Dispersion
We have seen in this post that the DFT can be used at the receiver to demultiplex the OFDM signal. For an OFDM channel with $N$ subchannels, we simply take $N$ samples spread evenly over the symbol length $T$ and perform the DFT on these samples. The $N$ outputs of the DFT then correspond to the signals in each subchannel. This works perfectly in a back-to-back configuration (where we pop the receiver input right onto the transmitter output). Here, the symbols in all subchannels are aligned in time and to the DFT window, just as they are output by the inverse DFT at the transmitter. This is shown in Fig. 1a. Now let’s assume that we send the signal over a dispersive channel – i.e. a channel that introduces different group delays in some form or another. In wireless applications, there is multi-path dispersion, where fractions of the original signal take different paths through the air to get to the receiver, by e.g. being reflected or diffracted. In multi-mode fibers there is mode dispersion, where light in each mode takes a different amount of time to get to the receiver. In single-mode optical fibers, there is (almost) no mode dispersion, but chromatic dispersion causes each frequency component (and thus each subchannel) to have a different propagation velocity. In all cases, parts of the original OFDM symbol appear at the receiver outside the DFT window of length $T$ – either delayed copies or the fast/slow subchannels. This also means that parts of the neighboring symbols appear inside the DFT window. This is shown in Fig. 1b and c.
Fig. 1: When a cyclic prefix becomes necessary; a) illustrates the ideal alignment of symbols in each subchannel at the transmitter output (disregarding spectral overlap for now) b) symbol alignment with two-path propagation, c) symbol alignment with accumulated chromatic dispersion. The DFT window is also shown. Parts of neighboring symbols (hatched) will interfere with calculation of the DFT whenever there is dispersion.
If we let the window $T$ start with the arrival of the fastest subchannel at $t=0$, the slower subchannels would at this moment still carry the remainders of the previous symbol, since they have been delayed. Thus, these subchannels would have a symbol transition within the DFT window, and parts of both symbols would be mixed in the DFT outputs corresponding to the respective subchannels. On the subchannel level, this leads to inter-symbol interference (ISI). Its effect on the constellation diagram of a 16QAM-modulated subchannel is shown in Fig. 2 for various amounts of relative time shift between neighboring subchannels.
Fig. 2: Constellation diagrams of a single OFDM subchannel with various amounts of time shift between neighboring subchannels (cf. Fig. 1c); a) all subchannels aligned as in Fig. 1a, b) 0.01 T delay between neighboring subchannels, c) 0.02 T delay between subchannels. Modulation format within each subchannel was 16QAM, and there were 7 subchannels total in the OFDM channel.
Another way to describe the interference that occurs due to dispersion in OFDM channels is by looking at the DFT spectrum of a single symbol, as we did in this post, in Figs. 3 and 4. If a symbol transition occurs within the DFT window for some subchannel, the DFT spectrum of that subchannel will no longer be a single Dirac delta peak, but will have components that are spread out over many neighboring subchannel “slots.” This is shown in Fig. 3 for different amounts of misalignment between DFT window and subchannel symbols. Due to the nature of this interference it may be called inter-channel interference (ICI).
Fig. 3: Single-symbol OFDM spectra of a single subchannel; a) ideal case with no symbol transition within the DFT window, b) symbol transition at T/4 due to e.g. group velocity dispersion, c) symbol transition at T/2.
How the Cyclic Prefix Works
A way to avoid this is to extend the duration of each symbol in Fig. 1 without increasing the length $T$ of the DFT window at the receiver. This is shown in Fig. 4.
Fig. 4: Illustration of cyclic prefix principle. By making the OFDM symbol longer than the DFT window, the window can be shifted so that there is again no interference from neighboring symbols (hatched) during calculation of the DFT.
Such an extension is easily done since the contribution to the OFDM symbol of a single subchannel $k$ is just a continuous wave with complex amplitude $c_{k}$ at a carrier frequency $\omega_{k}$ – see also equation (10) in the basics post. We simply need to hold amplitude $c_k$ of the subcarrier oscillation constant a bit longer, say $T’$ with $T’ > T$. This reduces the symbol rate from the minimum possible $R = \Delta f = 1/T$ to the somewhat lower $R’ = 1/T’$. Now some people will argue that since
$$\Delta f \ne \frac{1}{T’}$$
we no longer have an OFDM signal and we’re just labeling it OFDM to get funded by using popular buzzwords in proposals. Not so. The subchannels $k$ are still orthogonal over the integration interval $T$, i.e. orthogonality condition (1) from this post still holds. The subchannel signals within that window have not changed (except maybe for a phase shift due to the dispersion – I’ll talk in another post how that can be fixed) and we still need the DFT of length $T$ at the receiver for demultiplexing. If we changed the DFT window length to $T’$, the orthogonality of the subchannels would be lost since (1) would be no longer fulfilled. To paraphrase that, since we are still using the DFT to demultiplex the signal, and the DFT essentially computes the orthogonality integral for each subchannel simultaneously, we have an OFDM signal. The ability to trade effective data rate for dispersion robustness is actually one of the strong points of OFDM and has been made use of for a long time.
As we will see, the additional signal processing at the transmitter required for dispersion robustness is very little compared to the processing required for (adaptive) dispersion compensation in a coherent transmitter. However, when including the (inverse) DFT operations in the Tx and Rx in our complexity calculation, the signal processing expenditure is almost equal – this was presented very nicely by Spinnler at ECOC a while back [1].
Generation of the Cyclic Prefix
In most implementations, the inverse DFT at the transmitter is computed using digital signal processors. Since the receiver DFT window length is fixed at $T$ (since this determines the subchannel spacing $\Delta f$), we cannot just make the subchannel symbols a bit longer. In each clock cycle, the inverse DFT is computed over the $N$ subchannel inputs to generate an OFDM symbol, and there is no way to compute additional fractional symbol slots this way. However, since on the subchannel level an OFDM symbol consists of a full number $k$ of (sampled) complex oscillations (see this post), the samples necessary to extend the subchannel symbol in time will actually look like the first few samples of the subchannel symbol due to the periodicity of the sine, cosine, and complex exponential. We can then extend all subchannels simultaneously by copying the first few samples of the whole inverse DFT-computed OFDM symbol to its end, or prepending the final few samples to its beginning, as shown in Fig. 5. This cyclic continuation, which gave the cyclic prefix its name. Since it’s called prefix, I assume that data usually gets prepended but I have no idea why it couldn’t be a cyclic postfix either.
Fig. 5: Illustration of the cyclic postfix. The symbols of Fig. 1b in the basics post are extended cyclically, since each subchannel symbol consists of a full number of sine or cosine oscillations.
With a cyclic prefix/postfix of a length corresponding at least to the offset between the slowest and fastest subchannel, the received subchannel will again be unaffected by any dispersion, as shown in Fig. 6. Due to the different start time of the DFT window (in this case shifted by $0.07 T$) there a phase shift appears in the constellation diagram. This phase shift is different for each subchannel – due to the dispersion – but can be easily corrected.
Fig. 6: Constellation diagram of the received signal with the same dispersion as Fig. 2b, but with sufficient cyclic prefix. Transmission data rate can be traded for received signal quality.
Spectrum
The power spectral density (PSD) distribution also changes. This occurs since the symbol rate within the subchannels is reduced with a cyclic prefix while keeping the subchannel frequency separation $\Delta f$ constant. Namely, the spectrum is no longer square as shown in this post, but acquires dips between the subchannels, as these now overlap less. This is shown in Fig. 7 for cyclic prefixes of 25 and 50 percent of the original symbol length $T$. For large cyclic prefixes, the spectra start looking similar to regular WDM spectra. However, the subchannels still overlap significantly, as shown by the dashed line showing the PSD of a single subchannel. Therefore, the DFT must still be used for demultiplexing instead of simple filtering, as explained before.
Fig. 7: Spectra of OFDM channels with and without cyclic prefixes; a) OFDM spectrum without CP, b) spectrum with CP of T/4, c) spectrum with CP of T/2. Even with large CP, the subchannel spectra still overlap significantly and can only be ideally recovered using the DFT instead of filtering. This post describes how the spectra were calculated.
[1] B. Spinnler, “Complexity of algorithms for digital coherent receivers,” in 35th European Conference of Optical Communication (ECOC), September 2009, paper 7.3.6.
