Before we go further into the signal processing topics that started with the cyclic prefix, we shall take a look at how we can generate OFDM signals ready for transmission and then, next, how to get them into the fiber. By now you should have a firm grasp on what an OFDM signal is. If not, take a look here and here.

There is no single, unique way of generating and transmitting OFDM over fiber. Luckily so, since otherwise reading these heaps on OFDM papers would have been really boring. To give an (incomplete) overview of some of the methods, Fig. 1 shows a possible hierarchy that would have made old Aristotle proud$^1$.

At the root “node” OFDM can be separated into optically and electrically multiplexed OFDM. In the electrical version (let’s call that E-OFDM) the multiplexing is usually done by using an inverse FFT in a digital signal processor (DSP). That the inverse FFT can be used for that purpose was shown in this post. In the optical version (O-OFDM) the subchannels are multiplexed optically, generally by combining spectrally overlapping signals in an optical coupler. There are various methods to achieve this, although some only generate “OFDM-like” signals whose subchannels are not all modulated independently.

#### Electrically multiplexed OFDM (E-OFDM)

The output of an inverse FFT is generally complex-valued even if the inputs were real-valued. This is a consequence of the asymmetry of the signal spectrum which is defined by the inputs to the FFT$^2$. This is shown in Fig. 2. Since there is no such thing as a complex electrical current, the DSP will output two real-valued signals, one for the real part and one for the imaginary part of the OFDM signal. These two output signals can be used to directly drive an optical IQ-modulator which can imprint the complex OFDM signal directly onto a continuous light source. The disadvantage of this approach is that the optical components needed are still quite costly, and people started looking for ways to work with only a single electrical drive signal.

##### real-valued baseband signals

The most simple way is to adjust the inputs into the inverse FFT in such a way that corresponding positive and negative frequency components are conjugate complex. The resulting output of the inverse FFT must then be real-valued, as illustrated in Fig. 3 and as can be derived from the definition of the real part of a complex number,

$$c_n(t) \exp\bigl(i \omega_n t\bigr) + c_n^*(t) \exp\bigl(-i \omega_n t\bigr) = 2 \Re \bigl\lbrace c_n(t) \exp\bigl(i \omega_n t\bigr) \bigr\rbrace\tag{1}$$

Hence, when $c_n = c_{-n}^* = c_{N-n}^*$ (the latter equality results from the cyclic nature of the *N*-point DFT, see also footnote 2), the resulting signal will be real-valued. While this is one of the easiest ways to generate real-valued OFDM signals and needs only a single digital-to-analog converter (DAC), it wastes half of the DSP bandwidth. Wasting half the DSP capacity is usually not a good way to set capacity records.

##### radio-frequency IQ-modulation

A more elegant, though slightly more expensive, way to turn complex-valued signals into something real (I stole that from some pop song), is to use an electrical IQ-modulator which “mixes” the OFDM signal up to an intermediate radio frequency. This is basically the same as the optical IQ-modulator, but wholly in the electrical domain. But we didn’t talk about the optical IQ-modulator yet, either, so here’s the principle of IQ-modulation:

To modulate two electrical drive signals, $b^{re}(t)$ and $b^{im}(t)$ onto an intermediate frequency, we use the setup of Fig. 4. Mathematically,

$$I(t) = b^{re}(t) \cos\bigl(\omega t\bigr) - b^{im}(t) \sin\bigl(\omega t\bigr)\tag{2}$$

Now if you have paid attention so far and always read the post footnotes, this will look familiar. Looking at footnote 2 of the first post, we can write the signal $I(t)$ as

$$I(t) = \Re\Bigl\lbrace \Bigl[b^{re}(t) + i\, b^{im}(t)\Bigr] \exp\bigl( i \omega t \bigr) \Bigr\rbrace$$

which should explain the naming of the variables. Obviously, the signal $I(t)$ must be real-valued and can be used to drive an intensity modulator, such as an electroabsorption modulator or field modulator such as a Mach-Zehnder modulator. However, you could also use it to drive a phase modulator or frequency modulator, and this has all been done, I think. The spectrum of $I(t)$ is shown in Fig. 5. While we have used the full capacity of the DSP, we need two DACs (one for $b^{re}$ and one for $b^{im}$) and a large modulator bandwidth due to the intermediate frequency. And of course the IQ-mixer. The modulator bandwidth is used more efficiently by doing the IQ-upmixing directly onto the optical carrier frequency, without an intermediate frequency, but this requires an optical IQ-modulator.

##### optical IQ-modulation

The optical IQ-modulator, shown in Fig. 6, does basically the same as the electrical IQ-modulator and as was described with (2), except that in this case the carrier frequency $\omega$ is not intermediate but the optical frequency of the laser used to transmit the OFDM signal. This is usually around 190 THz or so, and the bandwidth of the OFDM signal is just a tiny fraction of the full spectrum. This makes IQ-modulation a bit easier from the technical viewpoint. Here’s why:

To upmix a complex input signal (comprising two real-valued signals) onto a frequency $\omega$, we need a cosine and a sine function of $\omega$, as given in (2). Usually, one would use the same source for both, which outputs, say, the cosine oscillation at $\omega$ (the light amplitude, being a real-world electric field, must always be real-valued - working with the complex exponential function makes one forget this sometimes). To get the sine oscillation, we simply need to shift the phase of the cosine signal by $\pi/2$ (see Fig. 6). The optical path length needed to achieve such a phase shift is

$$L_{\pi/2}(\omega) = \frac{\lambda(\omega)}{4 n} \approx \frac{0.4\,\mu\text{m}}{n}$$

where $\lambda(\omega) = 2\pi \, c / \omega$ is the free-space wavelength and $n$ is the refractive index of the waveguide used. Such small optical path lengths are usually implemented by just heating a waveguide or using the electro-optical effect to change its refractive index $n$ a bit until it fits. Anyway, the point here is that $\lambda_0$ will be very nearly the same for all frequency components of the tiny OFDM band around the central $\omega$ up to a few decimals, so we need not worry about the frequency dependence of $L_{\pi/2}$. With electrical upmixing, on the other hand, the OFDM band may span an octave or more (a band from some $\omega_\text{min}$ to an $\omega_\text{max} \ge 2 \omega_\text{min}$), which means that $L_{\pi/2}$ should also span an octave or more. Even though my RF knowledge is somewhat basic, I think this might be a challenge. But that’s not part of this overview.

Anyway, optical IQ-modulators have been available for a few years now and are, at least in the labs, common equipment. Since there is no longer any redundancy in the (one-sided) optical spectrum – we only consider the right side of the spectra in Fig. 5 when the carrier frequency is optical, as sort of explained around eq. (6) of this post – the spectral efficiency will be higher with the optical IQ-modulator by a factor of 2 over the above approaches, but they are much more expensive than “regular” Mach-Zehnder or electroabsorption modulators. Like almost everywhere, one has to make a trade-off between performance and cost.

#### Optically multiplexed OFDM (O-OFDM)

The optical multiplexing of OFDM signals is actually fairly simple. The demultiplex is somewhat more involved, but we will discuss this another time. The OFDM symbol $C(t)$ was defined in eq. (10) of this post as

$$C(t) = \sum_{k=0}^{N-1} C_k(t) = \sum_{k=0}^{N-1} c_k \cdot \exp\bigl(i\omega_k t\bigr)$$

Hence, if we have a bunch of optical oscillators operating at the various required subcarrier frequencies $\omega_k$, we must “simply” impart the signals $c_k$ onto each of them, **synchronously**. This last word here is really important, since we need to have all subchannel synced at the receiver to perform the DFT over the OFDM symbol.$^3$ What makes this a little bit difficult is the need for all subcarrier sources to be locked in frequency for a duration of at least a symbol, but likely much longer, and also constant in phase over an OFDM symbol length.

There are several methods to generate the required optical subcarriers. The easiest, “brute-force” method is to use $N$ tunable laser diodes and to lock them to the subcarrier frequencies. However, with each laser having finite phase noise and frequency drift, it is very difficult to keep all phases constant for anything but the shortest OFDM symbols – and short symbols usually means few subchannels, which makes OFDM rather useless. Another approach is to use a single source and expand its output into a frequency comb, separate the comb into its spectral lines, modulate these lines separately, and combine all these subchannel signals to form the OFDM signal. This is shown in Fig. 7. There are again several methods to generate the comb, like mode-locked lasers possibly succeeded by a spectral expansion using e.g. four-wave mixing, or the very nifty recirculating frequency shifter [1], which we shall not discuss at the moment.

To modulate all subchannels independently, an optical transmitter is needed **for each subchannel**, including the necessary processing electronics, digital-to-analog converters, and modulators. The exact requirements on these parts depend on the modulation format and symbol rate used. For the quadrature amplitude modulation formats, which offer very high spectral efficiency and are easily generated in the DSP in the E-OFDM methods detailed above, the output signals are complex-valued, both in each subchannel and in the combined OFDM channel. To use these in O-OFDM, we need two DACs and an optical IQ-modulator. Not the cheapest solution. But since we need to spend a lot of money on this equipment, we at least get to modulate the subcarriers at much higher symbol rates. We are no longer limited by electronics to Mbaud, but easily can do Gbaud. These high symbol rates and the associated large subchannel spacing, on the other hand, make the OFDM signal very sensitive to group-velocity dispersion (GVD).

In the lab, we usually modulate the same signals onto all even-numbered and all odd-numbered subchannels so that for the experiments one only needs a total of two transmitters, but this does not transfer into “real” communication links.

Another drawback of this approach is the power loss in the $N:1$ subchannel combiner. Such a combiner inherently must have very high losses (reciprocity is the magic word) and only $1/N$ of the power in each subchannel will make it into the OFDM signal. Again in the demo lab, when the number of transmitters is 2 no matter how large $N$, these losses will be around 3 dB, which is acceptable. In a 32-subchannel system, this would be 15 dB at least. Sure, one can always amplify the signals again, but does so with degrading signal-to-noise ratio.

However, O-OFDM is relatively new and the requirements shown here may not be the final word.

Read more on OFDM:

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**1** Please note that this is only one way of structuring the OFDM methods that appeared very sensible to me. Everyone seems to have their own way of doing this and naming things. Fig. 1 is just a reference for the remainder of these posts.

**2** At this point, looking at the figures, one may wonder where these negative frequency components come from that make the spectrum either symmetric or not. By definition of the OFDM symbol,

$$C(t) = \sum_{k=0}^{N-1} c_k \cdot \exp\bigl(i\omega_k t\bigr)$$

there are only positive frequency components in the signal. The reason lies in the sampling. Since the $N$-point DFT has an output consisting of $N$ time samples, the highest contained frequency component is, according to Nyquist,

$$f_\text{Nyquist} = \frac{1}{2\,\Delta t} = \frac{N}{2} \Delta f = \frac{\omega_{N/2-1}}{2\pi}$$

where $1/\Delta t$ and $\Delta f$ are the sample rate and subchannel spacing, respectively. It is quite easily imaginable by looking at the counter-clockwise rotating phasor in the complex plane. For all frequencies $\gt \omega_{N/2-1}$, the phasor rotates more than a half circle from one sample to the next and it rather looks like it (slowly) rotating clockwise instead, see Fig. 8. At $\omega_{N}$ it would rotate a full circle between samples and this could not be distinguished from it not rotating at all. This is the underlying reason for the periodicity of the spectrum of a sampled signal and the whole aliasing problem. Anyway, it makes more sense to show anything over $\omega_{N/2-1}$ to have negative frequencies, since it makes the whole symmetry thing and any estimations of required bandwidth easier.

**3** One advantage of O-OFDM is that (ideally) we can fix the timing for each subchannel separately (since there is a separate modulator for each subchannel). Hence, in the presence of GVD we can avoid the need for an extra cyclic prefix by delaying the faster subchannels right at the transmitter so that they will arrive simultaneously at the receiver where we can then perform the DFT without the need for GVD compensation or cyclic extensions. This does require, however, that the subchannel symbols are not significantly distorted by GVD which then limits the effectiveness of this approach.

**Update**: I had a look at this in MATLAB and it seems that the pulse distortions do limit the effectiveness of this. However, when simultaneously pre-compensating the pulse shapes (in each subchannel separately), this works excellent.

[1] S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” in *European Conference on Optical Communication* (ECOC), September 2009, paper PD2.6.

last posts in OFDM:

Einige Links scheinen nicht mehr zu funktionieren. Ganz ober zB …here and here… sowie unter Figure 1 der Link …in this post…

Tolles Tutorial btw.

Danke für den Tip, hab’s verbessert…