We analyze the stability properties of the Synchrosqueezing transform, a time-frequency signal analysis method that can identify and extract oscillatory components with time-varying frequency and amplitude. We show that Synchrosqueezing is robust to bounded perturbations of the signal and to Gaussian white noise. These results justify its applicability to noisy or nonuniformly sampled data that is ubiquitous in engineering and the natural sciences. We also describe a practical implementation of Synchrosqueezing and provide guidance on tuning its main parameters.As a case study in the geosciences, we examine characteristics of a key paleoclimate change in the last 2.5 million years, where Synchrosqueezing provides significantly improved insights.

We propose a new approach for studying the notion of the instantaneous frequency of a signal. We build on ideas from the Synchrosqueezing theory of Daubechies, Lu and Wu and consider a variant of Synchrosqueezing, based on the short-time Fourier transform, to precisely define the instantaneous frequencies of a multi-component AM-FM signal. We describe an algorithm to recover these instantaneous frequencies from the uniform or nonuniform samples of the signal and show that our method is robust to noise. We also consider an alternative approach based on the conventional, Hilbert transform-based notion of instantaneous frequency to compare to our new method. We use these methods on several test cases and apply our results to a signal analysis problem in electrocardiography.

We consider the recovery of real-valued bandlimited functions from the absolute values of their samples, possibly spaced nonuniformly. We show that such a reconstruction is always possible if the function is sampled at more than twice its Nyquist rate, and may not necessarily be possible if the samples are taken at less than twice the Nyquist rate. In the case of uniform samples, we also describe an FFT-based algorithm to perform the reconstruction. We prove that it converges exponentially rapidly in the number of samples used and examine its numerical behavior on some test cases.

We describe two applications of machine learning in the context of IP (Internet Protocol) /Optical networks. The first one allows agile management of resources in a core IP/Optical network by using machine learning for short-term and longterm prediction of traffic flows. It also allows joint global optimization of IP and optical layers using colorless/directionless (CD) ROADMs (Reconfigurable Optical Add Drop Multiplexers). Multilayer coordination allows for significant cost savings, flexible new services to meet dynamic capacity needs, and improved robustness by being able to proactively adapt to new traffic patterns and network conditions. The second application is important as we migrate our networks to Open ROADM networks, to allow physical routing without the need for detailed knowledge of optical parameters. We discuss a proof-of-concept study, where detailed performance data for established wavelengths in an existing ROADM network is used for machine learning to predict the optical performance of each wavelength. Both applications can be efficiently implemented by using a SDN (Software Defined Network) controller.

Abstract. In this paper we consider the following problem of phase retrieval: Given a collection of real-valued band-limited functions {ψ λ } λ∈Λ ⊂ L 2 (R d ) that constitutes a semi-discrete frame, we ask whether any real-valued function f ∈ L 2 (R d ) can be uniquely recovered from its unsigned convolutions {|f * ψ λ |} λ∈Λ .We find that under some mild assumptions on the semi-discrete frame and if f has exponential decay at ∞, it suffices to know |f * ψ λ | on suitably fine lattices to uniquely determine f (up to a global sign factor).We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of L 2 (R d ), d = 1, 2, we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.