\form#0:$x$
\form#1:$\omega$
\form#2:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \]
\form#3:$[\theta r]$
\form#4:$\psi=\psi(y_{1:t},u_{1:t})$
\form#5:$u_t$
\form#6:$e_t$
\form#7:\[ e_t \sim \mathcal{N}(0,1). \]
\form#8:$ y_t $
\form#9:$\theta,r$
\form#10:$ dt = [y_t psi_t] $
\form#11:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \]
\form#12:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \]
\form#13:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\]
\form#14:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \]
\form#15:$\psi$
\form#16:$w=[w_1,\ldots,w_n]$
\form#17:$\theta_i$
\form#18:$\Theta$
\form#19:$\Theta = [\theta_1,\ldots,\theta_n,w]$
\form#20:$A=Ch' Ch$
\form#21:$Ch$
\form#22:\[M = L'DL\]
\form#23:$L$
\form#24:$D$
\form#25:$V = V + w v v'$
\form#26:$C$
\form#27:$V = C*V*C'$
\form#28:$V = C'*V*C$
\form#29:$V$
\form#30:$x= v'*V*v$
\form#31:$x= v'*inv(V)*v$
\form#32:$U$
\form#33:$A'D0 A$
\form#34:$L'DL$
\form#35:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$
\form#36:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \]
\form#37:$ f(rvc) = \int f(rv,rvc) d\ rv $
\form#38:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \]
\form#39:$f_i(x)$
\form#40:$f(x)$
\form#41:\[ f(\theta_t | d_1,\ldots,d_t) = \frac{f(y_t|\theta_t,\cdot) f(\theta_t|d_1,\ldots,d_{t-1})}{f(y_t|d_1,\ldots,d_{t-1})} \]
\form#42:$y_t$
\form#43:$ c_t $
\form#44:\[ f(\theta_t | c_t, d_1,\ldots,d_t) \propto f(y_t,\theta_t|c_t,\cdot, d_1,\ldots,d_{t-1}) \]
\form#45:$x=$
\form#46:$ x $
\form#47:$ f_x()$
\form#48:$ [x_1 , x_2 , \ldots \ $
\form#49:$ f_x(rv)$
\form#50:$x \sim epdf(rv|cond)$
\form#51:$ t $
\form#52:$ t+1 $
\form#53:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $
\form#54:$t$
\form#55:$[y_{t} y_{t-1} ...]$
\form#56:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$
\form#57:$ f(x_t|x_{t-1}) $
\form#58:$ f(d_t|x_t) $
\form#59:$p$
\form#60:$p\times$
\form#61:$n$
\form#62:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \]
\form#63:$\gamma=\sum_i \beta_i$
\form#64:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \]
\form#65:$\beta$
\form#66:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \]
\form#67:$mu=A*rvc+mu_0$
\form#68:$\mu$
\form#69:$k$
\form#70:$\alpha=k$
\form#71:$\beta=k/\mu$
\form#72:$\mu/\sqrt(k)$
\form#73:$ \mu $
\form#74:$ k $
\form#75:$ \alpha=\mu/k^2+2 $
\form#76:$ \beta=\mu(\alpha-1)$
\form#77:$ \mu/\sqrt(k)$
\form#78:$l$
\form#79:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\]
\form#80:$\mathcal{I}$
\form#81:$\alpha$
\form#82:$ \Psi $
\form#83:$ \nu $
\form#84:$ \nu-p-1 $
\form#85:$w$
\form#86:$x^{(i)}, i=1..n$
\form#87:$f(x) = a$
\form#88:$f(x) = Ax+B$
\form#89:$f(x,u)$
\form#90:$f(x,u) = Ax+Bu$
\form#91:$f(x0,u0)$
\form#92:$A=\frac{d}{dx}f(x,u)|_{x0,u0}$
\form#93:$u$
\form#94:$A=\frac{d}{du}f(x,u)|_{x0,u0}$
\form#95:$ f(D) $
\form#96:\[ f(a,b,c) = f(a|b,c) f(b) f(c) \]
\form#97:$ f(a|b,c) $
\form#98:$ f(b) $
\form#99:$ f(c) $
\form#100:\begin{eqnarray} x_t &= &A x_{t-1} + B u_{t} + v_t,\\ y_t &= &C x_{t} + D u_{t} + w_t, \end{eqnarray}
\form#101:$ x_t $
\form#102:$ A, B, C, D$
\form#103:$v_t, w_t$
\form#104:$Q, R$
\form#105:\begin{eqnarray} x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ y_t &= &h( x_{t} , u_{t}) + w_t, \end{eqnarray}
\form#106:$ g(), h() $
\form#107:\[ y_t = \theta' \psi_t + \rho e_t \]
\form#108:$[\theta,\rho]$
\form#109:$\psi_t$
\form#110:$\mathcal{N}(0,1)$
\form#111:\[ V_t = \sum_{i=0}^{n} \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} \]
\form#112:\[ \nu_t = \sum_{i=0}^{n} 1 \]
\form#113:$ \theta_t , r_t $
\form#114:\[ V_t = \phi V_{t-1} + \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \]
\form#115:\[ \nu_t = \phi \nu_{t-1} + 1 + (1-\phi) \nu_0 \]
\form#116:$ \phi $
\form#117:$ \phi \in [0,1]$
\form#118:\[ \mathrm{win_length} = \frac{1}{1-\phi}\]
\form#119:$ \phi=0.9 $
\form#120:$ V_0 , \nu_0 $
\form#121:$ V_t , \nu_t $
\form#122:$ \phi<1 $
\form#123:$ [d_1, d_2, \ldots d_t] $
\form#124:$\theta$
\form#125:$\mathbf{X}$
\form#126:$n \times n$
\form#127:\[ \mathbf{X} = \mathbf{F}^T \mathbf{F} \]
\form#128:$\mathbf{F}$
\form#129:\[ \mathbf{X} = \mathbf{F}^H \mathbf{F} \]
\form#130:\[ \det(\mathbf{X}) = \det(\mathbf{P}^T \mathbf{L}) \det(\mathbf{U}) = \det(\mathbf{P}^T) \prod(\mathrm{diag}(\mathbf{U})) \]
\form#131:$ \pm 1$
\form#132:$\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$
\form#133:$\mathbf{v}_i, \: i=0, \ldots, n-1$
\form#134:$\mathbf{A}$
\form#135:\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]
\form#136:$ \mathbf{Y} \mathbf{X} = \mathbf{I}$
\form#137:$Ax=b$
\form#138:$A$
\form#139:$AX=B$
\form#140:$m \times n$
\form#141:$m \geq n$
\form#142:$m \leq n$
\form#143:\[ \mathbf{X} = \mathbf{P}^T \mathbf{L} \mathbf{U} , \]
\form#144:$\mathbf{L}$
\form#145:$\mathbf{U}$
\form#146:$\mathbf{P}$
\form#147:\[ \mathbf{A} = \mathbf{Q} \mathbf{R} , \]
\form#148:$\mathbf{Q}$
\form#149:$m \times m$
\form#150:$\mathbf{R}$
\form#151:$\mathbf{A}=\mathbf{Q}\mathbf{R}$
\form#152:$\mathbf{A}^{T}\mathbf{A}=\mathbf{R}^{T}\mathbf{R}$
\form#153:\[ \mathbf{A} \mathbf{P} = \mathbf{Q} \mathbf{R} , \]
\form#154:$\mathbf{A}^{H}\mathbf{A}=\mathbf{R}^{H}\mathbf{R}$
\form#155:$ \mathbf{A} $
\form#156:\[ \mathbf{U} \mathbf{T} \mathbf{U}^{T} = \mathbf{A} \]
\form#157:$ \mathbf{U} $
\form#158:$ \mathbf{T} $
\form#159:$ \mathbf{U}^{T} $
\form#160:$ 2 \times 2 $
\form#161:\[ \mathbf{U} \mathbf{T} \mathbf{U}^{H} = \mathbf{A} \]
\form#162:$ \mathbf{U}^{H} $
\form#163:$s$
\form#164:\[ \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \]
\form#165:$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$
\form#166:\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T \]
\form#167:$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} $
\form#168:\[ \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \]
\form#169:\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H \]
\form#170:$\mathbf{s}$
\form#171:\[ J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{ (-1)^{k} }{k! \Gamma(\nu+k+1) } \left(\frac{x}{2}\right)^{\nu+2k} \]
\form#172:$\nu$
\form#173:$ 0 < x < \infty $
\form#174:\[ Y_{\nu}(x) = \frac{J_{\nu}(x) \cos(\nu\pi) - J_{-\nu}(x)}{\sin(\nu\pi)} \]
\form#175:\[ I_{\nu}(x) = i^{-\nu} J_{\nu}(ix) \]
\form#176:\[ K_{\nu}(x) = \frac{\pi}{2} i^{\nu+1} [J_{\nu}(ix) + i Y_{\nu}(ix)] \]
\form#177:\[ \mathbf{X} = \mathbf{X}^H \]
\form#178:\[ \mathbf{X}^H = \mathbf{X}^{-1} \]
\form#179:$n+|K| \times n+|K|$
\form#180:$n = min(r, c)$
\form#181:$r \times c$
\form#182:$n-1$
\form#183:\[ \int_a^b f(x) dx \]
\form#184:\[ x \sim \Gamma(\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1} \exp(-\beta x) \]
\form#185:$\alpha=1$
\form#186:$\Theta(n\log n)$
\form#187:$\Theta(n^2)$
\form#188:$g(x) = x^{10} + x^9 + x^8 + x^6 + x^5 + x^3 + 1$
\form#189:$ r(t) $
\form#190:\[ r(t) = a(t) * s(t), \]
\form#191:$ s(t) $
\form#192:$ a(t) $
\form#193:$ \|a(t)\| $
\form#194:\[ R(\tau) = E[a^*(t) a(t+\tau)] = J_0(2 \pi f_\mathrm{max} \tau), \]
\form#195:$ f_\mathrm{max} $
\form#196:\[ f_\mathrm{max} = \frac{v}{\lambda} = \frac{v}{c_0} f_c. \]
\form#197:$ c_0 $
\form#198:$ f_c $
\form#199:$ f_\mathrm{max} T_s $
\form#200:$ T_s $
\form#201:$ R(\tau) $
\form#202:\[ h(t) = \sum_{k=0}^{N_\mathrm{taps}-1} a_k \exp (-j \theta_k ) \delta(t-\tau_k), \]
\form#203:$ N_\mathrm{taps} $
\form#204:$ a_k $
\form#205:$ \tau_k $
\form#206:$ \theta_k $
\form#207:$ k^{th} $
\form#208:\[ \mathbf{a} = [a_0, a_1, \ldots, a_{N_\mathrm{taps}-1}] \]
\form#209:\[ \mathbf{\tau} = [\tau_0, \tau_1, \ldots, \tau_{N_\mathrm{taps}-1}], \]
\form#210:$ \tau_0 = 0 $
\form#211:$ \tau_0 < \tau_1 < \ldots < \tau_{N_\mathrm{taps}-1} $
\form#212:$ h(t) $
\form#213:$ \tau_k = d_k T_s $
\form#214:$ d_k $
\form#215:\[ \rho \exp(2 \pi f_\rho t + \theta_\rho), \]
\form#216:$ \rho $
\form#217:$ f_\rho $
\form#218:$ \theta_\rho $
\form#219:$ f_\rho = 0.7 f_\mathrm{max} $
\form#220:\[ \tilde \mu_i(t) = \sum_{n=1}^{N_i} c_{i,n} \cos(2\pi f_{i,n} t + \theta_{i,n}) \]
\form#221:$ c_{i,n} $
\form#222:$ f_{i,n} $
\form#223:$ \theta_{i,n} $
\form#224:$ N_i \rightarrow \infty $
\form#225:\[ \tilde \mu(t) = \tilde \mu_1(t) + j \tilde \mu_2(t) \]
\form#226:$ N_i $
\form#227:$ N_\mathrm{fft} $
\form#228:\[ h(t) = \sum_{k=0}^{N_\mathrm{taps}-1} a_k \exp (-j \theta_k) \delta(t-\tau_k), \]
\form#229:$ N_{taps} $
\form#230:$ \mathbf{a} $
\form#231:$ \mathbf{\tau} $
\form#232:$N_0/2$
\form#233:$N_0$
\form#234:$ f_{norm} = f_{max} T_{s} $
\form#235:$ f_{max} $
\form#236:$ T_{s} $
\form#237:\[ \max_{p_0,...,p_{n-1}} \sum_{i=0}^{n-1} \log\left(1+p_i\alpha_i\right) \]
\form#238:\[ \sum_{i=0}^{n-1} p_i \le P \]
\form#239:$\alpha_0,...,\alpha_{n-1}$
\form#240:$p_0,...,p_{n-1}$
\form#241:$O(n^2)$
\form#242:$2^{K-1}$
\form#243:$ H = [H_{1} H_{2}] $
\form#244:$ H_{2} $
\form#245:$ [H_{1} H_{2}][I; G'] = 0 $
\form#246:\[ L = \log \frac{P(b=0)}{P(b=1)} \]
\form#247:\[ \mbox{QLLR} = \mbox{round} \left(2^{\mbox{Dint1}}\cdot \mbox{LLR}\right) \]
\form#248:\[ 2^{-(Dint1-Dint3)} \]
\form#249:\[ \log(\exp(a)+\exp(b)) \]
\form#250:\[ \mbox{sign}(a) * \mbox{sign}(b) * \mbox{min}(|a|,|b|) + f(|a+b|) - f(|a-b|) \]
\form#251:\[ f(x) = \log(1+\exp(-x)) \]
\form#252:\[r_k = c_k s_k + n_k,\]
\form#253:$c_k$
\form#254:$s_k$
\form#255:$n_k$
\form#256:$M = 2^k$
\form#257:$k = 1, 2, \ldots $
\form#258:$\{-(\sqrt{M}-1), \ldots, -3, -1, 1, 3, \ldots, (\sqrt{M}-1)\}$
\form#259:$\sqrt{2(M-1)/3}$
\form#260:$(1, 0)$
\form#261:$M = 4$
\form#262:$M = 2$
\form#263:$0 \rightarrow 1+0i$
\form#264:$1 \rightarrow -1+0i$
\form#265:$0 \rightarrow 1$
\form#266:$1 \rightarrow -1$
\form#267:$\{-(M-1), \ldots, -3, -1, 1, 3, \ldots, (M-1)\}$
\form#268:$ \sqrt{(M^2-1)/3}$
\form#269:\[\log \left( \frac{P(b_i=0|r)}{P(b_i=1|r)} \right) = \log \left( \frac{\sum_{s_i \in S_0} \exp \left( -\frac{|r_k - s_i|^2}{N_0} \right)} {\sum_{s_i \in S_1} \exp \left( -\frac{|r_k - s_i|^2}{N_0} \right)} \right) \]
\form#270:$d_0 = |r_k - s_0|$
\form#271:$d_1 = |r_k - s_1|$
\form#272:\[\frac{d_1^2 - d_0^2}{N_0}\]
\form#273:$c_k = 1$
\form#274:$L_c$
\form#275:\[\log \left( \frac{P(b_i=0|r)}{P(b_i=1|r)} \right) = \log \left( \frac{\sum_{s_i \in S_0} \exp \left( -\frac{|r_k - c_k s_i|^2}{N_0} \right)} {\sum_{s_i \in S_1} \exp \left( -\frac{|r_k - c_k s_i|^2}{N_0} \right)} \right) \]
\form#276:$d_0 = |r_k - c_k s_0|$
\form#277:$d_1 = |r_k - c_k s_1|$
\form#278:$r_k$
\form#279:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Im\{r_k \exp \left(j \frac{\Pi}{4} \right) \}\]
\form#280:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Re\{r_k \exp \left(j \frac{\Pi}{4} \right) \}\]
\form#281:$r$
\form#282:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Im\{r_k c_k \exp \left(j \frac{\Pi}{4} \right) \}\]
\form#283:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Re\{r_k c_k \exp \left(j \frac{\Pi}{4} \right) \}\]
\form#284:$c$
\form#285:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{4 \Re\{r\}} {N_0}\]
\form#286:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{4 \Re\{r c^{*}\}}{N_0}\]
\form#287:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{4 r}{N_0}\]
\form#288:$c = 1$
\form#289:\[ y = Hx+e \]
\form#290:$n_r\times n_t$
\form#291:$y$
\form#292:$n_r$
\form#293:$n_t$
\form#294:$e$
\form#295:\[ G = \left[ \begin{array}{cc} H_r & -H_i \\ H_i & H_r \end{array} \right] \]
\form#296:\[ \log \left( \frac {\sum_{s:b_k=0} \exp(-x^2) P(s)} {\sum_{s:b_k=1} \exp(-x^2) P(s)} \right) \]
\form#297:\[ \log \left( \frac {\sum_{s:b_k=0} \exp (-x^2) P(s)} {\sum_{s:b_k=1} \exp (-x^2) P(s)} \right) \]
\form#298:\[ LLR(k) = \log \left( \frac {\sum_{s:b_k=0} \exp \left( -\frac{|y - Hs|^2}{2\sigma^2} \right) P(s)} {\sum_{s:b_k=1} \exp \left( -\frac{|y - Hs|^2}{2\sigma^2} \right) P(s)} \right) \]
\form#299:$H = \mbox{diag}(h)$
\form#300:$|y-Hs|$
\form#301:\[ LLR(k) = \log \left( \frac {\sum_{s:b_k=0} \exp \left( -\frac{|y - Hs|^2}{\sigma^2} \right) P(s)} {\sum_{s:b_k=1} \exp \left( -\frac{|y - Hs|^2}{\sigma^2} \right) P(s)} \right) \]
\form#302:\[ \mbox{min} |y - Hs| \]
\form#303:$n_r\times 1$
\form#304:$ \alpha $
\form#305:\[ p(t) = \frac{\sin(\pi t / T)}{\pi t / T} \frac{\cos(\alpha \pi t / T)}{1 - (2 \alpha t / T)^2} \]
\form#306:\[ p(t) = \frac{4 \alpha}{\pi \sqrt{T}} \frac{\cos((1+\alpha)\pi t / T) + T \sin((1-\alpha)\pi t / T) / (4 \alpha t) }{1 - (4 \pi t / T)^2} \]
\form#307:$2^m$
\form#308:$2^m-1$
\form#309:$N = 2^{deg} - 1$
\form#310:$deg = \{ 5, 7, 8, 9 \}$
\form#311:$L \times N$
\form#312:\[ r_k = h_k c_k + w_k \]
\form#313:$h_k$
\form#314:$\{-\sqrt{E_c},+\sqrt{E_c}\}$
\form#315:$w_k$
\form#316:\[ z_k = \hat{h}_k^{*} r_k \]
\form#317:$\hat{h}_k^{*}$
\form#318:\[ L_c = 4\sqrt{E_c} / {N_0} \]
\form#319:\[ s(1), p_{1,1}(1), p_{1,2}(1), \ldots , p_{1,n_1}(1), p_{2,1}(1), p_{2,2}(1), \ldots , p_{2,n_2}(1), s(2), \ldots \]
\form#320:$s(n)$
\form#321:$p_{l,k}(n)$
\form#322:\[ t_1(1), pt_{1,1}(1), pt_{1,2}(1), \ldots , pt_{1,n_1}(1), \ldots pt_{1,n_1}(m) \]
\form#323:$f(\mathbf{x})$
\form#324:$\mathbf{x}$
\form#325:\[ \left\| \mathbf{f}'(\mathbf{x})\right\|_{\infty} \leq \varepsilon_1 \]
\form#326:\[ \left\| d\mathbf{x}\right\|_{2} \leq \varepsilon_2 (\varepsilon_2 + \| \mathbf{x} \|_{2} ) \]
\form#327:$\varepsilon_1 = 10^{-4}$
\form#328:$\varepsilon_2 = 10^{-8}$
\form#329:$\mathbf{h}$
\form#330:\[ \varphi(\alpha) = f(\mathbf{x} + \alpha \mathbf{h}) \]
\form#331:$\alpha_s$
\form#332:$f$
\form#333:\[ \phi(\alpha_s) \leq \varphi(0) + \alpha_s \rho \varphi'(0) \]
\form#334:\[ \varphi'(\alpha_s) \geq \beta \varphi'(0),\: \rho < \beta \]
\form#335:$\rho = 10^{-3}$
\form#336:$\beta = 0.99$
\form#337:\[ \| \varphi(\alpha_s)\| \leq \rho \| \varphi'(0) \| \]
\form#338:\[ b-a \leq \beta b, \]
\form#339:$\left[a,b\right]$
\form#340:$\beta = 10^{-3}$
\form#341:$a_1$
\form#342:$a_2$
\form#343:$\epsilon$
\form#344:\[ y(n) = b(0)*x(n) + b(1)*x(n-1) + ... + b(N)*x(n-N) \]
\form#345:\[ a(0)*y(n) = x(n) - a(1)*y(n-1) - ... - a(N)*y(n-N) \]
\form#346:\[ a(0)*y(n) = b(0)*x(n) + b(1)*x(n-1) + \ldots + b(N_b)*x(n-N_b) - a(1)*y(n-1) - \ldots - a(N_a)*y(n-N_a) \]
\form#347:$max(N_a, n_b) - 1$
\form#348:$\pi$
\form#349:$N>n$
\form#350:$N = 4 n$
\form#351:$R(k) = 0, \forall \|k\| > m$
\form#352:$2(m+n)$
\form#353:$N+1$
\form#354:\[ p_0 x^N + p_1 x^{N-1} + \ldots + p_{N-1} x + p_N \]
\form#355:\[ T(x) = \left\{ \begin{array}{ll} \cos(n\arccos(x)),& |x| \leq 0 \\ \cosh(n\mathrm{arccosh}(x)),& x > 1 \\ (-1)^n \cosh(n\mathrm{arccosh}(-x)),& x < -1 \end{array} \right. \]
\form#356:$X$
\form#357:$N$
\form#358:\[ X(k) = \sum_{j=0}^{N-1} x(j) e^{-2\pi j k \cdot i / N} \]
\form#359:\[ x(j) = \frac{1}{N} \sum_{k=0}^{N-1} X(k) e^{2\pi j k \cdot i / N} \]
\form#360:\[ X(k) = w(k) \sum_{j=0}^{N-1} x(j) \cos \left(\frac{(2j+1)k \pi}{2N} \right) \]
\form#361:\[ x(j) = \sum_{k=0}^{N-1} w(k) X(k) \cos \left(\frac{(2j+1)k \pi}{2N} \right) \]
\form#362:$w(k) = 1/sqrt{N}$
\form#363:$k=0$
\form#364:$w(k) = sqrt{2/N}$
\form#365:$k\geq 1$
\form#366:$i$
\form#367:\[ w_i = 0.54 - 0.46 \cos(2\pi i/(n-1)) \]
\form#368:\[ w_i = 0.5(1 - \cos(2\pi (i+1)/(n+1)) \]
\form#369:\[ w_i = 0.5(1 - \cos(2\pi i/(n-1)) \]
\form#370:\[ w_i = 0.42 - 0.5\cos(2\pi i/(n-1)) + 0.08\cos(4\pi i/(n-1)) \]
\form#371:\[ w_i = w_{n-i-1} = \frac{2(i+1)}{n+1} \]
\form#372:\[ w_i = w_{n-i-1} = \frac{2i+1}{n} \]
\form#373:\[ W[k] = \frac{T_M\left(\beta \cos\left(\frac{\pi k}{M}\right) \right)}{T_M(\beta)},k = 0, 1, 2, \ldots, M - 1 \]
\form#374:$ \mathbf{x} $
\form#375:\[ m_r = \mathrm{E}[x-\mu]^r = \frac{1}{n} \sum_{i=0}^{n-1} (x_i - \mu)^r \]
\form#376:\[ \gamma_1 = \frac{\mathrm{E}[x-\mu]^3}{\sigma^3} \]
\form#377:$\sigma$
\form#378:\[ \gamma_1 = \frac{k_3}{{k_2}^{3/2}} \]
\form#379:\[ k_2 = \frac{n}{n-1} m_2 \]
\form#380:\[ k_3 = \frac{n^2}{(n-1)(n-2)} m_3 \]
\form#381:$m_2$
\form#382:$m_3$
\form#383:\[ \gamma_2 = \frac{\mathrm{E}[x-\mu]^4}{\sigma^4} - 3 \]
\form#384:\[ \gamma_2 = \frac{k_4}{{k_2}^2} \]
\form#385:\[ k_4 = \frac{n^2 [(n+1)m_4 - 3(n-1){m_2}^2]}{(n-1)(n-2)(n-3)} \]
\form#386:$m_4$
\form#387:\[ \gamma_2 = \frac{\mathrm{E}[x-\mu]^4}{\sigma^4} \]
\form#388:$ w_{new} = [ \alpha \cdot w_{A} ~~~ \beta \cdot w_{B} ]^T $
\form#389:$ w_{new} $
\form#390:$ w_{A} $
\form#391:$ w_{B} $
\form#392:$ \alpha = K_A / (K_A + KB_in) $
\form#393:$ \beta = 1-\alpha $
\form#394:$ K_A $
\form#395:$ KB_in $
\form#396:$ -\frac{D}{2}\log(2\pi) -\frac{1}{2}\log(|\Sigma|) $
\form#397:$ D $
\form#398:$ |\Sigma| $
\form#399:$ \Sigma $