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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. $\endgroup$
    – user28715
    Jun 28, 2019 at 13:08
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    $\begingroup$ I suggest you stick to a single authority like O&S $\endgroup$
    – user28715
    Jun 28, 2019 at 13:12

2 Answers 2

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As you may also know, these are just letters of English alphabet which, by themselves, have no meaning or natural implication into anything other than what you have arbitrarily, or traditionally, or conventionally imposed on them. Hence, the following list describes most typically what's imposed, by convention of DSP tradition of the Oppenheim flavor, to some of them. They should be used consistently and carefuly, to mean in the expected way, unless otherwise stated explicitly.

  • $\omega$ may represent continuous-time or discrete-time frequencies, in radians-per-second or radians-per-sample respectively.

  • When continuous-time and discrete-time signals are used simultaneously (such as in sampling theorem and its associated relations), 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 if physical attribution is made.

  • $\omega = 2 \pi f$ interprets $f$ the same (frequency in Hertz) if $\omega$ was used for continuous-time radian-per-second frequency. Otherwise, if $\omega$ is used for discrete-time radian-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 is denoted as the sampling rate: $F_s = 1/T_s$ in samples- per-second. If $F_s$ is considered as the sampling frequency, then $F_s$ is interpreted in Hz.

  • Integer $k$ is mostly used as an index of the sequence $X[k]$ which is the DFT of a discrete-time sequence $x[n]$. It's also used to indicate the continuous-time Fourier series coefficient indexing as in $a_k$. Also, it's used typically as the dummy variable of the discrete-time convolution summation.

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  • $\begingroup$ Fat32, but the equations are right ? $\endgroup$ Jul 9, 2019 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, 2019 at 12:32
  • $\begingroup$ A couple are letters from the Greek alphabet, not the English alphabet! $\endgroup$ Jul 30, 2023 at 15:02
  • $\begingroup$ @DilipSarwate yes $\omega$ (and its capital $\Omega$) are so. $\endgroup$
    – Fat32
    Jul 31, 2023 at 7:47
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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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