In this first paper of the series, we present initial results of newly upgraded Giant Meterwave Radio Telescope (uGMRT) observation of European Large-Area ISO Survey-North 1 (ELAIS-N1) at 325 MHz with 32 MHz bandwidth. Precise measurement of fluctuations in Galactic and extragalactic foreground emission as a function of frequency as well as angular scale is necessary for detecting redshifted 21-cm signal of neutral hydrogen from Cosmic Dawn, Epoch of Reionization (EoR) and post-reionization epoch. Here, for the first time we have statistically quantified the Galactic and extragalactic foreground sources in the ELAIS-N1 field in the form of angular power spectrum using the newly developed Tapered Gridded Estimator (TGE). We have calibrated the data with and without direction-dependent calibration techniques. We have demonstrated the effectiveness of TGE against the direction dependent effects by using higher tapering of field of view (FoV). We have found that diffuse Galactic synchrotron emission (DGSE) dominates the sky, after point source subtraction, across the angular multipole range $ 1115 leqslant mathcal{ell} leqslant 5083 $ and $ 1565 leqslant mathcal{ell} leqslant 4754 $ for direction-dependent and -independent calibrated visibilities respectively. The statistical fluctuations in DGSE has been quantified as a power law of the form $mathcal{C}_{mathcal{ell}}= A mathcal{ell}^{-beta} $. The best fitted values of (A, $beta$) are ($ 62 pm 6$ $mK^{2}$, $2.55 pm 0.3 $) and ($ 48 pm 4$ $mK^{2}$, $2.28 pm 0.4 $ ) for the two different calibration approaches. For both the cases, the power law index is consistent with the previous measurements of DGSE in other parts of sky.
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