# Quantized Constant-Q Gabor Atoms for Sparse Binary Representations of Cyber-Physical Signatures

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

#### 1.1. Binary Representations of Time and Frequency

#### 1.2. Binary Representations of Energy and Information in Cyber-Physical Systems

^{−15}to 2

^{15}– 1. However, one may only be interested in the lower frequency components of the raw data, so one would implement a lowpass anti-aliasing filter before decimation. Such filters often require floating point arithmetic in double precision (52 bit mantissa re IEEE 754 at the time of this writing) to reduce instability. Therefore, the precision of the resulting lowpass filtered data would exceed the specification of the original 16-bit integral input. However, the theoretical dynamic range of the system would not exceed the specification of the integer 16 physical input. Furthermore, data compression can be more efficient on floats than integers, which leads us to the topic of fractional bits as a measure of CPS amplitude, power, and information.

## 2. Methods

#### 2.1. Transforming Time and Frequency to Scale

#### 2.2. Binary Quantized Constant-Q Gabor Atoms

#### 2.3. Quantum Order

#### 2.4. Continuous Wavelet Transform Deconstruction and Reconstruction

#### 2.5. Wavelet Information and Entropy

## 3. Discussion: Explosion Signature

## 4. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## Disclaimer

## Appendix A. Generalized Constant Q Bands

## Appendix B. The Gabor Atom

## Appendix C. The Q of the Quantum Wavelet

## Appendix D. The Gabor Box

## Appendix E. The Gabor Family

## Appendix F. The Analytic Function of a Blast Pulse

## References

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**Figure 1.**Analytic signal from mathematical equation, computation with SciPy Hilbert, and the continuous wavelet transfer (CWT) reconstruction. (

**a**) Real part; (

**b**) imaginary part. The wavelets were evaluated in binary bands (N = 1) and constructed around the target frequency of 6.3 Hz, which scales frequency and time. The real input waveform and its computed Hilbert transform are displayed in blue at the zero frequency.

**Figure 2.**Wavelet reconstruction with binary bands. (

**a**) Real part; (

**b**) imaginary part. The Equation waveform has no noise and is not filtered, whereas Hilbert has Gaussian noise and has been anti-aliased filtered.

**Figure 4.**Wavelet decomposition with 1/3 octave bands, with CWT amplitudes scaled by the reconstruction coefficients. (

**a**) Real part; (

**b**) imaginary part. As with Figure 1, the input waveform is displayed at the zero frequency.

**Figure 5.**Wavelet decomposition in order 3 binary bands, raw CWT amplitudes. (

**a**) Real part; (

**b**) imaginary part.

**Figure 6.**Shannon entropy in order 3 bands from raw CWT amplitudes. (

**a**) Real part; (

**b**) imaginary part.

**Figure 9.**Superposition of largest SNR entropy coefficients per band using all twenty 1/3 octave bands. (

**a**) Real part; (

**b**) imaginary part. The noise standard deviation is one bit below the signal’s. Dimensionality is reduced to the number of coefficients and their corresponding time shifts.

**Figure 10.**Superposition of largest coefficients per band within 4 bits of the peak SNR entropy. (

**a**) Real part; (

**b**) imaginary part. Dimensionality is further reduced by applying the cutoff.

**Figure 11.**(

**a**) Real part and (

**b**) imaginary part of the original and reconstructed waveform. Increasing the noise amplitude so that its variance is the same as the signal variance still permitted reconstruction from the superposition of the largest atoms per band.

**Figure 12.**(

**a**) Real part and (

**b**) imaginary part of the original and reconstructed waveform. Increasing the noise standard deviation is one bit above the signal standard deviation also allowed reconstruction from the quantum wavelet superposition.

N | ${\mathit{Q}}_{\mathit{N}}$ | ${\mathit{M}}_{\mathit{N}}$ |
---|---|---|

1 | 1.4142 | 2.3548 |

3 | 4.3185 | 7.1907 |

6 | 8.6514 | 14.4055 |

12 | 17.3099 | 28.8229 |

24 | 34.6235 | 57.6519 |

48 | 69.2488 | 115.3067 |

96 | 138.4984 | 230.6150 |

^{1}Dyadic base, G = 2.

**Table 2.**Exact and approximate quality factor Q for standard fractional octave bands of order N

^{1}.

N | ${\mathit{Q}}_{\mathit{N}}$ | ${\mathit{Q}}_{\mathit{N}}\text{}\approx \text{}\sqrt{2}\mathit{N}$ |
---|---|---|

1 | 1.4142 | 1.4142 |

3 | 4.3185 | 4.2426 |

6 | 8.6514 | 8.4853 |

12 | 17.3099 | 16.9706 |

24 | 34.6235 | 33.9411 |

48 | 69.2488 | 67.8823 |

96 | 138.4984 | 135.7645 |

^{1}Dyadic base, G = 2.

${\mathit{Q}}_{\mathit{N}}$ | $\mathit{N}\text{}\approx \text{}{\mathit{Q}}_{\mathit{N}}/\sqrt{2}$ | ${\mathit{M}}_{\mathit{N}}$ |
---|---|---|

1 | 0.7071 | 1.6651 |

2 | 1.4142 | 3.3302 |

4 | 2.8284 | 6.6604 |

8 | 5.6569 | 13.3209 |

16 | 11.3137 | 26.6417 |

32 | 22.6274 | 53.2835 |

64 | 45.2548 | 106.5670 |

128 | 90.5097 | 213.1340 |

^{1}Dyadic base, G = 2.

${\mathit{M}}_{\mathit{N}}$ | $~{\mathit{Q}}_{\mathit{N}}$ | $\mathit{N}$ |
---|---|---|

1 | 0.600561204 | 0.4246609 |

2 | 1.201122409 | 0.8493218 |

4 | 2.402244818 | 1.698643601 |

5 | 3.002806022 | 2.123304501 |

6 | 3.603367226 | 2.547965401 |

8 | 4.804489635 | 3.397287201 |

^{1}Dyadic base, G = 2.

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Garcés, M.A. Quantized Constant-Q Gabor Atoms for Sparse Binary Representations of Cyber-Physical Signatures. *Entropy* **2020**, *22*, 936.
https://doi.org/10.3390/e22090936

**AMA Style**

Garcés MA. Quantized Constant-Q Gabor Atoms for Sparse Binary Representations of Cyber-Physical Signatures. *Entropy*. 2020; 22(9):936.
https://doi.org/10.3390/e22090936

**Chicago/Turabian Style**

Garcés, Milton A. 2020. "Quantized Constant-Q Gabor Atoms for Sparse Binary Representations of Cyber-Physical Signatures" *Entropy* 22, no. 9: 936.
https://doi.org/10.3390/e22090936