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| "qid": "part-00025-8bdd594e-bf2f-4dee-8277-9c33e8468d28-c000_4210_multi", | |
| "messages": [ | |
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| "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.", | |
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| "content": "\\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 substitude $n=n-N$ in the $2^{nd}$}}\\\\ &= \\sum_{n=0}^{N-1} {x_{2N}[n] e^{-j\\frac{2\\pi}{N}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}{N}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 $e^{-2j\\frac\\pi N}$ with $W_N$ 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 $X[k]$ of $x[n]$, at $\\frac k2$}}\\\\ &= 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] , &\\scriptstyle{\\text{k=0,2,4,...,2N-2 }}\\\\ 0 , &\\scriptstyle{\\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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| "images": [ | |
| "./asset/images/question_part-00025-8bdd594e-bf2f-4dee-8277-9c33e8468d28-c000_4210_multi.png" | |
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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$$\nX_{2N}[k] = 0 ~~,~~ \\text{for}~~ k = 1,3,5...,2N-1\n$$", | |
| "edit_acc": 0.14628297362110312 | |
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