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?


  • $\begingroup$ 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. $\endgroup$ – robert bristow-johnson Jun 12 '19 at 18:46
  • $\begingroup$ I am looking for how they related in terms of equations above? $\endgroup$ – Natalie Johnson Jun 15 '19 at 11:45

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