## 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