H(ω) and H(s) as transfer functions

[PainterGuy]

TL;DR Summary

I'm confused about the differences between H(ω) and H(s). The textbook defines both as transfer functions though the transfer function is mostly reserved for H(s).

Hi,

I'm confused between H(ω) and H(s) as transfer functions. The textbook defines both as transfer functions though the term transfer function is mostly reserved for H(s) as far as I can tell. I have read that poles and zeroes of H(s) are helpful in determining the stability. Are poles and zeros of H(ω) also related to stability? Can it be said that H(s) determines the stability during transient state and H(ω) during steady state?

Could you please give it a look and comment on it?

H(s): https://imagizer.imageshack.com/img923/9010/wI1aGK.jpg
H(ω): https://imagizer.imageshack.com/img924/2056/xg4ug1.jpg

Thank you!

 

[marcusl]

H(s) is the Laplace transform of the impulse response as noted towards the bottom of your page and is defined in the complex plane, $s=\sigma+j\omega$. H(ω), on the other hand, is the Fourier transform of the impulse response. It is a subset of H(s), because it is H(s) evaluated along the imaginary or $j\omega$ axis.

Poles don't usually lie on that axis, however, so you can only determine their position and, therefore, evaluate system stability, from the Laplace transform. Zeroes likewise can lie anywhere in the complex plane, so must be found from the Laplace transform.