# Proper notation for frequency response between $H(\omega)$ and $H(j\omega)$

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!

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.

• Damn it, I missed the question you linked! Thank you very much for your answer and for the link. Jun 21, 2022 at 13:02
• 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). Jun 21, 2022 at 13:08

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.

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