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 ≤ N1-δ, where δ > 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.

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.

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.