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

Real-Time Digital Signal Processing. Sen M Kuo, Bob H Lee Copyright # 2001 John Wiley & Sons Ltd ISBNs:0-470-84137-0 4Hardback); 0-470-84534-1 4Electronic) 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: sina sina, A:1a cosa cosa, A:1b sina b sinacosb cosasinb, A:2a cosa b cosacosb sinasinb, A:2b 2sinasinb cosa b cosa b, A:3a 2cosacosb cosab cosa b, A:3b 2sinacosb sina b sina b, A:3c 2sinbcosa sina b sina b, A:3d sina sinb 2sinabcosa b, A:4a cosa cosb 2cosabcosa b, A:4b cosacosb 2sina bsina b, A:4c 446 APPENDIX A: SOME USEFUL FORMULAS sin2a 2sinacosa, A:5a cos2a 2cos2 a 1 12sin2 a, A:5b r sin a 11 cosa, A:6a r cos a 11 cosa, A:6b sin2 a cos2 a 1, A:7a sin2 a 11 cos2a , A:7b cos2 a 11 cos2a , A:7c eja cosa jsina, A:8a sina 2jeja eja, A:8b cosa 1eja eja: A:8c In Euler`s theorem 4A.8), j 1. 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 N1 n 1 xN n0 1 x This is a widely used identity. For example, x 1: A:9 N1 N1 ej!n ej!n j! : A:10 n0 n0 If the magnitude of x is less than 1, the infinite geometric series converges to X n0 x 1x , x < 1: A:11 COMPLEX VARIABLES 447 A.3 Complex Variables A complex number z can be expressed in rectangular 4Cartesian) form as z x jy Rez jImz : A:12 Since the complex number z represents the point 4x, y) in the two-dimensional plane, it can be drawn as a vector illustrated in Figure A.1. The horizontal coordinate x is called the real part, and the vertical coordinate y is the imaginary part. As shown in Figure A.1, the vector zalso can be definedby its length 4radius) r and its direction 4angle) y. The x and y coordinates of the vector are given by x rcosy and y rsiny: A:13 Therefore the vector z can be expressed in polar form as z rcosy jrsiny rejy, A:14 where p r z x2 y2 A:15 is the magnitude of the vector z and y tan1 x A:16 is its phase in radians. The basic arithmetic operations for two complex numbers, z1 x1 jy1 and z2 x2 jy2, are listed as follows: z1 z2 x1 x2 jy1 y2, z1z2 x1x2 y1y2 jx1y2 x2y1 r1r2ejy1y2, A:17 A:18a A:18b Im[z] y (x, y) r q 0 x Re[z] Figure A.1 Complex numbers represented as a vector 448 APPENDIX A: SOME USEFUL FORMULAS z1 x1x2 y1y2 jx2y1 x1y2 2 2 2 2 r1 ejy1y2: 2 A:19a A:19b Note that addition and subtraction are straightforward in rectangular form, but is difficult in polar form. Division is simple in polar form, but is complicated in rectangu-lar form. The complex arithmetic of the complex number x can be listed as z x jy rejy, A:20 where * denotes complex-conjugate operation. In addition, zz z2, A:21 z1 1 1ejy, A:22 zN rNejNy: A:23 The solution of zN 1 A:24 are zk ejyk ej2pk=N, k 0,1, ...,N 1: A:25 As illustrated in Figure A.2, these N solutions are equally spaced around the unit circle z 1. The angular spacing between them is y 2p=N. Im[z] ej(2π/N) Re[z] |z|=1, unit circle Figure A.2 Graphical display of the Nth roots of unity, N 8 VECTOR CONCEPTS 449 A.4 Impulse Functions The unit impulse function dt can be defined as dt 1, if t 0 if t 0. A:26 Thus we have … dtdt 1 A:27 and … dt t0xtdt xt0, A:28 where t0 is a real number. A.5 Vector Concepts Vectors and matrices are often used in signal analysis to represent the state of a system at a particular time, a set of signal values, and a set of linear equations. The vector concepts can be applied to effectively describe a DSP algorithm. For example, define an L1 coefficient vector as a column vector b b0 b1 ...bL1 T, A:29 where T denotes the transpose operator and the bold lower case character is used to denote a vector. We further define an input signal vector at time n as xn xn xn 1...xn L 1 T: A:30 The output signal of FIR filter defined in 43.1.16) can be expressed in vector form as L1 yn blxnl bTxn xTnb: A:31 l0 Therefore, the linear convolution of an FIR filter can be described as the inner 4or dot) product of the coefficient and signal vectors, and the result is a scalar y4n). If we further define the coefficient vector a a1 a2 aM T A:32 ... - tailieumienphi.vn