2.3 Mathematical Tools
49
and its rate
˙
x(t) = x 0 ω sin(ω t)
(2.89)
is obtained in the complex plane, see Fig. 2.14. Here
ω :=
2 π
T
and T
(2.90)
t
˙
x(t)
ω
= sin(ωt)
+1
0
−1
t
x(t) = cos(ωt)
+1
0
−1
e
iωt
ωt
Fig. 2.14 Complex representation x(t) of a harmonically oscillating signal x(t) = cos(ω t),
whereby ω := 2 π/T and T denote the angular frequency and the period duration, respectively.
The real and imaginary parts
x(t)
and
x(t)
relate to the signal x(t) and its scaled rate
˙
x(t)/ω, respectively