| 1 | | \form#0:$x$ |
| 2 | | \form#1:$\omega$ |
| 3 | | \form#2:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \] |
| 4 | | \form#3:$[\theta r]$ |
| 5 | | \form#4:$\psi=\psi(y_{1:t},u_{1:t})$ |
| 6 | | \form#5:$u_t$ |
| 7 | | \form#6:$e_t$ |
| 8 | | \form#7:\[ e_t \sim \mathcal{N}(0,1). \] |
| 9 | | \form#8:$ y_t $ |
| 10 | | \form#9:$\theta,r$ |
| 11 | | \form#10:$ dt = [y_t psi_t] $ |
| 12 | | \form#11:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \] |
| 13 | | \form#12:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \] |
| 14 | | \form#13:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\] |
| 15 | | \form#14:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \] |
| 16 | | \form#15:$\psi$ |
| 17 | | \form#16:$w=[w_1,\ldots,w_n]$ |
| 18 | | \form#17:$\theta_i$ |
| 19 | | \form#18:$\Theta$ |
| 20 | | \form#19:$\Theta = [\theta_1,\ldots,\theta_n,w]$ |
| 21 | | \form#20:$A=Ch' Ch$ |
| 22 | | \form#21:$Ch$ |
| 23 | | \form#22:\[M = L'DL\] |
| 24 | | \form#23:$L$ |
| 25 | | \form#24:$D$ |
| 26 | | \form#25:$V = V + w v v'$ |
| 27 | | \form#26:$ |
| 28 | | \form#27:$V = C*V*C'$ |
| 29 | | \form#28:$V = C'*V*C$ |
| 30 | | \form#29:$V$ |
| 31 | | \form#30:$x= v'*V*v$ |
| 32 | | \form#31:$x= v'*inv(V)*v$ |
| 33 | | \form#32:$U$ |
| 34 | | \form#33:$A'D0 A$ |
| 35 | | \form#34:$L'DL$ |
| 36 | | \form#35:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$ |
| 37 | | \form#36:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \] |
| 38 | | \form#37:$ f(rvc) = \int f(rv,rvc) d\ rv $ |
| 39 | | \form#38:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \] |
| 40 | | \form#39:$f_i(x)$ |
| 41 | | \form#40:$f(x)$ |
| 42 | | \form#41:\[ f(\theta_t | d_1,\ldots,d_t) = \frac{f(y_t|\theta_t,\cdot) f(\theta_t|d_1,\ldots,d_{t-1})}{f(y_t|d_1,\ldots,d_{t-1})} \] |
| 43 | | \form#42:$y_t$ |
| 44 | | \form#43:$ c_t $ |
| 45 | | \form#44:\[ f(\theta_t | c_t, d_1,\ldots,d_t) \propto f(y_t,\theta_t|c_t,\cdot, d_1,\ldots,d_{t-1}) \] |
| 46 | | \form#45:$x=$ |
| 47 | | \form#46:$ x $ |
| 48 | | \form#47:$ f_x()$ |
| 49 | | \form#48:$ [x_1 , x_2 , \ldots \ $ |
| 50 | | \form#49:$ f_x(rv)$ |
| 51 | | \form#50:$x \sim epdf(rv|cond)$ |
| 52 | | \form#51:$ t |
| 53 | | \form#52:$ t+1 $ |
| 54 | | \form#53:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $ |
| 55 | | \form#54:$t$ |
| 56 | | \form#55:$[y_{t} y_{t-1} ...]$ |
| 57 | | \form#56:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$ |
| 58 | | \form#57:$ f(x_t|x_{t-1}) $ |
| 59 | | \form#58:$ f(d_t|x_t) $ |
| 60 | | \form#59:$p |
| 61 | | \form#60:$p\times$ |
| 62 | | \form#61:$n$ |
| 63 | | \form#62:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \] |
| 64 | | \form#63:$\gamma=\sum_i \beta_i$ |
| 65 | | \form#64:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \] |
| 66 | | \form#65:$\beta$ |
| 67 | | \form#66:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \] |
| 68 | | \form#67:$mu=A*rvc+mu_0$ |
| 69 | | \form#68:$\mu$ |
| 70 | | \form#69:$ |
| 71 | | \form#70:$\alpha=k$ |
| 72 | | \form#71:$\beta=k/\mu$ |
| 73 | | \form#72:$\mu/\sqrt(k)$ |
| 74 | | \form#73:$ \mu $ |
| 75 | | \form#74:$ k $ |
| 76 | | \form#75:$ \alpha=\mu/k^2+2 $ |
| 77 | | \form#76:$ \beta=\mu(\alpha-1)$ |
| 78 | | \form#77:$ \mu/\sqrt(k)$ |
| 79 | | \form#78:$l$ |
| 80 | | \form#79:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\] |
| 81 | | \form#80:$\mathcal{I}$ |
| | 1 | \form#0:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \] |
| | 2 | \form#1:$[\theta r]$ |
| | 3 | \form#2:$\psi=\psi(y_{1:t},u_{1:t})$ |
| | 4 | \form#3:$u_t$ |
| | 5 | \form#4:$e_t$ |
| | 6 | \form#5:\[ e_t \sim \mathcal{N}(0,1). \] |
| | 7 | \form#6:$ y_t $ |
| | 8 | \form#7:$\theta,r$ |
| | 9 | \form#8:$ dt = [y_t psi_t] $ |
| | 10 | \form#9:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \] |
| | 11 | \form#10:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \] |
| | 12 | \form#11:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\] |
| | 13 | \form#12:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \] |
| | 14 | \form#13:$\psi$ |
| | 15 | \form#14:$w=[w_1,\ldots,w_n]$ |
| | 16 | \form#15:$\theta_i$ |
| | 17 | \form#16:$\Theta$ |
| | 18 | \form#17:$\Theta = [\theta_1,\ldots,\theta_n,w]$ |
| | 19 | \form#18:$A=Ch' Ch$ |
| | 20 | \form#19:$Ch$ |
| | 21 | \form#20:\[M = L'DL\] |
| | 22 | \form#21:$L$ |
| | 23 | \form#22:$D$ |
| | 24 | \form#23:$V = V + w v v'$ |
| | 25 | \form#24:$C$ |
| | 26 | \form#25:$V = C*V*C'$ |
| | 27 | \form#26:$V = C'*V*C$ |
| | 28 | \form#27:$V$ |
| | 29 | \form#28:$x$ |
| | 30 | \form#29:$x= v'*V*v$ |
| | 31 | \form#30:$x= v'*inv(V)*v$ |
| | 32 | \form#31:$U$ |
| | 33 | \form#32:$A'D0 A$ |
| | 34 | \form#33:$L'DL$ |
| | 35 | \form#34:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$ |
| | 36 | \form#35:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \] |
| | 37 | \form#36:$ f(rvc) = \int f(rv,rvc) d\ rv $ |
| | 38 | \form#37:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \] |
| | 39 | \form#38:$f_i(x)$ |
| | 40 | \form#39:$f(x)$ |
| | 41 | \form#40:\[ f(\theta_t | d_1,\ldots,d_t) = \frac{f(y_t|\theta_t,\cdot) f(\theta_t|d_1,\ldots,d_{t-1})}{f(y_t|d_1,\ldots,d_{t-1})} \] |
| | 42 | \form#41:$y_t$ |
| | 43 | \form#42:$ c_t $ |
| | 44 | \form#43:\[ f(\theta_t | c_t, d_1,\ldots,d_t) \propto f(y_t,\theta_t|c_t,\cdot, d_1,\ldots,d_{t-1}) \] |
| | 45 | \form#44:$x=$ |
| | 46 | \form#45:$ x $ |
| | 47 | \form#46:$ f_x()$ |
| | 48 | \form#47:$ [x_1 , x_2 , \ldots \ $ |
| | 49 | \form#48:$ f_x(rv)$ |
| | 50 | \form#49:$x \sim epdf(rv|cond)$ |
| | 51 | \form#50:$ t $ |
| | 52 | \form#51:$ t+1 $ |
| | 53 | \form#52:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $ |
| | 54 | \form#53:$t$ |
| | 55 | \form#54:$[y_{t} y_{t-1} ...]$ |
| | 56 | \form#55:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$ |
| | 57 | \form#56:$ f(x_t|x_{t-1}) $ |
| | 58 | \form#57:$ f(d_t|x_t) $ |
| | 59 | \form#58:$p$ |
| | 60 | \form#59:$p\times$ |
| | 61 | \form#60:$n$ |
| | 62 | \form#61:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \] |
| | 63 | \form#62:$\gamma=\sum_i \beta_i$ |
| | 64 | \form#63:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \] |
| | 65 | \form#64:$\beta$ |
| | 66 | \form#65:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \] |
| | 67 | \form#66:$mu=A*rvc+mu_0$ |
| | 68 | \form#67:$\mu$ |
| | 69 | \form#68:$k$ |
| | 70 | \form#69:$\alpha=k$ |
| | 71 | \form#70:$\beta=k/\mu$ |
| | 72 | \form#71:$\mu/\sqrt(k)$ |
| | 73 | \form#72:$ \mu $ |
| | 74 | \form#73:$ k $ |
| | 75 | \form#74:$ \alpha=\mu/k^2+2 $ |
| | 76 | \form#75:$ \beta=\mu(\alpha-1)$ |
| | 77 | \form#76:$ \mu/\sqrt(k)$ |
| | 78 | \form#77:$l$ |
| | 79 | \form#78:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\] |
| | 80 | \form#79:$\mathcal{I}$ |
| | 81 | \form#80:$\theta$ |
| 125 | | \form#124:$\theta$ |
| 126 | | \form#125:$\mathbf{X}$ |
| 127 | | \form#126:$n \times n$ |
| 128 | | \form#127:\[ \mathbf{X} = \mathbf{F}^T \mathbf{F} \] |
| 129 | | \form#128:$\mathbf{F}$ |
| 130 | | \form#129:\[ \mathbf{X} = \mathbf{F}^H \mathbf{F} \] |
| 131 | | \form#130:\[ \det(\mathbf{X}) = \det(\mathbf{P}^T \mathbf{L}) \det(\mathbf{U}) = \det(\mathbf{P}^T) \prod(\mathrm{diag}(\mathbf{U})) \] |
| 132 | | \form#131:$ \pm 1$ |
| 133 | | \form#132:$\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ |
| 134 | | \form#133:$\mathbf{v}_i, \: i=0, \ldots, n-1$ |
| 135 | | \form#134:$\mathbf{A}$ |
| 136 | | \form#135:\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \] |
| 137 | | \form#136:$ \mathbf{Y} \mathbf{X} = \mathbf{I}$ |
| 138 | | \form#137:$Ax=b$ |
| 139 | | \form#138:$A$ |
| 140 | | \form#139:$AX=B$ |
| 141 | | \form#140:$m \times n$ |
| 142 | | \form#141:$m \geq n$ |
| 143 | | \form#142:$m \leq n$ |
| 144 | | \form#143:\[ \mathbf{X} = \mathbf{P}^T \mathbf{L} \mathbf{U} , \] |
| 145 | | \form#144:$\mathbf{L}$ |
| 146 | | \form#145:$\mathbf{U}$ |
| 147 | | \form#146:$\mathbf{P}$ |
| 148 | | \form#147:\[ \mathbf{A} = \mathbf{Q} \mathbf{R} , \] |
| 149 | | \form#148:$\mathbf{Q}$ |
| 150 | | \form#149:$m \times m$ |
| 151 | | \form#150:$\mathbf{R}$ |
| 152 | | \form#151:$\mathbf{A}=\mathbf{Q}\mathbf{R}$ |
| 153 | | \form#152:$\mathbf{A}^{T}\mathbf{A}=\mathbf{R}^{T}\mathbf{R}$ |
| 154 | | \form#153:\[ \mathbf{A} \mathbf{P} = \mathbf{Q} \mathbf{R} , \] |
| 155 | | \form#154:$\mathbf{A}^{H}\mathbf{A}=\mathbf{R}^{H}\mathbf{R}$ |
| 156 | | \form#155:$ \mathbf{A} $ |
| 157 | | \form#156:\[ \mathbf{U} \mathbf{T} \mathbf{U}^{T} = \mathbf{A} \] |
| 158 | | \form#157:$ \mathbf{U} $ |
| 159 | | \form#158:$ \mathbf{T} $ |
| 160 | | \form#159:$ \mathbf{U}^{T} $ |
| 161 | | \form#160:$ 2 \times 2 $ |
| 162 | | \form#161:\[ \mathbf{U} \mathbf{T} \mathbf{U}^{H} = \mathbf{A} \] |
| 163 | | \form#162:$ \mathbf{U}^{H} $ |
| 164 | | \form#163:$s$ |
| 165 | | \form#164:\[ \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \] |
| 166 | | \form#165:$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ |
| 167 | | \form#166:\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T \] |
| 168 | | \form#167:$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} $ |
| 169 | | \form#168:\[ \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \] |
| 170 | | \form#169:\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H \] |
| 171 | | \form#170:$\mathbf{s}$ |
| 172 | | \form#171:\[ J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{ (-1)^{k} }{k! \Gamma(\nu+k+1) } \left(\frac{x}{2}\right)^{\nu+2k} \] |
| 173 | | \form#172:$\nu$ |
| 174 | | \form#173:$ 0 < x < \infty $ |
| 175 | | \form#174:\[ Y_{\nu}(x) = \frac{J_{\nu}(x) \cos(\nu\pi) - J_{-\nu}(x)}{\sin(\nu\pi)} \] |
| 176 | | \form#175:\[ I_{\nu}(x) = i^{-\nu} J_{\nu}(ix) \] |
| 177 | | \form#176:\[ K_{\nu}(x) = \frac{\pi}{2} i^{\nu+1} [J_{\nu}(ix) + i Y_{\nu}(ix)] \] |
| 178 | | \form#177:\[ \mathbf{X} = \mathbf{X}^H \] |
| 179 | | \form#178:\[ \mathbf{X}^H = \mathbf{X}^{-1} \] |
| 180 | | \form#179:$n+|K| \times n+|K|$ |
| 181 | | \form#180:$n = min(r, c)$ |
| 182 | | \form#181:$r \times c$ |
| 183 | | \form#182:$n-1$ |
| 184 | | \form#183:\[ \int_a^b f(x) dx \] |
| 185 | | \form#184:\[ x \sim \Gamma(\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1} \exp(-\beta x) \] |
| 186 | | \form#185:$\alpha=1$ |
| 187 | | \form#186:$\Theta(n\log n)$ |
| 188 | | \form#187:$\Theta(n^2)$ |
| 189 | | \form#188:$g(x) = x^{10} + x^9 + x^8 + x^6 + x^5 + x^3 + 1$ |
| 190 | | \form#189:$ r(t) $ |
| 191 | | \form#190:\[ r(t) = a(t) * s(t), \] |
| 192 | | \form#191:$ s(t) $ |
| 193 | | \form#192:$ a(t) $ |
| 194 | | \form#193:$ \|a(t)\| $ |
| 195 | | \form#194:\[ R(\tau) = E[a^*(t) a(t+\tau)] = J_0(2 \pi f_\mathrm{max} \tau), \] |
| 196 | | \form#195:$ f_\mathrm{max} $ |
| 197 | | \form#196:\[ f_\mathrm{max} = \frac{v}{\lambda} = \frac{v}{c_0} f_c. \] |
| 198 | | \form#197:$ c_0 $ |
| 199 | | \form#198:$ f_c $ |
| 200 | | \form#199:$ f_\mathrm{max} T_s $ |
| 201 | | \form#200:$ T_s $ |
| 202 | | \form#201:$ R(\tau) $ |
| 203 | | \form#202:\[ h(t) = \sum_{k=0}^{N_\mathrm{taps}-1} a_k \exp (-j \theta_k ) \delta(t-\tau_k), \] |
| 204 | | \form#203:$ N_\mathrm{taps} $ |
| 205 | | \form#204:$ a_k $ |
| 206 | | \form#205:$ \tau_k $ |
| 207 | | \form#206:$ \theta_k $ |
| 208 | | \form#207:$ k^{th} $ |
| 209 | | \form#208:\[ \mathbf{a} = [a_0, a_1, \ldots, a_{N_\mathrm{taps}-1}] \] |
| 210 | | \form#209:\[ \mathbf{\tau} = [\tau_0, \tau_1, \ldots, \tau_{N_\mathrm{taps}-1}], \] |
| 211 | | \form#210:$ \tau_0 = 0 $ |
| 212 | | \form#211:$ \tau_0 < \tau_1 < \ldots < \tau_{N_\mathrm{taps}-1} $ |
| 213 | | \form#212:$ h(t) $ |
| 214 | | \form#213:$ \tau_k = d_k T_s $ |
| 215 | | \form#214:$ d_k $ |
| 216 | | \form#215:\[ \rho \exp(2 \pi f_\rho t + \theta_\rho), \] |
| 217 | | \form#216:$ \rho $ |
| 218 | | \form#217:$ f_\rho $ |
| 219 | | \form#218:$ \theta_\rho $ |
| 220 | | \form#219:$ f_\rho = 0.7 f_\mathrm{max} $ |
| 221 | | \form#220:\[ \tilde \mu_i(t) = \sum_{n=1}^{N_i} c_{i,n} \cos(2\pi f_{i,n} t + \theta_{i,n}) \] |
| 222 | | \form#221:$ c_{i,n} $ |
| 223 | | \form#222:$ f_{i,n} $ |
| 224 | | \form#223:$ \theta_{i,n} $ |
| 225 | | \form#224:$ N_i \rightarrow \infty $ |
| 226 | | \form#225:\[ \tilde \mu(t) = \tilde \mu_1(t) + j \tilde \mu_2(t) \] |
| 227 | | \form#226:$ N_i $ |
| 228 | | \form#227:$ N_\mathrm{fft} $ |
| 229 | | \form#228:\[ h(t) = \sum_{k=0}^{N_\mathrm{taps}-1} a_k \exp (-j \theta_k) \delta(t-\tau_k), \] |
| 230 | | \form#229:$ N_{taps} $ |
| 231 | | \form#230:$ \mathbf{a} $ |
| 232 | | \form#231:$ \mathbf{\tau} $ |
| 233 | | \form#232:$N_0/2$ |
| 234 | | \form#233:$N_0$ |
| 235 | | \form#234:$ f_{norm} = f_{max} T_{s} $ |
| 236 | | \form#235:$ f_{max} $ |
| 237 | | \form#236:$ T_{s} $ |
| 238 | | \form#237:\[ \max_{p_0,...,p_{n-1}} \sum_{i=0}^{n-1} \log\left(1+p_i\alpha_i\right) \] |
| 239 | | \form#238:\[ \sum_{i=0}^{n-1} p_i \le P \] |
| 240 | | \form#239:$\alpha_0,...,\alpha_{n-1}$ |
| 241 | | \form#240:$p_0,...,p_{n-1}$ |
| 242 | | \form#241:$O(n^2)$ |
| 243 | | \form#242:$2^{K-1}$ |
| 244 | | \form#243:$ H = [H_{1} H_{2}] $ |
| 245 | | \form#244:$ H_{2} $ |
| 246 | | \form#245:$ [H_{1} H_{2}][I; G'] = 0 $ |
| 247 | | \form#246:\[ L = \log \frac{P(b=0)}{P(b=1)} \] |
| 248 | | \form#247:\[ \mbox{QLLR} = \mbox{round} \left(2^{\mbox{Dint1}}\cdot \mbox{LLR}\right) \] |
| 249 | | \form#248:\[ 2^{-(Dint1-Dint3)} \] |
| 250 | | \form#249:\[ \log(\exp(a)+\exp(b)) \] |
| 251 | | \form#250:\[ \mbox{sign}(a) * \mbox{sign}(b) * \mbox{min}(|a|,|b|) + f(|a+b|) - f(|a-b|) \] |
| 252 | | \form#251:\[ f(x) = \log(1+\exp(-x)) \] |
| 253 | | \form#252:\[r_k = c_k s_k + n_k,\] |
| 254 | | \form#253:$c_k$ |
| 255 | | \form#254:$s_k$ |
| 256 | | \form#255:$n_k$ |
| 257 | | \form#256:$M = 2^k$ |
| 258 | | \form#257:$k = 1, 2, \ldots $ |
| 259 | | \form#258:$\{-(\sqrt{M}-1), \ldots, -3, -1, 1, 3, \ldots, (\sqrt{M}-1)\}$ |
| 260 | | \form#259:$\sqrt{2(M-1)/3}$ |
| 261 | | \form#260:$(1, 0)$ |
| 262 | | \form#261:$M = 4$ |
| 263 | | \form#262:$M = 2$ |
| 264 | | \form#263:$0 \rightarrow 1+0i$ |
| 265 | | \form#264:$1 \rightarrow -1+0i$ |
| 266 | | \form#265:$0 \rightarrow 1$ |
| 267 | | \form#266:$1 \rightarrow -1$ |
| 268 | | \form#267:$\{-(M-1), \ldots, -3, -1, 1, 3, \ldots, (M-1)\}$ |
| 269 | | \form#268:$ \sqrt{(M^2-1)/3}$ |
| 270 | | \form#269:\[\log \left( \frac{P(b_i=0|r)}{P(b_i=1|r)} \right) = \log \left( \frac{\sum_{s_i \in S_0} \exp \left( -\frac{|r_k - s_i|^2}{N_0} \right)} {\sum_{s_i \in S_1} \exp \left( -\frac{|r_k - s_i|^2}{N_0} \right)} \right) \] |
| 271 | | \form#270:$d_0 = |r_k - s_0|$ |
| 272 | | \form#271:$d_1 = |r_k - s_1|$ |
| 273 | | \form#272:\[\frac{d_1^2 - d_0^2}{N_0}\] |
| 274 | | \form#273:$c_k = 1$ |
| 275 | | \form#274:$L_c$ |
| 276 | | \form#275:\[\log \left( \frac{P(b_i=0|r)}{P(b_i=1|r)} \right) = \log \left( \frac{\sum_{s_i \in S_0} \exp \left( -\frac{|r_k - c_k s_i|^2}{N_0} \right)} {\sum_{s_i \in S_1} \exp \left( -\frac{|r_k - c_k s_i|^2}{N_0} \right)} \right) \] |
| 277 | | \form#276:$d_0 = |r_k - c_k s_0|$ |
| 278 | | \form#277:$d_1 = |r_k - c_k s_1|$ |
| 279 | | \form#278:$r_k$ |
| 280 | | \form#279:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Im\{r_k \exp \left(j \frac{\Pi}{4} \right) \}\] |
| 281 | | \form#280:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Re\{r_k \exp \left(j \frac{\Pi}{4} \right) \}\] |
| 282 | | \form#281:$r$ |
| 283 | | \form#282:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Im\{r_k c_k \exp \left(j \frac{\Pi}{4} \right) \}\] |
| 284 | | \form#283:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Re\{r_k c_k \exp \left(j \frac{\Pi}{4} \right) \}\] |
| 285 | | \form#284:$c$ |
| 286 | | \form#285:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{4 \Re\{r\}} {N_0}\] |
| 287 | | \form#286:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{4 \Re\{r c^{*}\}}{N_0}\] |
| 288 | | \form#287:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{4 r}{N_0}\] |
| 289 | | \form#288:$c = 1$ |
| 290 | | \form#289:\[ y = Hx+e \] |
| 291 | | \form#290:$n_r\times n_t$ |
| 292 | | \form#291:$y$ |
| 293 | | \form#292:$n_r$ |
| 294 | | \form#293:$n_t$ |
| 295 | | \form#294:$e$ |
| 296 | | \form#295:\[ G = \left[ \begin{array}{cc} H_r & -H_i \\ H_i & H_r \end{array} \right] \] |
| 297 | | \form#296:\[ \log \left( \frac {\sum_{s:b_k=0} \exp(-x^2) P(s)} {\sum_{s:b_k=1} \exp(-x^2) P(s)} \right) \] |
| 298 | | \form#297:\[ \log \left( \frac {\sum_{s:b_k=0} \exp (-x^2) P(s)} {\sum_{s:b_k=1} \exp (-x^2) P(s)} \right) \] |
| 299 | | \form#298:\[ LLR(k) = \log \left( \frac {\sum_{s:b_k=0} \exp \left( -\frac{|y - Hs|^2}{2\sigma^2} \right) P(s)} {\sum_{s:b_k=1} \exp \left( -\frac{|y - Hs|^2}{2\sigma^2} \right) P(s)} \right) \] |
| 300 | | \form#299:$H = \mbox{diag}(h)$ |
| 301 | | \form#300:$|y-Hs|$ |
| 302 | | \form#301:\[ LLR(k) = \log \left( \frac {\sum_{s:b_k=0} \exp \left( -\frac{|y - Hs|^2}{\sigma^2} \right) P(s)} {\sum_{s:b_k=1} \exp \left( -\frac{|y - Hs|^2}{\sigma^2} \right) P(s)} \right) \] |
| 303 | | \form#302:\[ \mbox{min} |y - Hs| \] |
| 304 | | \form#303:$n_r\times 1$ |
| 305 | | \form#304:$ \alpha $ |
| 306 | | \form#305:\[ p(t) = \frac{\sin(\pi t / T)}{\pi t / T} \frac{\cos(\alpha \pi t / T)}{1 - (2 \alpha t / T)^2} \] |
| 307 | | \form#306:\[ p(t) = \frac{4 \alpha}{\pi \sqrt{T}} \frac{\cos((1+\alpha)\pi t / T) + T \sin((1-\alpha)\pi t / T) / (4 \alpha t) }{1 - (4 \pi t / T)^2} \] |
| 308 | | \form#307:$2^m$ |
| 309 | | \form#308:$2^m-1$ |
| 310 | | \form#309:$N = 2^{deg} - 1$ |
| 311 | | \form#310:$deg = \{ 5, 7, 8, 9 \}$ |
| 312 | | \form#311:$L \times N$ |
| 313 | | \form#312:\[ r_k = h_k c_k + w_k \] |
| 314 | | \form#313:$h_k$ |
| 315 | | \form#314:$\{-\sqrt{E_c},+\sqrt{E_c}\}$ |
| 316 | | \form#315:$w_k$ |
| 317 | | \form#316:\[ z_k = \hat{h}_k^{*} r_k \] |
| 318 | | \form#317:$\hat{h}_k^{*}$ |
| 319 | | \form#318:\[ L_c = 4\sqrt{E_c} / {N_0} \] |
| 320 | | \form#319:\[ s(1), p_{1,1}(1), p_{1,2}(1), \ldots , p_{1,n_1}(1), p_{2,1}(1), p_{2,2}(1), \ldots , p_{2,n_2}(1), s(2), \ldots \] |
| 321 | | \form#320:$s(n)$ |
| 322 | | \form#321:$p_{l,k}(n)$ |
| 323 | | \form#322:\[ t_1(1), pt_{1,1}(1), pt_{1,2}(1), \ldots , pt_{1,n_1}(1), \ldots pt_{1,n_1}(m) \] |
| 324 | | \form#323:$f(\mathbf{x})$ |
| 325 | | \form#324:$\mathbf{x}$ |
| 326 | | \form#325:\[ \left\| \mathbf{f}'(\mathbf{x})\right\|_{\infty} \leq \varepsilon_1 \] |
| 327 | | \form#326:\[ \left\| d\mathbf{x}\right\|_{2} \leq \varepsilon_2 (\varepsilon_2 + \| \mathbf{x} \|_{2} ) \] |
| 328 | | \form#327:$\varepsilon_1 = 10^{-4}$ |
| 329 | | \form#328:$\varepsilon_2 = 10^{-8}$ |
| 330 | | \form#329:$\mathbf{h}$ |
| 331 | | \form#330:\[ \varphi(\alpha) = f(\mathbf{x} + \alpha \mathbf{h}) \] |
| 332 | | \form#331:$\alpha_s$ |
| 333 | | \form#332:$f$ |
| 334 | | \form#333:\[ \phi(\alpha_s) \leq \varphi(0) + \alpha_s \rho \varphi'(0) \] |
| 335 | | \form#334:\[ \varphi'(\alpha_s) \geq \beta \varphi'(0),\: \rho < \beta \] |
| 336 | | \form#335:$\rho = 10^{-3}$ |
| 337 | | \form#336:$\beta = 0.99$ |
| 338 | | \form#337:\[ \| \varphi(\alpha_s)\| \leq \rho \| \varphi'(0) \| \] |
| 339 | | \form#338:\[ b-a \leq \beta b, \] |
| 340 | | \form#339:$\left[a,b\right]$ |
| 341 | | \form#340:$\beta = 10^{-3}$ |
| 342 | | \form#341:$a_1$ |
| 343 | | \form#342:$a_2$ |
| 344 | | \form#343:$\epsilon$ |
| 345 | | \form#344:\[ y(n) = b(0)*x(n) + b(1)*x(n-1) + ... + b(N)*x(n-N) \] |
| 346 | | \form#345:\[ a(0)*y(n) = x(n) - a(1)*y(n-1) - ... - a(N)*y(n-N) \] |
| 347 | | \form#346:\[ a(0)*y(n) = b(0)*x(n) + b(1)*x(n-1) + \ldots + b(N_b)*x(n-N_b) - a(1)*y(n-1) - \ldots - a(N_a)*y(n-N_a) \] |
| 348 | | \form#347:$max(N_a, n_b) - 1$ |
| 349 | | \form#348:$\pi$ |
| 350 | | \form#349:$N>n$ |
| 351 | | \form#350:$N = 4 n$ |
| 352 | | \form#351:$R(k) = 0, \forall \|k\| > m$ |
| 353 | | \form#352:$2(m+n)$ |
| 354 | | \form#353:$N+1$ |
| 355 | | \form#354:\[ p_0 x^N + p_1 x^{N-1} + \ldots + p_{N-1} x + p_N \] |
| 356 | | \form#355:\[ T(x) = \left\{ \begin{array}{ll} \cos(n\arccos(x)),& |x| \leq 0 \\ \cosh(n\mathrm{arccosh}(x)),& x > 1 \\ (-1)^n \cosh(n\mathrm{arccosh}(-x)),& x < -1 \end{array} \right. \] |
| 357 | | \form#356:$X$ |
| 358 | | \form#357:$N$ |
| 359 | | \form#358:\[ X(k) = \sum_{j=0}^{N-1} x(j) e^{-2\pi j k \cdot i / N} \] |
| 360 | | \form#359:\[ x(j) = \frac{1}{N} \sum_{k=0}^{N-1} X(k) e^{2\pi j k \cdot i / N} \] |
| 361 | | \form#360:\[ X(k) = w(k) \sum_{j=0}^{N-1} x(j) \cos \left(\frac{(2j+1)k \pi}{2N} \right) \] |
| 362 | | \form#361:\[ x(j) = \sum_{k=0}^{N-1} w(k) X(k) \cos \left(\frac{(2j+1)k \pi}{2N} \right) \] |
| 363 | | \form#362:$w(k) = 1/sqrt{N}$ |
| 364 | | \form#363:$k=0$ |
| 365 | | \form#364:$w(k) = sqrt{2/N}$ |
| 366 | | \form#365:$k\geq 1$ |
| 367 | | \form#366:$i$ |
| 368 | | \form#367:\[ w_i = 0.54 - 0.46 \cos(2\pi i/(n-1)) \] |
| 369 | | \form#368:\[ w_i = 0.5(1 - \cos(2\pi (i+1)/(n+1)) \] |
| 370 | | \form#369:\[ w_i = 0.5(1 - \cos(2\pi i/(n-1)) \] |
| 371 | | \form#370:\[ w_i = 0.42 - 0.5\cos(2\pi i/(n-1)) + 0.08\cos(4\pi i/(n-1)) \] |
| 372 | | \form#371:\[ w_i = w_{n-i-1} = \frac{2(i+1)}{n+1} \] |
| 373 | | \form#372:\[ w_i = w_{n-i-1} = \frac{2i+1}{n} \] |
| 374 | | \form#373:\[ W[k] = \frac{T_M\left(\beta \cos\left(\frac{\pi k}{M}\right) \right)}{T_M(\beta)},k = 0, 1, 2, \ldots, M - 1 \] |
| 375 | | \form#374:$ \mathbf{x} $ |
| 376 | | \form#375:\[ m_r = \mathrm{E}[x-\mu]^r = \frac{1}{n} \sum_{i=0}^{n-1} (x_i - \mu)^r \] |
| 377 | | \form#376:\[ \gamma_1 = \frac{\mathrm{E}[x-\mu]^3}{\sigma^3} \] |
| 378 | | \form#377:$\sigma$ |
| 379 | | \form#378:\[ \gamma_1 = \frac{k_3}{{k_2}^{3/2}} \] |
| 380 | | \form#379:\[ k_2 = \frac{n}{n-1} m_2 \] |
| 381 | | \form#380:\[ k_3 = \frac{n^2}{(n-1)(n-2)} m_3 \] |
| 382 | | \form#381:$m_2$ |
| 383 | | \form#382:$m_3$ |
| 384 | | \form#383:\[ \gamma_2 = \frac{\mathrm{E}[x-\mu]^4}{\sigma^4} - 3 \] |
| 385 | | \form#384:\[ \gamma_2 = \frac{k_4}{{k_2}^2} \] |
| 386 | | \form#385:\[ k_4 = \frac{n^2 [(n+1)m_4 - 3(n-1){m_2}^2]}{(n-1)(n-2)(n-3)} \] |
| 387 | | \form#386:$m_4$ |
| 388 | | \form#387:\[ \gamma_2 = \frac{\mathrm{E}[x-\mu]^4}{\sigma^4} \] |
| 389 | | \form#388:$ w_{new} = [ \alpha \cdot w_{A} ~~~ \beta \cdot w_{B} ]^T $ |
| 390 | | \form#389:$ w_{new} $ |
| 391 | | \form#390:$ w_{A} $ |
| 392 | | \form#391:$ w_{B} $ |
| 393 | | \form#392:$ \alpha = K_A / (K_A + KB_in) $ |
| 394 | | \form#393:$ \beta = 1-\alpha $ |
| 395 | | \form#394:$ K_A $ |
| 396 | | \form#395:$ KB_in $ |
| 397 | | \form#396:$ -\frac{D}{2}\log(2\pi) -\frac{1}{2}\log(|\Sigma|) $ |
| 398 | | \form#397:$ D $ |
| 399 | | \form#398:$ |\Sigma| $ |
| 400 | | \form#399:$ \Sigma $ |
