## 3 Simulation Framework
### The Space-Time Engine  
Realistic system simulation involves generating wireless channel realizations according to a pre-defined, realistic channel model, which approximates the typical propagation conditions encountered in the scenarios of interest well. The channel fading is described by three components: large-scale fading consisting of a path loss due to distance-dependent average attenuation of the channel; a local random fluctuation around the mean signal level due to shadowing effects; and a small-scale fading due to the multipath and diffuse propagation effects [10]. The path loss over a wireless link is generated according to:  
\[L(dB)=L_{0}(d_{0})(dB)+10n\log(d/d_{0})+S(dB) \tag{1}\]  
where \(d_{0}\) is a reference distance, \(L_{0}(d_{0})\) is the average path loss at the reference distance, \(d\) is the link distance and \(S\) is a zero-mean Gaussian random variable with standard deviation \(\sigma_{S}\), expressed in  
Figure 2: Main components of the system simulation framework. dB [10]. The latter represents the effect of the shadowing on the link budget. The space-time engine allows us to take the spatial correlation of the shadow fading into account, _i.e._, it enables the generation of non-independent shadow fading samples for geometrically closely-spaced wireless links according to some analytic or measurement model, such as exponential or the model described in [11]. There is an additional challenge in designing system simulation for a frequency-hopping mesh network that is not well addressed in the literature. Being such a system, WirelessHART employs a slot-by-slot frequency hopping scheme, _i.e._, a transmission and an eventual retransmission between the same network nodes will take place on different RF channels. A realistic model needs to account for a varying degree of correlation between the fading processes experienced by a single link on different carrier frequencies at closely-spaced time instants. Intuitively, a large degree of frequency correlation (high channel coherence bandwidth) will render the frequency hopping ineffective at combating long, deep fades, whereas a low frequency correlation is beneficial, as successive transmission attempts experience nearly independent channel realizations, thus yielding to frequency diversity. The existing literature does not provide reliable data on the frequency correlation in industrial environments. Measurement results for large indoor hotspots indicate that typical channels exhibit significant (\(>\)0.5...0.7) correlation over a bandwidth of 100 MHz for line-of-sight and 20 MHz for non-line-of-sight scenarios, respectively [12]. Thus, real channels might show significant correlation over multiple 5 MHz-wide 802.15.4 channels in an industrial indoor environment. To address all of the mentioned challenges, it is necessary to adopt a low-complexity analytical multipath fading model that allows tight control of temporal fading correlation, specification of a certain channel delay profile and that provides small-scale fading realizations at multiple carrier frequencies, in which the frequency correlation is determined by the temporal support of the power delay profile. The basis of the model is the channel simulation model presented in [13]. The channel is described by a discrete tapped-delay line model:  
\[h_{i}(t,\tau)=\sum_{l=0}^{L-1}\mu_{l}^{(i)}(t)\delta(\tau-\tau_{l}) \tag{2}\]  
where \(h_{i}(t,\tau)\) is the time-variant channel impulse response on carrier frequency index \(i\), \(L\) is the number of channel taps, \(\mu_{l}^{(i)}\) is the time-variant tap coefficient that characterizes the fading on the carrier frequency \(i\) and \(\tau_{l}\) is the propagation delay of the \(l\)-th channel tap. It is reasonable to assume flat fading, as long as the root mean square (rms) delay spread is much less than the signaling interval (for example, Channel D would be flat fading for 802.15.4 g).