## the Art of Nyquist WDM

Nyquist WDM is a (in my opin­ion) quite promis­ing method to gen­er­ate future “super­chan­nels” with bit rates beyond 500 Gbps by con­cate­nat­ing mul­ti­ple chan­nels very close to each oth­er spec­tral­ly after fil­ter­ing each of them very, very tight­ly with a fil­ter that approx­i­mates a rec­tan­gu­lar trans­fer func­tion [1]. Even Chang Liu point­ed out (twice) at ECOC that his no-guard-inter­val OFDM thing is inher­ent­ly com­pat­i­ble with the Nyquist WDM approach. I guess this makes Nyquist WDM a bit of a hype, since the spec­tral con­cate­na­tion of OFDM chan­nels is not new and has been pub­lished as (orthog­o­nal) band mul­ti­plex­ing years ago [2].

Any­way, to be able to put “reg­u­lar” non-OFDM chan­nels very close to each oth­er, their spec­trum needs to be fil­tered very tight­ly. The tight­est pos­si­ble spec­trum which con­tains all infor­ma­tion at the sam­pling points is rec­tan­gu­lar between the (pos­i­tive and neg­a­tive) Nyquist fre­quen­cies – in this case this is the fre­quen­cy $f_\mathrm{Nyquist} = 1 / 2T = f_T / 2$ where $T$ is the sym­bol rate. Every­thing out­side that is in some way redun­dant in a sin­gle chan­nel. The guys from Poli­to achieved a chan­nel cen­ter fre­quen­cy sep­a­ra­tion of $1.1/T$ by using a Fin­is­ar Wave­Shaper device [1] – not quite the cheap­est way to do that, even though it can also add high­pass fil­ter­ing to com­pen­sate for a pos­si­ble inline low­pass characteristic$^1$. Their trans­fer func­tion is shown in Fig. 1.

Fig. 1: Nyquist WDM sin­gle chan­nel fil­ter trans­fer func­tion accord­ing to [1].

A sim­pler and cheap­er way to do that would be to use some real-time pre­pro­cess­ing in the trans­mit­ter or – for those who can’t pro­gram their own FPGAs – an arbi­trary wave­form gen­er­a­tor to demon­strate the con­cept. How­ev­er, elec­tron­ic fil­ter­ing was not much more than a foot­note in the var­i­ous pre­sen­ta­tions on Nyquist WDM at ECOC. It wouldn’t even take much pro­cess­ing. The required wave­forms for each input could be stored in a look-up table and the wave­forms for all the sym­bols then just need to be summed just before being out­put. Sounds easy enough.

#### Symbol Length vs. Channel Spectrum

So what do these wave­forms look like? Well, to obtain a rec­tan­gu­lar sig­nal PSD, we need sinc wave­forms for each sym­bol. The sinc func­tion decays rather slow­ly and extends (ide­al­ly) over infi­nite­ly many sym­bol slots. How­ev­er, we can trun­cate the infi­nite­ly long sym­bols to extend only over a finite num­ber of sym­bol slots (that’s where the sum­ming before the final out­put comes in). The trun­ca­tion will of course affect the spec­trum, which will no longer be rec­tan­gu­lar. It’s quite sim­ple to cal­cu­late the expect­ed PSD using the time-domain sym­bols and the pro­ce­dure out­lined in foot­note 1 of this post. In the time domain, the sig­nal is

$$E(t) = \sum_{n=-\infty}^{\infty} c_n \, \mathrm{sinc}\biggl(\pi\frac{t - nT}{T}\biggr) \cdot \Pi\biggl(\frac{t-nT}{kT}\biggr)\tag{1}$$

where $\mathrm{sinc} x = \sin x / x$, $\Pi(t/\tau)$ is a rec­tan­gu­lar win­dow of width $\tau$ cen­tered on $t=0$, and $c_n$ is the data encod­ed on sym­bol $n$. Hence, the sinc func­tion is trun­cat­ed to a length of $k$ sym­bols. A typ­i­cal out­put sequence $E(t)$ for $c_n \in \lbrace -1, 1 \rbrace$ is shown in Fig. 2, togeth­er with the shape of a sin­gle sym­bol for a sym­bol length of $8T$ ($k=8$). Note that the sym­bol time form is zero at the cen­ter (the ide­al sam­pling point) of all neigh­bor­ing sym­bols.

Fig.2: Typ­i­cal Nyquist WDM time sig­nal as super­po­si­tion of the mod­u­lat­ed sig­nals for mul­ti­ple sym­bols. The sam­pling instants yield­ing the orig­i­nal BPSK sequence are shown as mark­ers. A sin­gle Nyquist WDM sym­bol with sym­bol length 8T is also shown for ref­er­ence.

Giv­en (1), the PSD can be cal­cu­lat­ed as

$$\mathrm{PSD}(f) = \tilde E(f)^* \tilde E(f) \prop­to \Bigl[kT \,\mathrm{sinc}\bigl(\pi kT \cdot f\bigr) * \Pi\bigl(T\cdot f\bigr)\Bigr]^2\tag{2}$$

where the “reg­u­lar” $*$ means con­vo­lu­tion, the super­script $^*$ means com­plex con­ju­ga­tion, and the $\Pi$ func­tion describes a rec­tan­gu­lar win­dow of width $2\pi\,T^{-1}$. The whole thing scales with the aver­age pow­er in the data sym­bols $c_n$, where­fore there is a pro­por­tion­al­i­ty rela­tion instead of an equal­i­ty. I asked trusty old Math­e­mat­i­ca to do the con­vo­lu­tion for me. Fig. 3 shows the so-cal­cu­lat­ed spec­tra for dif­fer­ent val­ues of $k$. Clear­ly, the longer the allot­ted time win­dow over the sinc func­tion, the clos­er the spec­trum will be to rec­tan­gu­lar. How­ev­er, the short­est time win­dow of $4T$ has a spec­trum that is already about as good as the Wave­Shaper of Fig. 1. Also, the spec­tra look sim­i­lar to the OFDM spec­tra in this post, which also become more rec­tan­gu­lar as the num­ber of sub­chan­nels (sam­ples per sym­bol) is increased – by com­par­ing (2) to the math­e­mat­i­cal descrip­tion of an OFDM spec­trum we see that there are fun­da­men­tal dif­fer­ences.

The (rec­tan­gu­lar) win­dow­ing caus­es side­lobes to appear which are about 25 dB below peak. These will cause some crosstalk when pack­ing such chan­nels close togeth­er to form these super­chan­nels. One way to sup­press these side­lobes with­out increas­ing the sym­bol length unnec­es­sar­i­ly is to use a non-rec­tan­gu­lar win­dow func­tion in the time domain. There are many such func­tions out there, some of which are bet­ter than oth­ers. Fig. 4 shows the spec­trum when using the Ham­ming win­dow (my per­son­al favorite for no par­tic­u­lar rea­son)

$$w(t) = 0.54 + 0.46 \cos\biggl(2\pi \frac{t}{k}\biggr) \quad \text{for} \quad -\frac{kT}{2} \le t \lt \frac{kT}{2}\tag{3}$$

The side­lobes are sig­nif­i­cant­ly decreased and crosstalk should be much less of a prob­lem. The win­dow func­tion can be changed as part of the stored wave­forms in the real-time pre­proces­sor and is thus eas­i­ly imple­ment­ed.

Fig.3: Spec­tra of DSP-gen­er­at­ed Nyquist WDM chan­nels using var­i­ous sym­bol lengths (tru­ca­tion of the sinc func­tion), where fT/2 is the Nyquist fre­quen­cy. The ticks on the ver­ti­cal axis have a dis­tance of 10 dB.

Fig.4: Spec­trum of a DSP-gen­er­at­ed Nyquist WDM chan­nel using a sym­bol length of 8T, win­dowed using a Ham­ming win­dow. The ticks on the ver­ti­cal axis have a dis­tance of 10 dB.

#### Sampled Signals

So far we have dealt with con­tin­u­ous sig­nals and talked about stor­ing wave­forms dig­i­tal­ly in an FPGA/ASIC, which doesn’t make much sense. We can only store sam­pled sig­nals, which neces­si­tates slight changes in our rea­son­ing. In short, sam­pling the wave­forms makes the cor­re­spond­ing spec­trum peri­od­ic and these peri­od­ic spec­tra may even over­lap, as results from basic Fouri­er the­o­ry. The free spec­tral range of that peri­od­ic­i­ty depends on the sam­pling rate, the width of the main “lobe” does not (unless we under­sam­ple). We can thus con­trol how much basi­cal­ly “emp­ty” spec­trum appears between two peri­od­ic lobes.

If we sam­ple the sinc wave­forms at the sym­bol rate (where we need­ed only a sin­gle sam­ple per sym­bol since all oth­er sam­ples are zero$^2$) the spec­tra would over­lap or at least “con­nect” and we would get a sin­gle con­tin­u­ous spec­trum whose shape then only depends on the sam­ple func­tion. For rec­tan­gu­lar sam­ples of width $T$, this is shown in Fig. 3 of this post. By over­sam­pling – just as is done in OFDM – we cause the image spec­tra to “dis­con­nect” and cre­ate an arbi­trar­i­ly sized (deter­mined by the amount of over­sam­pling) space, or gap, between the spec­tral lobes which can then be used to remove the unwant­ed image spec­tra using com­mon elec­tri­cal low-pass fil­ters. This prin­ci­ple is shown in Fig. 5. The zero of the envelop­ing sinc func­tion, which results from the rec­tan­gu­lar time sam­ples used, nec­es­sar­i­ly occurs exact­ly in the mid­dle of the image spec­tra. If there is enough room between the main lobes to allow fil­ter­ing (dashed curve), the image spec­tra are com­plete­ly removed (bot­tom part of the fig­ure) and we are left with an almost rec­tan­gu­lar chan­nel spec­trum, obtained with­out any Wave­Shaper devices and only with a sec­ond-order Gauss­ian fil­ter and a bit (1.8×) of over­sam­pling. Steep­er fil­ters need less over­sam­pling, shal­low­er fil­ters (e.g. Bessel-Thom­son fil­ters) may need more. If we had want­ed, we could even have intro­duced a high-pass trans­fer func­tion to pre-com­pen­sate sub­se­quent fil­ters by mod­i­fy­ing the stored wave­form accord­ing­ly.

Fig.5: Spec­tra of DSP-gen­er­at­ed Nyquist WDM chan­nels with finite (1.8×) over­sam­pling (top). Dashed line shows a 2nd-order Gauss­ian fil­ter trans­fer func­tion, bot­tom fig­ure shows the opti­cal­ly fil­tered Nyquist WDM out­put. The ticks on the ver­ti­cal axis have a dis­tance of 10 dB.

#### Implementation

The imple­men­ta­tion in an FPGA/ASIC would be quite straight­for­ward. The sam­pled wave­forms for each sym­bol could be stored in a look-up table, to be read out and added before being out­put. Alter­na­tive­ly, one could just store val­ues of the sinc func­tion and do a bit more pro­cess­ing for each sym­bol, but requir­ing less stor­age space. Over­sam­pling would not need to increase the imple­men­ta­tion com­plex­i­ty sig­nif­i­cant­ly. For 2× over­sam­pling, only the size of the entries in the look-up tables changes. For 1.5× over­sam­pling, we would addi­tion­al­ly need each pos­si­ble sym­bol twice – one ver­sion cen­tered at the sym­bol cen­ter and one appro­pri­ate­ly shift­ed, which we would alter­nate from sym­bol to sym­bol. Oth­er, more odd, over­sam­pling fac­tors would require some­what more exten­sive tables or the use of sinc tables. An FPGA that is capa­ble of gen­er­at­ing OFDM sig­nals should be more than suf­fi­cient for this.

A sim­pler alter­na­tive that would work at least for lab­o­ra­to­ry work would be to pro­gram the cal­cu­lat­ed wave­forms into an arbi­trary wave­form gen­er­a­tor (AWG). Here we would not even be lim­it­ed in the length of the indi­vid­ual sym­bols since these need not be cal­cu­lat­ed in real time. For real-time trans­mis­sion this is not an option, though.

#### Symbol Rate and Spectral Efficiency

The nec­es­sary over­sam­pling comes at the cost of reduced sym­bol rate for a giv­en DAC sam­ple rate since the sam­ple rate deter­mines the spec­tral width that we have con­trol over. We can fill any part of the spec­trum with zeros, but these zeros are poten­tial data that is not being trans­mit­ted. On the oth­er hand we can fill the emp­ty part of the spec­trum with parts of the neigh­bor­ing chan­nels so that over­all we do not sac­ri­fice spec­tral effi­cien­cy. This is what’s so great about Nyquist WDM (even though it shouldn’t be called that when using a Wave­Shaper – then it’s just very dense WDM).

I won­der how long we’ll have to wait until we find an exper­i­men­tal imple­men­ta­tion of this…

1 Sim­ply fil­ter­ing a mod­u­lat­ed sig­nal is how­ev­er not the same as gen­er­at­ing a rec­tan­gu­lar spec­trum. This would only work if the input to the fil­ter was a (Dirac) pulse sequence, which would yield the required sinc sig­nals at its out­put. Since the fil­ter input will in gen­er­al be some mod­u­lat­ed sig­nal, the out­put will be the con­vo­lu­tion of the sinc func­tion with what­ev­er is input, which can become a bit messy, as now the sinc zeros are no longer aligned to the sam­pling points of the neigh­bor­ing sym­bols. This is shown in Fig. 6, in which the sig­nal of Fig. 2 is cre­at­ed by fil­ter­ing 50% RZ-shaped puls­es. Clear­ly, there is some vari­a­tion in the sam­pling lev­els that was not there in Fig. 2.

Fig.6: Nyquist WDM time sig­nal obtained from rec­tan­gu­lar fil­ter­ing of an RZ-mod­u­lat­ed sig­nal.

The fil­ter­ing thing works best when the input puls­es are nar­row and thus the spec­trum very wide. This should be inter­est­ing for the OTDM folks who like to work with very short puls­es…

2 Since the sinc is zero at the sam­pling points of all neigh­bor­ing sym­bols, we only need a sin­gle sam­ple per sym­bol. This should ring a bell, as it is the same as NRZ-mod­u­lat­ed sig­nalling. And indeed, since there is no guard inter­val between the spec­tral images which appear as a result of the sam­pled time sig­nal, all we see is one con­tigu­ous mod­u­lat­ed spec­trum, which is exact­ly what we would expect from either NRZ mod­u­la­tion or Nyquist WDM with­out over­sam­pling.

[1] G. Gavi­o­li, E. Tor­ren­go, G. Bosco, A. Care­na, V. Cur­ri, V. Miot, P. Pog­gi­oli­ni, M. Bel­monte, F. Forghieri, C. Muzio, S. Pici­ac­cia, A. Brin­ciot­ti, A. La Por­ta, C. Lezzi, S. Savory, and S. Abrate, “Inves­ti­ga­tion of the impact of ultra-nar­row car­ri­er spac­ing on the trans­mis­sion of a 10-car­ri­er 1Tb/s super­chan­nel,” in Con­fer­ence on Opti­cal Fiber Com­mu­ni­ca­tion (OFC), March 2010, paper OThD3.
[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.

last posts in Nyquist WDM:

• christian wrote:

Great post. I would add a few sim­u­la­tions with real sig­nals and BER results and pub­lish this.

• Yaisog Bonegnasher wrote:

Yeah, but pub­lish­ing some­thing takes much more time than writ­ing a blog post…

• christian wrote:

you’re a post-doc right? give this to one of your stu­dents and make them do the hard work for you.

• Look­ing at the sig­nals a bit more in detail, we find that the sig­nals with elec­tron­i­cal­ly gen­er­at­ed sym­bols are no longer ful­ly orthog­o­nal, as shown in Fig. 2, after they have been low-pass fil­tered to get rid of the image spec­tra after dig­i­tal-to-ana­log con­ver­sion. They rather resem­ble the Nyquist WDM sig­nals as shown in Fig. 6. Hence, there is no inher­ent advan­tage of using one method over the oth­er, except that we don’t need a Wave­Shaper device for the method pro­posed here (instead we need high-res­o­lu­tion DACs, which are not nec­es­sar­i­ly cheap, either).

• […] or less the final act of my career as an aca­d­e­mic researcher, I put the ideas of the post the Art of Nyquist WDM into a sub­mis­sion to Pho­ton­ics Tech­nol­ogy Let­ters – with a few extra […]

• Oscar Gaete wrote:

for gen­er­at­ing sinc puls­es it is even eas­i­er to per­form an FFT over a par­al­lel set of data, pad some zeros (for over­sam­pling) and then iFFT. You send the sam­ples to the ADC and voila…the data is con­veyed by sinc puls­es. Of course you would still need a guard band or pre­fix because you are gen­er­at­ing in a block/parallel fash­ion.
In my OFC11 paper I added some pics of sinc eye diags ;-) (if you’d like to take a look)
again..cool post