In this study, we demonstrate two different methods to generate one-dimensional photonic crystals (PCs) based on aperiodic dielectric gratings and investigate their optical filtering characteristics. The first grating design approach relies on a function formed by the summation of two cosine functions that exhibit different spatial frequencies corresponding to predefined reciprocal lattice vectors (RLVs) and hence generating a grating function by placing the refractive index layer boundaries at the zero-crossing locations of this function. The second design approach starts with the discretization of the total grating thickness with layers of equal thicknesses and each layer's refractive index is selected to maximize the magnitude of the Fourier transform of the grating function at the spatial frequency locations corresponding to predefined RLVs. The non-periodic dielectric multilayers are exposed to time domain calculations utilizing finite-difference time domain method. Using numerical calculations, transmission properties of the designs are investigated and the free adjustability of the photonic bandgap locations in the spectra is demonstrated for both methods. With these advantages, both methods prove to be practical solutions for the design of optical filters based on one-dimensional PCs utilizing aperiodic index modulation.