tailieunhanh - Real-Time Digital Signal Processing - Appendix A: Some Useful Formulas

Tham khảo sách 'real-time digital signal processing - appendix a: some useful formulas', kỹ thuật - công nghệ, kĩ thuật viễn thông phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 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 Appendix A Some Useful Formulas This appendix briefly summarizes some basic formulas of algebra that will be used extensively in this book. Trigonometric Identities Trigonometric identities are often required in the manipulation of Fourier series transforms and harmonic analysis. Some of the most common identities are listed as follows sin a sin a A. la cos a cos a sin a sin a cos fi cos a sin fi cos a cos a cos fi sin a sin fi 2 sin a sin fi cos a cos a 2 cos a cos P cos a P cos a P 2 sin a cos P sin a sin a 2 sin p cos a sin a sin a sin a sin p 2 sin f cos f 7 2 2 n A - 0 cos a cos p 2 cos I - cos I - a o P VP 0 cos a cos p 2 sm - sm - 2 2 446 APPENDIX A SOME USEFUL FORMULAS sin 2a 2 sin a cos a cos 2a 2 cos2 a 1 1 2 sin2 a A. 5b An sm - i - 1 cos a V 2 2 a Az . pAC I I r _L pAÇ P 1 Czwo 1 _ I _ I T 1 Czwo IX 1 27 V 2 2 sin2 a cos2 a 1 sin2 a 1 cos 2a cos2 a 1 cos 2a C 2 _ cos a j sjn sin a 1 eja - e 2 2 cos a e2 e 2 . In Euler s theorem j v l. The basic concepts and manipulations of complex number will be reviewed in Section . Geometric Series The geometric series is used in discrete-time signal analysis to evaluate functions in closed form. Its basic form is E V X 1. A 1 A w 0 This is a widely used identity. For example 1 N 1 i p j N n 0 n 0 1 A. 10 If the magnitude of x is less than 1 the infinite geometric series converges to Ê x TT7 M 1-n 1 A n Q COMPLEX VARIABLES 447 Complex Variables A complex number z can be expressed in rectangular Cartesian form as z x j y Re z jlm z . A. 12 Since the complex number z represents the point x y in the two-dimensional plane it can be drawn as a vector illustrated in Figure . The horizontal coordinate x is called the real part and the vertical coordinate y