i salute the community!

while continuing my work on signals and systems, I got stuck at a point. .

the chart below : sin(80nt), that is, it belongs to [sin(80\pi t)].





What happens to the phase graph when the signal sin(80πt) is multiplied by -j?


I know that j causes a +90° phase shift, but how does the phase graph change with -j? What should be the new phase values?

 

i salute the community!

while continuing my work on signals and systems, I got stuck at a point. .

the chart below : sin(80nt), that is, it belongs to [sin(80\pi t)].


View attachment 345842


What happens to the phase graph when the signal sin(80πt) is multiplied by -j?


I know that j causes a +90° phase shift, but how does the phase graph change with -j? What should be the new phase values?

Multiplying by -j is a -90° phase shift. In fact, multiplying by any complex exponential is a scaling and rotating operation (though if the magnitude of the complex number is 1, then it's just a rotation).

If you need to convince yourself of this, recall any complex number, \(z\), can be written as:
\[z = \left|z\right| e^{j\phi_z}\]
Where \(\phi_z\) is the angle of the complex number. If we have some complex number with unitary magnitude, \(r\) (so the magnitude of \(r\) is 1), so \(r = \exp(j\phi_r)\), then multiplying the two results in:

\[z\cdot r = \left|z\right|e^{j\phi_z}e^{j\phi_r} = \left|z\right|e^{j\left(\phi_z+\phi_r\right)}\]

In the case of your question specifically, \(\pm j = \exp\left(\pm j\frac{\pi}{2}\right)\), so multiplying by \(\pm j\) is the same as a 90-degree rotation, with the direction depending on the sign.

 

Multiplying by -j is a -90° phase shift. In fact, multiplying by any complex exponential is a scaling and rotating operation (though if the magnitude of the complex number is 1, then it's just a rotation).

If you need to convince yourself of this, recall any complex number, \(z\), can be written as:
\[z = \left|z\right| e^{j\phi_z}\]
Where \(\phi_z\) is the angle of the complex number. If we have some complex number with unitary magnitude, \(r\) (so the magnitude of \(r\) is 1), so \(r = \exp(j\phi_r)\), then multiplying the two results in:

\[z\cdot r = \left|z\right|e^{j\phi_z}e^{j\phi_r} = \left|z\right|e^{j\left(\phi_z+\phi_r\right)}\]

In the case of your question specifically, \(\pm j = \exp\left(\pm j\frac{\pi}{2}\right)\), so multiplying by \(\pm j\) is the same as a 90-degree rotation, with the direction depending on the sign.

hmm, I think what I understand from what you said is this:
- if -j means a -90 degree phase shift

when the negative phase is π/2, it becomes 90 - 90 = 0
when the positive phase is -π/2, it becomes -90 - 90 = -180 (π)

did I understand you correctly?

 

If I understood your work correctly and what you're asking, then yes: you understood correctly. The leftmost point in your spectrum would end up as pi/2-pi/2=0 for its phase, and the rightmost point in your spectrum would be -pi/2-pi/2 = -pi - both precisely as you calculated.

 

If I understood your work correctly and what you're asking, then yes: you understood correctly. The leftmost point in your spectrum would end up as pi/2-pi/2=0 for its phase, and the rightmost point in your spectrum would be -pi/2-pi/2 = -pi - both precisely as you calculated.

I am really grateful to you for helping me with such a difficult to understand issue. God help you.
finally, I will ask this: if j is +90, if it is jsin, a +90 degree phase will be added on both sides, right?

 

I am really grateful to you for helping me with such a difficult to understand issue. God help you.
finally, I will ask this: if j is +90, if it is jsin, a +90 degree phase will be added on both sides, right?

That's correct!