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### 12.1    Simulation of continuous-time systems

#### 12.1.1    Simulation by approximation

Design methods based on continuous-discrete time changes actually consist of constructing a discrete-time simulator of a linear differential equation. This method provides satisfying results because the simulated systems are linear. Many precautions would have been needed had they not.

Exercise 12.1 illustrates the implementation of an RC filter simulator subjected to a periodic input.

Exercise 12.1 (Full-wave rectifier and simulation)

Consider a full-wave rectifier followed by an RC filter (Figure 12.1).

Figure 12.1Full-wave rectifier

1. The input signal s(t) = Asin(2πF0t) with F0 = 50 Hz is fed to the rectifier. Determine the Fourier series expansion of the rectified signal x(t) = |s(t)|. What is the amplitude of the continuous component of x(t)?
2. The output voltage y(t) of the RC filter verifies the differential equation:

Using the properties of the Fourier transform, determine the expression of this filter’s complex gain H(F).

3. 1/RC is chosen to be much greater than F0 so that only the continuous component and the first harmonics remain in the output signal. What is, in this case, the expression of y(t)?
4. We wish to simulate this system. In order to do so, we perform ...

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