| 1 | \form#0:$x$ |
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| 2 | \form#1:$\omega$ |
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| 3 | \form#2:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \] |
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| 4 | \form#3:$[\theta r]$ |
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| 5 | \form#4:$\psi=\psi(y_{1:t},u_{1:t})$ |
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| 6 | \form#5:$u_t$ |
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| 7 | \form#6:$e_t$ |
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| 8 | \form#7:\[ e_t \sim \mathcal{N}(0,1). \] |
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| 9 | \form#8:$ y_t $ |
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| 10 | \form#9:$\theta,r$ |
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| 11 | \form#10:$ dt = [y_t psi_t] $ |
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| 12 | \form#11:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \] |
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| 13 | \form#12:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \] |
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| 14 | \form#13:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\] |
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| 15 | \form#14:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \] |
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| 16 | \form#15:$\psi$ |
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| 17 | \form#16:$w=[w_1,\ldots,w_n]$ |
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| 18 | \form#17:$\theta_i$ |
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| 19 | \form#18:$\Theta$ |
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| 20 | \form#19:$\Theta = [\theta_1,\ldots,\theta_n,w]$ |
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| 21 | \form#20:$A=Ch' Ch$ |
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| 22 | \form#21:$Ch$ |
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| 23 | \form#22:\[M = L'DL\] |
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| 24 | \form#23:$L$ |
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| 25 | \form#24:$D$ |
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| 26 | \form#25:$V = V + w v v'$ |
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| 27 | \form#26:$C$ |
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| 28 | \form#27:$V = C*V*C'$ |
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| 29 | \form#28:$V = C'*V*C$ |
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| 30 | \form#29:$V$ |
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| 31 | \form#30:$x= v'*V*v$ |
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| 32 | \form#31:$x= v'*inv(V)*v$ |
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| 33 | \form#32:$U$ |
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| 34 | \form#33:$A'D0 A$ |
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| 35 | \form#34:$L'DL$ |
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| 36 | \form#35:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$ |
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| 37 | \form#36:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \] |
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| 38 | \form#37:$ f(rvc) = \int f(rv,rvc) d\ rv $ |
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| 39 | \form#38:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \] |
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| 40 | \form#39:$f_i(x)$ |
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| 41 | \form#40:$f(x)$ |
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| 42 | \form#41:\[ f(\theta_t | d_1,\ldots,d_t) = \frac{f(y_t|\theta_t,\cdot) f(\theta_t|d_1,\ldots,d_{t-1})}{f(y_t|d_1,\ldots,d_{t-1})} \] |
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| 43 | \form#42:$y_t$ |
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| 44 | \form#43:$ c_t $ |
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| 45 | \form#44:\[ f(\theta_t | c_t, d_1,\ldots,d_t) \propto f(y_t,\theta_t|c_t,\cdot, d_1,\ldots,d_{t-1}) \] |
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| 46 | \form#45:$x=$ |
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| 47 | \form#46:$ x $ |
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| 48 | \form#47:$ f_x()$ |
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| 49 | \form#48:$ [x_1 , x_2 , \ldots \ $ |
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| 50 | \form#49:$ f_x(rv)$ |
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| 51 | \form#50:$x \sim epdf(rv|cond)$ |
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| 52 | \form#51:$ t $ |
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| 53 | \form#52:$ t+1 $ |
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| 54 | \form#53:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $ |
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| 55 | \form#54:$t$ |
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| 56 | \form#55:$[y_{t} y_{t-1} ...]$ |
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| 57 | \form#56:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$ |
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| 58 | \form#57:$ f(x_t|x_{t-1}) $ |
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| 59 | \form#58:$ f(d_t|x_t) $ |
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| 60 | \form#59:$p$ |
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| 61 | \form#60:$p\times$ |
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| 62 | \form#61:$n$ |
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| 63 | \form#62:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \] |
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| 64 | \form#63:$\gamma=\sum_i \beta_i$ |
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| 65 | \form#64:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \] |
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| 66 | \form#65:$\beta$ |
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| 67 | \form#66:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \] |
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| 68 | \form#67:$mu=A*rvc+mu_0$ |
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| 69 | \form#68:$\mu$ |
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| 70 | \form#69:$k$ |
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| 71 | \form#70:$\alpha=k$ |
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| 72 | \form#71:$\beta=k/\mu$ |
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| 73 | \form#72:$\mu/\sqrt(k)$ |
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| 74 | \form#73:$ \mu $ |
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| 75 | \form#74:$ k $ |
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| 76 | \form#75:$ \alpha=\mu/k^2+2 $ |
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| 77 | \form#76:$ \beta=\mu(\alpha-1)$ |
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| 78 | \form#77:$ \mu/\sqrt(k)$ |
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| 79 | \form#78:$l$ |
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| 80 | \form#79:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\] |
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| 81 | \form#80:$\mathcal{I}$ |
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| 82 | \form#81:$\alpha$ |
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| 83 | \form#82:$ \Psi $ |
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| 84 | \form#83:$ \nu $ |
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| 85 | \form#84:$ \nu-p-1 $ |
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| 86 | \form#85:$w$ |
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| 87 | \form#86:$x^{(i)}, i=1..n$ |
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| 88 | \form#87:$f(x) = a$ |
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| 89 | \form#88:$f(x) = Ax+B$ |
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| 90 | \form#89:$f(x,u)$ |
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| 91 | \form#90:$f(x,u) = Ax+Bu$ |
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| 92 | \form#91:$f(x0,u0)$ |
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| 93 | \form#92:$A=\frac{d}{dx}f(x,u)|_{x0,u0}$ |
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| 94 | \form#93:$u$ |
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| 95 | \form#94:$A=\frac{d}{du}f(x,u)|_{x0,u0}$ |
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| 96 | \form#95:$ f(D) $ |
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| 97 | \form#96:\[ f(a,b,c) = f(a|b,c) f(b) f(c) \] |
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| 98 | \form#97:$ f(a|b,c) $ |
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| 99 | \form#98:$ f(b) $ |
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| 100 | \form#99:$ f(c) $ |
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| 101 | \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} |
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| 102 | \form#101:$ x_t $ |
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| 103 | \form#102:$ A, B, C, D$ |
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| 104 | \form#103:$v_t, w_t$ |
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| 105 | \form#104:$Q, R$ |
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| 106 | \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} |
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| 107 | \form#106:$ g(), h() $ |
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| 108 | \form#107:\[ y_t = \theta' \psi_t + \rho e_t \] |
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| 109 | \form#108:$[\theta,\rho]$ |
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| 110 | \form#109:$\psi_t$ |
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| 111 | \form#110:$\mathcal{N}(0,1)$ |
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| 112 | \form#111:\[ V_t = \sum_{i=0}^{n} \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} \] |
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| 113 | \form#112:\[ \nu_t = \sum_{i=0}^{n} 1 \] |
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| 114 | \form#113:$ \theta_t , r_t $ |
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| 115 | \form#114:\[ V_t = \phi V_{t-1} + \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \] |
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| 116 | \form#115:\[ \nu_t = \phi \nu_{t-1} + 1 + (1-\phi) \nu_0 \] |
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| 117 | \form#116:$ \phi $ |
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| 118 | \form#117:$ \phi \in [0,1]$ |
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| 119 | \form#118:\[ \mathrm{win_length} = \frac{1}{1-\phi}\] |
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| 120 | \form#119:$ \phi=0.9 $ |
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| 121 | \form#120:$ V_0 , \nu_0 $ |
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| 122 | \form#121:$ V_t , \nu_t $ |
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| 123 | \form#122:$ \phi<1 $ |
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| 124 | \form#123:$ [d_1, d_2, \ldots d_t] $ |
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| 125 | \form#124:$\theta$ |
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| 126 | \form#125:$\mathbf{X}$ |
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| 127 | \form#126:$n \times n$ |
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| 128 | \form#127:\[ \mathbf{X} = \mathbf{F}^T \mathbf{F} \] |
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| 129 | \form#128:$\mathbf{F}$ |
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| 130 | \form#129:\[ \mathbf{X} = \mathbf{F}^H \mathbf{F} \] |
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| 131 | \form#130:\[ \det(\mathbf{X}) = \det(\mathbf{P}^T \mathbf{L}) \det(\mathbf{U}) = \det(\mathbf{P}^T) \prod(\mathrm{diag}(\mathbf{U})) \] |
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| 132 | \form#131:$ \pm 1$ |
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| 133 | \form#132:$\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ |
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| 134 | \form#133:$\mathbf{v}_i, \: i=0, \ldots, n-1$ |
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| 135 | \form#134:$\mathbf{A}$ |
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| 136 | \form#135:\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \] |
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| 137 | \form#136:$ \mathbf{Y} \mathbf{X} = \mathbf{I}$ |
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| 138 | \form#137:$Ax=b$ |
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| 139 | \form#138:$A$ |
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| 140 | \form#139:$AX=B$ |
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| 141 | \form#140:$m \times n$ |
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| 142 | \form#141:$m \geq n$ |
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| 143 | \form#142:$m \leq n$ |
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| 144 | \form#143:\[ \mathbf{X} = \mathbf{P}^T \mathbf{L} \mathbf{U} , \] |
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| 145 | \form#144:$\mathbf{L}$ |
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| 146 | \form#145:$\mathbf{U}$ |
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| 147 | \form#146:$\mathbf{P}$ |
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| 148 | \form#147:\[ \mathbf{A} = \mathbf{Q} \mathbf{R} , \] |
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| 149 | \form#148:$\mathbf{Q}$ |
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| 150 | \form#149:$m \times m$ |
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| 151 | \form#150:$\mathbf{R}$ |
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| 152 | \form#151:$\mathbf{A}=\mathbf{Q}\mathbf{R}$ |
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| 153 | \form#152:$\mathbf{A}^{T}\mathbf{A}=\mathbf{R}^{T}\mathbf{R}$ |
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| 154 | \form#153:\[ \mathbf{A} \mathbf{P} = \mathbf{Q} \mathbf{R} , \] |
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| 155 | \form#154:$\mathbf{A}^{H}\mathbf{A}=\mathbf{R}^{H}\mathbf{R}$ |
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| 156 | \form#155:$ \mathbf{A} $ |
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| 157 | \form#156:\[ \mathbf{U} \mathbf{T} \mathbf{U}^{T} = \mathbf{A} \] |
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| 158 | \form#157:$ \mathbf{U} $ |
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| 159 | \form#158:$ \mathbf{T} $ |
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| 160 | \form#159:$ \mathbf{U}^{T} $ |
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| 161 | \form#160:$ 2 \times 2 $ |
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| 162 | \form#161:\[ \mathbf{U} \mathbf{T} \mathbf{U}^{H} = \mathbf{A} \] |
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| 163 | \form#162:$ \mathbf{U}^{H} $ |
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| 164 | \form#163:$s$ |
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| 165 | \form#164:\[ \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \] |
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| 166 | \form#165:$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ |
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| 167 | \form#166:\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T \] |
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| 168 | \form#167:$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} $ |
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| 169 | \form#168:\[ \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \] |
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| 170 | \form#169:\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H \] |
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| 171 | \form#170:$\mathbf{s}$ |
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| 172 | \form#171:\[ J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{ (-1)^{k} }{k! \Gamma(\nu+k+1) } \left(\frac{x}{2}\right)^{\nu+2k} \] |
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| 173 | \form#172:$\nu$ |
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| 174 | \form#173:$ 0 < x < \infty $ |
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| 175 | \form#174:\[ Y_{\nu}(x) = \frac{J_{\nu}(x) \cos(\nu\pi) - J_{-\nu}(x)}{\sin(\nu\pi)} \] |
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| 176 | \form#175:\[ I_{\nu}(x) = i^{-\nu} J_{\nu}(ix) \] |
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| 177 | \form#176:\[ K_{\nu}(x) = \frac{\pi}{2} i^{\nu+1} [J_{\nu}(ix) + i Y_{\nu}(ix)] \] |
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| 178 | \form#177:\[ \mathbf{X} = \mathbf{X}^H \] |
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| 179 | \form#178:\[ \mathbf{X}^H = \mathbf{X}^{-1} \] |
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| 180 | \form#179:$n+|K| \times n+|K|$ |
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| 181 | \form#180:$n = min(r, c)$ |
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| 182 | \form#181:$r \times c$ |
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| 183 | \form#182:$n-1$ |
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| 184 | \form#183:\[ \int_a^b f(x) dx \] |
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| 185 | \form#184:\[ x \sim \Gamma(\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1} \exp(-\beta x) \] |
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| 186 | \form#185:$\alpha=1$ |
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| 187 | \form#186:$\Theta(n\log n)$ |
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| 188 | \form#187:$\Theta(n^2)$ |
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| 189 | \form#188:$g(x) = x^{10} + x^9 + x^8 + x^6 + x^5 + x^3 + 1$ |
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| 190 | \form#189:$ r(t) $ |
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| 191 | \form#190:\[ r(t) = a(t) * s(t), \] |
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| 192 | \form#191:$ s(t) $ |
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| 193 | \form#192:$ a(t) $ |
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| 194 | \form#193:$ \|a(t)\| $ |
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| 195 | \form#194:\[ R(\tau) = E[a^*(t) a(t+\tau)] = J_0(2 \pi f_\mathrm{max} \tau), \] |
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| 196 | \form#195:$ f_\mathrm{max} $ |
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| 197 | \form#196:\[ f_\mathrm{max} = \frac{v}{\lambda} = \frac{v}{c_0} f_c. \] |
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| 198 | \form#197:$ c_0 $ |
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| 199 | \form#198:$ f_c $ |
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| 200 | \form#199:$ f_\mathrm{max} T_s $ |
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| 201 | \form#200:$ T_s $ |
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| 202 | \form#201:$ R(\tau) $ |
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| 203 | \form#202:\[ h(t) = \sum_{k=0}^{N_\mathrm{taps}-1} a_k \exp (-j \theta_k ) \delta(t-\tau_k), \] |
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| 204 | \form#203:$ N_\mathrm{taps} $ |
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| 205 | \form#204:$ a_k $ |
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| 206 | \form#205:$ \tau_k $ |
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| 207 | \form#206:$ \theta_k $ |
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| 208 | \form#207:$ k^{th} $ |
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| 209 | \form#208:\[ \mathbf{a} = [a_0, a_1, \ldots, a_{N_\mathrm{taps}-1}] \] |
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| 210 | \form#209:\[ \mathbf{\tau} = [\tau_0, \tau_1, \ldots, \tau_{N_\mathrm{taps}-1}], \] |
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| 211 | \form#210:$ \tau_0 = 0 $ |
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| 212 | \form#211:$ \tau_0 < \tau_1 < \ldots < \tau_{N_\mathrm{taps}-1} $ |
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| 213 | \form#212:$ h(t) $ |
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| 214 | \form#213:$ \tau_k = d_k T_s $ |
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| 215 | \form#214:$ d_k $ |
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| 216 | \form#215:\[ \rho \exp(2 \pi f_\rho t + \theta_\rho), \] |
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| 217 | \form#216:$ \rho $ |
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| 218 | \form#217:$ f_\rho $ |
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| 219 | \form#218:$ \theta_\rho $ |
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| 220 | \form#219:$ f_\rho = 0.7 f_\mathrm{max} $ |
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| 221 | \form#220:\[ \tilde \mu_i(t) = \sum_{n=1}^{N_i} c_{i,n} \cos(2\pi f_{i,n} t + \theta_{i,n}) \] |
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| 222 | \form#221:$ c_{i,n} $ |
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| 223 | \form#222:$ f_{i,n} $ |
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| 224 | \form#223:$ \theta_{i,n} $ |
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| 225 | \form#224:$ N_i \rightarrow \infty $ |
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| 226 | \form#225:\[ \tilde \mu(t) = \tilde \mu_1(t) + j \tilde \mu_2(t) \] |
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| 227 | \form#226:$ N_i $ |
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| 228 | \form#227:$ N_\mathrm{fft} $ |
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| 229 | \form#228:\[ h(t) = \sum_{k=0}^{N_\mathrm{taps}-1} a_k \exp (-j \theta_k) \delta(t-\tau_k), \] |
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| 230 | \form#229:$ N_{taps} $ |
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| 231 | \form#230:$ \mathbf{a} $ |
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| 232 | \form#231:$ \mathbf{\tau} $ |
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| 233 | \form#232:$N_0/2$ |
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| 234 | \form#233:$N_0$ |
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| 235 | \form#234:$ f_{norm} = f_{max} T_{s} $ |
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| 236 | \form#235:$ f_{max} $ |
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| 237 | \form#236:$ T_{s} $ |
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| 238 | \form#237:\[ \max_{p_0,...,p_{n-1}} \sum_{i=0}^{n-1} \log\left(1+p_i\alpha_i\right) \] |
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| 239 | \form#238:\[ \sum_{i=0}^{n-1} p_i \le P \] |
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| 240 | \form#239:$\alpha_0,...,\alpha_{n-1}$ |
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| 241 | \form#240:$p_0,...,p_{n-1}$ |
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| 242 | \form#241:$O(n^2)$ |
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| 243 | \form#242:$2^{K-1}$ |
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| 244 | \form#243:$ H = [H_{1} H_{2}] $ |
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| 245 | \form#244:$ H_{2} $ |
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| 246 | \form#245:$ [H_{1} H_{2}][I; G'] = 0 $ |
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| 247 | \form#246:\[ L = \log \frac{P(b=0)}{P(b=1)} \] |
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| 248 | \form#247:\[ \mbox{QLLR} = \mbox{round} \left(2^{\mbox{Dint1}}\cdot \mbox{LLR}\right) \] |
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| 249 | \form#248:\[ 2^{-(Dint1-Dint3)} \] |
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| 250 | \form#249:\[ \log(\exp(a)+\exp(b)) \] |
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| 251 | \form#250:\[ \mbox{sign}(a) * \mbox{sign}(b) * \mbox{min}(|a|,|b|) + f(|a+b|) - f(|a-b|) \] |
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| 252 | \form#251:\[ f(x) = \log(1+\exp(-x)) \] |
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| 253 | \form#252:\[r_k = c_k s_k + n_k,\] |
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| 254 | \form#253:$c_k$ |
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| 255 | \form#254:$s_k$ |
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| 256 | \form#255:$n_k$ |
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| 257 | \form#256:$M = 2^k$ |
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| 258 | \form#257:$k = 1, 2, \ldots $ |
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| 259 | \form#258:$\{-(\sqrt{M}-1), \ldots, -3, -1, 1, 3, \ldots, (\sqrt{M}-1)\}$ |
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| 260 | \form#259:$\sqrt{2(M-1)/3}$ |
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| 261 | \form#260:$(1, 0)$ |
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| 262 | \form#261:$M = 4$ |
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| 263 | \form#262:$M = 2$ |
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| 264 | \form#263:$0 \rightarrow 1+0i$ |
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| 265 | \form#264:$1 \rightarrow -1+0i$ |
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| 266 | \form#265:$0 \rightarrow 1$ |
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| 267 | \form#266:$1 \rightarrow -1$ |
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| 268 | \form#267:$\{-(M-1), \ldots, -3, -1, 1, 3, \ldots, (M-1)\}$ |
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| 269 | \form#268:$ \sqrt{(M^2-1)/3}$ |
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| 270 | \form#269:\[\log \left( \frac{P(b_i=0|r)}{P(b_i=1|r)} \right) = \log \left( \frac{\sum_{s_i \in S_0} \exp \left( -\frac{|r_k - s_i|^2}{N_0} \right)} {\sum_{s_i \in S_1} \exp \left( -\frac{|r_k - s_i|^2}{N_0} \right)} \right) \] |
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| 271 | \form#270:$d_0 = |r_k - s_0|$ |
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| 272 | \form#271:$d_1 = |r_k - s_1|$ |
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| 273 | \form#272:\[\frac{d_1^2 - d_0^2}{N_0}\] |
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| 274 | \form#273:$c_k = 1$ |
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| 275 | \form#274:$L_c$ |
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| 276 | \form#275:\[\log \left( \frac{P(b_i=0|r)}{P(b_i=1|r)} \right) = \log \left( \frac{\sum_{s_i \in S_0} \exp \left( -\frac{|r_k - c_k s_i|^2}{N_0} \right)} {\sum_{s_i \in S_1} \exp \left( -\frac{|r_k - c_k s_i|^2}{N_0} \right)} \right) \] |
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| 277 | \form#276:$d_0 = |r_k - c_k s_0|$ |
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| 278 | \form#277:$d_1 = |r_k - c_k s_1|$ |
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| 279 | \form#278:$r_k$ |
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| 280 | \form#279:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Im\{r_k \exp \left(j \frac{\Pi}{4} \right) \}\] |
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| 281 | \form#280:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Re\{r_k \exp \left(j \frac{\Pi}{4} \right) \}\] |
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| 282 | \form#281:$r$ |
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| 283 | \form#282:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Im\{r_k c_k \exp \left(j \frac{\Pi}{4} \right) \}\] |
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| 284 | \form#283:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{2 \sqrt{2}}{N_0} \Re\{r_k c_k \exp \left(j \frac{\Pi}{4} \right) \}\] |
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| 285 | \form#284:$c$ |
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| 286 | \form#285:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{4 \Re\{r\}} {N_0}\] |
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| 287 | \form#286:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{4 \Re\{r c^{*}\}}{N_0}\] |
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| 288 | \form#287:\[\log \left( \frac{P(b=0|r)}{P(b=1|r)} \right) = \frac{4 r}{N_0}\] |
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| 289 | \form#288:$c = 1$ |
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| 290 | \form#289:\[ y = Hx+e \] |
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| 291 | \form#290:$n_r\times n_t$ |
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| 292 | \form#291:$y$ |
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| 293 | \form#292:$n_r$ |
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| 294 | \form#293:$n_t$ |
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| 295 | \form#294:$e$ |
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| 296 | \form#295:\[ G = \left[ \begin{array}{cc} H_r & -H_i \\ H_i & H_r \end{array} \right] \] |
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| 297 | \form#296:\[ \log \left( \frac {\sum_{s:b_k=0} \exp(-x^2) P(s)} {\sum_{s:b_k=1} \exp(-x^2) P(s)} \right) \] |
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| 298 | \form#297:\[ \log \left( \frac {\sum_{s:b_k=0} \exp (-x^2) P(s)} {\sum_{s:b_k=1} \exp (-x^2) P(s)} \right) \] |
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| 299 | \form#298:\[ LLR(k) = \log \left( \frac {\sum_{s:b_k=0} \exp \left( -\frac{|y - Hs|^2}{2\sigma^2} \right) P(s)} {\sum_{s:b_k=1} \exp \left( -\frac{|y - Hs|^2}{2\sigma^2} \right) P(s)} \right) \] |
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| 300 | \form#299:$H = \mbox{diag}(h)$ |
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| 301 | \form#300:$|y-Hs|$ |
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| 302 | \form#301:\[ LLR(k) = \log \left( \frac {\sum_{s:b_k=0} \exp \left( -\frac{|y - Hs|^2}{\sigma^2} \right) P(s)} {\sum_{s:b_k=1} \exp \left( -\frac{|y - Hs|^2}{\sigma^2} \right) P(s)} \right) \] |
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| 303 | \form#302:\[ \mbox{min} |y - Hs| \] |
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| 304 | \form#303:$n_r\times 1$ |
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| 305 | \form#304:$ \alpha $ |
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| 306 | \form#305:\[ p(t) = \frac{\sin(\pi t / T)}{\pi t / T} \frac{\cos(\alpha \pi t / T)}{1 - (2 \alpha t / T)^2} \] |
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| 307 | \form#306:\[ p(t) = \frac{4 \alpha}{\pi \sqrt{T}} \frac{\cos((1+\alpha)\pi t / T) + T \sin((1-\alpha)\pi t / T) / (4 \alpha t) }{1 - (4 \pi t / T)^2} \] |
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| 308 | \form#307:$2^m$ |
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| 309 | \form#308:$2^m-1$ |
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| 310 | \form#309:$N = 2^{deg} - 1$ |
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| 311 | \form#310:$deg = \{ 5, 7, 8, 9 \}$ |
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| 312 | \form#311:$L \times N$ |
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| 313 | \form#312:\[ r_k = h_k c_k + w_k \] |
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| 314 | \form#313:$h_k$ |
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| 315 | \form#314:$\{-\sqrt{E_c},+\sqrt{E_c}\}$ |
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| 316 | \form#315:$w_k$ |
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| 317 | \form#316:\[ z_k = \hat{h}_k^{*} r_k \] |
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| 318 | \form#317:$\hat{h}_k^{*}$ |
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| 319 | \form#318:\[ L_c = 4\sqrt{E_c} / {N_0} \] |
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| 320 | \form#319:\[ s(1), p_{1,1}(1), p_{1,2}(1), \ldots , p_{1,n_1}(1), p_{2,1}(1), p_{2,2}(1), \ldots , p_{2,n_2}(1), s(2), \ldots \] |
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| 321 | \form#320:$s(n)$ |
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| 322 | \form#321:$p_{l,k}(n)$ |
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| 323 | \form#322:\[ t_1(1), pt_{1,1}(1), pt_{1,2}(1), \ldots , pt_{1,n_1}(1), \ldots pt_{1,n_1}(m) \] |
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| 324 | \form#323:$f(\mathbf{x})$ |
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| 325 | \form#324:$\mathbf{x}$ |
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| 326 | \form#325:\[ \left\| \mathbf{f}'(\mathbf{x})\right\|_{\infty} \leq \varepsilon_1 \] |
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| 327 | \form#326:\[ \left\| d\mathbf{x}\right\|_{2} \leq \varepsilon_2 (\varepsilon_2 + \| \mathbf{x} \|_{2} ) \] |
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| 328 | \form#327:$\varepsilon_1 = 10^{-4}$ |
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| 329 | \form#328:$\varepsilon_2 = 10^{-8}$ |
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| 330 | \form#329:$\mathbf{h}$ |
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| 331 | \form#330:\[ \varphi(\alpha) = f(\mathbf{x} + \alpha \mathbf{h}) \] |
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| 332 | \form#331:$\alpha_s$ |
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| 333 | \form#332:$f$ |
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| 334 | \form#333:\[ \phi(\alpha_s) \leq \varphi(0) + \alpha_s \rho \varphi'(0) \] |
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| 335 | \form#334:\[ \varphi'(\alpha_s) \geq \beta \varphi'(0),\: \rho < \beta \] |
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| 336 | \form#335:$\rho = 10^{-3}$ |
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| 337 | \form#336:$\beta = 0.99$ |
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| 338 | \form#337:\[ \| \varphi(\alpha_s)\| \leq \rho \| \varphi'(0) \| \] |
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| 339 | \form#338:\[ b-a \leq \beta b, \] |
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| 340 | \form#339:$\left[a,b\right]$ |
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| 341 | \form#340:$\beta = 10^{-3}$ |
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| 342 | \form#341:$a_1$ |
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| 343 | \form#342:$a_2$ |
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| 344 | \form#343:$\epsilon$ |
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| 345 | \form#344:\[ y(n) = b(0)*x(n) + b(1)*x(n-1) + ... + b(N)*x(n-N) \] |
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| 346 | \form#345:\[ a(0)*y(n) = x(n) - a(1)*y(n-1) - ... - a(N)*y(n-N) \] |
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| 347 | \form#346:\[ a(0)*y(n) = b(0)*x(n) + b(1)*x(n-1) + \ldots + b(N_b)*x(n-N_b) - a(1)*y(n-1) - \ldots - a(N_a)*y(n-N_a) \] |
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| 348 | \form#347:$max(N_a, n_b) - 1$ |
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| 349 | \form#348:$\pi$ |
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| 350 | \form#349:$N>n$ |
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| 351 | \form#350:$N = 4 n$ |
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| 352 | \form#351:$R(k) = 0, \forall \|k\| > m$ |
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| 353 | \form#352:$2(m+n)$ |
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| 354 | \form#353:$N+1$ |
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| 355 | \form#354:\[ p_0 x^N + p_1 x^{N-1} + \ldots + p_{N-1} x + p_N \] |
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| 356 | \form#355:\[ T(x) = \left\{ \begin{array}{ll} \cos(n\arccos(x)),& |x| \leq 0 \\ \cosh(n\mathrm{arccosh}(x)),& x > 1 \\ (-1)^n \cosh(n\mathrm{arccosh}(-x)),& x < -1 \end{array} \right. \] |
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| 357 | \form#356:$X$ |
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| 358 | \form#357:$N$ |
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| 359 | \form#358:\[ X(k) = \sum_{j=0}^{N-1} x(j) e^{-2\pi j k \cdot i / N} \] |
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| 360 | \form#359:\[ x(j) = \frac{1}{N} \sum_{k=0}^{N-1} X(k) e^{2\pi j k \cdot i / N} \] |
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| 361 | \form#360:\[ X(k) = w(k) \sum_{j=0}^{N-1} x(j) \cos \left(\frac{(2j+1)k \pi}{2N} \right) \] |
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| 362 | \form#361:\[ x(j) = \sum_{k=0}^{N-1} w(k) X(k) \cos \left(\frac{(2j+1)k \pi}{2N} \right) \] |
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| 363 | \form#362:$w(k) = 1/sqrt{N}$ |
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| 364 | \form#363:$k=0$ |
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| 365 | \form#364:$w(k) = sqrt{2/N}$ |
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| 366 | \form#365:$k\geq 1$ |
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| 367 | \form#366:$i$ |
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| 368 | \form#367:\[ w_i = 0.54 - 0.46 \cos(2\pi i/(n-1)) \] |
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| 369 | \form#368:\[ w_i = 0.5(1 - \cos(2\pi (i+1)/(n+1)) \] |
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| 370 | \form#369:\[ w_i = 0.5(1 - \cos(2\pi i/(n-1)) \] |
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| 371 | \form#370:\[ w_i = 0.42 - 0.5\cos(2\pi i/(n-1)) + 0.08\cos(4\pi i/(n-1)) \] |
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| 372 | \form#371:\[ w_i = w_{n-i-1} = \frac{2(i+1)}{n+1} \] |
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| 373 | \form#372:\[ w_i = w_{n-i-1} = \frac{2i+1}{n} \] |
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| 374 | \form#373:\[ W[k] = \frac{T_M\left(\beta \cos\left(\frac{\pi k}{M}\right) \right)}{T_M(\beta)},k = 0, 1, 2, \ldots, M - 1 \] |
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| 375 | \form#374:$ \mathbf{x} $ |
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| 376 | \form#375:\[ m_r = \mathrm{E}[x-\mu]^r = \frac{1}{n} \sum_{i=0}^{n-1} (x_i - \mu)^r \] |
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| 377 | \form#376:\[ \gamma_1 = \frac{\mathrm{E}[x-\mu]^3}{\sigma^3} \] |
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| 378 | \form#377:$\sigma$ |
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| 379 | \form#378:\[ \gamma_1 = \frac{k_3}{{k_2}^{3/2}} \] |
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| 380 | \form#379:\[ k_2 = \frac{n}{n-1} m_2 \] |
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| 381 | \form#380:\[ k_3 = \frac{n^2}{(n-1)(n-2)} m_3 \] |
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| 382 | \form#381:$m_2$ |
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| 383 | \form#382:$m_3$ |
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| 384 | \form#383:\[ \gamma_2 = \frac{\mathrm{E}[x-\mu]^4}{\sigma^4} - 3 \] |
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| 385 | \form#384:\[ \gamma_2 = \frac{k_4}{{k_2}^2} \] |
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| 386 | \form#385:\[ k_4 = \frac{n^2 [(n+1)m_4 - 3(n-1){m_2}^2]}{(n-1)(n-2)(n-3)} \] |
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| 387 | \form#386:$m_4$ |
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| 388 | \form#387:\[ \gamma_2 = \frac{\mathrm{E}[x-\mu]^4}{\sigma^4} \] |
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| 389 | \form#388:$ w_{new} = [ \alpha \cdot w_{A} ~~~ \beta \cdot w_{B} ]^T $ |
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| 390 | \form#389:$ w_{new} $ |
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| 391 | \form#390:$ w_{A} $ |
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| 392 | \form#391:$ w_{B} $ |
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| 393 | \form#392:$ \alpha = K_A / (K_A + KB_in) $ |
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| 394 | \form#393:$ \beta = 1-\alpha $ |
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| 395 | \form#394:$ K_A $ |
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| 396 | \form#395:$ KB_in $ |
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| 397 | \form#396:$ -\frac{D}{2}\log(2\pi) -\frac{1}{2}\log(|\Sigma|) $ |
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| 398 | \form#397:$ D $ |
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| 399 | \form#398:$ |\Sigma| $ |
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| 400 | \form#399:$ \Sigma $ |
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