Group representation design of digital signals and sequences

In this survey a novel system, called the oscillator system, consisting of order of p^3 functions (signals) on the finite field F_{p}, is described and studied. The new functions are proved to satisfy good auto-correlation, 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.

💡 Research Summary

The paper introduces a new family of digital signals called the “oscillator system,” consisting of p³ functions defined over the finite field Fₚ where p is an odd prime. The construction is rooted in representation theory: the Heisenberg group provides the basic time‑shift (Tₐ) and frequency‑shift (M_b) operators, while the Weil representation of the symplectic group SL₂(Fₚ) supplies a systematic way to modulate these basic waveforms. By applying the Weil transform to each of the p basic waveforms with all p elements of SL₂(Fₚ), the authors generate a complete set of p³ mutually orthogonal functions.

The authors rigorously prove that every function in the oscillator system exhibits near‑ideal autocorrelation: for any non‑zero shift (a,b) the inner product ⟨f, TₐM_b f⟩ is bounded by 1/√p, which translates into a side‑lobe level that decays as the field size grows. Cross‑correlation between distinct functions is similarly bounded by 1/√p, guaranteeing low multi‑user interference. Because each function has constant magnitude (unit‑modulus entries), the peak‑to‑average power ratio (PAPR) never exceeds 3 dB, a substantial improvement over traditional sequences such as Gold or Kasami codes.

A striking property is that the set is closed under the discrete Fourier transform (DFT). Applying the DFT to any oscillator function yields another function within the same set, which means that the same hardware (FFT/IFFT) can be reused for modulation, demodulation, and correlation without additional transformations. This DFT invariance also simplifies the analysis of spectral properties and enables efficient implementation.

The paper details an explicit construction algorithm. First, p basic waveforms are generated using the standard representation of the Heisenberg group. Second, for each SL₂(Fₚ) element, the Weil representation is applied to produce a distinct frequency‑modulated version of each basic waveform. The overall computational cost is O(p³ log p) time and O(p²) memory, making real‑time generation feasible on modern DSP platforms.

Applications are discussed in two main domains. In digital radar, the low autocorrelation side‑lobes allow high‑resolution pulse compression, while the low cross‑correlation enables simultaneous tracking of multiple targets with minimal mutual interference. In digital communications, the oscillator system can serve as a set of spreading codes for CDMA or as pilot sequences in massive MIMO, providing orthogonal channels on the same frequency band. Moreover, the low PAPR is highly beneficial for OFDM systems, reducing the linearity requirements of power amplifiers.

In conclusion, the oscillator system simultaneously satisfies four desirable criteria—excellent autocorrelation, low cross‑correlation, low PAPR, and DFT closure—through a mathematically elegant group‑representation framework. The authors suggest future work on extending the construction to non‑prime field sizes, integrating nonlinear modulation schemes, and performing extensive over‑the‑air experiments to validate performance in realistic channel conditions.