PhaseJumps: fast computation of zeros from planar grid samples

Abstract

We consider complex-valued functions on the complex plane and the task of computing their zeros from samples taken along a finite grid. We introduce PhaseJumps, an algorithm based on comparing changes in the complex phase and local oscillations among grid neighboring points. The algorithm is applicable to possibly non-analytic input functions, and also computes the direction of phase winding around zeros. PhaseJumps provides a first effective means to compute the zeros of the short-time Fourier transform of an analog signal with respect to a general analyzing window, and makes certain recent signal processing insights more widely applicable, overcoming previous constraints to analytic transformations. We study the performance of (a variant of) PhaseJumps under a stochastic input model motivated by signal processing applications and show that the input instances that may cause the algorithm to fail are fragile, in the sense that they are regularized by additive noise (smoothed analysis). Precisely, given samples of a function on a grid with spacing $δ$, we show that our algorithm computes zeros with accuracy $\sqrtδ$ in the Wasserstein metric with failure probability $O\big(\log^2(\tfrac{1}δ) δ\big)$, while numerical experiments suggests even better performance.

Figures (8)

• Figure 1: Phase values of a complex-valued function. Left: In applications to time-frequency analysis (see Section \ref{['sec_stft']}), phases fluctuate moderately near the origin but strongly away from it. The twisted shifts \ref{['eq_ts']} re-center the function while moderating phase fluctuations, as in the center of the image. Right: A zoom onto the pixel values close to a potential zero near the center. By calculating the argument change along the black square, we can deduce that a zero is likely within the box. This is justified because phase fluctuates moderately along the black box. On the other hand, along the dotted vertical segment, there is a large argument jump.
• Figure 2: The STFT of a Gaussian signal in additive complex white noise using the first Hermite function as window (Left: Modulus, Right: Phase). Zeros with positive charge are marked by $\times$ and zeros with negative charge by $\ocircle$.
• Figure 3: The squared absolute value of the STFT of (the same realization of) complex white noise using a Gaussian (left) or first Hermite function (right) as window, together with their zeros of positive (crosses) and negative (circles) charge. On the left-hand side, the magnitude increases locally at the same rate in all directions; on the right-hand side, zeros lie along "low-magnitude curves" (approximate nodal curves), which explains the difficulty in their computation. The positively (resp. negatively) charged zeros on the right-hand side correspond exactly to the saddle-points (resp. local maxima) of the function on the left-hand side.
• Figure 4: Illustration of the different grids. Black circles mark the elements of the fine grid $\Lambda_L$, red squares mark the elements of the coarse grid $\Delta$, blue lines mark the set $M_{L+1}$. The thick blue lines illustrate the part of $M_{L+1}$ originating from the element $\lambda \in \Delta$ highlighted by the filled red square. The illustration is based on $\delta=1/100$, and thus $\delta^*=6/100$ and $\delta^{**}=10/100$.
• Figure 5: Empirical charge (left) and number of zeros (right) based on 100 simulations of the STFT of complex white noise with first Hermite window $g(t) = t e^{-t^2/2}$, using the algorithms Twisted PhaseJumps (PJ) and Twisted PhaseJumps-coarse (PJC) for different values of $\delta$.

Theorems & Definitions (27)

• Remark 2.1
• Theorem 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof