Digital Signal Processing Reference
In-Depth Information
The geometric progression (GP) series sum of the form
=
N
ar
n
=
a
+
ar
+
ar
2
++
ar
N
S
n
=
0
is used frequently in this text while dealing with the discrete-time signals. Note
that the factor
r
can be real, imaginary, or complex.
A.4 Limits and diff
erential calculus
t
→∞
t
−α
ln
t
=
0
,
lim
Re(
α
)
>
0
t
→
0
t
α
ln
t
=
0
,
lim
Re(
α
)
>
0
L' H ˆopital's Rule:
x
′
(
t
)
y
′
(
t
)
If lim
t
→
a
x
(
t
)
=
lim
t
→
a
y
(
t
)
=
0 or lim
t
→
a
x
(
t
)
=
lim
t
→
a
y
(
t
)
=∞,
and lim
has a
t
→
a
x
′
(
t
)
y
′
(
t
)
x
(
t
)
y
(
t
)
finite value, then lim
t
→
a
=
lim
t
→
a
d
d
t
1
g
(
t
)
1
g
2
(
t
)
d
g
(
t
)
d
t
=−
d
d
t
h
(
t
)
g
(
t
)
1
g
2
(
t
)
g
(
t
)
d
h
(
t
)
d
t
−
h
(
t
)
d
g
(
t
)
d
t
=
A.5 Indefinite inte
grals
u
d
v
=
uv
−
v
d
u
d
f
d
t
f
(
t
)
g
(
t
)d
t
=
f
(
t
)
g
(
t
)d
t
−
g
(
t
)d
t
d
t
1
a
cos
at
d
t
=
sin
at
+
C
,
a
=
0
sin
at
d
t
=−
1
a
cos
at
+
C
,
a
=
0
t
2
sin 2
at
4
a
cos
2
at
d
t
=
+
+
C
,
a
=
0
t
2
sin 2
at
4
a
sin
2
at
d
t
=
−
+
C
,
a
=
0
1
a
2
t
cos
at
d
t
=
(cos
at
+
at
sin
at
)
+
C
,
a
=
0
1
a
2
t
sin
at
d
t
=
(sin
at
−
at
cos
at
)
+
C
,
a
=
0
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