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The definitions of the Fourier transform and inverse Fourier transform I learned in college were

$$ F(j\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t}\ dt $$ $$ f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(j\omega)e^{j\omega t} d\omega $$

The salient characteristics of this convention are

  • Non-unitary transform; frequency-domain units are radians (variable is $\omega$)
  • "Time-domain" units are in time (variable is $t$)
  • Function transforms are denoted by capital letters ($F$ vs. $f$)
  • The $j$ in $F(j\omega)$ strictly denotes that the function is a Fourier transform
  • And of course, the usual EE convention that $j=\sqrt{-1}$.

Nowadays I use a much different convention, essentially that used on the wikipedias:

$$ \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-j2\pi\xi x}dx $$ $$ f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{j2\pi\xi x} d\xi $$ The characteristics of this convention are

  • Unitary transform; frequency-domain units are normalized frequency (variable is $\xi$)
  • "Time-domain" units are unitless (variable is $x$)
  • Function transforms wear hats ($\hat{f}$ vs. $f$)
  • Variables in the Greek alphabet denote transformed variables in the Latin alphabet ($\xi$ vs. $x$)

I greatly prefer this convention for several reasons.

  1. Using a unitary convention greatly increases the symmetry and clarity of Fourier duals: compare
    • $\mathrm{rect}(x)\leftrightarrow\mathrm{sinc}(\xi)$,
    • $\mathrm{sinc}(x)\leftrightarrow\mathrm{rect}(\xi)$ to
    • $\mathrm{rect}(t)\leftrightarrow\mathrm{sinc}(\frac{\omega}{2\pi})$,
    • $\mathrm{sinc}(t)\leftrightarrow\mathrm{rect}(\frac{\omega}{2\pi})$.
  2. Using $x$ instead of $t$ for the "time-domain" variable makes the equations more agnostic of one's problem domain. This makes it much easier to analogize about 2D image processing concepts in terms of 1D signal processing concepts, without the cognitive dissonance of using $t$ as a variable representing distance, or having to change variables around when going from one domain to the other.
  3. I find capital letters to be more useful for denoting discrete-valued variables/functions than for representing transformed functions.
  4. Using the hat more clearly denotes the Fourier transform as an operator that is applied to $f$, where the resulting function accepts a frequency-domain parameter $\xi$. Compare this to the far more ungainly $\mathcal{F}\{f\}$, the "traditional" way I learned in college to denote the Fourier transform as an operator, which just seemed way too confusing to me at the time ($\mathcal{F}{f(t)}$ vs $\mathcal{F}\{f\}(t)$ vs $\mathcal{F}\{f\}(\omega)$ etc.)
  5. I generally found that doing signal processing analysis with radians just sprinkled a lot more $\pi$s around than I felt necessary. Using units of normalized frequency makes a lot more sense to me, particularly when working through problems involving sampling theory.

Of course, it would be quite vain of me to consider my choice of convention to be superior to that used by others. But I'm having a hard time coming up with good reasons to prefer the convention I originally learned in college (ie, reasons that don't involve tradition).

Presently, I can think of one decent reasons for preferring the "traditional" convention: Using the non-unitary transform, and the $F(j\omega)$ parameter notation, greatly improves notaional consistency with the Laplace transform. Also, hats might be easier to lose/confuse than capital letters.

Can anybody think of other reasons to prefer the "traditional" (non-unitary) convention? Is this "traditional" convention the same as what you learned a signal processing course (if you took one)? Which convention do you prefer?

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    $\begingroup$ Questions soliciting personal opinions are not really constructive for this site. The answer is that it really does not matter what your convention is, as long as you're defining it correctly, using it consistently and in a lot of cases, are sticking to the common notation used in your field. The important thing is to not invent crazy new notations to be intentionally obtuse. I'm not sure how personal preferences and opinions are useful in any of this... $\endgroup$ – Lorem Ipsum Oct 31 '11 at 20:06
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    $\begingroup$ I can understand the desire to avoid mere opinion, but I do think there is a legitimate question as to why traditional conventions are what they are: it is unlikely that they are defined only as historical accidents. I would be willing to rewrite this question to avoid soliciting opinions, and to focus on the question of how these decisions of convention/notation in signal processing literature came about in the first place. $\endgroup$ – rtollert Oct 31 '11 at 20:26
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    $\begingroup$ You forgot to replace all the 2π with τ. :D $\endgroup$ – endolith Oct 31 '11 at 20:40
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    $\begingroup$ @endolith You beat me to it :) $\endgroup$ – datageist Oct 31 '11 at 21:19
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    $\begingroup$ One place where the unitary form is often used is in communications textbooks. Communications engineers like Hertz, so $x(t)$ transforming to $X(f)$ is more intuitive than to $X(\omega)$. $\endgroup$ – Jason R Nov 1 '11 at 3:22
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The choice of convention should be the one most appropriate (or familiar) for the audience you are trying to communicate with.

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One thing about using x(t) for a signal is the parallel between

  • $y = x^2$

and

  • $y(t) = x(t)^2$

where x is still an input and y is still an output, just in this case they're signals instead of numbers.

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