# Theory of Faults (ToF): Numerical Quality Management in Complex Systems
## 3 Development
### Dsp  
In this section, we trace the path of control signals. The specifics of control signals, such as LTI, are relevant to the signals and their filters but not to the monitored system, which is non-linear and complex. The communication effects are superimposed on the complex system, adding to information load in the info-system. Definition: On construction sites, for each discrete unit of time, a work order is issued to workers, suppliers, and subcontractors, specifying work, means, and materials. Each specific assemblage constitutes an \"**activity**\". When the execution of an activity is completed, a signal that delivers the information \"DONE\" is produced. QC may approve = \"Ok\",  
or \"Reject\" = to be rectified prior to approval,  
or not approve = \"Rework\". The QC rulings are DSP control operators (portrayed in Table 1): \begin{tabular}{l l} \(>\) & Pass & (\"Ok\"), \\ \(>>\) & Reject = & one **feedback** loop & (1 FB) (\"Reject\"), \\ \(>>>\) & Rework = double **feedback** loop & (2 FB) (\"Rework\"). \\ \end{tabular}   
Definition: A \"**Fault**\" is what causes a reject or a rework in the info-system. A \"**Defect**\" is a small fault that is small enough not to cause an info-system response. Together they are referred to as faults. Change orders cause rework in the info-system; therefore, change orders are faults. Designs \"published\" are faults. The DSP operators are called \"**Filters**.\" The filters are Linear Time-Invariant (LTI). A unit signal is represented here by the Kronecker delta \(\delta\) [0] = 1. The control signals \(\delta\)[n] are LTI; signals only accumulate, scale, and time-shift. Time units n, \(n\in N\), a discrete time set. \(\delta[n]\) signals entering the 1 FB loop produce a constant outcome of 1, 1, 1, 1... as long as it is not terminated. The signal is augmented to the system's current data load. The 2 FB loops are amplification sequences generated by the accumulation of previous gains. The 2 FB loops cause a Fibonacci amplification: 1, 1, 2, 3, 5, 8, 13... [55]. The magnitude of the faults is determined by power functions. QC must scale the control signal in accordance with the fault's magnitude; this process may be biased by competence and interest. In our simulation, the control signals are directly scaled by a power function. The PDF determines the recurrence of the signal. The scaled and timed signals produced by QC pass through the filters, where they are scaled, shifted, and accumulated; the outgoing signal is fed into the information system, where the information constraint determines the eventual outcome (numeric experiment A.3). The response of LTI systems are functions called \"**Impulse Response**\", annotated h[n]. LTI systems are injectively determined by their impulse responses. The outcome y[n] of a specific fault signal vector \(x[n]\) passing through the LTI is the convolution (denoted by \"\(*\)\") of the impulse response h[n] and the fault signal vector \(x[n]\):  
\[y[n]=x[n]*h[n]=\sum\
olimits_{k=0}^{K-1}h[k]x[n-k] \tag{18}\]  
This entails breaking down the vector to its respective spectrum (=K phases), a procedure made simple by choosing a \(\delta[n]\) signal. The time domain is as follows:  
\[x[n]=\sum\
olimits_{k\in z}\delta[n-k] \tag{19}\]  
The frequency domain is the Fourier transform of the time domain, or Z transform for discrete signals. The lowercase variables stand for the time domain x(t), y(t), h[n], etc. the uppercase (X(s), Y(s), H[z]) stand for the transform:  
\[\mathbf{X}(\omega)=\frac{2\pi}{N}{\sum\
olimits_{k\in z}\delta\bigg{[}\omega -\frac{2\pi}{N}k\bigg{]}} \tag{20}\]  
A specific phase (\(\frac{2\pi}{N}k\)) matches a specific activity (k).