­per­sion, where frac­tions of the orig­i­nal sig­nal take dif­fer­ent paths through the air to get to the receiv­er, by e.g. being reflect­ed or dif­fract­ed. In mul­ti-mode fibers there is mode dis­per­sion, where light in each mode takes a dif­fer­ent amount of time to get to the receiv­er. In sin­gle-mode opti­cal fibers, there is (almost) no mode dis­per­sion, but chro­mat­ic dis­per­sion caus­es each fre­quen­cy com­po­nent (and thus each sub­chan­nel) to have a dif­fer­ent prop­a­ga­tion veloc­i­ty. In all cas­es, parts of the orig­i­nal OFDM sym­bol appear at the receiv­er out­side the DFT win­dow of length $T$ – either delayed copies or the fast/slow sub­chan­nels. This also means that parts of the neigh­bor­ing sym­bols appear inside the DFT win­dow. This is shown in Fig. 1b and c.

Fig. 1: When a cyclic pre­fix becomes nec­es­sary; a) illus­trates the ide­al align­ment of sym­bols in each sub­chan­nel at the trans­mit­ter out­put (dis­re­gard­ing spec­tral over­lap for now) b) sym­bol align­ment with two-path prop­a­ga­tion, c) sym­bol align­ment with accu­mu­lat­ed chro­mat­ic dis­per­sion. The DFT win­dow is also shown. Parts of neigh­bor­ing sym­bols (hatched) will inter­fere with cal­cu­la­tion of the DFT when­ev­er there is dis­per­sion.

If we let the win­dow $T$ start with the arrival of the fastest sub­chan­nel at $t=0$, the slow­er sub­chan­nels would at this moment still car­ry the remain­ders of the pre­vi­ous sym­bol, since they have been delayed. Thus, these sub­chan­nels would have a sym­bol tran­si­tion with­in the DFT win­dow, and parts of both sym­bols would be mixed in the DFT out­puts cor­re­spond­ing to the respec­tive sub­chan­nels. On the sub­chan­nel lev­el, this leads to inter-sym­bol inter­fer­ence (ISI). Its effect on the con­stel­la­tion dia­gram of a 16QAM-mod­u­lat­ed sub­chan­nel is shown in Fig. 2 for var­i­ous amounts of rel­a­tive time shift between neigh­bor­ing sub­chan­nels.

Fig. 2: Con­stel­la­tion dia­grams of a sin­gle OFDM sub­chan­nel with var­i­ous amounts of time shift between neigh­bor­ing sub­chan­nels (cf. Fig. 1c); a) all sub­chan­nels aligned as in Fig. 1a, b) 0.01 T delay between neigh­bor­ing sub­chan­nels, c) 0.02 T delay between sub­chan­nels. Mod­u­la­tion for­mat with­in each sub­chan­nel was 16QAM, and there were 7 sub­chan­nels total in the OFDM chan­nel.

Anoth­er way to describe the inter­fer­ence that occurs due to dis­per­sion in OFDM chan­nels is by look­ing at the DFT spec­trum of a sin­gle sym­bol, as we did in this post, in Figs. 3 and 4. If a sym­bol tran­si­tion occurs with­in the DFT win­dow for some sub­chan­nel, the DFT spec­trum of that sub­chan­nel will no longer be a sin­gle Dirac delta peak, but will have com­po­nents that are spread out over many neigh­bor­ing sub­chan­nel “slots.” This is shown in Fig. 3 for dif­fer­ent amounts of mis­align­ment between DFT win­dow and sub­chan­nel sym­bols. Due to the nature of this inter­fer­ence it may be called inter-chan­nel inter­fer­ence (ICI).

Fig. 3: Sin­gle-sym­bol OFDM spec­tra of a sin­gle sub­chan­nel; a) ide­al case with no sym­bol tran­si­tion with­in the DFT win­dow, b) sym­bol tran­si­tion at T/4 due to e.g. group veloc­i­ty dis­per­sion, c) sym­bol tran­si­tion at T/2.

How the Cyclic Prefix Works

A way to avoid this is to extend the dura­tion of each sym­bol in Fig. 1 with­out increas­ing the length $T$ of the DFT win­dow at the receiv­er. This is shown in Fig. 4.

Fig. 4: Illus­tra­tion of cyclic pre­fix prin­ci­ple. By mak­ing the OFDM sym­bol longer than the DFT win­dow, the win­dow can be shift­ed so that there is again no inter­fer­ence from neigh­bor­ing sym­bols (hatched) dur­ing cal­cu­la­tion of the DFT.

Such an exten­sion is eas­i­ly done since the con­tri­bu­tion to the OFDM sym­bol of a sin­gle sub­chan­nel $k$ is just a con­tin­u­ous wave with com­plex ampli­tude $c_{k}$ at a car­ri­er fre­quen­cy $\omega_{k}$ – see also equa­tion (10) in the basics post. We sim­ply need to hold ampli­tude $c_k$ of the sub­car­ri­er oscil­la­tion con­stant a bit longer, say $T’$ with $T’ > T$. This reduces the sym­bol rate from the min­i­mum pos­si­ble $R = \Delta f = 1/T$ to the some­what low­er $R’ = 1/T’$. Now some peo­ple will argue that since

$$\Delta f \ne \frac{1}{T’}$$

we no longer have an OFDM sig­nal and we’re just label­ing it OFDM to get fund­ed by using pop­u­lar buzz­words in pro­pos­als. Not so. The sub­chan­nels $k$ are still orthog­o­nal over the inte­gra­tion inter­val $T$, i.e. orthog­o­nal­i­ty con­di­tion (1) from this post still holds. The sub­chan­nel sig­nals with­in that win­dow have not changed (except maybe for a phase shift due to the dis­per­sion – I’ll talk in anoth­er post how that can be fixed) and we still need the DFT of length $T$ at the receiv­er for demul­ti­plex­ing. If we changed the DFT win­dow length to $T’$, the orthog­o­nal­i­ty of the sub­chan­nels would be lost since (1) would be no longer ful­filled. To para­phrase that, since we are still using the DFT to demul­ti­plex the sig­nal, and the DFT essen­tial­ly com­putes the orthog­o­nal­i­ty inte­gral for each sub­chan­nel simul­ta­ne­ous­ly, we have an OFDM sig­nal. The abil­i­ty to trade effec­tive data rate for dis­per­sion robust­ness is actu­al­ly one of the strong points of OFDM and has been made use of for a long time.

As we will see, the addi­tion­al sig­nal pro­cess­ing at the trans­mit­ter required for dis­per­sion robust­ness is very lit­tle com­pared to the pro­cess­ing required for (adap­tive) dis­per­sion com­pen­sa­tion in a coher­ent trans­mit­ter. How­ev­er, when includ­ing the (inverse) DFT oper­a­tions in the Tx and Rx in our com­plex­i­ty cal­cu­la­tion, the sig­nal pro­cess­ing expen­di­ture is almost equal – this was pre­sent­ed very nice­ly by Spinnler at ECOC a while back [1].

Generation of the Cyclic Prefix

In most imple­men­ta­tions, the inverse DFT at the trans­mit­ter is com­put­ed using dig­i­tal sig­nal proces­sors. Since the receiv­er DFT win­dow length is fixed at $T$ (since this deter­mines the sub­chan­nel spac­ing $\Delta f$), we can­not just make the sub­chan­nel sym­bols a bit longer. In each clock cycle, the inverse DFT is com­put­ed over the $N$ sub­chan­nel inputs to gen­er­ate an OFDM sym­bol, and there is no way to com­pute addi­tion­al frac­tion­al sym­bol slots this way. How­ev­er, since on the sub­chan­nel lev­el an OFDM sym­bol con­sists of a full num­ber $k$ of (sam­pled) com­plex oscil­la­tions (see this post), the sam­ples nec­es­sary to extend the sub­chan­nel sym­bol in time will actu­al­ly look like the first few sam­ples of the sub­chan­nel sym­bol due to the peri­od­ic­i­ty of the sine, cosine, and com­plex expo­nen­tial. We can then extend all sub­chan­nels simul­ta­ne­ous­ly by copy­ing the first few sam­ples of the whole inverse DFT-com­put­ed OFDM sym­bol to its end, or prepend­ing the final few sam­ples to its begin­ning, as shown in Fig. 5. This cyclic con­tin­u­a­tion, which gave the cyclic pre­fix its name. Since it’s called pre­fix, I assume that data usu­al­ly gets prepend­ed but I have no idea why it couldn’t be a cyclic post­fix either.

Fig. 5: Illus­tra­tion of the cyclic post­fix. The sym­bols of Fig. 1b in the basics post are extend­ed cycli­cal­ly, since each sub­chan­nel sym­bol con­sists of a full num­ber of sine or cosine oscil­la­tions.

With a cyclic prefix/postfix of a length cor­re­spond­ing at least to the off­set between the slow­est and fastest sub­chan­nel, the received sub­chan­nel will again be unaf­fect­ed by any dis­per­sion, as shown in Fig. 6. Due to the dif­fer­ent start time of the DFT win­dow (in this case shift­ed by $0.07 T$) there a phase shift appears in the con­stel­la­tion dia­gram. This phase shift is dif­fer­ent for each sub­chan­nel – due to the dis­per­sion – but can be eas­i­ly cor­rect­ed.

Fig. 6: Con­stel­la­tion dia­gram of the received sig­nal with the same dis­per­sion as Fig. 2b, but with suf­fi­cient cyclic pre­fix. Trans­mis­sion data rate can be trad­ed for received sig­nal qual­i­ty.

Spectrum

The pow­er spec­tral den­si­ty (PSD) dis­tri­b­u­tion also changes. This occurs since the sym­bol rate with­in the sub­chan­nels is reduced with a cyclic pre­fix while keep­ing the sub­chan­nel fre­quen­cy sep­a­ra­tion $\Delta f$ con­stant. Name­ly, the spec­trum is no longer square as shown in this post, but acquires dips between the sub­chan­nels, as these now over­lap less. This is shown in Fig. 7 for cyclic pre­fix­es of 25 and 50 per­cent of the orig­i­nal sym­bol length $T$. For large cyclic pre­fix­es, the spec­tra start look­ing sim­i­lar to reg­u­lar WDM spec­tra. How­ev­er, the sub­chan­nels still over­lap sig­nif­i­cant­ly, as shown by the dashed line show­ing the PSD of a sin­gle sub­chan­nel. There­fore, the DFT must still be used for demul­ti­plex­ing instead of sim­ple fil­ter­ing, as explained before.

Fig. 7: Spec­tra of OFDM chan­nels with and with­out cyclic pre­fix­es; a) OFDM spec­trum with­out CP, b) spec­trum with CP of T/4, c) spec­trum with CP of T/2. Even with large CP, the sub­chan­nel spec­tra still over­lap sig­nif­i­cant­ly and can only be ide­al­ly recov­ered using the DFT instead of fil­ter­ing. This post describes how the spec­tra were cal­cu­lat­ed.

 

[1] B. Spinnler, “Com­plex­i­ty of algo­rithms for dig­i­tal coher­ent receivers,” in 35th Euro­pean Con­fer­ence of Opti­cal Com­mu­ni­ca­tion (ECOC), Sep­tem­ber 2009, paper 7.3.6.

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  • “…we’re just label­ing it OFDM to get fund­ed by using pop­u­lar buzz­words in pro­pos­als…”

    you can make a post on OFDM real­ly enter­tain­ing!

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