To radiate electromagnetic energy from a single point of a finite difference time domain (FDTD) grid, there are typically two general classes of electromagnetic wave sources; the soft source which consists of impressing a current, and the hard source which consists of impressing an electric field. The physical meaning of the soft source is well understood and its analytical solution is known, whereas there is no analytical solution for the hard source excitation. Nevertheless, many FDTD works utilize the hard source for its practicality. A novel aspect is that the derivation of a field radiated from the hard source towards the free space is identical to the field radiated from the soft source, provided that a certain relationship holds between the source excitations. This provides us with an analytical solution for the field radiated from the hard source. The assessment of accuracy is then considered for a wide band field radiated from a punctual source into frequency-dependent FDTD Debye media. The quantification of the deviation of the waveform observed in the FDTD space from the analytical solution is demonstrated. The numerical experiments with this quantification show that the waveform observed with the soft source excitation matches the one with the hard source excitation when the minimum wavelength to the spatial discretization ratio is greater than 10. It turns out that the soft source outperforms the hard source when the minimum wavelength relative to the spatial discretization is less than 10 in the case of lossless media. Equivalent accuracy is achievable for both lossless and lossy media even when the minimum wavelength to the spatial discretization ratio is lower than 10 due to the loss tangent which absorbs the spurious frequencies related to the numerical noise.