In this series:

Recently, while working on some improvements to Nutaq’s orthogonal frequency-division multiplexing (OFDM) reference design, I faced a very frustrating but instructive problem: inter-symbol interference (ISI). ISI is usually generated when transmitting in a multipath fading channel. In this kind of channel, multiple copies of the transmitted signal is received at different time intervals, which causes interference.

For simplicity, let’s assume that we have an OFDM transmission of two OFDM symbols with two loaded subcarriers. In a multipath fading channel, the received signal can be illustrated as follows: Because we have a delayed copy of the original signal coming through our receive antenna, we can see that a portion of the first OFDM symbol creates interference with the second one, thus changing its amplitude and phase. On the low frequency subcarrier, this effect is less dramatic but on the second subcarrier there is no way to recuperate the information. The immediate fix to solve our problem, the cyclic prefix, was first presented by Peled and Ruiz in 1980

 in their paper regarding orthogonality problems in OFDM systems. A cyclic prefix is a copy of a portion of the end of an OFDM symbol which is put in front of each transmitted OFDM symbol: One immediate consequence of this approach is a reduction in data rates. However, it greatly reduces the ISI problem, as shown on the following figure: We can easily see that the cyclic prefix of length TG on the second OFDM symbol absorbs the delayed portion of the first OFDM symbol. By eliminating the cyclic prefix portion of each OFDM symbol before decoding, all the samples processed by the FFT algorithm will belong to only one OFDM symbol, thus solving our ISI problem at the cost of a reduced data rate.

So why am I introducing this result? Simply to mention that ISI is removed if and only if the length of the cyclic prefix TG is long enough! This value usually depends on the channel model used for a specific standard. As an example, in 802.11a, the length of the cyclic prefix is one quarter of an OFDM symbol (and the value used in Nutaq’s OFDM reference design).  Using this value, one should have no problem doing a simple loopback over the air, right? Not exactly… ISI can also come from your design, via finite impulse response (FIR) filters.

If you have questions, don't worry – everything will be clearer after reading part 2 of this blog series.

## Reference

 Peled, Abraham, and Antonio Ruiz. "Frequency domain data transmission using reduced computational complexity algorithms." Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP'80.. Vol. 5. IEEE, 1980.