Synthetic Signal Shaping

Analytic-gradient Phase Optimization over Cross-Spectral Density Constraints and Arbitrary Memoryless Functionals

Abstract

We present a phase-domain method for generating multi-input synthetic vibration signals that match a prescribed cross-spectral density (CSD) together with a user-selected set of higher-order diagonal and cross moments. The CSD is enforced structurally by pre-multiplying random-phase, unit-modulus, frequency-domain signals with a Cholesky factor of the target CSD, following the frequency-domain synthesis of Shinozuka and Jan (1972) and its extensions to partially coherent and non-Gaussian MIMO outputs (Smallwood and Paez 1993; Smallwood 1997; Cui et al. 2011); this leaves the remaining phase degrees of freedom unconstrained. We show that this freedom is enough to shape the channelwise moments – extending the IFFT phase-manipulation idea of Steinwolf (1993, 2011, 2012) – and, in addition, the joint cross-moments (co-skewness, co-kurtosis) that govern coincident-peak excursions and hence multi-axial fatigue damage (Saathoff et al. 2006). The key result is a single Fourier-domain identity: the gradient of any differentiable memoryless functional of the synthesized signal, with respect to the free phases, reduces to one forward Fourier transform of its pointwise derivative, projected through the conjugated unit-modulus source spectrum u* and the conjugated frequency response H*. Every target used in this paper – diagonal and cross moments, endpoint values, a crest-factor surrogate – is an instance of this one identity, at a cost of at most m FFTs per gradient evaluation for a target of order m, independent of the number of channels; the MIMO case therefore requires no machinery beyond the single-channel one. The resulting problem is smooth and is solved with SciPy's L-BFGS-B (Byrd et al. 1995; Virtanen et al. 2020) using the analytic gradient directly, found in a head-to-head comparison to need an order of magnitude fewer loss evaluations than NLopt's CCSAQ (Svanberg 2002; Johnson 2007–present) on the warm-started continuation used for crest minimization, while matching or improving the achieved optimum. When the target statistics are estimated from a measured multi-axial record the targets are realizable by construction – the record itself is a witness – which is the primary operating regime for the method; we demonstrate it on a 12-channel wheel-hub force/moment record measured on a test track, reproducing all 30 moment targets alongside the full CSD, and show empirically that the per-evaluation cost scales tractably to 32 channels and 560 simultaneous targets. A complete, unit-tested reference implementation – including Slepian multitaper CSD estimation and analytic-gradient verification against central differences – is open-sourced at https://github.com/luchp/mimoshape; every figure and table in this paper is regenerated from it with fixed seeds.