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Short version: which notation is formally correct between $H(\omega)$ and $H(j\omega)$?

Disclaimer: I know that no one cares in the industry. I am asking out of curiosity and care for rigorous definitions.

Longer version: I have a question regarding proper notation of the argument for transfer functions and frequency responses.

In the frequency domain, both of them are usually written choosing the letter $H$ with the Laplace variable $s=\sigma + j\omega$ as their argument, given as $s$ in the transfer function and $s\big|_{\sigma=0}=j\omega$ for the frequency response.

I would be tempted to write these functions as $H(s)$ and $H(j \omega)$, but, by taking example on complex analysis, I know we usually write complex valued function of a complex variables using real and imaginary parts of the complex variable.

For example, the square of a complex number $z$ is written as the sum of two real valued functions $u$ and $v$, taking its real and imaginary parts $a$ and $b$ as arguments like the following: $$z^2=u(a,b)+iv(a,b)=a^2-b^2 + i2ab$$

From experience, I have never (understandably) seen $H(s)$ written like $H(\sigma, \omega)$, which I have no problem with. However, $H(j \omega)$ bothers me; why do we keep the imaginary number $j$ inside the argument? Is it notation abuse? Which one is formally correct?

Thank you!

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2 Answers 2

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There's nothing that could be correct or wrong about the notations $H(j\omega)$ and $H(\omega)$, because it's just a matter of convention. The reason why some people use $H(j\omega)$ is that you can use the same function that is used in the Laplace domain, i.e., $H(s)$ evaluated at $s=j\omega$.

But: if you use $H(s)$ to denote a function in the Laplace domain, then it would be clearly wrong to use $H(\omega)$ as the corresponding frequency response. You'd need to define a new function $\tilde{H}(\omega)=H(j\omega)$.

If you never refer to the complex variable $s$ of the Laplace transform domain, then there isn't really any good reason to use $H(j\omega)$, even though it's certainly not wrong to do so.

Also take a look at this question and its answers.

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  • $\begingroup$ Damn it, I missed the question you linked! Thank you very much for your answer and for the link. $\endgroup$
    – chuchvara
    Jun 21, 2022 at 13:02
  • $\begingroup$ I will accept this post as an answer, but I would love to have the perspective of a mathematician to have the most rigorous definition (not that there is anything wrong with your explanation). $\endgroup$
    – chuchvara
    Jun 21, 2022 at 13:08
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I'm trying a guess (but that's all it s). To distinguish between complex and real numbers I put a arrow on top of the complex ones.

  1. $\vec{H}(\vec{s}) $ : means $\vec{H}$ is a complex function of a complex variable $\vec{s}$
  2. $\vec{H}(\omega) $: means $\vec{H}$ is a complex function of a real variable $\omega$
  3. $\vec{H}(j\omega) $: means $\vec{H}$ is a complex function of a purely imaginary variable whose imaginary part is $\omega, \omega \in \mathbb{R}$ and whose real part is $0$.

So in this interpretation, both 2 and 3 are correct. Something that's a function of purely imaginary variable can always also be written as a function of a real variable.

You could argue that $H(j\omega)$ contains a little more information by implying that "wherever $\omega$ shows up in the formula it shows up together with an $j$". You would not expect to see something like $H(j\omega) = 3j\omega - 2\omega^3$

Whether that's actually true or whether it makes any practical difference, I don't know for sure.

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  • $\begingroup$ I have also seen a similar distinction, but with complex variables underlined, rather than having an arrow on top. Thanks $\endgroup$
    – chuchvara
    Jun 21, 2022 at 13:05

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