Sampling. Part I

Modern control systems are typically implemented in computers that work with periodic samples of signals, that is discrete-time signals, rather than continuous-time signals. A system that produces samples of a given signal is a linear system, albeit a time-varying one. In this post, which starts a series exploring the issue of sampling and discretization, we delve into the nature of a system that can produce such signal samples. More will follow.

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More on the Fourier transform

In a previous post we explored some relationships between the Laplace and the Fourier transforms. In this post we address a common question: “when is the Fourier transform of a signal, $X(j\omega)$, equal to its Laplace transform, $X(s)$, evaluated at $s = j \omega$?” The short answer is: when the region of convergence of the corresponding Laplace transform contains the imaginary axis.

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The Fourier and the Laplace transforms

Because feedback requires careful consideration of stability and causality, the transform tool of choice is almost always the Laplace transform. Yet, in many applications it is the Fourier transform that arises most naturally.

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Is that a constant? Or is it a delta?

This might be a stupid question.. but oh well. So the inverse laplace of a constant is the dirac delta function times the constant.

With a proportional controller, K(s) = Kp, the inverse laplace of the controller would be the delta function. Is the delta function asymptotically stable?


As it turns out, this is not a stupid question at all! It hits right at the heart of what is and how to represent a dynamic system.

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