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zhongyufeng02
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Track binary files with Git LFS
Browse files- .gitattributes +1 -0
- asset/images/question_part-00014-8bdd594e-bf2f-4dee-8277-9c33e8468d28-c000_27959_all.png +3 -0
- asset/images/question_part-00014-8bdd594e-bf2f-4dee-8277-9c33e8468d28-c000_9402_single_15.png +3 -0
- asset/images/question_part-00025-8bdd594e-bf2f-4dee-8277-9c33e8468d28-c000_4210_multi.png +3 -0
- asset/test_jsons/line-level.json +1 -1
- asset/test_jsons/page-level.json +1 -1
- asset/test_jsons/paragraph-level.json +1 -1
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@@ -33,3 +33,4 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
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"model_answer": "\\begin{align}\n\\frac{d}{dt}X_1 + \\frac{d}{dt}X_2 &= \\left(AX_1 + B\\begin{bmatrix}c_1\\cos\\omega_1 t\\\\d_1\\cos\\omega_1 t\\end{bmatrix}\\right) + \\left(AX_2 + B\\begin{bmatrix}c_2\\cos\\omega_2 t\\\\d_2\\cos\\omega_2 t\\end{bmatrix}\\right)\\\\ &= AX_1 + AX_2 + B\\begin{bmatrix}c_1\\cos\\omega_1 t\\\\d_1\\cos\\omega_1 t\\end{bmatrix} + B\\begin{bmatrix}c_2\\cos\\omega_2 t\\\\d_2\\cos\\omega_2 t\\end{bmatrix}\\\\ &= A(X_1 + X_2) + B\\left(\\begin{bmatrix}c_1\\cos\\omega_1 t\\\\d_1\\cos\\omega_1 t\\end{bmatrix} + \\begin{bmatrix}c_2\\cos\\omega_2 t\\\\d_2\\cos\\omega_2 t\\end{bmatrix}\\right)\\\\ &= AX + B\\begin{bmatrix}u_1(t)\\\\u_2(t)\\end{bmatrix}\\\\ &= \\frac{d}{dt}X.\n\\end{align}",
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"edit_acc": 0.1109350237717908
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"model_answer": "\\begin{align}\n\\frac{d}{dt}X_1 + \\frac{d}{dt}X_2 &= \\left(AX_1 + B\\begin{bmatrix}c_1\\cos\\omega_1 t\\\\d_1\\cos\\omega_1 t\\end{bmatrix}\\right) + \\left(AX_2 + B\\begin{bmatrix}c_2\\cos\\omega_2 t\\\\d_2\\cos\\omega_2 t\\end{bmatrix}\\right)\\\\ &= AX_1 + AX_2 + B\\begin{bmatrix}c_1\\cos\\omega_1 t\\\\d_1\\cos\\omega_1 t\\end{bmatrix} + B\\begin{bmatrix}c_2\\cos\\omega_2 t\\\\d_2\\cos\\omega_2 t\\end{bmatrix}\\\\ &= A(X_1 + X_2) + B\\left(\\begin{bmatrix}c_1\\cos\\omega_1 t\\\\d_1\\cos\\omega_1 t\\end{bmatrix} + \\begin{bmatrix}c_2\\cos\\omega_2 t\\\\d_2\\cos\\omega_2 t\\end{bmatrix}\\right)\\\\ &= AX + B\\begin{bmatrix}u_1(t)\\\\u_2(t)\\end{bmatrix}\\\\ &= \\frac{d}{dt}X.\n\\end{align}",
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"edit_acc": 0.1109350237717908
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asset/test_jsons/page-level.json
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"model_answer": "$$\n\\sigma^2 = \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} |H(f)|^2 \\, \\mathrm{d}f\n$$\n\\begin{align}\n\\sigma^2 &= \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} |H(f)|^2 \\, \\mathrm{d}f \\\\ &= \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} |H(\\omega/2\\pi)|^2 \\, \\mathrm{d}(\\omega/2\\pi) \\\\ &= \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} |H(\\omega)|^2 \\, \\mathrm{d}\\omega \\\\ &= \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} \\frac{1}{(\\omega\\tau)^2 + 1} \\, \\mathrm{d}\\omega \\\\ &= \\frac{2\\pi\\tau}{2\\pi} \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} \\frac{1}{(\\omega\\tau)^2 + 1} \\, \\mathrm{d}(\\omega\\tau) \\\\ &= \\frac{1}{2\\pi\\tau} \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} \\frac{1}{u^2 + 1} \\, \\mathrm{d}u \\\\ &= \\frac{N_0}{4\\pi} \\left[ \\tan^{-1} u \\right]_{-\\infty}^{\\infty} \\\\ &= \\frac{N_0}{4\\pi} \\left[ \\pi/2 - (-\\pi/2) \\right] \\\\ &= \\frac{N_0}{4\\pi}\n\\end{align}\n$$\ny[k]=x(kT)\\iff R_y[k]=R_x(kT)\\tag{1}\n$$\n$$\nS_y(e^{j\\omega T})=\\frac{1}{T}\\sum_{k=-\\infty}^{\\infty}S_x\\left(\\omega-\\frac{2\\pi k}{T}\\right)\\tag{2}\n$$",
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"edit_acc": 0.15040650406504066
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"model_answer": "$$\n\\sigma^2 = \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} |H(f)|^2 \\, \\mathrm{d}f\n$$\n\\begin{align}\n\\sigma^2 &= \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} |H(f)|^2 \\, \\mathrm{d}f \\\\ &= \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} |H(\\omega/2\\pi)|^2 \\, \\mathrm{d}(\\omega/2\\pi) \\\\ &= \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} |H(\\omega)|^2 \\, \\mathrm{d}\\omega \\\\ &= \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} \\frac{1}{(\\omega\\tau)^2 + 1} \\, \\mathrm{d}\\omega \\\\ &= \\frac{2\\pi\\tau}{2\\pi} \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} \\frac{1}{(\\omega\\tau)^2 + 1} \\, \\mathrm{d}(\\omega\\tau) \\\\ &= \\frac{1}{2\\pi\\tau} \\int_{-\\infty}^{\\infty} \\frac{N_0}{2} \\frac{1}{u^2 + 1} \\, \\mathrm{d}u \\\\ &= \\frac{N_0}{4\\pi} \\left[ \\tan^{-1} u \\right]_{-\\infty}^{\\infty} \\\\ &= \\frac{N_0}{4\\pi} \\left[ \\pi/2 - (-\\pi/2) \\right] \\\\ &= \\frac{N_0}{4\\pi}\n\\end{align}\n$$\ny[k]=x(kT)\\iff R_y[k]=R_x(kT)\\tag{1}\n$$\n$$\nS_y(e^{j\\omega T})=\\frac{1}{T}\\sum_{k=-\\infty}^{\\infty}S_x\\left(\\omega-\\frac{2\\pi k}{T}\\right)\\tag{2}\n$$",
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"edit_acc": 0.15040650406504066
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"model_answer": "\\begin{align}\n\\text{Start with defining 2N-Point DFT of } x_{2N}[n] \\ldots \\\\ X_{2N}[k] &= \\sum_{n=0}^{2N-1} x_{2N}[n] e^{-j \\frac{2\\pi}{2N} nk} \\\\ \\\\ &\\text{first split the sum and then substitute } n = n-N \\text{ in the } 2^{nd} \\\\ &= \\sum_{n=0}^{N-1} x_{2N}[n] e^{-j \\frac{2\\pi}{2N} n (k/2)} + \\sum_{n=N}^{2N-1} x_{2N}[n] e^{-j \\frac{2\\pi}{2N} nk} \\\\ \\\\ &\\text{recognise } x_{2N}[n] = x_N[n], \\ x_{2N}[n+N] = x_N[n] \\\\ &= \\sum_{n=0}^{N-1} x_{2N}[n] e^{-j \\frac{2\\pi}{2N} n (k/2)} + \\sum_{n=0}^{N-1} x_{2N}[n+N] e^{-j \\frac{2\\pi}{2N} (n+N) k} \\\\ \\\\ &\\text{Expand the 2nd sum's multiplier factor} \\\\ &= \\sum_{n=0}^{N-1} x_N[n] e^{-j \\frac{2\\pi}{N} n (k/2)} + \\sum_{n=0}^{N-1} x_N[n] e^{-j \\frac{2\\pi}{N} (n+N) (k/2)} \\\\ \\\\ &\\text{Replace } e^{-2j\\frac{\\pi}{N}} \\text{ with } W_N \\text{ for simplicity} \\\\ &= \\sum_{n=0}^{N-1} x_N[n] W_N^{nk/2} + W_N^{Nk/2} \\sum_{n=0}^{N-1} x_N[n] W_N^{nk/2} \\\\ \\\\ &\\text{Recognise the sums as N-Point DFT } X[k] \\text{ of } x[n] \\text{, at } \\frac{k}{2} \\\\ &= X_N[k/2] + e^{-j \\pi k} X_N[k/2] \\\\ \\\\ &= X_N[k/2] \\cdot \\big( 1 + (-1)^k \\big) \\\\ \\\\ &= \\begin{cases} 2 X_N[k/2], & \\text{k = 0,2,4,...2N2} \\\\ 0, & \\text{k = 1,3,5,...2N-1} \\end{cases} \\\\\n\\end{align}\n$$\nX_{2N}[k] = 0 ~~,~~ \\text{for}~~ k = 1,3,5...,2N-1\n$$",
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"edit_acc": 0.14628297362110312
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"model_answer": "\\begin{align}\n\\text{Start with defining 2N-Point DFT of } x_{2N}[n] \\ldots \\\\ X_{2N}[k] &= \\sum_{n=0}^{2N-1} x_{2N}[n] e^{-j \\frac{2\\pi}{2N} nk} \\\\ \\\\ &\\text{first split the sum and then substitute } n = n-N \\text{ in the } 2^{nd} \\\\ &= \\sum_{n=0}^{N-1} x_{2N}[n] e^{-j \\frac{2\\pi}{2N} n (k/2)} + \\sum_{n=N}^{2N-1} x_{2N}[n] e^{-j \\frac{2\\pi}{2N} nk} \\\\ \\\\ &\\text{recognise } x_{2N}[n] = x_N[n], \\ x_{2N}[n+N] = x_N[n] \\\\ &= \\sum_{n=0}^{N-1} x_{2N}[n] e^{-j \\frac{2\\pi}{2N} n (k/2)} + \\sum_{n=0}^{N-1} x_{2N}[n+N] e^{-j \\frac{2\\pi}{2N} (n+N) k} \\\\ \\\\ &\\text{Expand the 2nd sum's multiplier factor} \\\\ &= \\sum_{n=0}^{N-1} x_N[n] e^{-j \\frac{2\\pi}{N} n (k/2)} + \\sum_{n=0}^{N-1} x_N[n] e^{-j \\frac{2\\pi}{N} (n+N) (k/2)} \\\\ \\\\ &\\text{Replace } e^{-2j\\frac{\\pi}{N}} \\text{ with } W_N \\text{ for simplicity} \\\\ &= \\sum_{n=0}^{N-1} x_N[n] W_N^{nk/2} + W_N^{Nk/2} \\sum_{n=0}^{N-1} x_N[n] W_N^{nk/2} \\\\ \\\\ &\\text{Recognise the sums as N-Point DFT } X[k] \\text{ of } x[n] \\text{, at } \\frac{k}{2} \\\\ &= X_N[k/2] + e^{-j \\pi k} X_N[k/2] \\\\ \\\\ &= X_N[k/2] \\cdot \\big( 1 + (-1)^k \\big) \\\\ \\\\ &= \\begin{cases} 2 X_N[k/2], & \\text{k = 0,2,4,...2N2} \\\\ 0, & \\text{k = 1,3,5,...2N-1} \\end{cases} \\\\\n\\end{align}\n$$\nX_{2N}[k] = 0 ~~,~~ \\text{for}~~ k = 1,3,5...,2N-1\n$$",
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"edit_acc": 0.14628297362110312
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