Okay, in reply. Yes, the FFT has been described several times over the last few centuries, but it never really achieved any recognition until Cooley & Tukey described it. (One of these authors actually seemed surprised at the recognition he gained from the FFT ...)As for hardware implementations of the FFT. The CSIRO FFT chip predates the WiFi by a significant amount of time. Probably not the first implementation in hardware, but undoubtedly one of the first ten. The FFT requires significant amounts of logic to implement, and certainly early FPGA's were did not have sufficient resources to implement it. Having grown up with much of this technology evolving as I worked with it, I don't have the impression that chips are (or were) as fast as some people here seem to think. You must remember that the state of the art in 1975 was the 8080 cpu ... roughly 10,000 gates. A decent FFT requires much more than that (and memory chips as well!) Whilst DSP's of the 80's (any computer chip actually) could do semi decent jobs in the FFT arena, they certainly couldn't do several megabytes a second, even for a 256 bin FFT. (Also, back in the late 80's / early 90's, DSP's cost an arm and a leg -- making them somewhat prohibitive for many applications!)

Yes, the technique is OFDM ... principally so that the symbol timing can be made relatively long, thus simplifying/improving the data recovery in the presence of echo's. (Simply integrating the data over time makes this an easy process to implement.) Digital communications really didn't begin to evolve until several technologies had evolved, not just FFT's, but also analog to digital converters that could operate successfully at multi MHz clock rates. In 1980, I purchased several 6 bit flash converters that would work at up to 30 MHz (results were pretty horrible by todays standards ...) --- these cost me $100 each. Now you can easily get a 100 MHz 14 bit converter for a few dollars, and the results are spectacular.
(OFDM requires at least two Analogue to digital converters on the Receive side ... plus high speed sine/cos generators to provide the mixers before the ADC's {to generate the real and imaginary components feeding the FFT ...}

So, what seems easy now (perhaps even obvious) was certainly ground breaking stuff only 20 or so years ago!