asset/test_jsons/page-level.json

[
    {
        "qid": "part-00014-8bdd594e-bf2f-4dee-8277-9c33e8468d28-c000_27959_all",
        "messages": [
            {
                "content": "<image> Analyze the image. Extract and output only the LaTeX formulas present in the image, in LaTeX code format. Ignore inline formulas, all other text, and do not include any explanations.",
                "role": "user"
            },
            {
                "content": "$$\n\\sigma^2 = \\int_{-\\infty}^\\infty \\frac{N_0}{2}|H(f)|^2\\,\\mathrm df\n$$\n\\begin{align}\n\\sigma^2 &= \\int_{-\\infty}^\\infty \\frac{N_0}{2}|H(f)|^2\\,df \\\\ &= \\int_{-\\infty}^\\infty \\frac{N_0}{2}|H(\\omega/2\\pi)|^2\\,d(\\omega/2\\pi) \\\\ &= \\frac{1}{2\\pi}\\int_{-\\infty}^\\infty \\frac{N_0}{2}|H(\\omega)|^2\\,d\\omega \\\\ &= \\frac{1}{2\\pi}\\int_{-\\infty}^\\infty \\frac{N_0}{2}\\frac{1}{(\\omega\\tau)^2+1}\\,d\\omega \\\\ &= \\frac{1}{2\\pi\\tau}\\int_{-\\infty}^\\infty \\frac{N_0}{2}\\frac{1}{(\\omega\\tau)^2+1}\\,d(\\omega\\tau) \\\\ &= \\frac{1}{2\\pi\\tau}\\int_{-\\infty}^\\infty \\frac{N_0}{2}\\frac{1}{u^2+1}\\, du \\\\ &= \\frac{N_0}{4\\pi\\tau}\\left[\\tan^{-1} u\\right]_{-\\infty}^{\\infty} \\\\ &= \\frac{N_0}{4\\pi\\tau}\\left[\\pi/2 - (-\\pi/2)\\right] \\\\ &= \\frac{N_0}{4\\tau},\n\\end{align}\n$$\ny[k]=x(kT)\\;\\Longleftrightarrow \\;R_y[k]=R_x(kT)\\tag{1}\n$$\n$$\nS_y(e^{j\\omega T})=\\frac{1}{T}\\sum_{k=-\\infty}^{\\infty}S_x\\left(\\omega-\\frac{2\\pi k}{T}\\right)\\tag{2}\n$$",
                "role": "assistant"
            }
        ],
        "images": [
            "./asset/images/question_part-00014-8bdd594e-bf2f-4dee-8277-9c33e8468d28-c000_27959_all.png"
        ],
        "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$$\nS_y(e^{j\\omega T})=\\frac{1}{T}\\sum_{k=-\\infty}^{\\infty}S_x\\left(\\omega-\\frac{2\\pi k}{T}\\right)\\tag{2}\n$$",
        "edit_acc": 0.15040650406504066
    }
]