This is my API for digital signal processing (DSP) with ARM Cortex®-M processor using CMSIS DSP software library. (Link for CMSIS DSP library website: http://www.keil.com/pack/doc/CMSIS/DSP/html/index.html)
This library contains DSP modules (example. FFT, window function, FIR filter, IIR filter, LMS filter etc.) which they're build based on using functions in CMSIS DSP library, but they're more friendly to be used.