How to connect $\omega$ in time domain to physical frequency?
There are several "frequency" variables in signal processing, the most common all being related by constant factors. I will use the notation from Digital Signal Processing by Proakis &amp; Manolakis:


$F$: "Ordinary" frequency in units of Hz. 
$\Omega$: Angular frqeuency, in units of radians/second; $\Omega = 2\pi F$
$f$: Normalized or "digital frequency". $f = F/F_s$, where $F_s$ is the sample rate in Hz. This is a formally unitless quantity, but sometimes its units are listed as being cycles/sample.  Sampling produces aliasing, which means that the spectrum of a sampled signal will be periodic with period 1 cycle/sample. For this reason DSP engineers only look at the spectrum of the signal in the principal period of $-1/2 \leq f &lt; 1/2$. 
$\omega$: Normalized radian frequency. $\omega = \Omega/F_s = 2\pi F/F_s$. Sometimes its units are listed as being radians/sample. Because of aliasing, it is only necessary to study the spectrum of a signal from $-\pi \leq \omega &lt; \pi$ in the digital domain. 


Other notions of the concept of "frequency" exist (such as instantaneous frequency, and a notion of an "instantaneous spectrum"), but these are more advanced concepts that you probably don't need to worry about yet.