# Entropy-Based Concentration and Instantaneous Frequency of TFDs from Cohen’s, Affine, and Reassigned Classes

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## Abstract

**:**

## 1. Introduction

#### 1.1. Time–Frequency Analysis

#### 1.1.1. Analytic Form of a Signal

#### 1.1.2. Instantaneous Frequency

#### 1.1.3. Group Delay

## 2. Affine TFRs with Respect to Cohen’s and Reassigned TFRs

- $k=2$—Affine Wigner representation (extended covariance along straight line paths);
- $k=1/2$—D-Flandrin representation (extended covariance along square-root-hyperbolic paths);
- $k=0$—Bertrand representation (extended covariance along hyperbolic paths);
- $k=-1$—Unterberger representation;
- $k=\pm 5$—Approximate affine Wigner representations (unsmoothed);
- $k=\pm \infty $—Margenau–Hill representation.

#### 2.1. Kernels

#### 2.2. Energy

#### 2.3. Covariance

#### 2.4. Marginals

#### 2.5. Interference Terms

#### 2.6. Reassignment Method

#### 2.7. Affine Class Distributions

#### 2.7.1. Scalogram

#### 2.7.2. Smoothed Pseudo-Affine Wigner Distribution

#### 2.7.3. Unitary Bertrand Distribution

#### 2.7.4. Unterberger Distribution

#### 2.7.5. D-Flandrin Distribution

#### 2.7.6. Other Affine Class Distributions

## 3. Examples and Simulation Results

#### 3.1. Examples of Synthetic Signals

#### 3.1.1. Examples of Mono-Component Noisy Signals

#### 3.1.2. Example of a Multi-Component Noisy Successive Signal

^{−5}and 136.99 × 10

^{−5}, respectively). From the affine class, AMWSC outperformed other TFRs achieving 76.27 × 10

^{−5}and 113.90 × 10

^{−5}for SNRs of 10 dB and 5 dB, respectively. From the reassigned class, for SNRs of 10 dB and 5 dB, RGSP outperformed other reassigned TFRs, achieving MSEs of 9.39 × 10

^{−5}and 39.73 × 10

^{−5}, respectively. When considering intensive noise scenarios, it is interesting to note that SP offers the most accurate IF estimation for 0 dB (304.51 × 10

^{−5}). For comparison, the worst-performing was AAU from the affine class, resulting in an IF estimation MSE of 1774.30 × 10

^{−5}. Similar conclusions can be drawn for the case of an SNR of −5 dB, where SP’s IF estimation MSE was 1125.70 × 10

^{−5}(negligibly outperformed by RPWV with an MSE of 1082.20 × 10

^{−5}). Again, as for 0 dB, AAU was the worst-performing with 2321.90 × 10

^{−5}. Thus, Cohen’s class resulted in the most accurate IF estimation for intensive noise SNRs for the noisy multi-component signal with successive parabolic and linear FM components, while the reassigned class of TFRs was the best-performing for low-noise environments. In both cases, the affine class was outperformed by the other two classes.

#### 3.1.3. Example of a Multi-Component Noisy Concurrent Signal

#### 3.2. Examples of Real-World Signals

#### 3.3. Computational Cost of Analyzed TFRs

^{®}Core™ i5-7300HQ CPU @ 2.50 GHz, 16 GB of DDR4-2666 (1333 MHz) RAM, and NVIDIA GeForce GTX 1050 Ti GPU with 4 GB GDDR5 of dedicated memory (7.9 GB total shared memory).

## 4. Results Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TFR | Time–Frequency Representation |

TFD | Time–Frequency Distribution |

TSR | Time–Scale Representation |

TSD | Time–Scale Distribution |

IF | Instantaneous Frequency |

TF | Time–Frequency |

FM | Frequency Modulation |

SNR | Signal-to-Noise Ratio |

MSE | Mean Squared Error |

SP | Spectrogram |

SPAWD | Smoothed Pseudo-Affine Wigner Distribution |

SPWV | Smoothed Pseudo-Wigner–Ville |

WV | Wigner–Ville |

AMWSC | Affine Morlet Wavelet Scalogram |

ASPWV | Affine Smoothed Pseudo-Wigner–Ville |

AUB | Affine Unitary Bertrand |

AUU | Affine Active Unterberger |

ADF | Affine D-Flandrin |

RSP | Reassigned Spectrogram |

RGSP | Reassigned Gabor Spectrogram |

RMSC | Reassigned Morlet Scalogram |

RPWV | Reassigned Pseudo-Wigner–Ville |

RSPWV | Reassigned Smoothed Pseudo-Wigner–Ville |

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**Figure 1.**Example of three elementary Gaussian atoms in time, frequency, and TF domains (spectrogram).

**Figure 2.**Example of three elementary Gaussian atoms in the time, frequency, and TF domains (Wigner-Ville) with interferences.

**Figure 4.**TFRs of a noisy mono-component sinusoidal FM signal with estimated IFs for the SNR of 5 dB.

**Figure 5.**TFRs of a noisy multi-component signal with successive parabolic and linear FM components for 5 dB SNR.

**Figure 6.**TFRs of a noisy multi-component signal with concurrent constant and parabolic FM components for the 5 dB SNR.

**Table 1.**Estimation of IF MSE for noisy mono-component linear FM signal for different SNR levels (averaged over 100 noise realizations), cell colors range from green for the best result to red for the worst result in each column respectively, 1 × 10

^{−5}.

10 dB | 5 dB | 0 dB | −5 dB | |
---|---|---|---|---|

SP | 1.58 | 2.39 | 111.24 | 1329.70 |

SPWV | 1.46 | 2.08 | 98.58 | 1320.50 |

WV | 1.64 | 34.19 | 231.40 | 918.89 |

AMWSC | 4.61 | 9.40 | 326.76 | 2039.20 |

ASPWV | 277.84 | 695.28 | 1789.10 | 2647.70 |

AUB | 31.88 | 77.71 | 706.32 | 2093.80 |

AAU | 75.61 | 280.15 | 1667.80 | 3137.30 |

ADF | 66.84 | 181.90 | 1249.60 | 2381.20 |

RSP | 11.88 | 35.94 | 497.45 | 1811.40 |

RGSP | 13.09 | 37.51 | 683.30 | 2014.00 |

RMSC | 22.85 | 88.66 | 609.31 | 1662.40 |

RPWV | 42.16 | 100.22 | 521.22 | 1288.90 |

RSPWV | 1.57 | 11.84 | 343.78 | 1645.20 |

**Table 2.**TFR Rényi entropy for a noisy mono-component linear FM signal for different SNR levels (averaged over 100 noise realizations), cell colors range from green for the best result to red for the worst result in each column respectively.

10 dB | 5 dB | 0 dB | −5 dB | |
---|---|---|---|---|

SP | 15.225 | 15.475 | 15.959 | 16.480 |

SPWV | 14.905 | 15.293 | 15.898 | 16.437 |

WV | 14.864 | 15.600 | 16.343 | 16.641 |

AMWSC | 13.774 | 14.242 | 15.078 | 15.883 |

ASPWV | 15.022 | 15.551 | 15.933 | 16.134 |

AUB | 14.193 | 15.071 | 15.875 | 16.232 |

AAU | 14.690 | 15.516 | 16.199 | 16.446 |

ADF | 14.301 | 15.191 | 15.919 | 16.187 |

RSP | 12.886 | 13.172 | 13.682 | 14.187 |

RGSP | 12.972 | 13.272 | 13.742 | 14.174 |

RMSC | 12.796 | 13.061 | 13.550 | 14.074 |

RPWV | 13.107 | 14.018 | 14.800 | 15.104 |

RSPWV | 12.240 | 12.854 | 13.684 | 14.349 |

**Table 3.**Estimation of IF MSE for a noisy mono-component sinusoidal FM signal for different SNR levels (averaged over 100 noise realizations), cell colors range from green for the best result to red for the worst result in each column respectively, 1 × 10

^{−4}.

10 dB | 5 dB | 0 dB | −5 dB | |
---|---|---|---|---|

SP | 6.88 | 10.28 | 36.54 | 153.62 |

SPWV | 3.09 | 6.05 | 34.86 | 150.80 |

WV | 187.12 | 153.31 | 135.68 | 150.92 |

AMWSC | 17.83 | 26.08 | 73.08 | 197.99 |

ASPWV | 11.58 | 60.63 | 150.89 | 217.60 |

AUB | 92.30 | 97.53 | 134.96 | 185.98 |

AAU | 99.05 | 127.31 | 187.59 | 245.58 |

ADF | 100.06 | 109.18 | 152.09 | 200.02 |

RSP | 1.79 | 8.54 | 72.52 | 197.80 |

RGSP | 1.06 | 8.68 | 90.08 | 217.16 |

RMSC | 7.93 | 24.19 | 89.60 | 197.35 |

RPWV | 2.74 | 13.09 | 65.93 | 145.99 |

RSPWV | 1.92 | 8.08 | 62.18 | 184.94 |

**Table 4.**TFR Rényi entropy for a noisy mono-component sinusoidal FM signal for different SNR levels (averaged over 100 noise realizations), cell colors range from green for the best result to red for the worst result in each column respectively.

10 dB | 5 dB | 0 dB | −5 dB | |
---|---|---|---|---|

SP | 15.947 | 16.111 | 16.381 | 16.654 |

SPWV | 15.724 | 15.973 | 16.300 | 16.586 |

WV | 16.118 | 16.355 | 16.569 | 16.690 |

AMWSC | 15.170 | 15.402 | 15.788 | 16.227 |

ASPWV | 15.416 | 15.871 | 16.080 | 16.240 |

AUB | 15.482 | 15.765 | 16.070 | 16.340 |

AAU | 15.615 | 15.956 | 16.269 | 16.518 |

ADF | 15.499 | 15.783 | 16.058 | 16.278 |

RSP | 13.579 | 13.720 | 13.998 | 14.301 |

RGSP | 13.368 | 13.587 | 13.923 | 14.249 |

RMSC | 13.522 | 13.689 | 13.991 | 14.207 |

RPWV | 14.127 | 14.578 | 14.957 | 15.152 |

RSPWV | 13.650 | 13.890 | 14.226 | 14.488 |

**Table 5.**Estimation of IF MSE for a noisy multi-component signal with successive parabolic and linear FM components for different SNR levels (averaged over 100 noise realizations), cell colors range from green for the best result to red for the worst result in each column respectively, 1 × 10

^{−5}.

10 dB | 5 dB | 0 dB | −5 dB | |
---|---|---|---|---|

SP | 127.12 | 166.02 | 304.51 | 1125.70 |

SPWV | 67.96 | 136.99 | 325.23 | 1160.50 |

WV | 1046.80 | 1068.60 | 1061.80 | 1206.60 |

AMWSC | 76.27 | 113.90 | 350.10 | 1501.40 |

ASPWV | 148.89 | 480.46 | 1292.90 | 1816.10 |

AUB | 684.84 | 783.60 | 1122.60 | 1546.40 |

AAU | 848.84 | 1098.20 | 1774.30 | 2321.90 |

ADF | 750.37 | 910.03 | 1337.60 | 1707.00 |

RSP | 22.18 | 65.28 | 456.63 | 1509.00 |

RGSP | 9.39 | 39.73 | 445.73 | 1513.40 |

RMSC | 9.84 | 45.78 | 524.60 | 1540.00 |

RPWV | 61.37 | 179.57 | 514.64 | 1082.20 |

RSPWV | 42.95 | 107.03 | 475.40 | 1487.00 |

**Table 6.**TFR Rényi entropy for a noisy multi-component signal with successive parabolic and linear FM components for different SNR levels (averaged over 100 noise realizations), cell colors range from green for the best result to red for the worst result in each column respectively.

10 dB | 5 dB | 0 dB | −5 dB | |
---|---|---|---|---|

SP | 19.616 | 19.839 | 20.226 | 20.519 |

SPWV | 19.379 | 19.696 | 20.148 | 20.449 |

WV | 20.097 | 20.301 | 20.518 | 20.637 |

AMWSC | 18.577 | 18.932 | 19.561 | 20.118 |

ASPWV | 19.184 | 19.742 | 20.108 | 20.267 |

AUB | 19.325 | 19.658 | 20.027 | 20.274 |

AAU | 19.542 | 19.897 | 20.258 | 20.478 |

ADF | 19.358 | 19.692 | 20.027 | 20.231 |

RSP | 16.639 | 16.916 | 17.416 | 17.869 |

RGSP | 16.288 | 16.642 | 17.217 | 17.733 |

RMSC | 16.105 | 16.491 | 17.122 | 17.662 |

RPWV | 17.575 | 18.256 | 18.858 | 19.097 |

RSPWV | 16.837 | 17.244 | 17.753 | 18.032 |

**Table 7.**Estimation of IF MSE for a noisy multi-component signal with concurrent constant and parabolic FM components for different SNR levels (averaged over 100 noise realizations), cell colors range from green for the best result to red for the worst result in each column respectively.

10 dB | 5 dB | 0 dB | −5 dB | |
---|---|---|---|---|

(a) First Component, 1 × 10^{−6} | ||||

SP | 5.26 | 22.41 | 1032.10 | 3914.40 |

SPWV | 7.60 | 38.28 | 1017.90 | 3877.20 |

WV | 4375.40 | 6032.50 | 8521.80 | 9397.40 |

AMWSC | 1.29 | 2.15 | 69.36 | 1446.50 |

ASPWV | 3384.30 | 3910.00 | 4724.80 | 5443.90 |

AUB | 4108.60 | 4593.10 | 5160.00 | 5430.20 |

AAU | 3774.00 | 3900.40 | 4066.40 | 3775.90 |

ADF | 4872.90 | 5352.50 | 5718.60 | 5528.00 |

RSP | 227.92 | 404.64 | 1942.20 | 4245.50 |

RGSP | 168.09 | 487.04 | 2243.00 | 4406.40 |

RMSC | 508.97 | 930.73 | 1785.50 | 4107.40 |

RPWV | 3648.10 | 4975.40 | 6204.10 | 6903.10 |

RSPWV | 45.29 | 193.54 | 1584.20 | 3961.30 |

(b) Second Component, 1 × 10^{−5} | ||||

SP | 5.16 | 8.04 | 136.88 | 483.73 |

SPWV | 5.76 | 8.80 | 133.50 | 467.15 |

WV | 237.34 | 306.31 | 405.21 | 447.99 |

AMWSC | 3.62 | 8.81 | 140.19 | 444.00 |

ASPWV | 400.74 | 450.91 | 557.93 | 635.48 |

AUB | 49.20 | 94.12 | 298.59 | 452.07 |

AAU | 19.29 | 54.31 | 273.11 | 445.41 |

ADF | 137.54 | 216.86 | 408.39 | 508.84 |

RSP | 26.17 | 52.88 | 288.19 | 624.45 |

RGSP | 26.73 | 73.70 | 371.42 | 700.51 |

RMSC | 20.90 | 39.49 | 277.02 | 600.25 |

RPWV | 183.87 | 251.13 | 398.47 | 508.63 |

RSPWV | 5.23 | 21.09 | 222.81 | 544.92 |

**Table 8.**TFR Rényi entropy for a noisy multi-component signal with concurrent linear and parabolic FM components for different SNR levels (averaged over 100 noise realizations), cell colors range from green for the best result to red for the worst result in each column respectively.

10 dB | 5 dB | 0 dB | −5 dB | |
---|---|---|---|---|

SP | 16.033 | 16.168 | 16.392 | 16.643 |

SPWV | 15.794 | 16.051 | 16.345 | 16.598 |

WV | 16.232 | 16.453 | 16.625 | 16.698 |

AMWSC | 14.642 | 14.946 | 15.497 | 16.125 |

ASPWV | 15.738 | 16.030 | 16.182 | 16.271 |

AUB | 15.239 | 15.710 | 16.122 | 16.342 |

AAU | 15.260 | 15.780 | 16.233 | 16.483 |

ADF | 15.287 | 15.737 | 16.099 | 16.277 |

RSP | 13.820 | 13.971 | 14.162 | 14.372 |

RGSP | 13.913 | 14.038 | 14.182 | 14.315 |

RMSC | 13.615 | 13.786 | 14.075 | 14.305 |

RPWV | 14.245 | 14.741 | 15.063 | 15.181 |

RSPWV | 13.308 | 13.774 | 14.223 | 14.506 |

**Table 9.**TFR Rényi entropy of an EEG P300 signal for the Cz electrode (averaged over 466 trials), cell colors range from green for the best result to red for the worst result in each column respectively.

Rényi Entropy | |
---|---|

SP | 13.414 |

SPWV | 13.251 |

WV | 14.366 |

AMWSC | 16.689 |

ASPWV | 16.200 |

AUB | 16.300 |

AAU | 16.302 |

ADF | 16.325 |

RSP | 11.369 |

RGSP | 11.230 |

RMSC | 10.178 |

RPWV | 11.782 |

RSPWV | 11.439 |

**Table 10.**Rényi entropy for the 2020 Zagreb earthquake vertical acceleration seismogram, cell colors range from green for the best result to red for the worst result in each column respectively.

Rényi Entropy | |
---|---|

SP | 18.961 |

SPWV | 18.871 |

WV | 19.831 |

AMWSC | 21.526 |

ASPWV | 20.956 |

AUB | 21.350 |

AAU | 21.417 |

ADF | 21.260 |

RSP | 16.636 |

RGSP | 16.441 |

RMSC | 15.084 |

RPWV | 18.099 |

RSPWV | 16.494 |

**Table 11.**Computation time for analyzed TFRs for tested signals (averaged over 100 realizations, SNR of 5 dB), cell colors range from green for the best result to red for the worst result in each column respectively, in milliseconds (ms).

Noisy Mono Component Linear FM | Noisy Mono Component Sinusoidal FM | Noisy Three Component Parabolic and Linear FM | Noisy Two Component Constant and Parabolic FM | EEG P300 Signal | 2020 Zagreb Earthquake | |
---|---|---|---|---|---|---|

Number of time samples | 128 | 128 | 384 | 128 | 128 | 487 |

SP | 1.6448 | 1.4987 | 7.4916 | 1.4124 | 1.4105 | 12.69 |

SPWV | 23.024 | 22.617 | 225.12 | 22.428 | 22.569 | 369.87 |

WV | 1.4677 | 1.3349 | 7.7009 | 1.2996 | 1.2905 | 13.539 |

AMWSC | 10.219 | 9.7795 | 60.675 | 9.7801 | 9.6643 | 93.012 |

ASPWV | 188.92 | 188.19 | 3122.5 | 188.94 | 189.51 | 4356.7 |

AUB | 284.9 | 283.59 | 2110.8 | 281.91 | 283.34 | 2476.9 |

AAU | 749.92 | 751.21 | 6348.1 | 745.8 | 748.24 | 7999 |

ADF | 250.64 | 250.61 | 1797.7 | 250.28 | 251.04 | 2049.4 |

RSP | 19.526 | 19.205 | 170.08 | 19.449 | 19.203 | 275.86 |

RGSP | 16.126 | 15.649 | 145.11 | 15.565 | 13.163 | 192.5 |

RMSC | 275.13 | 268.87 | 2650.7 | 273.55 | 269.98 | 4207.1 |

RPWV | 3.7773 | 3.4253 | 24.559 | 3.4197 | 3.4948 | 41.555 |

RSPWV | 44.623 | 43.586 | 412.51 | 43.931 | 43.878 | 670.22 |

**Table 12.**Computational time for analyzed TFRs for a noisy linear FM signal with different numbers of time samples (averaged over 100 realizations, SNR of 5 dB), cell colors range from green for the best result to red for the worst result in each column respectively, in milliseconds (ms).

Number of Time Samples | 32 | 64 | 128 | 256 | 512 |
---|---|---|---|---|---|

SP | 0.53308 | 0.68926 | 1.4145 | 3.6583 | 10.057 |

SPWV | 1.9574 | 5.7625 | 22.275 | 92.194 | 407.77 |

WV | 0.38185 | 0.47453 | 1.2714 | 2.968 | 11.174 |

AMWSC | 2.2736 | 3.9472 | 9.6087 | 28.944 | 104.7 |

ASPWV | 26.755 | 47.652 | 179.4 | 455.84 | 4537.6 |

AUB | 46.272 | 103.14 | 278.18 | 791.83 | 2500.5 |

AAU | 99.284 | 257.56 | 727.06 | 2276.1 | 8031.1 |

ADF | 42.68 | 94.99 | 244.54 | 681.51 | 2116.2 |

RSP | 2.2171 | 5.4668 | 18.87 | 70.807 | 285.32 |

RGSP | 2.1111 | 5.0283 | 15.18 | 51.456 | 201.96 |

RMSC | 16.188 | 63.321 | 261.1 | 1110.3 | 4600.3 |

RPWV | 0.99658 | 1.3783 | 3.3517 | 9.9509 | 38.288 |

RSPWV | 3.8662 | 11.514 | 42.46 | 169.57 | 721.94 |

**Table 13.**Computational time for analyzed TFRs of different dimensions for a noisy linear FM signal with 128 time samples (averaged over 100 realizations, SNR of 5 dB), cell colors range from green for the best result to red for the worst result in each column respectively, in milliseconds (ms).

TFR Dimension | $32\times 32$ | $64\times 64$ | $128\times 128$ | $256\times 256$ | $512\times 512$ |
---|---|---|---|---|---|

SP | 1.4193 | 1.3427 | 1.4466 | 1.6453 | 2.0954 |

SPWV | 23.058 | 22.478 | 22.415 | 23.576 | 24.098 |

WV | 1.0921 | 1.0986 | 1.2774 | 1.3618 | 1.5274 |

AMWSC | 3.1886 | 5.0576 | 9.607 | 19.048 | 37.809 |

ASPWV | 34.879 | 57.728 | 101.24 | 188.75 | 447.8 |

AUB | 19.101 | 28.757 | 48.061 | 88.604 | 160.95 |

AAU | 26.891 | 41.811 | 73.08 | 126.18 | 228.62 |

ADF | 16.001 | 25.113 | 43.079 | 75.009 | 141.32 |

RSP | 7.2236 | 11.604 | 19.851 | 38.007 | 72.442 |

RGSP | 5.8219 | 9.1319 | 16.085 | 29.338 | 57.913 |

RMSC | 66.308 | 134.42 | 269.46 | 542.5 | 1129.2 |

RPWV | 2.8339 | 2.7727 | 3.3549 | 4.8437 | 8.2814 |

RSPWV | 30.529 | 34.782 | 42.888 | 59.86 | 95.77 |

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**MDPI and ACS Style**

Bačnar, D.; Saulig, N.; Petrijevčanin Vuksanović, I.; Lerga, J.
Entropy-Based Concentration and Instantaneous Frequency of TFDs from Cohen’s, Affine, and Reassigned Classes. *Sensors* **2022**, *22*, 3727.
https://doi.org/10.3390/s22103727

**AMA Style**

Bačnar D, Saulig N, Petrijevčanin Vuksanović I, Lerga J.
Entropy-Based Concentration and Instantaneous Frequency of TFDs from Cohen’s, Affine, and Reassigned Classes. *Sensors*. 2022; 22(10):3727.
https://doi.org/10.3390/s22103727

**Chicago/Turabian Style**

Bačnar, David, Nicoletta Saulig, Irena Petrijevčanin Vuksanović, and Jonatan Lerga.
2022. "Entropy-Based Concentration and Instantaneous Frequency of TFDs from Cohen’s, Affine, and Reassigned Classes" *Sensors* 22, no. 10: 3727.
https://doi.org/10.3390/s22103727