A new two-dimensional modulation technique called Orthogonal Time Frequency Space (OTFS) modulation designed in the delay-Doppler domain is introduced. Through this design, which exploits full diversity over time and frequency, OTFS coupled with equalization converts the fading, time-varying wireless channel experienced by modulated signals such as OFDM into a time-independent channel with a complex channel gain that is roughly constant for all symbols. Thus, transmitter adaptation is not needed. This extraction of the full channel diversity allows OTFS to greatly simplify system operation and significantly improves performance, particular in systems with high Doppler, short packets, and large antenna arrays. Simulation results indicate at least several dB of block error rate performance improvement for OTFS over OFDM in all of these settings. In addition these results show that even at very high Dopplers (500 Km/h), OTFS approaches channel capacity through linear scaling of throughput with the MIMO order, whereas the performance of OFDM under typical design parameters breaks down completely.

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 the delay-Doppler parameters is the matched filter algorithm. 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(N2logN). In this paper we introduce a novel approach of designing sequences that allow faster channel estimation. Using group representation techniques we construct sequences, which enable us to introduce a new algorithm, called the flag method, that significantly improves the matched filter algorithm. The flag method finds m delay-Doppler parameters in O(mNlogN) operations. We discuss applications of the flag method to GPS, and radar systems.

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(N2log(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.

The cryo-electron microscopy (cryo-EM) reconstruction problem is to find the three-dimensional structure of a macromolecule given noisy versions of its two-dimensional projection images at unknown random directions. We introduce a new algorithm for identifying noisy cryo-EM images of nearby viewing angles. This identification is an important first step in three-dimensional structure determination of macromolecules from cryo-EM, because once identified, these images can be rotationally aligned and averaged to produce "class averages" of better quality. The main advantage of our algorithm is its extreme robustness to noise. The algorithm is also very efficient in terms of running time and memory requirements, because it is based on the computation of the top few eigenvectors of a specially designed sparse Hermitian matrix. These advantages are demonstrated in numerous numerical experiments.

A novel system, called the oscillator system, consisting of order of p3functions (signals) on the finite field Fp, with p an odd prime, is described and studied. The new functions are proved to satisfy good autocorrelation, cross-correlation, and low peak-to- average power ratio properties. Moreover, the oscillator system is closed under the operation of discrete Fourier transform. Applications of the oscillator system for discrete radar and digital communication theory are explained. Finally, an explicit algorithm to construct the oscillator system is presented.