By Patrick Anderson
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2. 3. 4. The system or signal in question is analog. The system or signal in question is Linear. The system or signal in question is Time-Invariant. The system or signal in question is causal. The transform is defined as such: [Laplace Transform] Laplace transform results have been tabulated extensively. If we have a linear differential equation in the time domain: With zero initial conditions, we can take the Laplace transform of the equation as such: And separating, we get: Inverse Laplace Transform This operation can be performed using this MATLAB command: The inverse Laplace Transform is defined as such: [Inverse Laplace Transform] The inverse transform converts a function from the Laplace domain back into the time domain.
From practice, the rule thumb is if the PSD of the input process is the same from hour to hour and day to day then the input PSD can be used and the above equation is valid. Manufacture Once the system has been properly designed we can prototype our system and test it. Assuming our analysis was correct and our design is good, the prototype should work as expected. Now we can move on to manufacture and distribute our completed systems. Classical Controls The classical method of controls involves analysis and manipulation of systems in the complex frequency domain.
Sampled Data Systems Ideal Sampler In this chapter, we are going to introduce the ideal sampler and the Star Transform. First, we need to introduce (or review) the Geometric Series infinite sum. The results of this sum will be very useful in calculating the Star Transform, later. Consider a sampler device that operates as follows: every T seconds, the sampler reads the current value of the input signal at that exact moment. The sampler then holds that value on the output for T seconds, before taking the next sample.