Differentiable Radar Ambiguity Functions: Mathematical Formulation and Computational Implementation

TL;DR

This paper introduces GRAF, a differentiable formulation of radar ambiguity functions, enabling gradient-based optimization and integration with machine learning for radar waveform design.

Contribution

It provides the first complete mathematical and computational framework for differentiable radar ambiguity functions, addressing key technical challenges.

Findings

Enables gradient flow through radar ambiguity computations

Provides efficient parallelized FFT implementation

Demonstrates practical applicability in radar system optimization

Abstract

The ambiguity function is fundamental to radar waveform design, characterizing range and Doppler resolution capabilities. However, its traditional formulation involves non-differentiable operations, preventing integration with gradient-based optimization methods and modern machine learning frameworks. This paper presents the first complete mathematical framework and computational implementation for differentiable radar ambiguity functions. Our approach addresses the fundamental technical challenges that have prevented the radar community from leveraging automatic differentiation: proper handling of complex-valued gradients using Wirtinger calculus, efficient computation through parallelized FFT operations, numerical stability throughout cascaded operations, and composability with arbitrary differentiable operations. We term this approach GRAF (Gradient-based Radar Ambiguity Functions), which reformulates the ambiguity function computation to maintain mathematical equivalence while enabling gradient flow through the entire pipeline. The resulting implementation provides a general-purpose differentiable ambiguity function compatible with modern automatic differentiation frameworks, enabling new research directions including neural network-based waveform generation with ambiguity constraints, end-to-end optimization of radar systems, and integration of classical radar theory with modern deep learning. We provide complete implementation details and demonstrate computational efficiency suitable for practical applications. This work establishes the mathematical and computational foundation for applying modern machine learning techniques to radar waveform design, bridging classical radar signal processing with automatic differentiation frameworks.