# What are $H(i\omega)$, $H(s)$ and $H(z)$ called?

What are the Fourier-, Laplace- and Z-transforms of the impulse response of a filter called? I've seen $H(s)$ referred to as the system function or transfer function, what about $H(i\omega)$ and the discretized $H(z)$?

• The nomenclature depends on which authority you cite. See some of the comments following my answer to this question – Dilip Sarwate Nov 8 '11 at 15:12
• @Dilip: Interesting, I think your comment there contains the answer (except for $H(z)$). – Andreas Nov 8 '11 at 15:31
• A comment made by someone else in that thread claims that $H(z)$ is also called the transfer function by some authorities. – Dilip Sarwate Nov 8 '11 at 18:35

I would consider them to all be 'transfer' or 'system' functions. Those two names are interchangable, but in control theory books I see then listed more as transfer functions rather than system functions.

The domain you are in, whether it be fourier, laplace, Z or time, isn't entirely relevant to the name in that respect.

• I agree with your assessment. I don't think nomenclature specifics are all that important, but I've typically heard the $H(j\omega)$ version called the "frequency response." – Jason R Nov 9 '11 at 2:49
• true... I would consider "Frequency response" another name for it. H(jw) is a transfer function, that more specifically is a frequency response – Fuzz Nov 9 '11 at 6:57