Delay-Doppler channel estimation with almost linear complexity: To Solomon Golomb for the occasion of his 80 birthday mazel tov

A fundamental task in wireless communication is channel estimation: Compute the channel parameters a signal undergoes while traveling from a transmitter to a receiver. In the case of delay-Doppler channel, a widely used method is the matched filter algorithm. It uses a pseudo-random waveform of length N, and, in case of non-trivial relative velocity between transmitter and receiver, its computational complexity is O(N² log(N)). In this paper we introduce a novel approach of designing waveforms that allow faster channel estimation. Using group representation techniques we construct waveforms, which enable us to introduce a new algorithm, called the flag method, that significantly improves the matched filter algorithm. The flag method finds the channel parameters in O(m · N log(N)) operations, for channel of sparsity of order m. We discuss applications of the flag method to GPS, and radar system as well.