Orthogonal Frequency Division Multiplexing continued [Under Modulation >> Digital modulation]
Tutorial from complex2real site gives a graphical example of process of OFDM (using BPSK). In brief, in OFDM, (digital) information signal is split in N streams and transmitted on N subcarriers (one stream on one subcarrier).
Mathematically this process is done with Inverse Fast Fourier Transform (IFFT) on transmitter side and Fast Fourier Transform (FFT) on receiving side.
| Fast Fourier Transform
Mathematician Fourier postulated that it is possible write a function f(x) as a sum of sine and cosine terms of increasing frequency. These summation is know as "Fourier series". Another view to look at this postulation is a regular or periodic signal (function) can be written down as sum of number of sinusoidal signals. Check out digram below as an illustration where a square waveform is tried to be reproduced with addition of sine and cosine waveforms. As more number of frequencies are added, we have more closer waveform matching to input waveform. This relate to Mathematical calculations known as "Fourier Analysis". 
Please check out excellent flash program from fourier-series.com for more visual insight. Mathematically it is possible to calculate characteristics (amplitude, frequency, and phase) of these individual signals. Once the characteristics of individual signals are known, it is in a way possible to extract signal of particular characteristics. Say use frequency filter to filter particular frequency. This mathematical process in a way gives "frequency domain representation" of "time domain" signals. And so the name "Fourier transform". Along with Fourier postulation, Euler's formula related to complex presentation of sinusoidal waveforms is used for simplified calculations. In Discrete FT, instead of input function being continuous, sampled input function is used as input - making the input values finite. Fast FT does the same as DFT, but using better techniques making it faster and so the name FFT. Ref: Wikipedia page on FFT and FFT tutorial from cs.otago.ac.nz |
How come IFFT on transmitter side and FFT on receiver side ?
A non-mathematical explanation would be: in OFDM, what we do on transmitter side is to sum the subcarrier (sinusoidal) frequencies (of various characteristics - result of their individual modulations). Resulting waveform is in a way Fourier series ! This is reverse (or inverse) of FFT and so transmitter side is done using IFFT.
But mathematically, FFT and IFFT are considered as "linear" processes and so reversible. This mean even if we exchange positions of FFT and IFFT - FFT on transmitter side and IFFT on receiver side - we achieve the same. Please check Tutorial from complex2real site for more on it.
FFT or IFFT size relate to time multiplexing of OFDM signals. Certain number of samples are transmitted and then the next set of samples. This number is FFT or IFFT size. Typical numbers being 256 (Fixed WiMAX/802.16), 512/1024 (Mobile WiMAX/802.16e). One set of samples sent in time presentation is known as OFDM symbol. OFDM symbols are separated by "Guard period" (typically copy of end of symbol).
OFDM References: In addition to Wikipedia page on OFDM, check out Thesis from Lawrey, and Complex2Real site OFDM Tutrorial. You may check book Fundamentals of WiMAX by Andrews, Ghosh, and Muhamed for OFDM details from WiMAX point of view.
We will take on OFDM model diagram in later article.
© Copyright Samir Amberkar 2010
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