## Optical OFDM - OFDM Basics

This post con­tin­ues the intro­duc­tion to opti­cal OFDM that I start­ed here.

In this post, we’ll dis­cuss what an OFDM sym­bol looks like and show, start­ing from the orthog­o­nal­i­ty con­di­tion, that the dis­crete Fouri­er trans­form can be used to demul­ti­plex an OFDM chan­nel into its sub­chan­nels. Using the Fouri­er trans­form, we also take a look at the OFDM spec­trum.

Basi­cal­ly OFDM is just plain old fre­quen­cy-divi­sion mul­ti­plex­ing (FDM) with the orthog­o­nal­i­ty con­di­tion (7), name­ly

$$\Delta f = \frac{1}{T}$$

where $\Delta f$ is the sub­car­ri­er sep­a­ra­tion and $T$ is the sym­bol dura­tion. Since the sym­bol rate $R = 1/T$, we can also write $\Delta f = R$.

In (O)FDM, mul­ti­ple sig­nals are trans­mit­ted using dif­fer­ent car­ri­er fre­quen­cies. This is just like piano music, where each tone rep­re­sents a sub­chan­nel and the notes cor­re­spond to data mod­u­la­tion – in this case sim­ple on-off key­ing (either a piano key is pressed or not). Math­e­mat­i­cal­ly, we can write this for a piano with $N$ keys (or an OFDM chan­nel with $N$ sub­chan­nels) as

$$C(t) = \sum_{k=0}^{N-1} C_k(t) = \sum_{k=0}^{N-1} c_k \cdot \exp\bigl(i\omega_k t\bigr)\tag{10}$$

where the $f_k = \omega_k/2\pi$ ful­fill the orthog­o­nal­i­ty con­di­tion above. Again, the $c_k$ are the actu­al encod­ed data and the $C_k(t)$ are the sub­chan­nel sym­bols. The $C_k(t)$ can either be num­bers in a proces­sor that we use to gen­er­ate our (O)FDM sig­nal or elec­tri­cal / opti­cal field quan­ti­ties which can sim­ply be super­posed as in (10) by com­bin­ing the fields from mul­ti­ple sources. The fig­ure below shows what the sub­chan­nel sym­bols (actu­al­ly, their real part) look like for the first few $k$, and also a super­po­si­tion of some such sym­bols.

Fig. 1a: (click to 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 sym­bol for c1 = c3 = c4 = 1 (bot­tom right). The sym­met­ric shape of each sym­bol is a result of the ck being real-val­ued, a prop­er­ty of the Fouri­er trans­form.

Fig. 1b: (click to enlarge) sub­chan­nel sym­bols with encod­ed data, c1 = 0.5 (ampli­tude mod­u­la­tion), c2 = -1 (inver­sion), c3 = i (phase shift), and the OFDM sym­bol result­ing from their super­po­si­tion. The sym­bol is no longer sym­met­ric due to the com­plex-val­ued spec­trum result­ing from c3.

For music, a trained ear will be able to dif­fer­en­ti­ate the tones and tell which notes were played, or “trans­mit­ted.” Sim­i­lar­ly, a “trained” FDM receiv­er will be able to demul­ti­plex the (O)FDM sig­nal. How does it do that? It can use the known wave­form of the trans­mit­ted sub­chan­nel sym­bols (that is his train­ing) and com­pare the sig­nal $C(t)$ to this wave­form using the orthog­o­nal­i­ty prop­er­ty of the sub­chan­nels,

$$b_n =\frac{1}{T} \intop_0^T C(t) \cdot \exp\bigl(-i \omega_n t\bigr) \quad \text{with} \quad b_n = \begin{cases} c_k & n = k \\ 0 & n \ne k\end{cases} \tag{11}$$

which is the orthog­o­nal­i­ty def­i­n­i­tion (1) that was giv­en in part 1. The inte­gral will be zero for all sub­chan­nels $k$ with $k \ne n$ and will yield the coef­fi­cient $c_k$ when $n = k$. This is illus­trat­ed below for $C(t)$ of Fig. 1 (bot­tom right) and $n = \lbrace 1,2 \rbrace$. For $n=1$, the inte­gral (area under the curve) is clear­ly greater than zero, while for $n=2$ it is (not so clear­ly) zero.

Fig. 2: (click to enlarge) illus­trates the mech­a­nism of extract­ing the orig­i­nal data from the OFDM sym­bol C accord­ing to (11); the coef­fi­cient bn cor­re­sponds to the area under the curve C · Cn, shown gray for n = 1 (top) and n = 2 (bot­tom).

The inte­gral (11) looks sus­pi­cious­ly sim­i­lar to the Fouri­er trans­form (FT) inte­gral, except for its inte­gral lim­its, which are $-\infty$ and $\infty$ in the case of the FT. More exact­ly, this par­tic­u­lar inte­gral is called a short-time Fouri­er trans­form (STFT, with a rec­tan­gu­lar win­dow), which is often used for spec­tro­grams, albeit usu­al­ly with a dif­fer­ent win­dow func­tion. Since we are only inter­est­ed in the Fouri­er coef­fi­cients at the $N$ dis­crete fre­quen­cies $\omega_n$, we can use the dis­crete Fouri­er trans­form (DFT) of order $N$ to cal­cu­late the $b_n$, which is a sim­ple sum instead of an inte­gral:

$$b_n = \frac{1}{N} \sum_{m=0}^{N-1} C(t_m) \exp(-i \omega_n t_m) \tag{12}$$

where the $C(t_m)$ are, per def­i­n­i­tion of the DFT, sam­ples of the received sig­nal at $N$ equidis­tant points with­in the trans­for­ma­tion inter­val $T$, so that $t_m = m \cdot T/N$. Thus, to dif­fer­en­ti­ate $N$ sub­chan­nels, we need to sam­ple the incom­ing OFDM sig­nal with a sam­ple rate that is (at least) $N$ times the sym­bol rate.

If we write the sub­chan­nel car­ri­er fre­quen­cies as $\omega_n = 2 \pi n\cdot \Delta f = 2 \pi n / T$, we can rewrite (12) as

$$b_n = \frac{1}{N} \sum_{m=0}^{N-1} C\biggl(m\frac{T}{N}\biggr) \exp\biggl(-i 2 \pi \, \frac{n\cdot m}{N}\biggr)\tag{13}$$

which is sim­ply a mul­ti­pli­ca­tion of the sig­nal sam­ples with some phase terms. A quick algo­rithm that makes use of the var­i­ous redun­dan­cies in cal­cu­lat­ing (8) for all $n$ simul­ta­ne­ous­ly is called the fast Fouri­er trans­form (FFT). It is this algo­rithm that is used in actu­al OFDM receivers to demul­ti­plex the OFDM chan­nel into its sub­chan­nels, after the incom­ing sig­nal has been sam­pled and dig­i­tized.

We can see a cou­ple of inter­est­ing prop­er­ties of an OFDM sig­nal by look­ing at its spec­trum. To obtain this, we need only solve the Fouri­er trans­form (6) for all $\omega$, not just the $\omega_n$. We’ll do it for a sin­gle sym­bol of a sin­gle sub­chan­nel first, with $C(t) = c_n \exp (i \omega_n t)$. The result­ing, sinc-shaped, spec­trum is shown below.$^1$ The sinc func­tion shape is a result of the rec­tan­gu­lar enve­lope of the puls­es that are used to mod­u­late the sub­chan­nel, which lead to the rec­tan­gu­lar win­dow in the STFT (11). Clear­ly (hope­ful­ly), the spec­trum has zeros at all $\omega_k$ except for $k = n$. Hence, the DFT spec­trum, which only gives the spec­tral con­tent at the $\omega_k$, will only have a sin­gle peak. This val­ue of this peak is (at least pro­por­tion­al to) the desired out­put $c_n$.

Fig. 3: (click to enlarge) spec­tra of a sin­gle sub­chan­nel, n = 2; con­tin­u­ous Fouri­er trans­form (top) and dis­crete Fouri­er trans­form (bot­tom).

Since the Fouri­er trans­form is lin­ear, we can deter­mine the spec­trum of a sin­gle sym­bol of the whole chan­nel by sim­ply adding the (com­plex) spec­tra of all sub­chan­nels. This is shown below for PAM-mod­u­lat­ed subchannels$^2$ (in which the $c_n$ are real-val­ued).

Fig. 4: (click to enlarge) wave­forms (top) and spec­tra (bot­tom) of the OFDM sym­bol with c1 = c4 = 1, c2 = 0.5, c3 = 0.75, and c5 = 0.25; con­tin­u­ous (left) and dis­crete (right) case. Note that the spec­trum of the N-point DFT is peri­od­ic with peri­od N, as has been hint­ed at.

Ini­tial­ly, here was a para­graph on the long-time spec­trum of an OFDM chan­nel (encom­pass­ing more than a sin­gle sym­bol), but I think I’ll turn this into a post of its own, even­tu­al­ly. I guess this is quite enough for a first entry on OFDM. Hope it wasn’t too con­fus­ing. By the way, the graph­ics look a bit bet­ter this time, because I made the actu­al curves with Math­e­mat­i­ca which is real­ly great for such stuff…

Note Equa­tion (10) describ­ing the trans­mit­ted sym­bol $C(t)$ also almost looks like an inverse DFT, except that the func­tions which are summed over are con­tin­u­ous. Since we only need $N$ sam­ples of the sym­bol for demul­ti­plex­ing, we could just gen­er­ate those $N$ sam­ples using the inverse DFT. This is what is usu­al­ly done in DSP proces­sors at the trans­mit­ter. These sam­ples are then con­vert­ed to ana­log wave­forms approx­i­mat­ing (10) using a dig­i­tal-to-ana­log con­vert­er (DAC) and sent over the fiber.
1 To obtain the real-val­ued sinc func­tion in this trans­for­ma­tion, we need to change the trans­for­ma­tion inter­val bound­aries to $(-T/2, T/2)$ (Her­mit­ian sym­met­ric func­tions have a real-val­ued Fouri­er trans­form). Oth­er­wise there will be a lin­ear phase super­posed on the sinc as a result of the time shift.
2 PAM stands for pulse ampli­tude mod­u­la­tion

last posts in OFDM:

• sehr schön. da hab ich ja mal ne lek­türe für ruhige stun­den. freu mich schon drauf, das mal zu lesen.

• He, es gibt ja wieder mal was inter­es­santes auss­er immer nur links :-) Sehr schön, OFDM wollte ich schon immer mal ver­ste­hen…

Btw, euer Sender ist aber nicht wirk­lich OFDM oder? ;-) Jeden­falls sehe ich da keine Orthog­o­nal­ität der Sub­träger am Sender.

Freu mich schon aufs Wieder­se­hen näch­ste Woche,
JoE

• Yaisog Bonegnasher wrote:

Das kommt auf die Def­i­n­i­tion von OFDM an… ein­er der näch­stens Posts wird sich damit beschäfti­gen. =)

• […] 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, […]

• Brian wrote:

Hi,
Thank you for the good infor­ma­tion.
I have a ques­tion about the phase infor­ma­tion in the spec­trum of an indi­vid­ual sub­chan­nel.
Q. Is phase infor­ma­tion of an indi­vid­ual sub­chan­nel shown in the spec­trum? How are two dif­fer­ent sub­chan­nels con­tain­ing dif­fer­ent phas­es dis­tin­guished from each oth­er when their cen­ter fre­quen­cies are the same?