The samples are going to come in via the FPGA, so that's why I was considering an FPGA. Trouble is, I don't have a lot of experience with embedded DSP beyond simple FIR filters, so while I get the theory, I am lacking in knowledge of the "tricks" that Dave mentioned for FFTs. I guessed (correctly I suppose?) that DSPs would be easier and there will probably have to be a processor of some sort on the board anyway.
I wish these hybrid ARM/FPGA SoCs weren't in such horrible BGA packages
They seem perfect.
I should clarify: I'm interested in the power spectral density aka real FFT, not all that interested in the phase.