Observe, not calculate

Calculus Level 4

L { f ( t ) } = 2 s 2 − 4 s 4 − 4 \large \mathcal{L}\{f(t)\} = \cfrac{2s^2-4}{s^4-4}

The equation above shows the Laplace transform of f ( t ) f(t) .

By observing the placement of the poles of this transform on the complex plane, determine whether f ( t ) f(t) is:

  • Purely sinusoidal
  • Purely exponential
  • Both
  • Neither

Hint: This video can help you understand how to observe the Laplace transforms of functions; I also recommend it to those who are familiar with them because it is interesting anyway

Hint 2: Maybe writing out a program to help with the problem could be useful...

A mix of both Purely sinusoidal ∞ \infty Neither Purely exponential

James Watson by James Watson , 16, United Kingdom

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1 solution

Tom Engelsman
Jul 24, 2020

Let F ( s ) = 2 s 2 − 4 s 4 − 4 = 2 s 2 − 2 ( s 2 − 2 ) ( s 2 + 2 ) = 2 ⋅ 2 s 2 + ( 2 ) 2 F(s) = \frac{2s^2-4}{s^4-4} = 2\frac{s^2-2}{(s^2-2)(s^2+2)} = \sqrt{2} \cdot \frac{\sqrt{2}}{s^2 + (\sqrt{2})^2} , which the inverse Laplace Transform computes to f ( t ) = 2 s i n ( 2 t ) . \boxed{f(t) = \sqrt{2} sin(\sqrt{2}t)}. Hence, it's purely sinusoidal.

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Good solution! However is there any way to come to this conclusion without performing an inverse Laplace transform?

James Watson - 10 months, 3 weeks ago

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Being an electrical engineer, I've worked with Laplace, Fourier, & z-Transforms for so long they're almost second-nature to me. Some of them become obvious after performing the necessary partial fractions, but others will simply require a Laplace Transform table (which are ubiquitous on the Web).

tom engelsman - 10 months, 3 weeks ago

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i guess performing the inverse transform on this one was easy to do; i'll have to make it less obvious next time because the aim of this problem was to "observe, not calculate".

James Watson - 10 months, 3 weeks ago

Yes, simply notice that the poles of the transform are complex valued. Complex valued poles always correspond to the oscillatory nature of the original signal (which are represented as linear combinations of sinusoids) and real valued poles always correspond to some exponential decay or growth of the signal and therefore give information about the stability of the system a Laplace transform governs.

Vincent Moroney - 10 months, 1 week ago

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perfect solution!

James Watson - 10 months, 1 week ago

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