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You have $$f\left(\mathbf x, u\right) = \begin{bmatrix}\frac{-1}{T}\tau+\frac{K}{T} u \\ \frac{\tau}{mr} \\ 0 \end{bmatrix} \tag a$$ From which you (eventually) derive \mathbf {A}_d=\begin{bmatrix} 1-\frac{\Delta T}{T} & 0 &0 \\ \frac{\Delta T}{m_{op} r} & 1 & \frac{-\tau_{op} \Delta T}{m_{op}^{2} r}\\ 0& 0 & 1 \end{bmatrix} \tag ... 2 The OP's updated working is incorrect. Following up what Hilmar suggested gives \begin{align} Y(t) &= a\left(X(t)\right)^2\\ &= a\left(S(t) + N(t)\right)^2\\ &= a\left(S(t)\right)^2 + 2aS(t)N(t) + a\left(N(t)\right)^2\\ &{\large\Downarrow}\\ E[Y(t)]&= aE\left[\left(S(t)\right)^2 \right] + 2aE\left[S(t)N(t)\right] + aE\left[\left(S(t)\... 2 Put your second equations into your first equation, express Y(t) as a function of S(t) and N(t) Apply the definition for mean and autocorrelation. Simplify and solve 2 This depends a bit on what you want to get our of the analysis Step 1: Instantaneous power You square the signal. That changes the spectrum considerably. In particularly you need to watch out for aliasing. Let's say your signal is a 15kHz sine wave sampled at 48 kHz. If you square this you get one component at DC (0 Hz) and another at 30 kHz. However, the ... 2 y[n] = \cos(\frac{5\pi}{8}n + \frac{\pi}{4}) is not a system, but a signal. Anyway, if you insist that you have a system with a fixed output y[n] = \cos(\frac{5\pi}{8}n + \frac{\pi}{4}), then this will be a non-linear system. Proof is easy: assume an arbitrary input x_1[n] to your system. The output will be y_1[n] = \cos(\frac{5\pi}{8}n + \frac{\pi}{4}... 2 Hard limits on what the output actuator can do has got to be the most common control non-linearity there is. The search term that'll get you hits on academic papers is "actuator saturation". But the moment I get outside linear controller design everything gets nebulous; discussions of a dozen different techniques with no discussion of how to ... 1 You may be overthinking this. If the input has length T_x and the impulse response of the LTI system has length T_h than the output will have have length T_x + T_h. If either input or impulse response are infinite so will be the output. That's all there is to it. In other words: if both input and impulse response have finite support so will have the ... 1 If superposition works, then independent mode/component extraction is of interest. Synchrosqueezing is well-suited for this task. Extracted features can ten be fed to an anomaly detection system - optionally with Gaussianization. Other methods can be applied to the extracted components as if they were individual signals, so the described approach is ... 1 When one nonlinear and one linear operation are suitably adapted to the problem, the nonlinear one is quite often applied first. Nonlinear processing is often applied to modify data such that more classical algorithms apply more easily, even when the original data does not follow the proper assumptions. Examples are: removing outliers/trimming data before ... 1 This technique is similar to a classical signal processing in seismic being an intendance of "first break picking". It was proposed in First arrival picking on common-offset trace collections for automatic estimation of static corrections, Françoise Coppens, 1985 (a former colleagues of mine). You may find the name STA/LTA (short-term average/long-... 1 Given the system I/O definition:y[n] = \mathcal{H}\{x[n]\} = x[n^2] \tag{1} $$you can easily show that it's a linear (but time-varying) system. Following the standard procedure let$$y_1[n] = \mathcal{H}\{x_1[n]\} = x_1[n^2] \tag{2.1}$$and$$y_2[n] = \mathcal{H}\{x_2[n]\} = x_2[n^2] \tag{2.2}$$then define$$x_3[n] = a x_1[n] + b x_2[n] \tag{3}$$and$$ ...