Digital Signal Processing Reference
In-Depth Information
Stopband gain vs. norm. freq. for C−LPF (
r
= 3 dB)
0
−10
n = 1
−20
n = 2
−30
−40
n = 3
−50
n = 4
−60
1
2
3
4
5
6
7 8 9
10
Normalized frequency,
ω
/
ω
c
Useful Definitions and Relations
Complex Fourier Series (FS):
x
ð
t
Þ¼
X
1
k
¼1
X
k
e
þ
j2pkf
o
t
;
where:
Z
T
o
X
o
¼
1
T
o
x
ð
t
Þ
dt
;
0
Z
T
o
X
k
¼
1
T
o
x
ð
t
Þ
e
j2pkf
o
t
dt
0
Trigonometric FS:
x
ð
t
Þ¼
a
o
þ
X
1
n
¼
1
a
n
cos
ð
2pnf
o
t
Þþ
b
n
sin
ð
2pnf
o
t
Þ;
where:
a
o
¼
X
o
;
a
n
¼
X
n
þ
X
n
;
b
n
¼
j
ð
X
n
X
n
Þ:
[Note that X
n
's are the coefficients of the Complex FS.]
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