Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
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sin2a 2 sin a cos a,
A:5a
cos2a 2 cos
2
a À 1 1 À 2 sin
2
a,
A:5b
sin
a
2
1
2
1 À cos a
r
,
A:6a
cos
a
2
1
2
1 cos a
r
,
A:6b
sin
2
a cos
2
a 1,
A:7a
sin
2
a
1
2
1 À cos2a,
A:7b
cos
2
a
1
2
1 cos2a,
A:7c
e
Æja
cos a Æ j sin a,
A:8a
sin a
1
2j
e
ja
À e
Àja
,
A:8b
cos a
1
2
e
ja
e
Àja
:
A:8c
In Euler's theorem (A.8), j
À1
p
. The basic concepts and manipulations of complex
number will be reviewed in Section A.3.
A.2 Geometric Series
The geometric series is used in discrete-time signal analysis to evaluate functions in
closed form. Its basic form is
X
NÀ1
n0
x
n
1 À x
N
1 À x
, x T 1:
A:9
This is a widely used identity. For example,
X
NÀ1
n0
e
Àj!n
X
NÀ1
n0
e
Àj!
n
1 À e
Àj!N
1 À e
Àj!
:
A:10
If the magnitude of x is less than 1, the infinite geometric series converges to
X
I
n0
x
n
1
1 À x
, jxj < 1:
A:11
446
APPENDIX A: SOME USEFUL FORMULAS
Summary :
sin2a 2 sin a cos a, A:5a cos2a 2 cos 2 a À 1 1 À 2 sin 2 a, A:5b sin a 2 1 2 1 À cos a r , A:6a cos a 2 1 2 1 cos a r , A:6b sin 2 a cos 2 a 1, A:7a sin 2 a 1 2 1 À cos2a, A:7b cos 2 a 1 2 1 cos2a, A:7c e Æja cos a Æ j sin a, A:8a sin a 1 2j e ja À e Àja , A:8b cos a 1 2 e ja e Àja : A:8c In Euler's theorem (A.8), j À1 p .
Tags :
cos,sin,geometric,series,nà1,cos2a,basic,àja,form,used,àjn,useful,less