[Complex exponential] Solutions must be wrong

[JJ91]

Homework Statement

Hi,

I would like to get feedback if my z-plot is accurate for the following complex exponentiala:

a=2*exp(j*∏*t)
b=2*exp(j*∏*-1.25)
c=1*exp(j*∏*t)
d=-j*exp(j*∏*t)

Further analysis:
a= -2 because cos(∏) = -1 and sin(∏)=0
b= actual complex number A*cos(∅)+j*sin(∅)
c= -1 because cos(∏) = -1 and sin(∏)=0
d=+j because -j*cos(∏)= +j and -j*j*sin(∏)=0

However according to solutions from other source my analysis is wrong.

I've decided to plot both graphs in MatLab so you can see the difference:
(my solution on the left)/(solution from other source on the right)

2.
A*exp(x)=A*cos(x)+A*j*sin(x)

3.
As stated in one, theory doesn't agree with solution found on the internet.
Also, in whatever form I put the exponential in the function I keep getting totally different results.
e.g.
[2*exp(j*pi), 2*exp(j*pi*-1.25), 1*exp(j*pi), -j*exp(j*pi)] would not give me similar amplitude as above for some reason.

P.S. Ignore the R{1} on the left graph.

 

[aralbrec]

I would like to get feedback if my z-plot is accurate for the following complex exponentiala:

a=2*exp(j*∏*t)
b=2*exp(j*∏*-1.25)
c=1*exp(j*∏*t)
d=-j*exp(j*∏*t)

This looks very confused. From the solution, the above looks like phasors and should be written as:

a = 2 e^jwt
b = 2 e^{j(wt - 1.25∏)}
c = 1 e^jwt
d = -j e^jwt

where w could be pi rad/s but I think is more likely a frequency variable.

First thing you need to notice is these are not fixed points in the Z plane -- they vary with t. So instead we view each phasor as a time-varying part multiplied by a fixed complex multiplier. Do this by factoring out e^jwt and plotting the multiplier in the Z plane. For example, A would be 2 times the time varying part e^jwt.

On your plot, each point is understood to be multiplied by e^jwt and as time varies, they all get rotated counterclockwise by wt radians but maintain fixed spatial relationships between them so that you can imagine rotating the entire sheet of paper the plot is on clockwise wt radians to find the location of each complex number for all time.

The solution provided looks like a phasor diagram for the above.

 

[Ray Vickson]

I would like to get feedback if my z-plot is accurate for the following complex exponentiala:

a=2*exp(j*∏*t)
b=2*exp(j*∏*-1.25)
c=1*exp(j*∏*t)
d=-j*exp(j*∏*t)

This looks very confused. From the solution, the above looks like phasors and should be written as:

a = 2 e^jwt
b = 2 e^{j(wt - 1.25∏)}
c = 1 e^jwt
d = -j e^jwt

where w could be pi rad/s but I think is more likely a frequency variable.

First thing you need to notice is these are not fixed points in the Z plane -- they vary with t. So instead we view each phasor as a time-varying part multiplied by a fixed complex multiplier. Do this by factoring out e^jwt and plotting the multiplier in the Z plane. For example, A would be 2 times the time varying part e^jwt.

On your plot, each point is understood to be multiplied by e^jwt and as time varies, they all get rotated counterclockwise by wt radians but maintain fixed spatial relationships between them so that you can imagine rotating the entire sheet of paper the plot is on clockwise wt radians to find the location of each complex number for all time.

The solution provided looks like a phasor diagram for the above.

As written,
b = 2 e^{-1.25 j \pi}. Is that really what is wanted?

RGV

 

[JJ91]

On your plot, each point is understood to be multiplied by e^jwt and as time varies, they all get rotated counterclockwise by wt radians but maintain fixed spatial relationships between them

That make sense, however, so if e.g. 2 is multiplied by e^jωt where ω is pi rad/sec then the result of this exponential is -1 (considering only real part of this complex exponential) thus the resulting value of this exponential is -2 but solution says 2.

The first graphs (subplot 1 and 2) shows a synthesis of those 4 exponential using MatLab function which is based on this formula (sinusoidal synthesis):

The question or the form of this exponentials is represented as follows in textbook:

 

[aralbrec]

however, so if e.g. 2 is multiplied by e^jωt where ω is pi rad/sec then the result of this exponential is -1 (considering only real part of this complex exponential) thus the resulting value of this exponential is -2 but solution says 2.

At what time are you supposing e^j∏t is equal to -1? I'd agree with you it equals -1 at time t=1, eg, but at time t=0 it equals +1. By factoring out the time component we can plot the constant part of the phasor in the Z plane. If A = 2 e^jwt and we agree to draw a diagram for t=0, then A will be plotted at (2,0) as in the solution. To find A at other times you have to multiply this position by e^jwt, the part we factored out. This has the effect of rotating the point A counter clockwise by wt radians. For time t=1 and w=∏, eg, we would multiply 2 by e^j∏ to find at time t=1, A=2*(-1) = -2 as you have been saying.

On the phasor diagram, you can add those plotted points as vectors to find out what overall sum of the signal will be, then multiply by e^j∏t and take the real part for the sinusoid.

But I think your discrepancy is coming from evaluating e^j∏t at time t=1 instead of t=0 if I've understood correctly.

 

[JJ91]

Just realized right now that I was over-thinking about that. Very basic, thanks for answer.