The Complex-Step Integral Transform

TL;DR

The paper introduces the Complex-Step-Integral-Transform (CSIT), a generalized operator that blends analytic continuation with derivative estimation and multi-scale smoothing to improve numerical differentiation, spectral analysis, and seismic signal processing. It provides existence and error bounds, analyzes operator properties, and demonstrates the method on analytical functions, the advection equation, and instantaneous-frequency estimation, highlighting phase preservation and suppression of high-wavenumber noise. CSIT is implemented via FFT or interpolation, offering a spectrum of accuracy and stability advantages over classical Fourier derivatives and standard Hilbert-transform-based approaches, particularly in non-periodic or under-resolved settings. The work shows that CSIT yields smoother, more robust attributes, with built-in regularization that benefits PDE solvers and seismic analysis, and outlines practical guidelines and future extensions to non-periodic grids and finite-element-inspired implementations.

Abstract

Building on the well-established connection between the Hilbert transform and derivative operators, and motivated by recent developments in complex-step differentiation, we introduce the Complex-Step Integral Transform (CSIT): a generalized integral transform that combines analytic continuation, derivative approximation, and multi-scale smoothing within a unified framework. A spectral analysis shows that the CSIT preserves phase while suppressing high-wavenumber noise, offering advantages over conventional Fourier derivatives. We discuss the roles of the real and imaginary step parameters, compare FFT-based and interpolation-based implementations, and demonstrate the method on the advection equation and instantaneous-frequency computation. Results show that the CSIT yields smoother, more robust attributes than Hilbert-based methods and provides built-in stabilization for PDE solvers. The CSIT thus represents a flexible alternative for numerical differentiation, spectral analysis, and seismic signal processing. The method opens several avenues for future work, including non-periodic implementations, adaptive parameter selection, and integration with local interpolation frameworks such as high-order Finite-Element methods.

Figures (6)

• Figure 1: Fourier amplitude response of the CSIT operator. Single CSIT $(H=0)$ uses only the $\tau$ integral, approximated by $\mathrm{Shi}(Z k)$. Double-averaged CSIT includes $\eta \in [-H,H]$ averaging, introducing the $\sin(kH)/(kH)$ modulation, which damps high-wavenumber modes.
• Figure 2: (a) Logistic function (Eq. \ref{['eq.Logistic_function']}) for $t\in[0,1]$ s, $t_0=0.5$ s, $k=100$, and $N=500$. (b) Comparison between the centered finite-difference and CSIT derivatives. (c) Pseudospectral derivative. (d) Relative errors with respect to the analytical derivative.
• Figure 3: Comparison of advection solutions using finite-difference, pseudospectral, and CSIT discretizations. The CSIT solution eliminates the parasitic mode observed in the other two schemes.
• Figure 4: Reconstruction of the instantaneous frequency for the chirp signal defined in eq. \ref{['eq.True_IF']} using two time-sampling rates $n_t$: (a) $n_t = 2500$ and (b) $n_t = 300$.
• Figure 5: (a) Location of the April 2010 Granada earthquake epicenter (red star) and seismological station used in this study (blue triangle). (b) Normalized vertical component velocity recording of the event.

Theorems & Definitions (7)

• Definition 2.1: Complex-Step Integral Transform (CSIT)
• Theorem 2.1: Existence and Taylor expansion for $H,Z\to 0$
• proof
• Remark 2.2
• Definition 4.1: Stieltjes Transform
• Definition 4.2: Stieltjes Inverse Transform
• Remark 4.1