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.
.png "subchannel symbols")
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
