39 * \f[ u(t)=u_c(t) + j u_s(t)\f] |
39 * \f[ u(t)=u_c(t) + j u_s(t)\f] |
40 * \f[ u_c(t) = \frac{2}{\sqrt{N}}\sum_{n=0}^{M}a_n\cos(\omega_n t+\phi_n)\f] |
40 * \f[ u_c(t) = \frac{2}{\sqrt{N}}\sum_{n=0}^{M}a_n\cos(\omega_n t+\phi_n)\f] |
41 * \f[ u_s(t) = \frac{2}{\sqrt{N}}\sum_{n=0}^{M}b_n\cos(\omega_n t+\phi_n)\f] |
41 * \f[ u_s(t) = \frac{2}{\sqrt{N}}\sum_{n=0}^{M}b_n\cos(\omega_n t+\phi_n)\f] |
42 * where |
42 * where |
43 * \f[ a_n=\left \{ \begin{array}{ll} |
43 * \f[ a_n=\left \{ \begin{array}{ll} |
44 * \sqrt{2}\cos\beta_0 & n=0 \ \ |
44 * \sqrt{2}\cos\beta_0 & n=0 \\ |
45 * 2\cos\beta_n & n=1,2,\ldots,M |
45 * 2\cos\beta_n & n=1,2,\ldots,M |
46 * \end{array} |
46 * \end{array} |
47 * \right .\f] |
47 * \right .\f] |
48 * \f[ b_n=\left \{ \begin{array}{ll} |
48 * \f[ b_n=\left \{ \begin{array}{ll} |
49 * \sqrt{2}\sin\beta_0 & n=0 \ \ |
49 * \sqrt{2}\sin\beta_0 & n=0 \\ |
50 * 2\sin\beta_n & n=1,2,\ldots,M |
50 * 2\sin\beta_n & n=1,2,\ldots,M |
51 * \end{array} |
51 * \end{array} |
52 * \right .\f] |
52 * \right .\f] |
53 * \f[ \beta_n=\left \{ \begin{array}{ll} |
53 * \f[ \beta_n=\left \{ \begin{array}{ll} |
54 * \frac{\pi}{4} & n=0 \ \ |
54 * \frac{\pi}{4} & n=0 \\ |
55 * \frac{\pi n}{M} & n=1,2,\ldots,M |
55 * \frac{\pi n}{M} & n=1,2,\ldots,M |
56 * \end{array} |
56 * \end{array} |
57 * \right .\f] |
57 * \right .\f] |
58 * \f[ \omega_n=\left \{ \begin{array}{ll} |
58 * \f[ \omega_n=\left \{ \begin{array}{ll} |
59 * 2\pi f_d & n=0 \ \ |
59 * 2\pi f_d & n=0 \\ |
60 * 2\pi f_d \cos\frac{2\pi n}{N} & n=1,2,\ldots,M |
60 * 2\pi f_d \cos\frac{2\pi n}{N} & n=1,2,\ldots,M |
61 * \end{array} |
61 * \end{array} |
62 * \right .\f] |
62 * \right .\f] |
63 * |
63 * |
64 * The parameter \f$f_d\f$ is the doppler frequency and \f$N=4M+2\f$ where |
64 * The parameter \f$f_d\f$ is the doppler frequency and \f$N=4M+2\f$ where |