I wonder why the phase range in DSP applications is between $-\pi$ and $+\pi$. For example, why is that not between $0$ and $2\pi$?
This question popped when I was reading about synchronization in digital communications.
It said instead of $\arg( Im(x) / Re(x) )$ we can use just $Im(x)$ because they are of the same sign because phase (here $x$) is always between $-\pi$ and $+\pi$ !!!
It is just a convention, but it is useful in some cases. For example, the phase of the DFT of a real discrete-time signal is odd only if the angles are expressed in the range $[-\pi, \pi)$. Sometimes you just have to adapt to the convention used by your tools -- for example, MATLAB functions like angle and atan2 return angles in $[-\pi, \pi)$.
Note that the frequency of a discrete-time signal, measured in radians per sample, is in the range $(-\pi, \pi]$, but in this case it has a different interpretation.
First, when you're talking angles, in DSP pretty much all angles are $\mod 2\pi$. So $2\pi \equiv 0$. Usually it's more convenient to keep angles on the interval $\left [-\pi, \pi \right )$, because we're usually most interested in angles around $0$. You don't have to do this, however -- if your problem at hand is easier to solve if your angle lies on $[-2\pi, 0)$, or $[0, 2\pi)$, or any such interval, by all means use that. Just be careful to point it out along the way, in case any innocent bystanders are trying to understand your thinking.
It's a judgement call, though -- sometimes if you're working with actual physically rotating objects, or you're otherwise dealing with a sequence of angular steps, you may want to treat a sequence of angles as continuous -- which is common enough that most math packages have an "unwrap" function hidden within them someplace.
This can even extend to phase-locked loops, where you may want a phase detector that, instead of reporting an angular error that jumps at the $-\pi$ to $\pi$ boundary, smoothly transitions from $\pi - \epsilon$ to $\pi + \epsilon$ (and the equivalent in the negative direction) and from $-2\pi$ or $2\pi$ to 0 -- such a detector is called a "phase-frequency" detector and nicely extends a loop's lock range.
I have thought about some points which could help find the answer:
1- I think there might be something related to $\operatorname{arctan}(x)$ which is continuous in $(-\pi/2 \ \ \pi/2 )$ but I am not sure how.
2- We almost always work with phase DIFFERENCE rather than the absolute phase itself. Phase difference could be both positive and negative. So, it might be better to consider a signed range than an unsigned one.
3- Because of the nonlinearity of the phase argument in the applicable functions (e.g. $\sin(\cdot)$, $\cos(\cdot)$, etc.), it would be better to work with the functions instead of the phase itself. Among these functions, sinusoidal-like functions show the advantage of having the same sign as their argument (phase). Plus, the range of the argument to keep this is $[-\pi, \pi)$.
It has to do with the unit circle in the i vs Re plane - instead of going counterclockwise by 360 degrees, we could equivalently go +/- 180 degrees.
In my experience with audio, this thought process allows one to minimize phase delay.
For example, let's say I've got a L and R audio signal arriving at some listening point in space, where (arbitrarily) the L signal arrives with a phase delay of 270 degrees relative to the R signal.
In this case, I could either (1) delay the R signal by 270 degrees, or (2) delay the L signal by 90 degrees.
As far as relative phase is concerned (not absolute phase), both methods create perfect phase coherence (0 and 360 phase difference respectively). However, delaying the L signal by 90 degrees induces less phase delay than a 270 degree phase delay. (The audible differences between 0 vs 360 degree phase difference is an entirely different discussion in psychoacoustics).
