Explanation of sinusoid phase angle

[magnetra]

sinusoidal phase angle

This link has a discussion on 'sinusoids.

https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

Midway thru the discussion, it states,

Since the sine function is periodic with period \[ 2\pi \], the initial phase \[ \phi \pm 2\pi\] is indistinguishable from \[ \phi\]. As a result, we may restrict the range of \[\phi\] to any length\[ 2\pi \] interval. When needed, we will choose

\[\displaystyle -\pi \leq \phi < \pi, \]

i.e., \[ \phi\in[-\pi,\pi)\]. You may also encounter the convention \[\phi\in[0,2\pi)\]

.

Why does it say \[\phi \in [-\pi,\pi)\] and not \[\phi \in [-\pi,\pi]\] ?

M

 

[doraemon]

phase angle

Hello!

It's probably a way to say that pi is not included, but -pi is.

Dora.

This link has a discussion on 'sinusoids.

https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

Midway thru the discussion, it states,

Since the sine function is periodic with period \[ 2\pi \], the initial phase \[ \phi \pm 2\pi\] is indistinguishable from \[ \phi\]. As a result, we may restrict the range of \[\phi\] to any length\[ 2\pi \] interval. When needed, we will choose

\[\displaystyle -\pi \leq \phi < \pi, \]

i.e., \[ \phi\in[-\pi,\pi)\]. You may also encounter the convention \[\phi\in[0,2\pi)\]

.

Why does it say \[\phi \in [-\pi,\pi)\] and not \[\phi \in [-\pi,\pi]\] ?

M

 

[magnetra]

sinusoid initial phase

That I got that part, -pi is included but pi isn't. I think i understand the reason for excluding pi.

Considering the general form [tex:3cf6c997bb]sin(\omega t + \phi)\]. When\[ \phi = -\pi\] our signal is \[sin(\omega t -\pi)\] and when\[ \phi = \pi\], the signal is \[sin(\omega t + \pi)\] which is equal to the previous signal \[sin(\omega t -\pi)\].

Hence the phase angle should be only in the range [-pi , pi) i.e pi need not be included, as when\[ \phi = \pi\] we are already back to the signal with \[ \phi = -\pi\]

I think I'm clear about this myself and made it clear for others too.

M

 

[FvM]

sinusoids phase

Clearly, the subtile details of interval definition should be considered in an exact mathematical presentation.
But I wonder, why you are presenting the discussion in the digital signal processing forum. Can you
give an example, where the difference matters for a real world signal processing problem?

 

[doraemon]

finding phase angle of sinusoid

Hello!

Basically the difference does not matter.
But when you are implementing an algorithm, do you usually do the same
calculation twice?
I guess not. And most of the time you iterate from 0 to 255, 0 to 511, etc.
(i.e not from 0 to 256 or 512).
What concerns me more is why he asks the question if he knows the reply.
Was it a quiz?

Dora.

Clearly, the subtile details of interval definition should be considered in an exact mathematical presentation.
But I wonder, why you are presenting the discussion in the digital signal processing forum. Can you
give an example, where the difference matters for a real world signal processing problem?

 

[magnetra]

how to find angles in sinusoids

Dora,
No it was not a quiz. At the time I posted the question, I was confused about it and hence was a 'question' for me. But after your first reply, I just realized the answer myself and hence explained it.

M