Understanding Laplace Transform's Spectrum of Damped Sinusoids

[muzialis]

Hi All,

in a previous post on the physical meaning of Laplace's Transform I found the following statement
" The fundamental Laplace transform pair is H(t), the Heaviside step function, and 1/s, its spectrum of damped sinusoids. Note that the spectrum is weighted towards low frequencies (1/abs(s) goes to zero as abs(s) goes to infinity), as one would expect for an excitation that begins but is never turned off".
I am struggling to understand where the spectrum of damped sinusoids comes from, I found this interesting.
Can anybody maybe help?
Thanks a lot

 

[NascentOxygen]

I am struggling to understand where the spectrum of damped sinusoids comes from, I found this interesting.

I interpret it thus:

Bear in mind that to relate a Laplace transform back to the frequency (sinusoid wave) domain, we map s → jω.

So 1/s → |1/ω| in magnitude, meaning amplitude of each component sinusoid is inversely proportional to its frequency.

Beyond this, I can't say much more.