# Is there a special reason why the principal range is defined as $-0.5\le F \le 0.5$?

According to Digital Signal Processing by Ashok Ambardar, The principal range is defined as follows:

Put another way, a DT sinusoid is periodic in frequency (has a periodic spectrum) with unit period. The range $-0.5\le F \le 0.5$ defines the principal period or principal range.

I see that the major rationale behind this definition is the unit period. If only for the purpose of uniqueness, it seems to me that any interval with a unit period surely meets the requirements. I also think that $0\le F \le1$ is more fit for the purpose, because it is more easy to remember and aesthetically beautiful. In conclusion, my question is:

• Is there a special reason why the principal range is defined that way?

## 1 Answer

The $1$-periodicity indeed guarantees that any unit interval will contain all the information. Beauty can be subjective, but remember that art and science in the Western world have been strongly influenced by the Greeks. In Polyclitus's Canon and the Idea of Symmetria, you can see that beauty secrets:

lay in the mastery of symmetria

With the $[-0.5,\,0.5]$ interval, you only have one number to remember, and the DC or $0$ frequency in the middle better underlines symmetry in quantities related to spectra.

Actually, beauty and simplicity put aside, the notion of "principal range" could be inherited from complex analysis, where the principal range of the phase is generally defined as $]-\pi,\,\pi]$. This is however a convention: any particular choice, defined in advance, can be called a principal range.

The $[0 ,\,1]$ frequency axis, often used by default by some FFT programs, is troublesome for many DSP learners, at the beginning. However, the $[0 ,\,1]$ is still used quite often.

Note that the interval can be (should) open at one end: $]-0.5,\,0.5]$ or $[0 ,\,1[$.