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The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states.We present an alternative approach to these important quantum systems based on the finite frame quantization.Finite systems of coherent states,usually called finite tight frames,can be defined in a natural way in the case of finite quantum systems.The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system.They are obtained by using a finite frame and a Klauder-Berezin-Toeplitz type quantization.The finite frames used in our approach are defined by starting from the finite Gaussian used by Mehta in its paper on the eigenvectors of the finite Fourier transform.In the three-dimensional case,the nine projectors corresponding to our frame are linearly independent and can be used in order to define an alternative description for qutrits.We present a more general class of finite Gaussians and compute explicitly the corresponding Fourier transforms and Wigner functions.