ty34 wrote:>> Tim, >> >> As you surmised there is an error. Euler's identity is e^(jx) =3D >> cos(x) +j*sin(x) >> >> However your comment about rotation is not totally incorrect. A >> complex phaser (two component vector written as a+jb) will be rotated >> by theta degrees when the phaser is multiplied by e^(j*theta). So >> when theta=3D90 degrees, e^(j*theta) is j. But don't confuse >> mutiplying a phaser with just multiplying one of its componets by a >> complex entity. >> >> IHTH, >> >> Clay > > Thanks again Clay. You're right - I saw > > x(n) = cos (n) + j sin(n) > > as a combination of two out-of-phase signals varying in the same > axis. Its clear to me now just how wrong that was.Yes, it's important to visualize / construct in some well-understood framework. There are a few that seem to help but probably also confuse if they aren't well-defined or visualized: cos(wt) varies with "t" and it's easy to visualize it plotted on a real line (its amplitude) vs. time ... so, a 2-D representation. cos(wt) + jsin(wt) also varies with "t" but is harder to visualize. So, we come up with some ways to deal with it: One way is to conjure up a complex plane (instead of a real line) so that we can plot the imaginary part amplitude on one axis and the real amplitude on the other axis. Then, we can plot a trajectory over time in a 3rd dimension. That's where the spiral trajectory comes in for a complex sinusoid. It's a rotating complex-valued vector of constant magnitude that moves perpendicular to it's plane of rotation (in time). The rate of rotation and advance are reflected in "wt". The next step might be to envision this 3-D figure looking perpendicular to the imaginary plane and straight down the time axis. Now some of the time variations are obscured as there is no longer a visible "t" axis. Here, the spiral trajectory becomes a circle. Further, it's helpful to freeze time in this display. These time-frozen vectors are usually referred to as "phasors". Because "w" is constant, the position of the vectors making up a single signal is constant. This allows us to visually see how signals of a single frequency add in phase - as it's the relative rotational position of the vectors that displays phase. This same display is handy for envisioning the sum of two sinusoids of different frequencies - if one is only willing to allow one or both of the phasors to rotate (they can both rotate according to their own "wt" or one can rotate relative to the other according to "(w1-w2)t" or "(w1 + w2)t" if the rotation is opposite. Then if we split the real and imaginary parts of the complex-valued vector into separate vector components, one is aligned with the real axis and one is aligned with the imaginary axis. Of course, pictures are better... Fred
Basic question - what does the signal e^jwt look like?
Started by ●March 30, 2009
Reply by ●March 31, 20092009-03-31
Reply by ●April 2, 20092009-04-02
On Mon, 30 Mar 2009 09:02:44 -0500, "ty34" <minesadab@hotmail.com> wrote:>Hi, I'm an electrical engineering undergraduate in my final year and I'm >still struggling to get my head around complex signals. > >I would be really grateful if someone could give me an answer to something >that has had me stumped for the couple of days. What would the signal > >x[n] = e^(j(pi/8)n) > >look like on a scope, compared to say, the signal x[n] = sin((pi/8)n)?Hi, Ya' might find something useful at: http://www.dspguru.com/info/tutor/QuadSignals.pdf Good Luck, [-Rick-]
Reply by ●June 3, 20092009-06-03
Tim, Sin (wt) is not equal to -j cos (wt). That is where you have got your assumption wrong. Substitute some value of wt to clarify the answer yourself.>>On Mar 30, 10:02=A0am, "ty34" <minesa...@hotmail.com> wrote: >>> Hi, I'm an electrical engineering undergraduate in my final year and >I'm >>> still struggling to get my head around complex signals. >>> >>> I would be really grateful if someone could give me an answer to >somethin= >>g >>> that has had me stumped for the couple of days. What would the signal >>> >>> x[n] =3D e^(j(pi/8)n) >>> >>> look like on a scope, compared to say, the signal x[n] =3D >sin((pi/8)n)? >> >>Well scopes display real valued signals. And the signal you propose is >>complex valued. Of course you may display the real part on one axis >>and the imaginary part of the other axis then the result is points on >>a circle. I doubt that is what you want ed though. Now you can do a >>three dimensional plot and then your points will be on a helix. >> >>IHTH, >> >>Clay >> > >Thanks very much, that is a great help - I can see what your sayingabout>it forming a circle in the complex plane and that you'd need twoseparate>mag vs time traces to represent it. > >The real source of my confusion is that i've somehow got it into my head >that the 'j' operator simply represents a phase shift of 90deg. > >ie. sin wt = -jcos(wt) > >But if that was the case, > >then e^jx = cos x + j (-j cos x) = 2 cos x > >and this is a non-complex signal that can be displayed on a scope. Can >anyone show me where I am going wrong? > >Tim >
