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I think I am looking at conflicting equations from a few sources, or maybe I just dont understand it. But please can someone help me with these fundamental equations that everyone should know

  • $ \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

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    $\begingroup$ It depends on the book you use. Oppenhiem and Schafer use cap Omega for continous time frequency and lower omega for discrete time. There are no formal conventions. $\endgroup$ – Stanley Pawlukiewicz Jun 28 at 13:08
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    $\begingroup$ I suggest you stick to a single authority like O&S $\endgroup$ – Stanley Pawlukiewicz Jun 28 at 13:12
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D ear N atalie, as you also know that these are just letters of English (org Greek) alphabet which by themselves have no meaning or explanations other than what you have arbitrarily imposed on them and the following is by convention what's being imposed on them in the mathematics, physics and the standard DSP literature as accepted unless otherwise stated explicitly...

  • $\omega$ can represent both continuous-time and discrete-time frequencies in radians-per-second and radians-per-sample respectively.

  • When continuous-time and discrete-time signals are used simultaneously (such as in sampling operations) then $\Omega$ represents continuous-time radians per second frequency and $\omega$ represents the discrete-time radians per sample frequency.

  • $\Omega = 2 \pi f$ interprets $f$ as the continuous-time frequency in Hertz.

  • $\omega = 2 \pi f$ interprets $f$ the same (as frequency in Hertz) if $\omega$ was used for continuous-time radians per second frequency, otherwise if $\omega$ is used for discrete-time radians per sample frequency, then $f$ has no such interpretation.

  • If a discrete-time signal $x[n]$ is periodic, then its (fundamental) period is indicated by the capital letter $N$ indicating a minimum integer larger than $1$ for which $x[n] = x[n+N]$ holds.

  • if a continuous-time signal $x(t)$ is periodic, then its (fundamental) period is indicated by the capital letter $T_0$ indicating a minimum real number larger than $0$ for which $x(t) = x(t+T_0)$ holds. This $T_0$ can also be taken as $T$.

  • When a continuous-time signal is uniformly sampled by an (ideal) periodic impulse train, then the period of sampling is indicated by the real number $T_s$ seconds per sample.

  • The reciprocal of the sampling period $T_s$ is denoted as the sampling rate: $F_s = 1/T_s$ in samples per second. If $F_s$ considered as the sampling frequency, then $F_s$ is interpreted in Hz.

  • Letter $k$ is mostly used as the frequency integer index for the DFT or DFS $X[k]$ of a discrete-time sequence $x[n]$. It's also used to indicate the continuous-time Fourier series coefficient indexing. It's also used to indicate convolution sums index in discrete-time.

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  • $\begingroup$ Fat32, but the equations are right ? $\endgroup$ – Natalie Johnson Jul 9 at 9:46
  • $\begingroup$ @NatalieJohnson imho they are wrong... You can get the right equations from Oppenheim, Haykin, Proakis etc, or from most answers provided here. $\endgroup$ – Fat32 Jul 9 at 12:32
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there's an ISO for that.

https://www.iso.org/obp/ui/#iso:std:iso:18431:-4:ed-1:v1:en

You must buy. No, I need not.

Anyway, as FAT32 says, they are just symbols. Which symbols stand for what is just convention, and as you have pointed out different "Authoritays" have slightly different conventions.

What is important is how the math works and that is what you understand. A common convention makes that easier, but it is not necessary, and more importantly it is not binding.

And I don't know if you plunk down a large chunk of coin you will get the answer you are seeking.

I would accept his answer and move on without the expectation of finding uniform conformity in the literature.

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