In this paper, we use Fourier analysis to study the superconvergence of the semi-discrete discontinuous Galerkin method for scalar linear advection equations in one spatial dimension. The error bounds and asymptotic errors are derived for initial discretization by $L_2$ projection, Gauss-Radau projection, and other projections proposed by Cao et. al. For pedagogical purpose, the errors are computed in two different ways. In the first approach, we compute the difference between the numerical solution and a special interpolation of the exact solution, and show that it consists of an asymptotic error of order $2k+1$ and a transient error of lower order. In the second approach, as by Chalmers and Krivodonova, we compute the error directly by decomposition into physical and nonphysical modes, and obtain agreement with the first approach. We then extend the analysis to vector conservation laws, solved using the Lax-Friedrichs flux. We prove that the superconvergence holds with the same order. The error bounds and asymptotic errors are demonstrated by various numerical experiments for scalar and vector advection equations.