Poles and zeros determine response.
The Laplace transform generalises Fourier to include growth/decay, defined for s=σ+jω. Converts ODEs into algebraic equations, includes initial conditions, and gives transfer function H(s). Region of convergence defines causality and stability; inverse via partial fractions.
L{e^{-at}u(t)}=1/(s+a).
Poles and zeros decide how a system responds in time, so reading them from a transfer function predicts speed and stability.
Poles and zeros determine response.
Poles and zeros decide how a system responds in time, so reading them from a transfer function predicts speed and stability.
Their locations set the natural modes, such as decay rate and oscillation frequency, and stable systems have poles in the left half of the s-plane.
They shape the response and can cancel or boost certain modes, so they affect how the input is passed to the output.
Typical applications include judging whether a control loop will oscillate by looking at pole locations, predicting the ringing of a filter from its poles and designing a notch by placing zeros at the unwanted frequency.
Study "Parseval's theorem" first (topic 17 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
Unilateral vs bilateral. Initial and final value theorems. Table of transforms.
L{e^{-at}u(t)}=1/(s+a).
The next topic is "Poles and zeros", which builds directly on this one.
It is classed as Beginner in Signals, Control & Comms (topic 18 of 101). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.
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