FFT Overlap in Matlab
Matlab normalizes in the inverse Fourier transform, so if you want to calculate the fft accurately, you have to normalize by the fft vector by the lenght of the fft itself.

This matlab script will demonstrate the normalization. There is a 1 seconds long 1000 Hz 
pure sine wave with amplitude 1. We will sample it with fs = 6000 Hz, and calculate the spectrum to it in 60 points.

After plotting, we expect two pure spikes at 1000 and -1000 Hz with amplitude 0.5.

fs = 6e3;
T = 1;
f = 1000;
x = sin(2*pi*f*(0:1/fs:T-1/fs));
Nfft = 60;
X = fft(x,Nfft)./Nfft;
stem((0:Nfft-1)*fs/Nfft-fs/2,abs(fftshift(X)))
title('Normalized spectrum')
xlabel('frequency')