Digital Signal Processing Reference
In-Depth Information
()
Chart of element
ary inte
gl
ra s
Chart of elementary derivatives
f
´
x
()
()
³
³
af xdx a fxdx
⋅
=
Function ( ) Derivative ´( )
konstant 0
1
f
x
f
x
c
=
n
+
1
x
xdx
n
xdx
³
³
³
³
³
n
=
x
x
+
=
1
n
n
−
1
nx
cos
sin
x
1
1
−
sin
xdx
=−
cos
x
x
x
2
x
x
edx
=
e
e
x
e
x
e
ax
ae
ax
sin
ax
cos
ax dx
=
a
sin
x
cos
x
cos
ax
cos
x
−
sin
x
³
s
in
ax dx
=−
a
sin
ax
a x
cos
ax
e
cos
ax
−
a
sin
ax
³
ax
edx
=
a
The absolute values of integration were left out
df t
()
(
)
<
≡
f t
()
The derivative of time is shown as a point , that is
dt
2 applications for differentiation
<
<
1
1
§
1
1
·
§
·
(
)
(
)
it
(
ω
)
−
it
(
ω
)
(
)
it
ω
−
it
ω
(
)
<
()
<
()
i
sin
ω
t
=
e
−
e
cos
ω
t
=
e
+
e
=
¨
¸
¨
¸
2
2
2
2
©
¹
©
¹
i
ω
−
i
ω
i
ω
i
ω
§ ·
i
(
)
(
)
(
)
(
)
it
ω
−
it
ω
it
ω
−
it
ω
()
()
i
ωω
cos
t
=
e
−
e
÷
i
ω
−
ωω
sin
t
=
e
−
e
⋅ −
¨
© ¹
2
2
2
2
ω
1
1
(
)
(
)
it
ω
−
it
ω
()
2
2
cos
ω
t
=
e
+
e
i
i
(
)
(
)
it
ω
−
it
ω
()
2
i
sin
ω
t
=−
e
+
e
i
=−
1
2
2
2
2
1
1
(
)
(
)
it
ω
−
it
ω
()
i
sin
ω
t
=
e
−
e
2
2
1 application for integration :
1
1
1
1
§
(
)
(
)
·
(
)
(
)
it
ω
−
it
ω
it
ω
−
it
ω
()
³
³
³
³
cos
ω
tdt
=
e
+
e
dt
=
e
dt
+
e
dt
¨
¸
2
2
2
2
©
¹
()
sin
ω
t
1
1
1
1
(
)
(
)
(
)
(
)
it
ω
−
it
ω
it
ω
−
it
ω
()
=
e
+
e
⋅
i
ω
sin
i
ω
t
=
e
−
e
(
)
ω
2
i
ω
2
−
i
ω
2
2
Abbildung 307:
Differentiation and integration of time-dependend complex values
In both charts above “hard and fast” rules for differentiation and integration are listed that play a role
within the topic. The FOURIER- integrals are examined and explained separately.
The applications refer to the derivation and integration of real part cos (Ȧ t) and imaginary part isin (Ȧ
t). Strangely, the imaginary part can be derived from the differentiated and also from the integrated real
part and vice versa.
NB: notation for the time-dependent derivation.
At this point the sine and the cosine oscillations are of importance in a physical sense.
They are physically real, but nevertheless the way they come into being, from a mathe-
matical point of view, can only be illustrated in the complex plane!
