# IFFT-based OFDM system

Please explain, mathematically, how can a bank of modulators be replaced with the IFFT in an OFDM system?

Say you want to transmit complex symbols $s_0,s_2,\ldots,s_{N-1}$ using frequency-division multiplexing. In the "bank of modulators" approach, you would assign a carrier frequency $f_k$ to each symbol and create the signal $$x(t) = \sum_{k=0}^{N-1} s_k \exp(j2\pi f_kt).$$

Contrast this with the definition of the inverse discrete Fourier transform: $$x[n] = \sum_{k=0}^{N-1} s_i \exp(j2\pi nk/N).$$

So, if you interpret the symbols as the magnitude and phase of each carrier, and choose the carriers as $f_k=k/N$, then you'll see that you can generate $x(t)$ simply by converting $x[n]$ to an analog signal.

Note that:

• I've explained how to transmit one block of symbols; extending this to multiple blocks has a slight complication and requires inserting a cyclic prefix to $x[n]$.
• I'm assuming a rectangular pulse shape, which results in sinc-shaped carriers in frequency. The beauty of this method is that the carriers are orthogonal even if they overlap. In practice, this means that you can recover the symbols $s_k$ by performing the DFT of $x(t)$.
• It is perfectly possible to use other pulse shapes with this method.

In order to understand the Pilots and Cyclic Prefix insertion you may want to have a look to a simpler OFDM blocks system. I hope this one: http://www.mathworks.com/matlabcentral/fileexchange/42902-lte-16qam in Simulink can help you. Let me know if you have any question about the model :)