Unit Problem in Designing a Filter for a Given Auto Correlation Function

Question

Given a WSS process with the following Auto Correlation Function:

$$ r\left ( \tau \right ) = {\sigma}^{2} {e}^{-\alpha \left | \tau \right |} $$

The Laplace Transform would be:

$$ R \left ( s \right ) = \mathfrak{L} \left \{ r \left ( \tau \right ) \right \} = \frac{-2 \alpha {\sigma} ^ {2}}{\left ( s - \alpha \right ) \left ( s + \alpha \right )} $$

Hence the filter would be of the form:

$$ H \left( s \right) = \frac{c}{s + \alpha} $$

My question is about units. In the Auto Correlation Function the units of $ \alpha $ are [Hz]. While in the filter form, assuming $ s = j \omega $ the units are [Rad / Sec]. How this conflict can be resolved? What a I missing?

Thanks.

Answer 1

Radians are considered to be dimensionless. See Are angles dimensionless? and Dimensionless quantity. They are considered to be pure numbers like pi.

So $\alpha$ is in Hz, which is a measure of 1/second, and $s$ is also considered to be measured per second.