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?



1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.