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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Appendix A
Some Useful Formulas
This appendix briefly summarizes some basic formulas of algebra that will be used
extensively in this book.
A.1 Trigonometric Identities
Trigonometric identities are often required in the manipulation of Fourier series, trans-
forms, and harmonic analysis. Some of the most common identities are listed as follows:
sinÀa À sin a,
A:1a
cosÀa cos a,
A:1b
sina Æ b sin a cos b Æ cos a sin b,
A:2a
cosa Æ b cos a cos b Ç sin a sin b,
A:2b
2 sin a sin b cosa À b À cosa b,
A:3a
2 cos a cos b cosa b cosa À b,
A:3b
2 sin a cos b sina b sina À b,
A:3c
2 sin b cos a sina b À sina À b,
A:3d
sin a Æ sin b 2 sin
a Æ b
2
cos
a Ç b
2
,
A:4a
cos a cos b 2 cos
a b
2
cos
a À b
2
,
A:4b
cos a À cos b À2 sin
a b
2
sin
a À b
2
,
A:4c
Real-Time Digital Signal Processing. Sen M Kuo, Bob H Lee
Copyright # 2001 John Wiley & Sons Ltd
ISBNs:0-470-84137-0 (Hardback); 0-470-84534-1 (Electronic)
Summary :
Some of the most common identities are listed as follows: sinÀa À sin a, A:1a cosÀa cos a, A:1b sina Æ b sin a cos b Æ cos a sin b, A:2a cosa Æ b cos a cos b Ç sin a sin b, A:2b 2 sin a sin b cosa À b À cosa b, A:3a 2 cos a cos b cosa b cosa À b, A:3b 2 sin a cos b sina b sina À b, A:3c 2 sin b cos a sina b À sina À b, A:3d sin a Æ sin b 2 sin a Æ b 2 cos a Ç b 2 , A:4a cos a cos b 2 cos a b 2 cos a À b 2 , A:4b cos a À cos b À2 sin a b 2 sin a À b 2 , A:4c Real-Time Digital Signal Processing.
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