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missing scale factor

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Steven G. Johnson 2023-05-01 08:48:02 -04:00
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commit 58931ea10a
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psets/pset6.ipynb
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@ -17,7 +17,7 @@
"source": [
"## Problem 1 (10+10 points)\n",
"\n",
"**(a)** In class, we showed that if $\\tilde{y} = \\bar{F} y, \\tilde{a} = \\bar{F} a, \\tilde{x} = \\bar{F} x$ where $F$ is the DFT matrix, then $y = a \\circledast x$ (circular convolution) implies $\\tilde{y} = \\tilde{a} \\, \\mbox{.*} \\, \\tilde{x}$ (element-wise product). Show that the reverse is also true: show that if $\\tilde{y} = \\tilde{a} \\circledast \\tilde{x}$, then $y = a \\, \\mbox{.*} \\, x$. \n",
"**(a)** In class, we showed that if $\\tilde{y} = \\bar{F} y, \\tilde{a} = \\bar{F} a, \\tilde{x} = \\bar{F} x$ where $F$ is the DFT matrix, then $y = a \\circledast x$ (circular convolution) implies $\\tilde{y} = \\tilde{a} \\, \\mbox{.*} \\, \\tilde{x}$ (element-wise product). Show that the reverse is also true: show that if $\\tilde{y} = \\tilde{a} \\circledast \\tilde{x}$, then $y = \\alpha (a \\, \\mbox{.*} \\, x) $ for some scale factor $\\alpha = ???$. \n",
"\n",
"**(b)** Differentiating through convolutions, in class, involved the \"reversal\" permutation $R$:\n",
"$$\n",