I have found that most commonly the DTFT is defined as: $X(\omega) = \sum_{n=-\infty}^{\infty} x[n]e^{-j \omega n}$.
However the class I am taking frequently uses the DTFT expressed in "normalized continuous frequency of discrete signals" units i.e. cycles/sample. I am trying to understand how to convert a DTFT pair using one units to other units.
For example, the DTFT pair with $\omega$ [radians/sample]:
$$x[n] = 1 \Leftrightarrow X(\omega) = 2 \pi \sum_{k=-\infty}^{\infty} \delta( \omega-2 \pi k)$$
My understanding that this pair with $\mu$ [cycles/sample] is given by: $$x[n] = 1 \Leftrightarrow X(\mu)= \sum_{k=-\infty}^{\infty} \delta( \mu -k)$$
I have honestly struggled to find many resources which make any mention of the latter DTFT.