# Beginner's guide to OFDM

As the title suggests, I'm a bit of a dummy, and utterly perplexed by the concept of OFDM, overlapping the carriers at the clock nulls and somehow magically not creating an unusable amount of cross carrier interference. What's the science (or sorcery, as is so often the case with wireless communications) behind this concept?

• I would change the title to something like "Beginner's...." – msm Oct 30 '16 at 10:07

In time domain, OFDM breaks one serial fast bit stream into many parallel slow bit streams. Then, these parallel slow bit streams are multiplied with orthogonal sinusoids, where orthogonality between two sinusoids is defined with a summation over a certain time interval as

$$\sum \limits _{n=0} ^{N-1} \cos 2\pi k f_0 n \cdot \cos 2\pi k' f_0 n = 0$$ when $k \neq k'$.

This process is illustrated in detail in Figure below with the help of an example.

Now for each such individual signal, the multi-path components arrive in not too distant future (just like a low rate stream). Hence, the equalizer design is easy having less spread paths and consequently less interference with future symbols, provided that we find a way to separate the subcarriers at the Rx.

Separating these bits at the Rx is easy: we can correlate (multiply sample by sample and sum) the composite signal with just one subcarrier. Utilizing their orthogonality property, contribution from all other subcarriers will cancel out to zero, while the contribution from the subcarrier with that frequency will pop out, scaled in amplitude by our modulation signal. The same procedure can be repeated for all other subcarriers.

Essentially, this is an operation of Discrete Fourier Transform (DFT) as

$$X[k] =\sum \limits _{n=0} ^{N-1} s[n] e^{-j2\pi \frac{k}{N}n}$$

which, for each $k$, will generate our modulate data.

With respect to frequency domain, the signal at the Rx is a product of the spectra of the Tx signal and the wireless channel. Thus, the channel will allow some frequencies to pass through unharmed while suppressing some others. The low data rate signal needs less manipulation by the Rx to get the original data back due to a narrower spectrum. Essentially for this kind of signal, the channel acts just as a single multiplier that can be equalized through estimating that number and dividing the Rx signal by that number. So the equalization reduces to a single division operation. On the other hand, a high data rate signal needs a lot of Rx processing to equalize the channel.

To solve this problem, what OFDM does in frequency domain is fairly simple. It just segments the available bandwidth into many parallel almost flat channels through utilizing those sinusoidal subcarriers. This slicing of the spectrum is drawn in Figure below, just like slicing a bread.

Hence, equalization for each narrow slice requires just a single division operation, rendering the computational load of the equalizer to a total of $N$ divisions.

• Those are some nice figures! – Robert L. May 5 '17 at 2:20