## 3 Implementation of the Soft Sensor in the Time Domain
### Application of the TVB Method to Obtain the Soft Sensor  
From a computational perspective, performing a full convolution of functions over the entire time range is impractical for use in a soft sensor. Moreover, it is unnecessary, as the impact of pressure data diminishes over time, evidenced by the convergence of the weighting function to zero for large times. [PERSON] and [PERSON] devised a method to compute convolutions efficiently between weighting functions and the time derivative of a discrete function [37]. This approach can similarly be applied to the convolution of a weighting function with a discrete function, as demonstrated in this paper. The weighting function is composed of exponential terms, and the discrete function is the difference between measured pressure signals. The method's efficiency lies in requiring only a few pressure data points while still delivering accurate results. The approach, termed the \"Trikha-Vardy-Brown\" (TVB) method, is an extension of a convolution simplification method initially introduced by [PERSON] [38]. In 1975, [PERSON] proposed a one-step scheme for the convolution integral, incorporating historical acceleration into parameters updated at each integration time step [38]. However, [PERSON]'s method had two limitations: first, the historical component lacked the generality needed for different types of flows, and second, it was only valid for very small time steps, which was impractical for real-world applications. In 1983, [PERSON] et al. improved [PERSON]'s technique by deriving a model with ten exponential terms, in contrast to the previously used three [39]. Although this approach required more memory, it enhanced [PERSON]'s method. By 1991, [PERSON] et al. had developed a hybrid model, combining full convolution for short durations with [PERSON]'s algorithm for longer times, using a sum of five exponential terms [40]. Later, [PERSON] proposed an enhanced stepping formula to correct the first-time step limitations of [PERSON]'s method. Assuming uniform acceleration, [PERSON]'s approach also employed five exponential terms [41]. Similarly, [PERSON] et al. improved the first-step accuracy by introducing an empirical decay factor and developed advanced exponential sum weighting functions applicable to both laminar and turbulent flows [42]. In 2002, [PERSON] and [PERSON] presented an alternative to [PERSON]'s one-step method by using a weighting function derived from the work of [PERSON] and [PERSON] [43], which they employed to study the evolution of specific integrals in unsteady flow [44]. More recently, [PERSON] refined [PERSON]'s method by correcting the erratic recursive formula used to compute unsteady wall shear stresses [45]. His results were validated using the computationally expensive formula from [PERSON] and [PERSON] in 2010 [46]. In this contribution, the TVB method was implemented in Matlab [47]. The derivation follows the authors' approach but is applied to the current equations. The method is suitable for convolving a weighting function of the form \(f(t)=\sum_{i=1}^{N}m_{i}e^{-n_{i}t}\) with the time derivative of a second function expressed as \(g(t)=\frac{\partial p(t)}{\partial t}\). Beginning with Equation (53), the flow rate can be divided into a stationary flow rate (according to the Law of Hagen and Poiseuille) and two dynamic flow rates corresponding to each pressure:  
\[Q_{2}(t_{\rm n})=Q_{2,\rm stat}(t_{\rm n})-[Q_{2,\rm dyn,1}(t_{\rm n})-Q_{2, \rm dyn,2}(t_{\rm n})]. \tag{55}\]  
The convolution integral for the stationary volumetric flow rate can be readily solved and is already presented in Equation (53). Consequently, the TVB method must be applied solely to the dynamic volumetric flow rates. The following derivation will exemplify this method for \(Q_{2,\rm dyn,1}\), while the procedure for \(Q_{2,\rm dyn,2}\) would follow similarly.