Why is phase range between $-\pi$ and $+\pi$ (instead of $0$ and $2\pi$)?

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.

• (+1) Afaik for frequencies Opp & Sch uses the range $[-\pi, \pi)$ (in addition to $[0, 2\pi)$) where the lower limit is included, (upper limit discluded)... Commented Apr 24, 2021 at 17:30
• Good point. And again a matter of preference :-)
– MBaz
Commented Apr 24, 2021 at 18:17

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 $$[-\pi, \pi)$$.

• I think your #2 is the most salient. Most often we are dealing with an angle increment, that angle can also decrement which is a negative increment. so a a bipolar value is, i think, more useful in general. Commented May 2, 2021 at 13:20

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).