Asymptotic Spectral Efficiency of Digital Transmission via Overcomplete Frames With Discrete, Finite, and Uniform Alphabets

It is recognized that, for digital symbols taken from a discrete and finite alphabet, perfect transmultiplexing at a signaling rate larger than the Nyquist rate can be achieved by modulation with overcomplete frame pulses. By invoking the spectral distribution theory of large random matrix, the spectral efficiency of digital transmission scheme via overcomplete frames in band-limited additive white Gaussian noise (AWGN) channel with discrete, finite, and uniform alphabets is investigated in this paper. It is shown that the proposed digital signaling scheme can asymptotically achieve the maximum spectral efficiency dictated by the Shannon capacity theorem for reliable transmission without employing signal shaping techniques. The extension to the case of non-white Gaussian noise/spectrally shaped channels is also considered. It is shown that the employment of Weyl-Heisenberg frames facilitates the optimal “water-filling” power allocation and rate adaptation by adjusting the signal amplitudes and energy-dispersion factors. Some numerical results are provided to support the theoretical finding.