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Localized kernel method for separation of linear chirps

Eric Mason, Sippanon Kitimoon, Hrushikesh Mhaskar

Abstract

The task of separating a superposition of signals into its individual components is a common challenge encountered in various signal processing applications, especially in domains such as audio and radar signals. A previous paper by Chui and Mhaskar proposes a method called Signal Separation Operator (SSO) to find the instantaneous frequencies and amplitudes of such superpositions where both of these change continuously and slowly over time. In this paper, we amplify and modify this method in order to separate chirp signals in the presence of crossovers, a very low SNR, and discontinuities. We give a theoretical analysis of the behavior of SSO in the presence of noise to examine the relationship between the minimal separation, minimal amplitude, SNR, and sampling frequency. Our method is illustrated with a few examples, and numerical results are reported on a simulated dataset comprising 7 simulated signals.

Localized kernel method for separation of linear chirps

Abstract

The task of separating a superposition of signals into its individual components is a common challenge encountered in various signal processing applications, especially in domains such as audio and radar signals. A previous paper by Chui and Mhaskar proposes a method called Signal Separation Operator (SSO) to find the instantaneous frequencies and amplitudes of such superpositions where both of these change continuously and slowly over time. In this paper, we amplify and modify this method in order to separate chirp signals in the presence of crossovers, a very low SNR, and discontinuities. We give a theoretical analysis of the behavior of SSO in the presence of noise to examine the relationship between the minimal separation, minimal amplitude, SNR, and sampling frequency. Our method is illustrated with a few examples, and numerical results are reported on a simulated dataset comprising 7 simulated signals.

Paper Structure

This paper contains 21 sections, 4 theorems, 52 equations, 18 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Mathematical foundations
  4. Constant amplitudes and frequencies
  5. Reduction to the constant parameter case
  6. Signal separation operator
  7. Algorithms
  8. Finding instantaneous frequencies
  9. Parameter estimation
  10. Refinements
  11. Data generation
  12. Choice of tunable parameters
  13. Parameter $\eta$ (minimal separation)
  14. Parameter $\Delta$ (half-length of each snippet)
  15. Selection of the threshold
  16. ...and 6 more sections

Key Result

Lemma 3.1

Let $\delta\in (0,1)$. There exist positive constants $C_1, C_2, C_3$, depending only on $H$ such that for $n\ge C_1(\ge 1)$, we have

Figures (18)

  • Figure 1: The flowchart diagram for our algorithm scheme.
  • Figure 2: The flowchart diagram for our algorithm scheme.
  • Figure 3: (Left) The raw signal $f(t)$ as given in \ref{['eq:nonstationary']} at sampling rate $1$ GHz. (Middle) The noised signal $F(t)=f(t)+\epsilon(t)$ at -10 dB SNR. (Right) The ground truth of $\phi'(t)$ from \ref{['eq:pulsephasedef']}.
  • Figure 4: (Top) $|\sigma_{n,k}(x)|$ for the interval $I_k$. (Bottom) SSO results at $t_k$. (Left) The plots for $t_k =7.5 \times 10^{-5}$ where the signals pass from the beginning through the end of the interval $I_k$. (Middle) The plots for $t_k =1 \times 10^{-5}$ where the signal starts within the interval $I_k$. (Right) The plots for $t_k =2.35 \times 10^{-5}$ where there are 2 signals crossover within the interval $I_k$.
  • Figure 5: (Left) The plot of threshold SSO result $\{\Lambda_{j,k}\}$ for $j=1,\ldots,J_k$, $k=1,\ldots,D$ by choosing $\Delta = 2\times 10^{-6}, D=2500, D_1=D/2, D_2=D/100,$ and $t_k$ are equidistant samples from 0 to $1\times 10^{-4}$. The peaks represented by blue dots result from noise, and may be subject to aliasing. Our sampling rate ensures that the signal of interest is not affected by aliasing. (Right) The plot of threshold SSO result after step 1 of Algorithm 2.
  • ...and 13 more figures

Theorems & Definitions (8)

  • Remark 3.1
  • Lemma 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Definition 3.1
  • Theorem 3.3
  • Remark 3.2
  • Remark 3.3