FFT the whole input vs FFT segment
Question #1: Are these 2 methods equivalent?


No, method #1 is taking an FFT of the entire sequence resulting in frequency bin widths of fs/N = 1024/4096 = 0.25 Hz. Method #2 is summing shorter FFTs at a frequency resolution (bin width) of 1024/512 = 2.0 Hz. For #2, if you divide by the number of FFTs after you sum them you are then averaging them. This is often times used to reduce the variance of the result.


  Question #2: Is the frequency resolution for both methods equal to 0.25Hz per bin?


No, they are 0.25 &amp; 2.0 Hz. The resolution, or bin width, of the FFT is fs/N.