Real-Time Fractional Fourier Sound Synthesis and Processing

Abstract

The fractional Fourier transform (FRFT) extends the classical FT by introducing a continuous order parameter that interpolates between time and frequency representations. While widely used for audio analysis, enhancement, and compression, its potential as a direct basis for sound synthesis, particularly in real-time, remains largely unexplored. This paper presents a real-time framework for sound synthesis and processing in the intermediate time–frequency representation defined by the FRFT, based on a computationally efficient approximation suitable for low-latency applications. The authors introduce several novel techniques, including α-processing, α-synthesis, α-convolution, α-filters, and α-ring modulation, which exploit the FRFT’s geometric interpretation as a rotation in the time–frequency plane. To facilitate direct auditory exploration, the authors formalize the real FRFT and analyze its mathematical and sonic properties. Validation shows that the implementation preserves the key properties of the FRFT with high numerical accuracy, while performance benchmarks confirm its real-time viability. Through sound examples and spectrographic analyses, it is demonstrated that processing in this intermediate representation enables novel timbral structures, dynamic frequency modulations, and hybrid behaviors unattainable with conventional Fourier-based methods. These results position the FRFT and real FRFT as practical and expressive tools for interactive sound synthesis and design.