Channel Estimation, Interpolation and Extrapolation in Doubly-dispersive Channels

The OTFS (Orthogonal Time Frequency Space) is widely acknowledged for its ability to combat Doppler spread in time-varying channels. In this paper, another advantage of OTFS over OFDM (Orthogonal Frequency Division Multiplexing) will be demonstrated: much reduced channel training overhead. Specifically, the sparsity of the channel in delay-Doppler (D-D) domain implies strong correlation of channel gains in time-frequency (T-F) domain, which can be harnessed to reduce channel training overhead through interpolation. An immediate question is how much training overhead is needed in doubly-dispersive channels? A conventional belief is that the overhead is only dependent on the product of delay and Doppler spreads, but we will show that it's also dependent on the T-F window size. The finite T-F window leads to infinite spreading in D-D domain, and aliasing will be inevitable after sampling in T-F domain. Two direct consequences of the aliasing are increased channel training overhead and interference. Another factor contributing to channel estimation error is the inter-symbol-carrier-interference (ISCI), resulting from the uncertainty principle. Both aliasing and ISCI are considered in channel modelling, a low-complexity algorithm is proposed for channel estimation and interpolation through FFT. A large T-F window is necessary for reduced channel training overhead and aliasing, but increases processing delay. Fortunately, we show that the proposed algorithm can be implemented in a pipeline fashion. Further more, we showed that data-aided channel tracking is possible in D-D domain to further reduce the channel estimation frequency, i.e., channel extrapolation. The impacts of aliasing and ISCI on channel interpolation error are analyzed.

Fast Signal Interpolation Through Zero-padding and FFT/IFFT

Based on the sampling theorem, interpolation should be conducted by employing the sinc functions as the kernels. Inspired by the fact that the discrete Fourier transform (DFT) is sampled from the discrete time Fourier transform, a fast signal interpolation algorithm based on zero-padding and fast Fourier transform (FFT) and inverse FFT (IFFT) is presented. This algorithm gives a good approximate of the ideal interpolation, in spite of the windowing effect. The fundamental difference of this algorithm and the ideal sinc interpolation is unveiled, and shown to be deeply rooted in the connection of the sinc function and the Dirichlet function.