Discrete Frequency Equation and Relation

I think I am looking at conflicting equations from a few sources, or maybe I just dont understand it.

• $$\Omega$$ is Discrete frequency

• $$N$$ is Discrete Period in number of samples

• $$\omega$$ is Continuous frequency

• $$T$$ is Continuous time period

• $$F_{s}$$ is Sampling frequency

• $$T_{s} = n$$ is Sampling Time Instant

• $$k$$ is Discrete harmonic index which goes from 0 to N/2 for a real signal

First equation forms I have seen $$\Omega = 2\pi f/F_{s} =\omega / F_{s} = \omega T_{s} = \omega n = 2\pi n / T$$ Second equation I have seen $$\Omega N = 2\pi k$$ $$\Omega = 2\pi k / N$$ Third equation I have seen $$\Omega_{k} = k F_{s} / N$$

First and second look the same with if $$k=T_{s}$$ and if $$N=T$$ and first and third seem contrary.

Can someone explain please? Also if you can point in in a direction where all this is neatly derived and defined?

Thanks

• Since you're using $\LaTeX$ so nicely, may I suggest sticking more to the de facto convention we get from Oppenheim and Schafer? There, $\Omega=2 \pi f$ is continuous angular frequency in the continuous-time domain. $f_\mathrm{s}=\frac{1}{T}$ is the sample rate. $\omega$ is continuous (but periodic) angular frequency in the discrete-time domain (the DTFT) and $\omega_k = 2 \pi \frac{k}{N}$ is discrete frequency (as in the DFT). Normally the relationship of angular frequencies of the continuous-time and discrete-time domains is $\omega = \Omega T$. $N$ is the size of data in the DFT. Jun 12, 2019 at 18:46
• I am looking for how they related in terms of equations above? Jun 15, 2019 at 11:45