Digital Signal Processing Reference
In-Depth Information
Sine and cosine oscillation in the complex plane
i
ϕ
cos
cos
ϕ
+
i
sin
ϕ
=
e
()
()
−
i
ϕ
−
ϕ
+
i
sin
−
ϕ
=
e
cos
cos: axis symmetry
() ()
()
−ϕ
=
cos
ϕ
due to axis symmetry of the cosines function
and sin
−ϕ
=−
i
sin
ϕ
due to point symmetry of the sine fu
nction
i
ϕ
-
cos
ϕ
+
i
sin
ϕ
=
e
½
°
°
i
ϕ
−
i
ϕ
+
results i
n 2 cos
ϕ
=+
ee
®
¾
−
i
ϕ
cos
ϕ
−
i
sin
ϕ
=
e
°
°
¯
¿
i
ϕ
−
i
ϕ
ee
+
1
1
1
(
)
i
ϕ
−
i
ϕ
i
ϕ
−
i
ϕ
c
os
ϕ
=
=
ee
+
=
e
+
e
sine: point symmetry
2
2
2
2
i
ϕ
-
cos
ϕ
+
i
sin
ϕ
=
e
½
°
°
i
ϕ
−
i
ϕ
+
results in 2 sin
i
ϕ
=
e
−
e
®
¾
−
i
ϕ
−
cos
ϕ
+
i
sin
ϕ
= −
e
°
°
¯
¿
i
ϕ
−
i
ϕ
ee
−
1
1
1
(
)
i
ϕ
−
i
ϕ
i
ϕ
−
i
ϕ
i
sin
ϕ
=
=
ee
−
=
e
−
e
2
2
2
2
Time dependent sine and cosine oscillation in the complex plane
(
)
(
)
it
ωϕ
+
−
it
ωϕ
+
0
0
(
)
(
)
e
+
e
1
1
it
ωϕ
+
−+
it
ωϕ
(
)
0
0
co
s
ωϕ
t
+
=
=
e
+
e
0
2
2
2
(
)
(
)
it
ωϕ
+
−
it
ωϕ
+
0
0
(
)
(
)
e
−
e
1
1
it
ωϕ
+
−
it
ωϕ
+
(
)
0
0
i
sin
ωϕ
t
+
=
=
e
−
e
0
2
2
2
Illustration 305:
Sine and cosine as linear combination of conjugated complex vectors e
i
ϕ
and e
-i
ϕ
Here we deduce from which complex vectors sine and cosine are composed. At any given time there are
two complex vectors (below, “frequency vectors”) that are bound together in an inseparable relation. The
one frequency vector is
conjugated complex
to the other. The different algebraic signs mean that the sum
of their angles is the same size at any given time, whereas with a time-change of the angle they rotate dia-
metrically.
The resulting imaginary component of both the diametrically rotating vectors is zero for any angle (see
Illustration 306 as well as Illustration 97 in Chapter 5). The imaginary component does not appear exter-
nally either visually or measurably, but exists in a causal sense.
