# should we encourage or expect posters to adopt the current common notation, $x[n]$, for discrete-time signals?

one of the things i remember that impressed me about my first exposure to Oppenheim & Schafer was the formal and rigorous use of math. one thing they did early on was to adopt a different notation, $x[n]$ for discrete-time signals that was different than for continuous-time $x(t)$. they may have been influenced by the C language notation for arrays, x[n], i dunno.

in the days of old (the 70s), there were DSP texts that used the standard math notation for discrete sequences (i.e. $x_n$), but the problem was when we would get a vector of signals in continuous-time, $x_k(t)$. if sampled, $x_k(nT)$, how was that notation nicely converted to a sequence. $x_{k,n}$ was not good and $x_k(n)$ was also not good.

should we encourage, expect, edit to maintain the usage of this notation?

• looks like $\LaTeX$ doesn't work in meta. – robert bristow-johnson May 8 '16 at 1:49
• Yeah, we tried to get it turned on in the early days, but SE wasn't having it. This post may give us a good example to justify it though. – datageist May 8 '16 at 3:43
• meta.dsp.stackexchange.com/questions/1212/… – datageist May 8 '16 at 3:52
• Ive never used that notation. And from a theoretical point of view, there is no need to indicate that you are measuring only discrete points intead of a continuum. You are just taking the value of x at the instant t regardless that t=dt n...... You only need to properly extend the $\delta$ framework. – Brethlosze Dec 23 '16 at 2:23

I don't strongly care about what variable is used to index the signal ($t$ versus $n$ and so on), but I do think it's an important notational distinction to use square brackets when referring to a discrete-time signal. I'm always sure to follow this convention because there's real potential for confusion as to whether a particular signal is defined in continuous or discrete time.

With that said, I do agree that $x[n]$ is the standard notation and should be preferred when possible. Not sure that it's worth editing a question over it, but it is the clearest way to express the concept. Likewise, for frequency-domain signals, $X[k]$ is, in my opinion, the most standard notation.

• i think $X[k]$ only for discrete frequency domain. like the DFT. for the DTFT it should be the same as the Z-transform with $z = e^{j \omega}$. $$X(e^{j\omega} = \sum\limits_{n=-\infty}^{\infty} x[n] e^{-j \omega n}$$ – robert bristow-johnson May 10 '16 at 13:57
• oh, i forgot, $\LaTeX$ doesn't work here. – robert bristow-johnson May 10 '16 at 13:58
• @robertbristow-johnson: Yes, I agree completely. – Jason R May 10 '16 at 14:09
• Yes, I agree: the issue is clarity. I don't think we should berate users to adhere to this, but I'm perfectly happy to edit questions for clarity --- especially since I've seen the discrete vs continuous confusion happen many times on this site. – Peter K. May 10 '16 at 16:35

I like using square brackets for discrete signals (arrays as in C), but you also changed my question to use $n$ instead of $t$, and I disagree with that... Fortran implicit typing aside, $t$ makes a fine integer (discrete) index/variable when $i$ and $j$ are already lost to representing imaginary numbers, and $n$ is conventionally the length not the index.

However, despite the fact that I agree with and like your convention for square brackets, I'd be wary of imposing these conventions on every new user that comes along. It's off-putting and turns people away. The stack exchange sites are already filled with pedantic jerks who don't add much value but run around enforcing rules, and they don't need another stick to swing at new users.

The beurocrats and bullies are the reason I create a new throw-away account every time I have a question. As soon as some jackass adds his "contribution" to my questions or answers, I get angry, close the window, forget the password, and don't come back for another year or so. Your changes to my notation were fine, but you can easily imagine this being abused by other people.

• +1 for The stack exchange sites are already filled with pedantic jerks who don't add much value but run around enforcing rules – Peter K. May 10 '16 at 13:06
• i try to do a little of both. there are two reason (maybe derived from the same reason) that i don't like integer $t$ for time. one, nearly all of the lit uses $n$, $m$, $k$, or $i$ (if there is no confusion with the imaginary unit) or $j$. happens to be the range that Fortran defaults to for integer variables (i know that anyone can explicitly override that). I, J, K, L, M, N are the default integer variables. the rest default to floats. most math textbooks use $i$, $j$, $k$, $n$ for counting variables in summations or series. – robert bristow-johnson May 10 '16 at 17:02

It is fine to encourage a particular style, but not to enforce it on every question/answer. It is not true that DSP texts would have converged to a single notation. I did today a search for "is a discrete time signal" (also accepts discrete-time with a hyphen) in Google Scholar on texts freely available in PDF form from 2006 to 2016, and collected statistics on the notation, only including from each article the first signal unambiguously presented as a discrete-time signal. In each article I also quickly looked for use of the imaginary unit, and recorded the symbol used. The following are the results accumulated in about two hours of work:

So in fact, most articles use the same parenthesis () notation for discrete-time signals as they would use for continuous-time signals. There may be some bias to this finding because the square brackets [] and a subscript are more self-explanatory. There were 2 articles not included in the statistics that avoided the distinction altogether by using the product of an integer discrete index and the sampling period as the index in the continuous-time signal.

Signal processing is used in multidisciplinary contexts so it seems like a stretch to expect unified notation.

• +1 for the work (even if i don't particularly like the results). – robert bristow-johnson Jun 6 '16 at 10:20
• I didn't see this until just now, but wow! Thanks, Olli. Good stuff. – Peter K. Nov 15 '16 at 9:43