How do you hand crank out magnitude and phase of a sum of complex exponentials. I know it’s in one of my books but I can’t find it for the life of me. I don’t want to put it in Mathcad because it comes out too messy to work with.
I did a Fourier Transform on a transfer function of an impulse train and have come up with e^(-jw5) +2e^(-jw6)+e^(-jw*7).
I know this is a fairly simple transformation but I can’t remember how to do this for the life of me. This is only a very small part of a much larger problem.
Can someone help me out with this or at least give me the knowledge to do the job on my own?
Thanks!
(no this isn’t homework by the way. Well, graded homework anyway. It’s a problem that the prof gave us out to try on our own. He gave us the final solution but not many of the steps to get there.)
My first thought is to turn them all into (a + bi) format, add them up that way to get a new number (c + di) and then return that to the exponential form, which will give you the overall magnitude and phase.
I can’t think of anything that doesn’t just boil down to this approach. Multiplication is easy with complex exponentials, but that doesn’t carry over to addition.
In general, as others have said, you can’t do much better than conversion to rectangular complex coordinates.
For this particular case, you can simplify things considerably. Factor the expression as exp(-6 j w) * [exp(j w) + 2 + exp(-j w)] and recognize the quantity […] as 4cos[sup]2/sup. This is an explicit factorization into magnitude and phase. (Of course this only comes out so nicely by chance, because your transfer function has such a nice form.)
Thanks. I just figured that out on my own a little bit ago, I swear! Lol. I remember the prof doing a similar problem in class. I came up with the same thing as you.