What is the notation of 'a single term in the DFT'

Question

I have a notation/terminology question: I am writing a paper in not-quite-my-area and can't figure out the right way to phrase/notate the following:

I have a discrete function $p[x]$, of which I can assume harmonic behaviour, so the function can be substituted by a single term of its Fourier series: $p[x] \rightarrow e^{j x X}$. I am trying to come up with a notation for this, or to find the conventional notation for this. I know that this is not the one:

$$\mathcal{F}_x \{ p[x] \} $$

as that commonly represent the whole Fourier Transform. So I am trying to come up with an operator (say $Q$) of which:

$$\begin{equation} \begin{aligned} Q\{ p[x] \} &= e^{jxX}\\ Q\{ p[x - 3] \} &= e^{-3jxX}\\ Q \{ \frac{\partial}{\partial x} p[x] \} &\approx e^{jxX} - e^{-jxX} \end{aligned} \end{equation}$$

et cetera. And I'd like to know if this operator has a name. Does such a thing exist?

Answer 1

The following book is the classic source for signal processing community:

Discrete-Time Signal Processing (2nd Edition) (Prentice-Hall Signal Processing Series) (10 January 1999), pp. 775-802 by Alan V. Oppenheim, Ronald W. Schafer, John R. Buck

Around the page 561, where the authors introduce Discrete Fourier Transform, they introduce the following notation:

$$ x[n]\mathop{\longleftrightarrow}^{\mathcal{DFT}} X[k]. $$

I cannot remember any operator like notation for DFT but you may come up with something like:

$$ \mathcal{DFT}\left\{x[n]\right\} = X[k]. $$

Answer 2

You can phrase it in terms of Z-transforms, but I think that's overkill. Let's use $n$ for the independent variable, then you have that $p[n]=e^{jnX}$ for some value of $X$. This says that the sequence is periodic with a particular form. You don't need an operator with properties you mention, they are immediate consequences of the definition of $p[n]$. If you generalize to Fourier series with more terms, the DFT or z-transform notations are your best bet.

Just say "assume $p$ is a sequence with the form....then it follows that these properties hold..." and continue on writing the paper.