| | 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 $ |
