## 2 GNSS Measurement Noise  
GNSS measurements, like all other observables, are burdened with measurement errors. Most of the measurement errors, with the highest value, such as the orbital and clock errors or atmospheric delays, are included in the observation functional model and taken into account in a deterministic manner. The rest of the errors can be included in the observation model using the laws of probability as stochastic quantities [56]. Then, their random behavior is assumed, and their values and mutual correlations are described as variances and covariances in the observation VC matrix. A significant portion of these errors is highly dependent on the GNSS receiver itself, its components performance, and the finite precision of measurements. This section presents the theoretical relationship between the signal strength, the measurement noise and the nominal precision of GNSS receiver performance. From the point of view of electrical signals processing, measurement random errors can be treated as thermal noise caused by random electrons movement in any of the electronic components of the GNSS receiver system with a temperature above 0 K [9]. Assuming thermal noise is white noise (with constant power spectral density) with a Gaussian distribution, its power for typical temperature \(T=290\) K and GPS L1 signal bandwidth \(B=2\) MHz can be expressed as [79]:  
\[P_{N}=kTB \tag{1}\]  
where \(k=1.3806\cdot 10^{-23}\) W \(\cdot\) s \(\cdot\) K\({}^{-1}\) is the Boltzmann constant. This results in a noise power of approx. \(8\cdot 10^{-15}\) W (\(-141.0\) dBW in decibels relative to one watt) or noise power spectral density of \(N_{0}=-204.0\)  dBW/Hz expressed per unit of bandwidth. However, the noise power value alone does not yield sufficient information about the signal quality and the resulting measurement errors. The key to the signal quality analysis is the received signal power, and its ratio to the noise power (referred to as SNR) when it is related to a given bandwidth (\(P_{N}\)) or C/N\({}_{0}\) if referred to the power per unit of bandwidth (\(N_{0}\)). Table 1 summarizes a simplified example of received signal power budget and thermal noise for C/A GPS observation with the minimum transmitted signal power of about 26.8 W. The received signal power on the earth's surface can be calculated after taking into account: the power path loss due to the spreading of the total signal energy over the entire surface area, satellite antenna gain caused by the amplification of the signal power towards the  
Earth in relation to the isotropic antenna, atmospheric attenuation and receiver antenna gain taking into account effective area of an omnidirectional receive antenna and gain of patch antenna relative to the isotropic. The presented values refer to the worst-case scenario, for a satellite elevation of about \(e=5^{\circ}\), for which the values of satellite signal spread angle of \(27.8^{\circ}\) and gain of the receiver antenna related to the isotropic one of \(-2\,\mathrm{d}\mathrm{B}\mathrm{i}\mathrm{c}\) were adopted [7; 79; 80]. The numbers in Table 1 for a very low elevation satellite illustrate the lower values of the received signal power that can be acquired. This signal power relates directly to the minimum received signal power that is a guaranteed minimum received power level, as specified in the Interface Control Document (ICD) for each of the GNSS systems, but may vary slightly depending on the signal power budget assumptions. Based on the minimum received signal power, compiled in [81], the nominal (theoretical) \(\mathrm{C}/\mathrm{N}_{0}\) for GNSS signals were determined. Figure 1 shows the \(\mathrm{C}/\mathrm{N}_{0}\) for individual systems and signals analyzed in the study, distinguishing between individual satellite blocks for GPS signals. Note that if the information on the power of a given signal was not publicly available (as in the case of GLONASS P-code or BeiDou 6l), values for other signals of the given system were adopted for the calculation.