The Laplace transform X(s) of a signal x(t) has four poles and an unknown number of zeroes. The signal x(t) is known to have an impulse at t=0. Determine what information, if any , this provides about the number of zeroes and their locations.

Jaden Easton
2021-09-14
Answered

The Laplace transform X(s) of a signal x(t) has four poles and an unknown number of zeroes. The signal x(t) is known to have an impulse at t=0. Determine what information, if any , this provides about the number of zeroes and their locations.

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hosentak

Answered 2021-09-15
Author has **100** answers

The numerator of the polynominal must be of the same of higher degree than the denominator polynominal because the system impulse at t=0. So, there must be four poles at minimum.

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How to work out the Laplace transform with respect to t of:

$$\mathrm{sin}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}$$

I know that the transform of $\mathrm{sin}\left(t\right)$ is $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{a}{{s}^{2}+{a}^{2}}}$, and transform of $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}\phantom{\rule{thinmathspace}{0ex}}$ is ${s}^{2}F\left(s\right)-s\phantom{\rule{thinmathspace}{0ex}}f\left(0\right)-s\phantom{\rule{thinmathspace}{0ex}}{f}^{\prime}\left(0\right)$

$$\mathrm{sin}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}$$

I know that the transform of $\mathrm{sin}\left(t\right)$ is $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{a}{{s}^{2}+{a}^{2}}}$, and transform of $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}\phantom{\rule{thinmathspace}{0ex}}$ is ${s}^{2}F\left(s\right)-s\phantom{\rule{thinmathspace}{0ex}}f\left(0\right)-s\phantom{\rule{thinmathspace}{0ex}}{f}^{\prime}\left(0\right)$

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(a)

(b)

(c)

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