Digital Signal Processing Reference
In-Depth Information
f(x)
=
cos(x)
0
f
(x)
=sin(x)
00
f
(x)
=cos(x)
000
f
(x)
=
sin(x)
0000
f
(x)
=
cos(x); and the above pattern repeats:
Replacing f(x) of equation 7.2 with cos(x):
x
2
+ ::: +
f
n
(0)
n!
x
n
+
f
n+1
(y)
cos(x) = cos(0)
sin(0)
1!
x
cos(0)
2!
(n + 1)!
x
n+1
:
Simplifying and substituting for x:
cos() = 1
2
2!
+
4
4!
+ :::
(7.4)
Now repeat equation 7.1, but notice how the real parts correspond to the cos()
function (equation 7.4), and that the imaginary parts correspond to the sin() func-
tion (equation 7.3).
e
j
= (1()
2
=2! +
4
=4! + :::) + j(
3
=3! +
5
=5! + :::)
This means that this equation can be simplied to:
e
j
= cos() + j sin()
which, of course, is Euler's formula.
7.5
Alternate Form of Euler's Equation
As we saw previously, Euler's equation is
e
j
= cos() + j sin():
What if happens to be a negative number? Let =, then replace with
in the above equation. Keep in mind that it does not really matter what symbol we
use, as long as we are consistent. On the righthand side, we can replace the sine and
cosine functions with a couple of trigonometric identities, that is, cos() = cos(),
and sin() =sin().
Therefore, Euler's equation also works for a negative
