## 6 Physical Layer Aspects  
Although this paper does not aim to provide the reader with a detailed analysis of PHY research, we feel that without briefly discussing PHY, the survey will not be complete. Not wanting to detail all the standards (e.g., IEEE 802.15.1, IEEE 802.11) used for WSNs, we solely discuss the IEEE 802.15.4 standard in this section, since most of the popular standards, such as ZigBee Pro, WirelessHART, and ISA100.11a, are built on top of this one. The IEEE 802.15.4 standard is developed for low data rate applications. The PHY of this standard operates in different frequency bands, mostly in the 2.4 GHz band, using Orthogonal Quadrature Phase Shift Keying (O-QPSK) modulation with a data rate of 250 kbps. No channel model has been proposed in this standard, so most of the analysis has been done with the channel model prescribed in the IEEE 802.11 standard. We can adjust this channel model to represent an industrial environment, which can be represented as:  
\[d= 10^{\frac{P_{t}-P_{r}-40.2-\psi}{20}},\text{ for }\ d<8\text{ m and} \tag{1}\] \[d= 8\times 10^{\frac{P_{t}-P_{r}-58.5-\psi}{33}},\text{ for }\ d>8\text{ m} \tag{2}\]  
where, \(P_{t}\) is the transmit power in dB; \(P_{r}\) is the received power in dB at distance \(d\); \(\psi\left[\text{dB}\right]\) is a Gauss distributed random variable with a certain mean and \(\sigma_{\psi}^{2}\)variance generated due to the shadowing effect. In the case of a home environment, \(\psi\) is generally considered as a zero mean random variable. However, in the industrial environment, we expect large shadowing due to the presence of heavy machinery, which typically causes a positive biased shadowing effect. The mean of the shadow fading can vary according to different industrial setups. The authors in [62] claim that the Signal-to-Noise Ratio (SNR) should not be used as a performance matrix to calculate the reliability of IEEE 802.15.4 communication links, as the relationship between SNR and the Packet Error Rate (PER) is non-linear. A small degradation in SNR beyond the threshold can shift the PER from 0 to 1. However, SNR has a direct relationship with the bit error probability and outage probability, which are two important criteria of interest in the performance analysis of IEEE 802.15.4 PHY. These performance matrices are dependent on the channel model. As most of the sensors and actuators in an industrial environment are immobile, we consider the channel as a slow fading channel, which has almost the same performance as an Additive White Gaussian Noise (AWGN) channel. In AWGN, the bit error probability for the QPSK modulation is defined as:  
\[P_{b}=Q\Big{(}\sqrt{2\gamma_{0}}\,\Big{)}, \tag{3}\]  
where, \(\gamma_{0}\) is the SNR of the received signal. The outage probability, \(P_{out}\) is the probability that the received signal's average SNR, \(\overline{\gamma}_{s}\) falls below the minimum required SNR, \(\gamma_{0}\) for the pre-defined acceptable communication performance [63]. Mathematically,  
\[P_{out}=p\Big{(}\overline{\gamma}_{s}<\gamma_{0}\,\Big{)}=\int\limits_{0}^{ \gamma_{0}}p_{\gamma_{0}}\,\big{(}\gamma\big{)}d\gamma \tag{4}\]  
From Equations (3) and (4) the required average SNR is:  
\[\overline{\gamma}_{s}=\frac{\Big{(}Q^{-1}\big{(}P_{b}\,\big{)}\Big{)}^{2}}{-2 \ln\big{(}1-P_{out}\,\big{)}} \tag{5}\]  
In AWGN, the carrier-to-noise ratio of the received signal can be expressed as:  
\[\left(\frac{C}{N}\right)_{dB}=\left(\frac{E_{b}}{N_{0}}\right)_{dB}+\left( \frac{f_{b}}{B}\right)_{dB}, \tag{6}\]  
where, \(\left(\frac{E_{b}}{N_{0}}\right)_{dB}=\overline{\gamma}_{b}\) is the SNR per bit; \(f_{b}\) is the channel data rate (net bitrate) and \(B\) is the channel bandwidth [64].