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Imagine transfer function obtained by Laplace transform, for example:

$G(s) = \dfrac{1}{s+1}$

Now, I would like to do some frequency analysis, so I replace the $s$ with $\omega i$ (let's consider this operation valid for this example).

What is the unit of the $\omega$? So far what I have seen, the $\omega$ is noted as frequency or angular velocity. I asked my colleagues and I got various answers:

  • rad/sec
  • Hz
  • no unit

What is correct and why? Does it depend on real variable passed to transform (if somebody uses different variable than time)?

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  • $\begingroup$ A transfer function relates an input to an output, so if the input and output have the same units, it would likely be unitless but if the transfer function was something like a transducer, the input and output would have different units so the transfer function would have units. Looking at your example, what would be the units associated with $1$ $\endgroup$
    – user28715
    Sep 14, 2017 at 14:18

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If you are dealing with the Laplace transform $G(s)$ of a time domain signal $g(t)$ and its evaluation on the imaginary axis to get the Fourier transform $G(j\omega)$ (assuming it exists) then the unit of your frequency $\omega$ is radians per second assuming the unit of the time was seconds.

It's relation to cyclic frequency is : $$\omega = 2 \pi f$$ where $f$ is the frequency in Hz (cycles per second).

On the other hand if the initial function was like $g(x)$ where $x$ was a spatial variable with unit of meters, then the transform domain frequency unit will be in radians per meter where its relation to space-frequency is still the same with $\omega = 2 \pi f$ where $f$ will have the unit of cycles per meter

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  • $\begingroup$ Thanks, it makes sense! Can you provide reference of further explanation why is it so? $\endgroup$
    – matousc
    Sep 14, 2017 at 15:04
  • $\begingroup$ basic physics books, electric circuit theory books and signals and systems theory books sparsely describe such terms. I don't remember a single reference devoted to the study of units. $\endgroup$
    – Fat32
    Sep 14, 2017 at 15:45
  • $\begingroup$ so if $t$ is years, $f$ is in Hz? $\endgroup$
    – user28715
    Sep 14, 2017 at 19:56
  • $\begingroup$ @StanleyPawlukiewicz of course not why? $t$ was is in seconds above (now I explicitly state it after the edit) By the way, OP is actually asking the unit of $w$ (frequency) variable of the Fourier transform, not the unit of the Transfer function, as your comment mentions about. My answer is about the unit of the frequency variable $w$ of the Fourier transform. $\endgroup$
    – Fat32
    Sep 14, 2017 at 20:35
  • $\begingroup$ $$ w \ne \omega $$ $\endgroup$
    – robert bristow-johnson
    Sep 14, 2017 at 20:49

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