Hello Everyone,

I tried very hard to find information that would help me to visualise a complex signal but didn't find much information anywhere.
I am hoping that this forum helps me out.
Consider a signal e^(jwt)
where w = 2*pi*f.
I know that the above can be written as cos(wt) + jsin(wt).
Both are sinusoidal waves with a phase shift of 90 degrees.
But how does the signal cos(wt)+jsin(wt) look like?
Is it obtained by adding the instantaneous values of cos and sin?
And I also heard that as these two signals are orthogonal they don't interfere with each other, then what does a + sign between them indicate in
the above equation.
Really appreciate your help.

Thanks,
NaveenKori

 

The proper visualization is a radius vector rotating on a unit circle. This unit circle is in the complex plane. As the radius vector rotates around the origin the coordinates of the endpoint are given by:

cos(ωt) + jsin(ωt)

The symbol j is the clue that the two parts of a complex number (point) cannot be combined into a single term. In algebra it would be like trying to add 2x to 3y. No matter how hard you try they cannot be combined. This is a classic form of a parametric equation.

 

Thank you very much Papabravo.

 

A complex signal is a variation whose reference origin moves perpendicular to the variation. The cos term sets the amplitude of variation. The sine term sets the rate of perpendicular movement, in reference the the cos movement.

You can think of it as setting the pitch of the variation. Or the pitch on a bolt thread. Or frequency.

Think of the origin moving along Z as the X rotates towards Y. It's a spiral stair case. It's a helix. The imaginary part, is the movement or rotation of the helix...in or out of the paper graph.

I don't know why the teach these concepts the way they do. They make it very confusing.

 

Thank u BR-549, Now I got a complete picture of how it looks like.

 

10-4.

 

Hello Everyone,

I tried very hard to find information that would help me to visualise a complex signal but didn't find much information anywhere.
I am hoping that this forum helps me out.
Consider a signal e^(jwt)
where w = 2*pi*f.
I know that the above can be written as cos(wt) + jsin(wt).
Both are sinusoidal waves with a phase shift of 90 degrees.
But how does the signal cos(wt)+jsin(wt) look like?
Is it obtained by adding the instantaneous values of cos and sin?
And I also heard that as these two signals are orthogonal they don't interfere with each other, then what does a + sign between them indicate in
the above equation.
Really appreciate your help.

Thanks,
NaveenKori

Hi there,

e^(jwt) graphed parametrically produces a helix. However, that's just a part of the entire signal which would be:

e^((a+jw)*t)

which is a exponential helix which is either increasing in diameter or decreasing in diameter.
In this case we can also write e^sT where s=a+jw and T is a time delay.

Several graphs are shown in the attachment.

Also, the addition sign just shows us that the two terms have to stay together which can be interpreted as a single quantity, even though we dont directly add them like we do with normal numbers. This is almost the same as when we have:
c=a+b

the a and b can not be added unless we know what numbers they are. When we have:
c=a+bj

however, the b can not be added directly even when we know what a and b are, but we can find the amplitude and phase shift with:

amplitude=sqrt(a^2+b^2)

and
phaseshift=atan2(b,a)

In the case of e^jw, we have:
a=cos(wt) and b=sin(wt)

so the amplitude is:
ampl=sqrt(cos(wt)^2+sin(wt)^2)=1

so the amplitude is 1. The phase shift is always 90 degrees because sin is always 90 degrees away from cos.

In signals e^-sT represents a pure delay in the signal, so if we have a signal V then if we have V*e^sT then the original signal V is delayed by T seconds.

See the attachment for a more clear picture of these things.

 

Thanks for the additional info and the graphs, MrAI

 

Hi,

You're welcome, and arent those graphs cool? I like the exponentially 'modulated' ones best. The ones with the minus 'a' part eventually becomes a singularity at t=infinity.

We could look at an example of the e^-sT form too as that creates a delay in the signal, which appears quite a bit in literature. The graphs are not as interesting though because they are just the original signal drawn a little bit later on the graph but have the same shape.

There are so many interesting polar plots even in just two dimensions that we dont see very often that are really cool. I might have to plot some of them and make up a collection and post them one day.

 

Yeah, those are cool, stuff like these are supposed to be provided in the books which will help students visualise signals.