FFT of two sets of samples vs FFT of sum of samples If I have two sets of samples $a$ and $b$ and I have already calculated $A = FFT(a)$ and $B = FFT(b)$ is it possible to calculate $FFT(a + b)$ without performing another FFT?
Fourier Transform is linear, hence if $\mathcal F[x(t)]=X(f)$ and $a$, $b$ are complex numbers, then:

$$\mathcal{F}[ax(t)+by(t)] = a X(f) + bY(f) $$

So in your case, simply sum the results together.