48
2 Preliminaries
Table 2.2 Real-valued causal signals x(t) in time domain and corresponding Laplace transformed
complex-valued signals X (s) in complex frequency domain
x =
⎧
⎨
⎩
x
+ i x
,
x 0 [cos α + i sin α],
x 0 e
i α
.
(2.85)
Here i denotes the imaginary unit, and x
:= x 0 cos α and x
:= x 0 sin α are the
real and imaginary parts of x. The length x 0 and angle α in the complex plane thus
compute from
[x 0 ]
2
:= [x
]
2
+ [x
]
2 and tan α := x
/x
.
(2.86)
If the angle α varies linear in time as α = ω t, the complex representation
x(t) =
⎧
⎨
⎩
x
(t) + i x
(t),
x 0 [cos(ω t) + i sin(ω t)],
x 0 e
i ω t
.
(2.87)
of a harmonically oscillating signal
x(t) = x 0 cos(ω t)
(2.88)
2 Preliminaries
Table 2.2 Real-valued causal signals x(t) in time domain and corresponding Laplace transformed
complex-valued signals X (s) in complex frequency domain
x =
⎧
⎨
⎩
x
+ i x
,
x 0 [cos α + i sin α],
x 0 e
i α
.
(2.85)
Here i denotes the imaginary unit, and x
:= x 0 cos α and x
:= x 0 sin α are the
real and imaginary parts of x. The length x 0 and angle α in the complex plane thus
compute from
[x 0 ]
2
:= [x
]
2
+ [x
]
2 and tan α := x
/x
.
(2.86)
If the angle α varies linear in time as α = ω t, the complex representation
x(t) =
⎧
⎨
⎩
x
(t) + i x
(t),
x 0 [cos(ω t) + i sin(ω t)],
x 0 e
i ω t
.
(2.87)
of a harmonically oscillating signal
x(t) = x 0 cos(ω t)
(2.88)