FFT representation means:
X=FFT{x}
where:
x time domain signal
X frequency domain signal, or spectrum
the Spectrum is a frequency representation of the time domain signal.
I'm not quite shure about the english therms. you shoud check some DSP book to be completely shure.
The weights W are choosen by you. Experimentally or based on some theory. They are just numbers between 0 and 1 and generally they mean that the non-important frequencies are suppressed.
Mainly I'm using Matlab where you may choose between many spectrum estimation technics, FFT, DCT, etc. I never needed to choose the radix, because this can result in a different cpu load (not important when developing the algorithm). When I'm programming a DSP in C I use a general FFT radix 2 algorithm from the TI's library optimized for TMS320C6711. You may obtain it too from their website.
Added after 3 minutes:
P.S. it doesn't mattter in which domain is the decimation you will need a bit reversal procedure anyway.