# Theory of Faults (ToF): Numerical Quality Management in Complex Systems
## 3 Development
### Dsp
The Z transforms that interest us are  
\[\mathbf{X}[z]=\sum\
olimits_{m=-\infty}^{+\infty}x[n]\mathbf{z}^{-n} \tag{21}\]  
\[\mathbf{H}[\mathbf{z}]\equiv\sum\
olimits_{k=-\infty}^{+\infty}\mathrm{h[k] }\mathbf{z}^{-\mathrm{k}} \tag{22}\]  
\[\mathbf{Y}[\mathbf{z}]=\mathrm{H[z]}\ \mathbf{X}[\mathbf{z}] \tag{23}\]  
For an eigenvector signal \(x[n]=\mathbf{z}^{n}\),  
\[y[n]=\sum\
olimits_{k=-\infty}^{+\infty}h[k]x[n-k]=\mathbf{z}^{n}{\sum \
olimits_{k=-\infty}^{+\infty}h[k]\mathbf{z}^{-k}} \tag{24}\]  
\[y[n]=H[\mathbf{z}]\mathbf{z}^{n}=\lambda\mathbf{z}^{n} \tag{25}\]  
H[\(\mathbf{z}\)] is the eigenvalue of \(\mathbf{z}^{n}\). H[\(\mathbf{z}\)] is called the Frequency Response. Any signal can be synthesized from eigenvectors:  
\[x[n]=\sum\
olimits_{k=-\infty}^{+\infty}\mathbf{X}[k]\mathbf{z}^{n} \tag{26}\]\[y[n]=\sum_{k=-\infty}^{+\infty}H_{k}[\mathbf{z}]\mathbf{X}[k]\mathbf{z}^{n}\]  
\(\mathbf{z}^{n}\) is the eigenvector of all LTIs. \(\mathbf{Z}\) may be any complex number in the region of convergence (ROC). \[x[n]=\sum\
olimits_{k\in z}\delta[n-k] \tag{19}\]0  
Therefore, \(\delta[n-K]\) are eigenvectors of all LTIs. The difference equation for the 1 FB loop is  
\[x[n]=\sum\
olimits_{k\in z}\delta[n-k] \tag{19}\]1  
\(Z\) transform on the difference equation yields  
\[x[n]=\sum\
olimits_{k\in z}\delta[n-k] \tag{19}\]2  
From Equations (22) and (28),  
\[x[n]=\sum\
olimits_{k\in z}\delta[n-k] \tag{19}\]3  
Table 1 summarizes the DSP tools used here. A delay in the signal causes a delay in the system response:  
\[x[n]=\sum\
olimits_{k\in z}\delta[n-k] \tag{19}\]4  
The phase shift is  
\[x[n]=\sum\
olimits_{k\in z}\delta[n-k] \tag{19}\]5  
The signal, a linear combinate of the \"k\" activities, has one frequency (Equation (20)). The outgoing signal has the same frequency, unaltered by the LTI systems (Equation (23)). Complex systems are not LTI; they do affect frequency. If site activities, subcontractors, or supply chains work on different periods, there are delays when a new period evolves to the lowest common denominator. When a difference in periods causes the system to be out of sync, stuttering halts in the work sequence occur, accompanied by constrained resources and collisions [56]. In Figure 7, we simulated a convolution to show its effect on a system like design offices. We created a vector of random generated numbers with an LN (\(\mu=4.8\), \(\sigma=1.6\)) distribution (Figure 7b) in green; this distribution was found in the Case Study 1; it \"averaged\" 27 system responses. The PDF was fed into the power function of Figure 3 to create a scaler for an eigenvector of \(\delta[n-k]\); this is the control signal of Figure 7a. The \begin{table}
\begin{tabular}{l l c} \hline
**The Operators** & **The Difference Equations**[55] & **Equation Numbers** \\ \hline
**Pass** & y[n] = x[n] & (31) \\ \hline
**Reject** & y[n] = x[n] + \(\rho_{1}\)y[n - 1] & (32) \\
**feedback loop** & y[n] = x[n] + \(\rho_{1}\)y[n - 1] + \(\rho_{2}\)y[n - 2] & (33) \\ \hline
**Reject** & \(H(z)=\frac{1}{1-z^{-1}}\) & (34) \\ \hline
**Rework** & \(H(z)=\frac{1}{1-\mathbf{z}^{-1}-\mathbf{z}^{-2}}\) & (35) \\ \hline \end{tabular}
\end{table} 
Table 1: The DSP control operators. signal was convoluted with an impulse response function in Figure 7b. We assumed a step function for \(h(t)\), which is an oversimplified impulse response. The convolution holds the meaning of a design workload generated by faults. Another distribution LN (\(\mu=5\), \(\sigma=1\)) was taken from the Figure 2 range. The results (Figure 7b in blue color) are dramatically different. This is another deterrent for modeling. The outcome is a doubling of the timeline (the exit signal abscise length is the sum of the incoming and impulse function abscissa's length), and the workload (ordinate) is stacked two degrees of magnitude higher than the entry magnitude. The amplification is the multiplying of magnitudes of \"\(x\)\" and \"\(h\)\" summed over the length of \"\(h\)\", that is, summed over the number activities.