And now for something completely different…

This will be the first of (hopefully) a series of posts that will be an introduction to optical OFDM (or what I have learned in the past half year or so about it). Optical OFDM has been somewhat of a hot topic in recent years in the community, but I think there are not many people who really understand what’s going on, including me. So by writing this down I am hoping to figure it out myself.

I’ll try to use as little math as possible, and will employ an awesome library called MathJax do display it where needed. So if you can’t read any of the math here, I have either configured the server badly or you need a better Internet browser. Also, if you spot any mistake, let me know in the comments.

But enough of the introduction and on with the OFDM…

OFDM means *orthogonal* frequency division multiplex, and it’s a means of **multiplexing** (and **not modulating**) multiple data streams. The difference between OFDM and “regular” optical FDM (or WDM, which, for our purposes, is the same thing) is obviously the orthogonality that must be involved in the former. So, in order to understand OFDM, one must understand what **orthogonality** means. Therefore we will start with a short (and slightly mathematical) reiteration of the principles of orthogonality.

Mathematically, the definition of orthogonality for two symbols $A_n$ and $A_m$ (which are basically functions defined in a time interval $0 \le t \le T$ that describe what the symbol looks like) is quite simple:

$$\frac{1}{T} \intop_0^{T} A_n(t) \cdot A_m^*(t)\, dt = 0 \quad \text{if}\quad m \ne n \tag{1}$$

where $^*$ denotes the complex conjugate and $1/T$ is a normalization that will become useful later, maybe even much later. The time $T$ is the duration of the symbols, and $R = T^{-1}$ is the symbol rate. The orthogonality of these symbols can be used for signal multiplexing based on time (TDM), frequency (FDM), phase (PDM), or polarization (PolDM). Shown below are examples for orthogonal symbols used for TDM and FDM. Phase division multiplex (PDM) is seldom called PDM, but is often seen in the form of IQ modulation, in which both quadratures (the sine and cosine component which are orthogonal in phase) are modulated simultaneously. PolDM makes use of the fact that symbols in orthogonal polarizations are, well, orthogonal (don’t get me started on polarization).

For FDM without PDM, the set of real-valued symbols can be written

$$A_n(t) = a_n \cos(\omega_n t) = a_n \cos\biggl(\frac{2\pi n t}{T_0}\biggr) \tag{2}$$

in which $a_n$ is the modulation data and $T_0$ is the period of the fundamental oscillation (the frequency for $n=1$, defining the frequency grid for the OFDM system). In a description of the transmitted signal, $a_n$ eventually becomes a discrete-time series of data, with double indices and whatnot. We will limit ourselves to a description of the symbols here. The cosine part of (2) determines the symbol shape. The orthogonality condition (1) then becomes

$$\frac{a_n a_m}{T} \intop_0^{T} \cos(\omega_n t) \cdot \cos(\omega_m t) \, dt = 0 \quad \text{if} \quad m \ne n \tag{3}$$

It is a property of the cosine that any two disjunct frequencies $\omega_m = 2\pi m / T_0$ and $\omega_n = 2\pi n / T_0$ are orthogonal over a suitable integration interval $T$, the minimum value of in turn which determines the required symbol length. We find$^1$ that the smallest $T$ for which (3) is fulfilled for arbitrary integers $m$ and $n$ is

$$T = \frac{T_0}{2} \tag{4}$$

That is, each symbol $n$ contains $n$ half oscillations.

Note that a possible phase term is intentionally missing in the argument of the cosine of $A_n$ since the presence of a phase difference between both symbols will destroy their orthogonality.

Since the symbols $A_n$ are real-valued, the modulation $a_n$ must also be real. This leaves pulse amplitude modulation (PAM) as only possible modulation format, with OOK and BPSK being special cases (in BPSK, possible values for $a_n$ are 1 and -1, which are both real). This concept, albeit a bit trivial, has recently been published in [1].

We can, on the other hand, make use of phase-division multiplex (PDM) and symbols that use both quadratures,

$$\begin{aligned}

A_n(t) &= a_n^\text{re} \cos(\omega_n t) - a_n^\text{im} \sin(\omega_n t)\\

&= \bigl|c_n\bigr| \cos\bigl( \omega_n t + \mathrm{arg}[c_n] \bigr)\\

&= \Re\bigl\lbrace c_n \exp(i \omega_n t) \bigr\rbrace \end{aligned}\tag{5}$$

where $c_n = a_n^\text{re} + i a_n^\text{im}$ is now a complex quantity and $\Re$ denotes the real part.$^2$ We define the complex optical field

$$C_n(t) = c_n \exp(i \omega_n t) \quad \text{with} \quad A_n(t) = \Re\bigl\lbrace C_n(t) \bigr\rbrace \tag{6}$$

since operating on the complex field saves us the hassle of dragging the projection $\Re$ onto the real axis around or having to deal with phase-shifted cosine functions. We need to keep in mind, however, that the actual symbols consist of only the real part of $C_n$. Also, this yields correct results only as long as we limit ourselves to linear operations on the field. The relation between $A_n$ and $C_n$ is illustrated in Fig. 2. A consequence of using both quadratures is that the length of the integration interval necessary for orthogonality of the symbols $C_1$ and $C_2$ increases two-fold to

$$T = T_0 \tag{7}$$

or alternatively, for the same symbol length $T$ the necessary frequency spacing between orthogonal frequencies doubles (this can easily be verified by inserting $C_n$ into the orthogonality condition). However, each symbol now carries two modulated data values $a^\text{re}$ and $a^\text{im}$, or equivalently $\bigl|c\bigr|$ and $\mathrm{arg}\bigl[c\bigr]$. Thus, the spectral efficiency remains constant. Furthermore, a possible phase difference between both symbols no longer affects their orthogonality (one can imagine it being simply absorbed into the complex data modulation coefficient $c_n$ which can be recovered at the receiver if the phase shift is known – this is not possible with the single quadrature). The indifference to the subcarrier phase is actually one of the advantages of OFDM over TDM – the latter needs very short optical pulses onto which data is modulated (see above). These pulses are very sensitive to phase variations within their spectrum.

The relation $T = T_0 = \Delta f^{-1}$ with $\Delta f = f_n - f_{n-1}$ above is then the **defining condition** for the symbol length (or frequency spacing) of “true” OFDM with complex modulation (I show in another post how the commonly used cyclic prefix sort of messes this up). If the symbols are shorter or the frequency spacing is chosen smaller, the modulation symbols will not be orthogonal and reception without crosstalk is not possible.

**1** The integral can easily be solved by using the trigonometric identity

$$\cos \theta \cos \varphi = \frac{\cos \bigl(\theta - \phi\bigr) + \cos \bigl( \theta + \phi \bigr)}{2} \tag{8}$$

and then integrating both terms (although I just hacked this into Mathematica).

**2** This relation is obtained by writing the sine and cosine in their exponential form:

\begin{align}

A &= a^\text{re} \cos\bigl(\omega t\bigr) - a^\text{im} \sin\bigl(\omega t\bigr)\\

&= a^\text{re} \frac{\exp\bigl(i\omega t\bigr) + \exp\bigl(-i\omega t\bigr)}{2} - a^\text{im} \frac{\exp\bigl(i\omega t\bigr) - \exp\bigl(-i\omega t\bigr)}{2i} \\

&= \frac{1}{2}\underbrace{\bigl( a^\text{re} + i a^\text{im}\bigr)}_{\textstyle c} \exp\bigl(i\omega t\bigr) + \frac{1}{2} \underbrace{\bigl(a^\text{re} - i a^\text{im}\bigr)}_{\textstyle c^*} \exp\bigl(-i\omega t\bigr)\\

&= \Re \bigl\lbrace c \, \exp\bigl(i\omega t\bigr) \bigr\rbrace \tag{9}

\end{align}

We can also use the following trigonometric identity:

$$a^\text{re} \cos\bigl(\omega t\bigr) - a^\text{im} \sin\bigl(\omega t\bigr) = \sqrt{\bigl(a^\text{re}\bigr)^2 + \bigl(a^\text{im}\bigr)^2} \cos\bigl(\omega t + \varphi\bigr)$$

with

$$\varphi = \text{atan2}\bigl(a^\text{im}, a^\text{re}\bigr) = \arg\bigl(a^\text{re} + i\, a^\text{im}\bigr)$$

which is thus just a different way of expressing (9) when $c = a^\text{re} + i\, a^\text{im}$.

[1] J. Zhao and A. D. Ellis, “A novel optical fast OFDM with reduced channel spacing equal to half of the symbol rate per carrier,” in *Conference on Optical Fiber Communication* (OFC). Optical Society of America, March 2010, paper OMR1.

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brilliant… :-)

your post makes me think one thing…in your 3rd eq. (yep numbering would be nice)… a_n*a_m is outside the integral because within one symbol period they are constant. May be there are other type of pulses that are orthogonal that may allow frequency separation of the subcarriers to be less than half a cosine oscillation…

Cheers!

Thanks. =)

I finally figured out how to add equation numbers manually and added them retroactively.

The data modulation $a_n$ and $a_m$ must per definition be constant within the integration interval, since modulation happens separately with symbol rate $R$ - this distinction between modulation and multiplex is very important, actually. Any pulse shaping beyond the cosine is a function of $t$ would show up as an additional factor in (2) and would then remain within the integral in (3). Such more complex pulse shapes are e.g. the basis for wavelets. But this is a bit beyond the scope of this article…

[…] enlarge) subchannel symbols A0 through A5, each encoding the data ck = 1, where A = Re{C} (see the previous post); the symbols differ by the number of full oscillations they describe. Also shown is the OFDM […]

Can you show the maths to obtain the result (4) from the integral (3)? Thanks.

Actually, you found an inaccuracy here. When I wrote this post, I had the definition $f_n = n$ with integer $n$ in the back of my head, for which things made sense. This wasn’t very clear, though, and I introduced the period $T_0$ of the fundamental symbol oscillation (basically corresponding to my “old” $f_1$) and defined $T$ in terms of that $T_0$ instead of a poorly defined frequency difference. Everything should now be consistent and footnote 1 should be sufficient to figure out how (4) comes about. I also made some small changes throughout the text to improve clarity.