I was looking for a FFT implementation in C. However, I am not looking for a huge library (like FFTW) but for a easy to use single C-file implementation. Unfortunately I haven't been able to find anything like this.
Can someone recommend a simple implementation?
Your best bet is KissFFT - as its name implies it's simple, but it's still quite respectably fast, and a lot more lightweight than FFTW. It's also free, whereas FFTW requires a hefty licence fee if you want to include it in a commercial product.