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The complex number z=ej2π/Nz=ej2π/N is illustrated in Figure 1. It lies
on the unit circle at angle θ=2π/Nθ=2π/N. When this number is raised to the
nthnth power, the result is zn=ej2πn/Nzn=ej2πn/N. This number is also illustrated in
Figure 1. When one of the complex numbers ej2πn/Nej2πn/N is raised to the NthNth
power, the result is
(ej2πn/N)N=ej2πn=1.(ej2πn/N)N=ej2πn=1.
(1)We say that ej2πn/Nej2πn/N is one of the NthNth roots of unity, meaning that ej2πn/Nej2πn/N
is one of the values of z for which
z
N
-
1
=
0
.
z
N
-
1
=
0
.
(2)There are N such roots, namely,
e
j
2
π
n
/
N
,
n
=
0
,
1
,
...
,
N
-
1
.
e
j
2
π
n
/
N
,
n
=
0
,
1
,
...
,
N
-
1
.
(3)As illustrated in Figure 2, the 12th12th roots of unity are uniformly distributed
around the unit circle at angles 2πn/122πn/12. The sum of all of the NthNth roots of unity is zero:
S
N
=
∑
n
=
0
N
-
1
e
j
2
π
n
/
N
=
0
.
S
N
=
∑
n
=
0
N
-
1
e
j
2
π
n
/
N
=
0
.
(4)This property, which is obvious from Figure 2, is illustrated in Figure 3,
where the partial sums Sk=∑n=0k-1ej2πn/NSk=∑n=0k-1ej2πn/N are plotted for k=1,2,...,Nk=1,2,...,N.
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:
S
k
=
∑
n
=
0
k
-
1
e
j
2
π
n
/
N
.
S
k
=
∑
n
=
0
k
-
1
e
j
2
π
n
/
N
.
(5)We can imbed this question in the more general question, is there an analytical
solution for the “geometric sum”
S
k
=
∑
n
=
0
k
-
1
z
n
?
S
k
=
∑
n
=
0
k
-
1
z
n
?
(6)The answer is yes, and here is how we find it. If z=1z=1, the answer is Sk=kSk=k.
If z≠1z≠1, we can premultiply
S
k
S
k
by
z
z and proceed as follows:
z
S
k
=
∑
n
=
0
k
-
1
z
n
+
1
=
∑
m
=
1
k
z
m
=
∑m=0k-1zm+zk-1
=
S
k
+
z
k
-
1
.
z
S
k
=
∑
n
=
0
k
-
1
z
n
+
1
=
∑
m
=
1
k
z
m
=
∑m=0k-1zm+zk-1
=
S
k
+
z
k
-
1
.
(7)From this formula we solve for the geometric sum:
S
k
=
1
-
z
k
1
-
z
z
≠
1
k
,
z
=
1
.
S
k
=
1
-
z
k
1
-
z
z
≠
1
k
,
z
=
1
.
(8)This basic formula for the geometric sum Sk is used throughout electromagnetic theory and system theory to solve problems in antenna design and spectrum analysis. Never forget it.
Find formulas for Sk=∑n=0k-1ejnθSk=∑n=0k-1ejnθ and for Sk=∑n=0k-1ej2π/Nn.Sk=∑n=0k-1ej2π/Nn.
Prove ∑n=0N-1ej2πn/N=0∑n=0N-1ej2πn/N=0.
Find formulas for the magnitude and phase of the partial sum Sk=∑n=0k-1ej2πn/NSk=∑n=0k-1ej2πn/N.
(MATLAB) Write a MATLAB program to compute and plot the partial sum Sk=∑n=0k-1ej2πn/NSk=∑n=0k-1ej2πn/N for k=1,2,...,Nk=1,2,...,N. You should observe Figure 3.
Solve the equation (z+1)3=z3(z+1)3=z3.
Find all roots of the equation z3+z2+3z-15=0z3+z2+3z-15=0.
Find
c
c so that (1+j)(1+j) is a root of the equation z17+2z15-c=0z17+2z15-c=0.
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