We have seen in this post that the DFT spectrum of a single OFDM symbol is a line spectrum with one line for each subchannel $k$ which carries the modulated data $c_{k}$. Within a single symbol there is no spectral overlap, which is the reason why OFDM works. What’s also interesting is the long-time (more than a single symbol) spectrum as seen e.g. by an optical spectrum analyzer (OSA) and comparing that to the spectra of “regular” modulation formats as well as between various implementations of OFDM.

The power spectral density (PSD) of the OFDM channel is shown below in logarithmic dB units.$^1$ The spectrum is almost rectangular, and the signal power is spread very evenly over the used bandwidth (similar to white noise), which is one of the advantages of OFDM. The first out-of-band sidelobes are always ~10.5 dB below the peak PSD value$^2$, which are an artifact of the sinc-shaped subchannel spectrum. The width of these sidelobes decreases with the spectral width of each subchannel and thus with increasing subchannel count, as can be seen in the figure.

This nearly rectangular shape also allows multiple WDM OFDM channels to be located spectrally close to each other. However, if not filtered away (and thus affecting a number of the outside channels) the considerable width of the out-of-band OFDM spectrum imposes a minimum spectral distance – unless of course their subchannels are mutually orthogonal (i.e. all subchannels of all OFDM channels fulfill the orthogonality condition). This has been dubbed orthogonal band multiplexed OFDM and was published e.g. in [2]. The advantage of this approach is that each OFDM band can be generated by a transmitter that needs only a fraction of the bandwidth required to generate the full-width OFDM band. At the receiver, optical filters can separate the different bands with each band being received in a different low-bandwidth circuit.

Fig. 3 compares the (unfiltered) spectra of an OFDM channel and a comparable (i.e. same total bit rate) “regularly” modulated channel. Both are assumed to carry the same modulation format and have rectangular pulses. The single channel can be seen as the limit of Fig. 1 for a small number of subchannels. Thus, the sidelobes consume a considerable part of the spectrum. However, these sidelobes can be filtered very generously, whereas such filtering is detrimental for the orthogonality required for the OFDM subchannels (the subchannel symbols must remain constant over the DFT window length $T$).

Finally, here is a picture of an “OFDM spectrum” often seen in the literature (albeit usually more colorful):

This is a graphical superposition of multiple subchannel spectra. It is the correct continuous Fourier transform for a single symbol of each subchannel by itself, but the composite spectrum of the OFDM signal will not just be the sum of all these curves. The Fourier spectrum of more symbols of a single subchannel would not even look like this, but would instead be a single delta peak in the limit of infinite symbols (see footnote 1 for a derivation). When dealing with random sequences you have to work with the PSD instead of the direct spectrum. Furthermore, the PSD is real-valued and has no negative values. The curves in Fig. 4 correspond also to the transfer functions of a DFT. Without going into too much mathematical detail, the sinc-shaped spectrum is a result of the rectangular time window of the DFT in e.g. (12) which is then frequency-shifted by the exponential term. Since the DFT transfer function is roughly the same as the square root of the PSD of each subchannel,$^3$ the DFT is the so-called matched filter to ideally demultiplex OFDM signals.

**1** Working out the continuous spectrum of a randomly modulated subchannel or channel is actually a bit tricky. Instead of looking at a single OFDM symbol, as in (10) from the basics post, we need to look at the sequence of all symbols in a single subchannel $k$. This can be written as

$$C_{k}(t) = \sum_{m = -\infty}^{\infty} c_{mk} \cdot \Pi_{T}(t-mT) \exp\bigl(i \omega_{k} [t-mT]\bigr)$$

where $c_{mk}$ is a random variable and $\Pi_{T}(t)$ is a rectangular window of width $T$, centered on $t=0$. The exponential function describes the (unmodulated) OFDM symbol shape in subchannel $k$. Because of the linearity of the Fourier transform, we can transform each term of the sum separately and have

$$\tilde C_{k}(\omega) = \sum_m \tilde C_{mk}(\omega)$$

with

$$ \tilde C_{mk}(\omega) = c_{mk} \, \mathrm{sinc}\Bigl(\frac{\omega - \omega_k}{2}\Bigr) \exp\bigl(-i \omega m T\bigr)$$

where we used both the time-shift and frequency-shift properties of the Fourier transform. Now the exponential term will add a rapidly varying phase to the spectrum for all $\omega \ne 2\pi n / T$ with integer $n$, and the sinc-function has zeros at all $\omega = 2\pi n / T$ except for $n=k$. The phases will generally be different for all $m$ so that when superposing infinitely many single-symbol continuous spectra, as required by the sum above, we will have something like a random walk with very many steps in the complex plane. This means that $\tilde C_{mk}(\omega)$ will have a zero average, but $\bigl|\tilde C_{mk}(\omega)\bigr|$ will on average increase with the square root of the number of steps. For each $\omega$, the random walk will be different due to the $\omega$-dependent phase term above. And that’s without assuming anything about the statistics of the $c_{mk}$. We could obtain something meaningful out of this by ensemble-averaging $\bigl|\tilde C_{mk}(\omega)\bigr|$, which would give us something proportional to the sinc-function after sufficiently long averaging (actually, its absolute value), but hardly anything like Fig. 4.

We can make use of the known autocorrelation properties of the random variable $c_{mk}$ to more easily get an equally useful quantity, the power spectral density (PSD), which is defined for a single subchannel as

$$P_k(\omega) = \tilde C_{k}(\omega) \, \tilde C_{k}^*(\omega)$$

The PSD tells us how much power (or energy, depending how you look at it) is contained in an infinitesimal spectral slice $d\omega$ and is thus quite useful. Inserting $\tilde C_{k}(\omega)$ from above, we have

$$P_k(\omega) = \mathrm{sinc}^2\Bigl(\frac{\omega - \omega_k}{2}\Bigr) \sum_m \sum_n c_{mk} c_{nk}^* \exp\bigl(-i \omega \bigl[m - n\bigr] T\bigr)$$

The term $\sum_n c_{mk} c_{nk}^*$ corresponds to the autocorrelation of the time-discrete variable $c_{mk}$ which is zero for all $n \ne m$ for a random variable. Hence we have

$$P_k(\omega) = \mathrm{sinc}^2\Bigl(\frac{\omega - \omega_k}{2}\Bigr) \sum_m c_{mk} c_{mk}^*$$

The PSD of a single subchannel is thus sinc$^2$-shaped and scales with the energy $\sum_m c_{mk} c_{mk}^*$ in the signal, as it should. If we now add more subchannels $l$ with independent random data, the PSD will be simply the sum of the individual subchannel PSDs (the reason being that the cross-correlation between $c_{k}$ and $c_{k}$ is zero for $k\ne l$ and so are the mixing terms that would appear in the product term of the above equation for multiple subchannels). This is what is shown in Fig. 1.

**2** Armstrong in [1] claims that the sidelobes are 13 dB below the peak, which is only true for a single subchannel (a sinc function), but not for the superposition of multiple subchannels, where all sidelobes add.

**3** Actually, the DFT transfer function must be periodic as a result of the sampling in the time domain. The sinc-shaped transfer function of the Fourier transform thus overlaps with its images in the neighboring periods, which is the same as “wrapping around” at the period edges. This alters the function shape at the edges of the frequency window a bit. This is shown in Fig. 5 for the center channel, which is largely unaffected, and an edge channel, which differs significantly on the opposite edge. However, for a single OFDM symbol, the transfer function zeros will still be at the right places, and everything is okay. We just need to take care that there is no signal at the next transfer function maximum (shown dashed) because this will interfere with our subchannel of interest.

[1] J. Armstrong, “OFDM for optical communications,” *Journal of Lightwave Technology*, vol. 27, no. 3, pp. 189-204, February 2009. http://dx.doi.org/10.1109/JLT.2008.2010061

[2] W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” *Optics Express*, vol. 16, no. 9, pp. 6378-6386, April 2008. http://dx.doi.org/10.1364/OE.16.006378

last posts in OFDM:

cool. dein blog ist mittlweile ‘ne richtig guten wissensquelle für nachrichtentechnische fragen. gefällt mir. ich würd sagen, das wird der einstieg zu deinem ersten buch, oder?

Zu OFDM gibt’s doch bestimmt schon jede Menge Bücher…

Hi. Your posts are very informative. However, i am not able to see the figures for this one as well as some other posts. I do see the figures in the orthogonality and basics sections. can you please fix that?