We analyse the Baryon Acoustic Oscillation (BAO) signal of the final Baryon Oscillation Spectroscopic Survey (BOSS) data release (DR12). Our analysis is performed in Fourier-space, using the power spectrum monopole and quadrupole. The dataset includes $1,198,006$ galaxies over the redshift range $0.2 < z < 0.75$. We divide this dataset into three (overlapping) redshift bins with the effective redshifts $zeff = 0.38$, $0.51$ and $0.61$. We demonstrate the reliability of our analysis pipeline using N-body simulations as well as $sim 1000$ MultiDark-Patchy mock catalogues, which mimic the BOSS-DR12 target selection. We apply density field reconstruction to enhance the BAO signal-to-noise ratio. By including the power spectrum quadrupole we can separate the line-of-sight and angular modes, which allows us to constrain the angular diameter distance $D_A(z)$ and the Hubble parameter $H(z)$ separately. We obtain two independent $1.6%$ and $1.5%$ constraints on $D_A(z)$ and $2.9%$ and $2.3%$ constraints on $H(z)$ for the low ($zeff=0.38$) and high ($zeff=0.61$) redshift bin, respectively. We obtain two independent $1%$ and $0.9%$ constraints on the angular averaged distance $D_V(z)$, when ignoring the Alcock-Paczynski effect. The detection significance of the BAO signal is of the order of $8sigma$ (post-reconstruction) for each of the three redshift bins. Our results are in good agreement with the Planck prediction within $Lambda$CDM. This paper is part of a set that analyses the final galaxy clustering dataset from BOSS. The measurements and likelihoods presented here are combined with others in~citet{Alam2016} to produce the final cosmological constraints from BOSS.
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