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arm9 is a type of ARMv5TE, which include dsp instructions. It can do FFT(I think even no dsp instructions FFT also can be done though very slow). What you should study carfully is if it meet your application, not if it can do FFT. Doing FFT ARM even with dsp instructions can not compared to DSP chips like TI320C6000.
I belive the ARM9 is good enough to do a FFT .The TI OMAP Kits and S/w tools are pretty expensive
I general FFT can be even implemented in any GPP when u have a simple algorithm which does FFT with a descent variance with a standard FFT results ( from matlab).
pls figure-out a simple FFT algorithm. As far as i know radix4 FFT is better to implement . the complexity comes in the butterfly design . you must be able to design a general butterfly macro which can acess the twiddles two memory regions( real and imag part) and do a radix4 process with the sampled signal form a circular buffer.
u must read a lot before u start and test u r algo and optimize it with MATLAB or similar tools