Recitations

Project1/LIGOMainProject.ipynb

@@ -1571,7 +1571,7 @@
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Recitation1/Recitation_01_learner.ipynb

@@ -35,6 +35,8 @@
     "\n",
     "There's also an optional section for the exponential distribution if you finish the above in time.\n",
     "\n",
+    "Note: had already worked on this but it didn't save!\n",
+    "\n",
     "<!--end-block-->"
    ]
   },
@@ -822,7 +824,7 @@
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Recitation12/MIT_8S50_recitation_12.ipynb

@@ -1 +1,193 @@
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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "j_OYjrski6wJ"
+   },
+   "source": [
+    "# MIT_8.S50 Recitation 12: Additional features of the Discrete Fourier Transform (DFT)\n",
+    "\n",
+    "Author: Juan Mena-Parra\n",
+    "\n",
+    "Date: December 08, 2021\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "u9C9AJMv4shG"
+   },
+   "source": [
+    "# Preliminaries\n",
+    "\n",
+    "QUESTION: What is the Fourier transform of $f(t)=\\sin(\\omega t)$? and of $f(t)=e^{i\\omega t}$?\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "jnxnxlpt49Pe"
+   },
+   "source": [
+    "# Part 1\n",
+    "\n",
+    "The discrete Fourier transform (DFT) of a discrete signal $f(n)$ of lenght $N$ is\n",
+    "\n",
+    "$F(k) = \\sum_{n=0}^{N-1} f(n)e^{-i2\\pi k n/N}$\n",
+    "\n",
+    "The inverse discrete Fourier transform (IDFT) is\n",
+    "\n",
+    "$x(n) = \\frac{1}{N}\\sum_{k=0}^{N-1} F(k)e^{i2\\pi k n/N}$\n",
+    "\n",
+    "For what follows, you may need to write functions that implement the DFT and IDFT of a function for given values of $k$ and $n$, respectively\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "executionInfo": {
+     "elapsed": 4,
+     "status": "ok",
+     "timestamp": 1641915586147,
+     "user": {
+      "displayName": "Juan David Mena",
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+      "userId": "04300883876663913307"
+     },
+     "user_tz": 300
+    },
+    "id": "KEi4u01GLuE-"
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "def DFT(f,k):\n",
+    "    fk = 0\n",
+    "    for i in range(len(f)):\n",
+    "        fk += f[i]*np.exp(-2*np.pi *k * i / len(f) * j)\n",
+    "    return fk"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "x9OqOQQmrbpW"
+   },
+   "source": [
+    "Consider the discrete function $x(n)=\\sin(2\\pi\\cdot 0.02n)$\n",
+    "\n",
+    "Plot $x(n)$\n",
+    "\n",
+    "QUESTION: What is the period of $x(n)$ in samples? This corresponds to an analog signal sampled at 100 Hz. What was the period of the analog signal in seconds?\n",
+    "\n",
+    "For $N=100$ compute the $X(k)=DFT\\{x(n)\\}$ for $k=0, \\cdots N-1$. Plot |X(k)| vs $k$. \n",
+    "\n",
+    "QUESTION: Is |X(k)| what you expected based on your intuition from the Fourier transform of a sine function of continuous time $t$? yes/no, why?\n",
+    "\n",
+    "Compute $y(n)=IDFT\\{X(k)\\}$ for $n=0, \\cdots N-1$. Plot $y(n)$ vs $n$.\n",
+    "\n",
+    "QUESTION: is $y(n)=x(n)$ from $n=0, \\cdots N-1$? yes/no, why?\n",
+    "\n",
+    "Compute $z(n)=IDFT\\{X(k)\\}$ for $n=0, \\cdots 3N-1$. Plot $z(n)$ vs $n$.\n",
+    "\n",
+    "QUESTION: is $z(n)=x(n)$ from $n=0, \\cdots 3N-1$? yes/no, why?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "executionInfo": {
+     "elapsed": 3,
+     "status": "ok",
+     "timestamp": 1641915636126,
+     "user": {
+      "displayName": "Juan David Mena",
+      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GitLOFy2FYeZTcocC3230KsRS0NCBCmzitpTP3zzw=s64",
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+     },
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+    },
+    "id": "zVKxHwN839vn"
+   },
+   "outputs": [],
+   "source": [
+    "# Add your plots here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "GEjvLbYqxt-C"
+   },
+   "source": [
+    "# Part 2\n",
+    "\n",
+    "Repeat the same procedure above and answer the same questions for $x(n)=\\sin(2\\pi\\cdot 0.025n)$\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "executionInfo": {
+     "elapsed": 156,
+     "status": "ok",
+     "timestamp": 1641915649096,
+     "user": {
+      "displayName": "Juan David Mena",
+      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GitLOFy2FYeZTcocC3230KsRS0NCBCmzitpTP3zzw=s64",
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+    },
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+   },
+   "outputs": [],
+   "source": [
+    "# Add your plots here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "dcOoJNYqmRbA"
+   },
+   "source": [
+    "# Part 3\n",
+    "\n",
+    "QUESTION: How does the DFT help with the inversion of $N^{-1}d$ for the CMB project? What are the trade-offs, if any?\n"
+   ]
+  }
+ ],
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