We use data from the first-year (Y1) observations of the DES collaboration to measure the galaxy angular power spectrum (APS), and search for its BAO feature using a template-fitting method. We test our methodology in a sample of 1800 DES Y1-like mock catalogs. The APS is measured with the pseudo-$C_ell$ method, using pixelized maps constructed from the mock catalogs and the DES mask. The covariance matrix of the $C_ell$s in these tests are also obtained from the mock catalogs. We use templates to model the measured spectra and estimate template parameters firstly from the $C_ell$s of the mocks using two different methods, a maximum likelihood estimator and a MCMC, finding consistent results with a good reduced $chi^2$. Robustness tests are performed to estimate the impact of different choices of settings used in our analysis. After these tests on mocks, we apply our method to a galaxy sample constructed from DES Y1 data specifically for LSS studies. This catalog comprises galaxies within an effective area of 1318 deg$^2$ and $0.6<z<1.0$. We fit the observed spectra with our optimized templates, considering models with and without BAO features. We find that the DES Y1 data favors a model with BAO at the $2.6,sigma$ C.L. with a best-fit shift parameter of $alpha=1.023pm 0.047$. However, the goodness-of-fit is somewhat poor, with $chi^2/$(dof) = 1.49. We identify a possible cause of this issue and show that using a theoretical covariance matrix obtained from $C_ell$s that are better adjusted to data results in an improved value of $chi^2/$(dof) = 1.36 which is similar to the value obtained with the real-space analysis. Our results correspond to a distance measurement of $D_A(z_{rm eff} = 0.81)/r_d = 10.65 pm 0.49$, consistent with the main DES BAO findings. This is a companion paper to the main DES BAO article showing the details of the harmonic space analysis.
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