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.
