## Optical OFDM - Orthogonality

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).

Fig. 1: (click to enlarge) exam­ples for pos­si­ble orthog­o­nal sym­bols as used for TDM (left) and FDM (right); any two sym­bols in each col­umn are orthog­o­nal accord­ing to the def­i­n­i­tion above.

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.

Fig. 2: (click to enlarge) rela­tion between a phase-divi­sion mul­ti­plexed sym­bol (also known as IQ mod­u­la­tion) to the com­plex field pha­sor, whose real part is equiv­a­lent to the PDM sym­bol; cal­cu­lat­ing with the com­plex field is often eas­i­er, but will yield the exact same results.

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.

last posts in OFDM:

• GagoX wrote:

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!

• Yaisog Bonegnasher wrote:

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 […]

• Jeffrey wrote:

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

• Marcus wrote:

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.