Almost linear complexity methods for delay-Doppler channel estimation
Alexander Fish, Shamgar Gurevich
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, i.e., a signal undergoes only delay and Doppler shifts, a widely used method to compute delay-Doppler parameters is the pseudo-random method. It uses a pseudo-random sequence of length \(N\); and, in case of non-trivial relative velocity between transmitter and receiver, its computational complexity is \(O(N^2 logN)\) arithmetic operations. In  the ﬂag method was introduced to provide a faster algorithm for delay-Doppler channel estimation. It uses specially designed flag sequences and its complexity is \(O(rN logN)\) for channels of sparsity \(r\). In these notes, we introduce the incidence and cross methods for channel estimation. They use triple-chirp and double-chirp sequences of length \(N\), correspondingly. These sequences are closely related to chirp sequences widely used in radar systems. The arithmetic complexity of the incidence and cross methods is \(O(N logN+r^3)\), and \(O(N logN+r^2)\), respectively.
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