<< Chapter < Page | Chapter >> Page > |
The complex number is illustrated in [link] . It lies on the unit circle at angle . When this number is raised to the power, the result is . This number is also illustrated in [link] . When one of the complex numbers is raised to the power, the result is
We say that is one of the roots of unity, meaning that is one of the values of z for which
There are N such roots, namely,
As illustrated in [link] , the roots of unity are uniformly distributed around the unit circle at angles . The sum of all of the roots of unity is zero:
This property, which is obvious from [link] , is illustrated in [link] , where the partial sums are plotted for .
These partial sums will become important to us in our study of phasors and light diffraction in "Phasors" and in our discussion of filters in "Filtering" .
Geometric Sum Formula. It is natural to ask whether there is an analytical expression for the partial sums of roots of unity:
We can imbed this question in the more general question, is there an analytical solution for the “geometric sum”
The answer is yes, and here is how we find it. If , the answer is . If , we can premultiply by and proceed as follows:
From this formula we solve for the geometric sum:
This basic formula for the geometric sum S k is used throughout electromagnetic theory and system theory to solve problems in antenna design and spectrum analysis. Never forget it.
Notification Switch
Would you like to follow the 'A first course in electrical and computer engineering' conversation and receive update notifications?