Almost Linear Complexity Methods for Delay-Doppler Channel Estimation

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^2logN) arithmetic operations. In [1] 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(rNlogN) 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(NlogN + r^3), and O(NlogN + r^2), respectively.Comment: 4 double column pages. arXiv admin note: substantial text overlap with arXiv:1309.372

On the diagonalization of the discrete Fourier transform

The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors for the DFT. The transition matrix from the standard basis to the canonical basis defines a novel transform which we call the discrete oscillator transform (DOT for short). Finally, we describe a fast algorithm for computing the discrete oscillator transform in certain cases.Comment: Accepted for publication in the journal "Applied and Computational Harmonic Analysis": Appl. Comput. Harmon. Anal. (2009), doi:10.1016/j.acha.2008.11.003. Key words: Discrete Fourier Transform, Weil Representation, Canonical Eigenvectors, Oscillator Transform, Fast Oscillator Transfor

Performance Estimates of the Pseudo-Random Method for Radar Detection

A performance of the pseudo-random method for the radar detection is analyzed. The radar sends a pseudo-random sequence of length $N$, and receives echo from $r$ targets. We assume the natural assumptions of uniformity on the channel and of the square root cancellation on the noise. Then for $r \leq N^{1-\delta}$, where $\delta > 0$, the following holds: (i) the probability of detection goes to one, and (ii) the expected number of false targets goes to zero, as $N$ goes to infinity.Comment: 5 pages, two figures, to appear in Proceedings of ISIT 2014 - IEEE International Symposium on Information Theory, Honolul