# The Evaluation of Noise Spectroscopy Tests

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Models: Analogies and Similarities

_{s}, where λ

_{s}[m] is the wavelength of the light from a light source. This procedure has been utilized by a number of researchers, such as the authors of [10,11] in solving models and comparing light systems.

_{s}(λ

_{s}denotes the wavelength of light from λ

_{s}∈ <λ

_{min};λ

_{max}>). We assume the basic reflection [-] and attenuation [-], without polarization of the wave. Two other variants are acceptable: V

_{λ}≠ f(λ) and V

_{λ}= f(λ). With V

_{λ}= f(λ), geometries and effects are considered in which there occur the polarization of light, diffraction in parts (l ≈ λ

_{s}), and light interferences. The former of the above cases, V

_{λ}≠ f(λ), can be solved via the duality between a light-related task and a thermal task with radiation.

_{T}[W·m

^{−1}·K

^{−1}] is the thermal conductivity, q [W·s·m

^{−3}] denotes the heat source (in the temperature field), Ω

_{T}represents the temperature model region, and T [K] is the temperature. For light, we have

_{s}[W·m

^{−1}·K

^{−1}] is the conductivity of light, q

_{s}[W·s·m

^{−3}] denotes the source of light (in the geometrical concept of light propagation), Ω

_{s}is the light model region, and ϕ

_{s}represents the scalar function [K]. We also have

^{−2}] is the luminous flux and E

_{s}[K·s·m

^{−1}] denotes the illuminance. In general terms, within Table 1 above, H

_{λ}[W·m

^{−1}] is monochromatic radiation, W

_{λ}[W·m

^{−1}] denotes radiation, λ [m] expresses wavelength, Φ

_{e}[W] is radiant flux, Φ

_{λ}[W·m

^{−1}] is monochromatic luminous flux, H

_{e}[W·m

^{−2}] is radiant exposure, I [W·sr

^{−1}] denotes luminous intensity, ω [sr]represents a spatial angle, I

_{e}[W·sr

^{−1}] is radiant intensity, L

_{e}[W·m

^{−2}·sr

^{−1}] denotes radiance, L

_{ϑ}[W·m

^{−2}·sr

^{−1}] is luminance, S [m

^{2}] is a plane, A [m

^{2}] is a plane, E

_{e}[K·s·m

^{−1}] represents irradiance, φ

_{T}[W·s·m

^{−2}] denotes thermal flux, φ [W·s·m

^{−2}] is luminous flux, K

_{m}[-] are the conversion constants of SI units and [lm] luminous flux, and V

_{λ}[-] denotes the function of the relative spectral sensitivity of a standard sensor.

## 3. Broadband Signal Quantities

_{b}[W] of a black body in relation to the temperature T [K]. We have

^{−1}] is the Stefan–Boltzmann constant. Then, for the general surface of a real body, the radiated power P

_{r}[W] is

^{2}] denotes the surface of the radiation emitting area. To evaluate tasks with the propagation of an electromagnetic wave (as a signal), and using the relationship between the modeled task and the wavelength x/λ in the interval x/λ ∈ 〈10

^{7};10

^{14}〉, the radiant intensity I depending on the temperature T (from Planck’s law, [13,14]) is expressed for a small body and radiation from a half sphere. We have

^{−1}] is the angular frequency of an electromagnetic wave, f [Hz] is the frequency, h [W·s

^{2}] is the Planck constant, and c [m·s

^{−1}] denotes the velocity of (white) light. The radiated power can then be written in the form for one half of a spherical object having an angle Θ:

_{r}can be graphically represented as the behavior for independent variables f and T, and this is shown in Figure 2, Figure 3, Figure 4 and Figure 5.

_{r}/∆f and the frequency f has to be captured as either the broadband task or individual analyses, then, given that the frequency range f extends over several decades or reaches above 3 GHz, we can observe a difference between the simplified Expressions (12) and (13) and the one derived from the radiation of hemisphere (9). By extension, when comparing the spectral power densities represented in Figure 6, Figure 7 and Figure 8, we can also notice marked differences in the behavior of the functions. According to the relevant simplified criterion (12), power spectral density does not depend on the signal frequency f, but, as further outlined in Formula (9), the spectral density for half a sphere depends on the frequency (in a nonlinear manner). Interestingly, a comparison between the above Equation (9) for a hemisphere and the function derived by Nyquist will reveal a difference in the power of the frequency that determines the power spectral density of the signal, or noise; this finding subsequently enables us to explicate certain effects related to UWB and noise spectral tasks or, by extension, their analyses, estimates, predictions, real measurements, and experimental results.

## 4. Experimental Comparison of Signal Properties for Noise Spectral Analyses

_{test}is conducted in the noise spectrometer area without the sample, and the spectrometer is employed to evaluate the variation of the signal with respect to the background. Thus, for example, the power of the noise signal was changed from 22 dBm (Figure 10a) through 30 dBm (Figure 10b), 70 dBm (Figure 10c), 70 dBm (Figure 10d), and 105 dBm (Figure 10e) to 127 dBm (Figure 10f). The results of the given test are indicated in Figure 10. The relevant experimental measurements make it possible to estimate the power rates to be achieved in a chain comprising a noise generator, an amplifier, and an antenna in order to secure such conditions for the given noise spectral analysis that will facilitate the cumulation of the required frequency characteristics of the tested samples.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The behavior of function ${\Psi}_{r}\left(f,T\right)$, with the frequency range of f ~0.1 MHz–1000 THz and temperature of T = 10–1000 K.

**Figure 2.**The behavior of radiated power P

_{r}[W]: (

**a**) the frequency range of f ~0.1 MHz–1000 THz and temperatures T between 10 and 1000 K; (

**b**) the behavior of radiated power P

_{r}[W] in the frequency range of f ~100 MHz–100 GHz and at temperatures of T = 10–1000 K.

**Figure 3.**The behavior of radiated power P

_{r}[W] in the frequency range of f ~10 MHz–10 GHz and at temperatures of T = 100–1000 K.

**Figure 4.**The behavior of radiated power P

_{r}[W] in the frequency range of f ~10 MHz–1 GHz and at temperatures of T = 100–1000 K.

**Figure 5.**The behavior of radiated power P

_{r}[W] in the frequency range of f ~10 MHz–1 GHz and at temperatures of T = 100–10,000 K (relation (11)).

**Figure 6.**The behavior of power spectral density P

_{r}/∆f [Ws]: (

**a**) the frequency range of f ~0.1 MHz–1000 THz and temperatures between 10 and 1000 K, according to relation (9); (

**b**) in the frequency range of f ~100 MHz–10 GHz and at temperatures of T = 100–1000 K, from Planck’s law of a black body, relation (9).

**Figure 7.**The behavior of power spectral density P

_{r}/∆f [Ws] in the frequency range of f ~100 MHz–10 GHz and at temperatures of T = 10–100 K, related to Johnson–Nyquist noise, relation (12).

**Figure 8.**The behavior of power spectral density P

_{r}/∆f [Ws]: (

**a**) the frequency range of f ~0.1 MHz–1000 THz and temperatures between 10 and 1000 K, according to the Nyquist–Planck Formula (13); (

**b**) related to the Nyquist–Planck Formula (13); the frequency range of f ~100 MHz–10 GHz and temperatures of T = 10–100 K.

**Figure 9.**The tested samples of periodic structures: the expected resonant frequencies of the first mode: (

**a**) resonator f

_{1}

^{1}= 58 MHz; (

**b**) resonator f

_{1}

^{2}= 83 MHz; (

**c**) resonator f

_{1}

^{3}= 200 MHz.

**Figure 10.**The test of the noise spectroscopy setup with respect to the power of the excitation signal at one frequency: (

**a**) the background checking, 198–202 MHz; the generator signal of 200 MHz–22 dBm, without an amplifier—number of samples: 2000; (

**b**) the background checking, 198–202 MHz; the generator signal of 200 MHz–30 dBm, without an amplifier—number of samples: 2000; (

**c**) the background checking, 198–202 MHz; the generator signal of 200 MHz–70 dBm, without an amplifier—number of samples: 10,000; (

**d**) the test generator: signal of 58 MHz–70 dBm (noise level signal); input: a +27 dBm amplifier; output: a two-stage electronic radio-frequency amplifier (ERA) ; (

**e**) the test generator: signal of 58 MHz–105 dBm (signal immersed in noise); input: a +27 dBm amplifier; output: a two-stage ERA—number of samples: 5000; (

**f**) the test generator: signal of 58 MHz–127 dBm (signal immersed in noise); input: a +27 dBm amplifier; output: a two-stage ERA—number of samples: 5000.

**Figure 11.**The testing of the samples from Figure 9 with noise spectroscopy: (

**a**) the noise generator: horizontal configuration; (

**b**) the noise generator: vertical configuration; (

**c**) the noise generator: tangential configuration; (

**d**) the noise generator: a TVA amplifier +27 dBm; an antenna–an antenna –a two-stage ERA amplifier–an Agilent spectrometer (Keysight Technologies, Inc., Westlake Village, CA, USA); a sample from Figure 9. f

_{1}

^{1}, horizontal position, 58 MHz; (

**e**) the noise generator: a TVA amplifier +27 dBm; an antenna–an antenna–a two-stage ERA amplifier–an Agilent spectrometer; a sample from Figure 9. f

_{1}

^{1}, vertical position; (

**f**) the noise generator: a time variant amplifier (TVA) + 27 dBm; an antenna–an antenna–a two-stage ERA amplifier–an Agilent spectrometer; a sample from Figure 9. f

_{1}

^{1}, tangential position; (

**g**) the noise generator: a TVA amplifier + 27 dBm; an antenna–an antenna–a two-stage ERA amplifier–an Agilent spectrometer; a sample from Figure 9. f

_{1}

^{2}, horizontal position; (

**h**) the noise generator: a TVA amplifier + 27 dBm; an antenna–an antenna–a two-stage ERA amplifier–an Agilent spectrometer; a sample from Figure 9. f

_{1}

^{3}, horizontal position.

**Figure 12.**The design and types of the samples tested for noise spectroscopy: (

**a**) the design; (

**b**) the fabricated unit; (

**c**) the frequency dependence without a capacitor; (

**d**) the frequency dependence with capacitors; (

**e**) a detailed view of the tuned structure; (

**f**) the resonance values measured over elementary resonators; (

**g**) the magnetic imaging (MI) lens designed with square resonators [16,17]; (

**h**) the fabricated MI-based unit; (

**i**) the resonant structure consisting of two plates with quasi-periodic resonators: f

_{r}= 58 MHz; (

**j**) and f

_{r}= 83 MHz.

Heat (λ ∈ <780;10,000> nm) | White Light (λ ∈ <440;780> nm) |
---|---|

Monochromatic (spectral) radiation [W·m^{−1}] ${H}_{\lambda}=\frac{d{W}_{\lambda}}{d\lambda}$ | Monochromatic (spectral) luminous flux [W·m^{−1}] ${\Phi}_{\lambda}=\frac{d{\Phi}_{e}}{d\lambda}$ |

Radiant flux [W] ${\Phi}_{e}={{\displaystyle \int}}_{S}{\mathrm{H}}_{e}dS$, ${H}_{e}={{\displaystyle \int}}_{0}^{\infty}{H}_{\lambda}d\lambda $ | Radiant flux [W] ${\Phi}_{e}=\underset{0}{\overset{\infty}{{\displaystyle \int}}}{\Phi}_{\lambda}d\lambda $ |

Radiant intensity [W·sr^{−1}] I$=\frac{d{\Phi}_{e}}{d\omega}$ | Luminous intensity [W·sr^{−1}] ${I}_{e}=\frac{d{\Phi}_{e}}{d\omega}$ |

Radiance [W·m^{−2}·sr^{−1}] ${L}_{e}=\frac{d{I}_{e}}{dScos\vartheta}$ | Luminance [W·m^{−2}·sr^{−1}] ${L}_{e}=\frac{d{I}_{\vartheta}}{dScos\vartheta}$ |

Radiant exposure [W·m^{−2}] ${H}_{e}=\frac{d{\Phi}_{e}}{dS}$ | Luminous exitance [W·m^{−2}] $H=\frac{d{\Phi}_{e}}{dS}$ |

Irradiance [K·s·m^{−1}] ${E}_{e}=\frac{d{\Phi}_{e}}{dA}$ | Illuminance [K·s·m^{−1}] ${E}_{s}=\frac{d{\Phi}_{e}}{dA}$ |

Thermal flux [W·s·m^{−2}] ${\varphi}_{T}={k}_{T}\mathrm{grad}T$ | Luminous flux [W·s·m^{−2}] $\varphi ={K}_{m}{{\displaystyle \int}}_{0}^{\infty}{V}_{\lambda}{\varphi}_{\lambda}d\lambda $ |

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**MDPI and ACS Style**

Fiala, P.; Drexler, P.; Nespor, D.; Szabo, Z.; Mikulka, J.; Polivka, J. The Evaluation of Noise Spectroscopy Tests. *Entropy* **2016**, *18*, 443.
https://doi.org/10.3390/e18120443

**AMA Style**

Fiala P, Drexler P, Nespor D, Szabo Z, Mikulka J, Polivka J. The Evaluation of Noise Spectroscopy Tests. *Entropy*. 2016; 18(12):443.
https://doi.org/10.3390/e18120443

**Chicago/Turabian Style**

Fiala, Pavel, Petr Drexler, Dusan Nespor, Zoltan Szabo, Jan Mikulka, and Jiri Polivka. 2016. "The Evaluation of Noise Spectroscopy Tests" *Entropy* 18, no. 12: 443.
https://doi.org/10.3390/e18120443