# linear to circular representation

Discussion in 'Math' started by sharanbr, Aug 15, 2015.

1. ### sharanbr Thread Starter Active Member

Apr 13, 2009
76
1
This topics has been probably discussed few times here.

I am a little confused as to how a linear signal can be represented in circular form.
Take for example a sine waveform represented as linear form. How can this be represented in circular form.
Unless, it is represented using two complex phasors. If this is the case, circular representation is in complex domain.

2. ### Roderick Young Member

Feb 22, 2015
409
169
I'm not sure exactly what you're asking but maybe it's this?

A sine wave can be plotted in Cartesian (x-y) coordinates, with the x-coordinate being time, and the y-coordinate being amplitude.

Because the sine wave is repetitive, it could also be plotted in polar coordinates, with theta being time, and rho being the amplitude.

3. ### BR-549 Distinguished Member

Sep 22, 2013
3,073
747
A signal is a 3D entity. Its true form is a spring.

When you REPRESENT(or look at) it from the side.........it looks like a sine wave.

When you REPRESENT it from the ends.......it looks like a circle.

4. ### WBahn Moderator

Mar 31, 2012
20,237
5,758

Perhaps give a specific example or two so that we can determine exactly what you are trying to get at.

5. ### sharanbr Thread Starter Active Member

Apr 13, 2009
76
1
Dear all,

I apologize for stating the question in such a confusing manner (my inability to articulate the question is related to my lack of good understanding of the topic itself).

It is mentioned that all periodic signals which we see as time vs amplitude plots can also be represented in circular form.
What I did not understand where the time variable is when it comes to circular representation.

But from the answers above, I know realize that my confusion is because I did not understand that a periodic signal just needs one cycle to
represent it completely which can be done by representing it in circular form. But this is true only if signal is periodic, else, it is not possible to convert a linear representation into circular form.

I am assuming that my above statement is true for the 2 cases,
1) when linear signal is aperiodic
2) when linear signal varies such that in a narrow window, it appears linear but as one extends the window, it is not linear. For example, a sine signal whose period is same for first 5 cycles and then period decreases for next 5 cycles and so on. It is essentially a aperiodic signal.

Kindly let me know if I have got this correct ...

6. ### BR-549 Distinguished Member

Sep 22, 2013
3,073
747
When you look at the circular form, the longitudinal motion of the signal, is out of, or into, the origin(paper or screen).

The sine wave form shows you variation with time.

The circular form shows you variation with rotation.

7. ### WBahn Moderator

Mar 31, 2012
20,237
5,758
Again, it would really help if you gave a concrete example of what you mean by a "linear representation" and what you mean by a "circular representation".

Let's keep it simple and use a pure sinusoidal signal with an amplitude of 10 V, a frequency of 1 kHz, and a phase angle of 30°. What is the linear representation and what is the circular representation for this signal.

8. ### amilton542 Active Member

Nov 13, 2010
496
64
The elementary trig' functions are circular, the only relationship you could demonstrate is by means of Pythagoras' theorem.

I feel confident to say the gap between your understanding of linear signals and trig' is through Fourier Analysis.

9. ### Papabravo Expert

Feb 24, 2006
11,154
2,180
We still do not have a precise definition of the kind of linear signal you are talking about.

I can make a cogent argument that neither the sin(x), nor the cos(x), which might represent certain signals, are linear functions.
sin (x + y) ≠ sin(x) + sin(y) ∀ x, y ∈ℝ
and sin(kx) ≠ ksin(x) ∀ x, k ∈ℝ

Those functions, sin(x) and cos(x), are however possible solutions to certain linear differential equations with constant coefficients.

Is that what you mean?

Last edited: Aug 24, 2015
10. ### WBahn Moderator

Mar 31, 2012
20,237
5,758
sin(x) is certainly not a linear function, but any function can potentially be the solution to a linear differential equation because linear is referring to the system, not the signal.

11. ### Papabravo Expert

Feb 24, 2006
11,154
2,180
I know that, and you know that, and I know that you know that. What I want is for the TS/OP to find a better way to articulate his question -- and I offered the best hint I could think of. I'm not sure he was talking about linear systems -- it was just a guess. There seems to be some deep meaning in the terms 'linear representation' and 'circular representation' but I can't quite get to the essentials of the concept. It's all just guesswork.

12. ### WBahn Moderator

Mar 31, 2012
20,237
5,758
Agreed. I've asked him more than once to give a concrete, explicit example of what HE means by both, and he won't.

13. ### MrAl Distinguished Member

Jun 17, 2014
3,747
791
Hello,

I can offer a guess here...

Circular functions are functions that describe points on a circle. For example:
x=r*cos(A)
y=r*sin(A)

for r a constant r=R, x and y are points on a circle with radius R.
There are other 'circular' relationships too because the points are always on the circle and any given point makes a certain angle A from the origin to the point.

But could you say that any function that is represented by sines and cosines (for example) is circular? The function itself may not be circular, but you could say that it could be made up of a sum of circular functions.

14. ### Papabravo Expert

Feb 24, 2006
11,154
2,180
@MrAl I know what a circular function is, and I know you know, but what is at issue here is the completely ambiguous nature of:
1. Linear signal
2. Linear representation
3. Circular representation
These terms referenced in the TS/OP's original post are completely without precise and meaningful definition. I cannot find a single usage in any of my references. Until or unless we have some clarity on this issue, I recommend that we ignore this thread.

15. ### MrAl Distinguished Member

Jun 17, 2014
3,747
791
Hi,

He he, you have a point there

BTW i am also on a fixed income.