{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Using the Fourier Series and the DFT\n",
    "\n",
    "This is a homework. Do the calculations by hand or using your calculator, but not a computer program.  I suggest you include it in your notebook, but you don't have to.  I just need to be able to look it over to make sure you figured it out okay.  Dr. Roth says you have a light homework load with only the notebooks.  This is due Friday. \n",
    "1. Find the complex Fourier series for a triangle wave with peak value $A = 10$ and frequency $1/T_0$ that starts ascending from zero at $t=-T_0/8$ and peaks at $3T_0/8$, then goes back down to zero at $7T_0/8$, and then begins a new period.  See the picture below. ![Triangle Wave](Triangle_Wave.png)  \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First we find the Fourier series coefficients, $$c_n = 1/T_0 \\int_{T_0} x(t) e^{-j2\\pi nt/T_0} dt$$  We use the python sympy package for symbolic mathematics."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\begin{cases} - \\frac{A T_{0} e^{- \\frac{7 i \\pi n}{4}}}{2 \\pi^{2} n^{2}} - \\frac{\\left(4 i \\pi^{2} A T_{0} n^{2} - 4 \\pi A T_{0} n\\right) e^{- \\frac{3 i \\pi n}{4}}}{8 \\pi^{3} n^{3}} & \\text{for}\\: n > -\\infty \\wedge n < \\infty \\wedge n \\neq 0 \\\\\\frac{A T_{0}}{4} & \\text{otherwise} \\end{cases} + \\begin{cases} - \\frac{A T_{0} e^{\\frac{i \\pi n}{4}}}{2 \\pi^{2} n^{2}} + \\frac{\\left(4 i \\pi^{2} A T_{0} n^{2} + 4 \\pi A T_{0} n\\right) e^{- \\frac{3 i \\pi n}{4}}}{8 \\pi^{3} n^{3}} & \\text{for}\\: n > -\\infty \\wedge n < \\infty \\wedge n \\neq 0 \\\\\\frac{A T_{0}}{4} & \\text{otherwise} \\end{cases}}{T_{0}}$"
      ],
      "text/plain": [
       "⎛⎧        -7⋅ⅈ⋅π⋅n                                   -3⋅ⅈ⋅π⋅n                 \n",
       "⎜⎪        ─────────                                  ─────────                \n",
       "⎜⎪            4       ⎛     2       2             ⎞      4                    \n",
       "⎜⎪  A⋅T₀⋅ℯ            ⎝4⋅ⅈ⋅π ⋅A⋅T₀⋅n  - 4⋅π⋅A⋅T₀⋅n⎠⋅ℯ                         \n",
       "⎜⎪- ─────────────── - ────────────────────────────────────────  for n > -∞ ∧ n\n",
       "⎜⎨         2  2                          3  3                                 \n",
       "⎜⎪      2⋅π ⋅n                        8⋅π ⋅n                                  \n",
       "⎜⎪                                                                            \n",
       "⎜⎪                            A⋅T₀                                            \n",
       "⎜⎪                            ────                                      otherw\n",
       "⎝⎩                             4                                              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "            ⎞   ⎛⎧        ⅈ⋅π⋅n                                  -3⋅ⅈ⋅π⋅n     \n",
       "            ⎟   ⎜⎪        ─────                                  ─────────    \n",
       "            ⎟   ⎜⎪          4     ⎛     2       2             ⎞      4        \n",
       "            ⎟   ⎜⎪  A⋅T₀⋅ℯ        ⎝4⋅ⅈ⋅π ⋅A⋅T₀⋅n  + 4⋅π⋅A⋅T₀⋅n⎠⋅ℯ             \n",
       " < ∞ ∧ n ≠ 0⎟   ⎜⎪- ─────────── + ────────────────────────────────────────  fo\n",
       "            ⎟ + ⎜⎨       2  2                        3  3                     \n",
       "            ⎟   ⎜⎪    2⋅π ⋅n                      8⋅π ⋅n                      \n",
       "            ⎟   ⎜⎪                                                            \n",
       "            ⎟   ⎜⎪                          A⋅T₀                              \n",
       "ise         ⎟   ⎜⎪                          ────                              \n",
       "            ⎠   ⎝⎩                           4                                \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "            T₀                                                                \n",
       "\n",
       "                        ⎞\n",
       "                        ⎟\n",
       "                        ⎟\n",
       "                        ⎟\n",
       "r n > -∞ ∧ n < ∞ ∧ n ≠ 0⎟\n",
       "                        ⎟\n",
       "                        ⎟\n",
       "                        ⎟\n",
       "                        ⎟\n",
       "      otherwise         ⎟\n",
       "                        ⎠\n",
       "─────────────────────────\n",
       "                         "
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sp\n",
    "sp.init_printing()\n",
    "t,n,T_0, A = sp.symbols(('t', 'n', 'T_0','A'))\n",
    "c_n  = 1/T_0*(sp.integrate(A*(t+T_0/8)/(T_0/2)*sp.exp(-sp.I*2*sp.pi*n*t/T_0),\\\n",
    "                           (t,-T_0/8,3*T_0/8)) + \\\n",
    "           sp.integrate(-A*(t-7*T_0/8)/(T_0/2)*sp.exp(-sp.I*2*sp.pi*n*t/T_0),\\\n",
    "                        (t,3*T_0/8,7*T_0/8)))\n",
    "c_n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.00000000000000\n",
      "-1.43289792062689 - 1.43289792062689*I\n",
      "0.e-125 + 0.e-126*I\n",
      "0.159210880069655 - 0.159210880069655*I\n",
      "-0.e-127 + 0.e-125*I\n"
     ]
    }
   ],
   "source": [
    "c_0 = c_n.subs({n:0, T_0:1, A:10})\n",
    "c_1 = c_n.subs({n:1, T_0:1, A:10})\n",
    "c_2 = c_n.subs({n:2, T_0:1, A:10})\n",
    "c_3 = c_n.subs({n:3, T_0:1, A:10})\n",
    "c_4 = c_n.subs({n:4, T_0:1, A:10})\n",
    "print(c_0.evalf())\n",
    "print(c_1.evalf())\n",
    "print(c_2.evalf())\n",
    "print(c_3.evalf())\n",
    "print(c_4.evalf())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Fourier series is then:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\sum_{n=-\\infty}^{\\infty} \\begin{cases} \\frac{\\left(- \\frac{A T_{0} e^{\\frac{i \\pi n}{4}}}{2 \\pi^{2} n^{2}} - \\frac{A T_{0} e^{- \\frac{7 i \\pi n}{4}}}{2 \\pi^{2} n^{2}} - \\frac{\\left(4 i \\pi^{2} A T_{0} n^{2} - 4 \\pi A T_{0} n\\right) e^{- \\frac{3 i \\pi n}{4}}}{8 \\pi^{3} n^{3}} + \\frac{\\left(4 i \\pi^{2} A T_{0} n^{2} + 4 \\pi A T_{0} n\\right) e^{- \\frac{3 i \\pi n}{4}}}{8 \\pi^{3} n^{3}}\\right) e^{\\frac{2 i \\pi n t}{T_{0}}}}{T_{0}} & \\text{for}\\: n > -\\infty \\wedge n < \\infty \\wedge n \\neq 0 \\\\\\frac{A e^{\\frac{2 i \\pi n t}{T_{0}}}}{2} & \\text{otherwise} \\end{cases}$"
      ],
      "text/plain": [
       "     ∞                                                                        \n",
       "___________                                                                   \n",
       "╲                                                                             \n",
       " ╲          ⎧⎛        ⅈ⋅π⋅n         -7⋅ⅈ⋅π⋅n                                  \n",
       "  ╲         ⎪⎜        ─────         ─────────                                 \n",
       "   ╲        ⎪⎜          4               4       ⎛     2       2             ⎞ \n",
       "    ╲       ⎪⎜  A⋅T₀⋅ℯ        A⋅T₀⋅ℯ            ⎝4⋅ⅈ⋅π ⋅A⋅T₀⋅n  - 4⋅π⋅A⋅T₀⋅n⎠⋅\n",
       "     ╲      ⎪⎜- ─────────── - ─────────────── - ──────────────────────────────\n",
       "      ╲     ⎪⎜       2  2            2  2                          3  3       \n",
       "       ╲    ⎪⎝    2⋅π ⋅n          2⋅π ⋅n                        8⋅π ⋅n        \n",
       "        ╲   ⎪─────────────────────────────────────────────────────────────────\n",
       "         ╲  ⎨                                                                T\n",
       "         ╱  ⎪                                                                 \n",
       "        ╱   ⎪                                                              2⋅ⅈ\n",
       "       ╱    ⎪                                                              ───\n",
       "      ╱     ⎪                                                                 \n",
       "     ╱      ⎪                                                           A⋅ℯ   \n",
       "    ╱       ⎪                                                           ──────\n",
       "   ╱        ⎪                                                                2\n",
       "  ╱         ⎩                                                                 \n",
       " ╱                                                                            \n",
       "╱                                                                             \n",
       "‾‾‾‾‾‾‾‾‾‾‾                                                                   \n",
       "   n = -∞                                                                     \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       " -3⋅ⅈ⋅π⋅n                                   -3⋅ⅈ⋅π⋅n ⎞                        \n",
       " ─────────                                  ─────────⎟  2⋅ⅈ⋅π⋅n⋅t             \n",
       "     4       ⎛     2       2             ⎞      4    ⎟  ─────────             \n",
       "ℯ            ⎝4⋅ⅈ⋅π ⋅A⋅T₀⋅n  + 4⋅π⋅A⋅T₀⋅n⎠⋅ℯ         ⎟      T₀                \n",
       "────────── + ────────────────────────────────────────⎟⋅ℯ                      \n",
       "                                3  3                 ⎟                        \n",
       "                             8⋅π ⋅n                  ⎠                        \n",
       "─────────────────────────────────────────────────────────────────  for n > -∞ \n",
       "₀                                                                             \n",
       "                                                                              \n",
       "⋅π⋅n⋅t                                                                        \n",
       "──────                                                                        \n",
       " T₀                                                                           \n",
       "                                                                              \n",
       "──────                                                                     oth\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "∧ n < ∞ ∧ n ≠ 0\n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "erwise         \n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "               \n",
       "               "
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sp.Sum(c_n*sp.exp(sp.I*2*sp.pi*n*t/T_0),(n, -sp.oo, sp.oo))\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Convert this to a Fourier series using only sine and cosines (real coefficients, not complex ones).  How do the $c_n$ compare to the coefficients, $a_n$ and $b_n$ in the ordinary Fourier series?\n",
    "\n",
    "First note that $c_n = c_{-n}^*$ if $x(t) \\in \\mathscr R$.  Then break the Fourier series into the sums for $n < 0$, for $n = 0$, and for $n > 0$.   $$x(t) = x(t+T_0) = \\sum_{n=-\\infty}^{\\infty}c_n e^{j2\\pi nt/T_0} = c_0 + \\sum_{n=-\\infty}^{-1} c_n e^{j2\\pi nt/T_0} + \\sum_{n=1}^{\\infty} c_n e^{j2\\pi nt/T_0}$$  Then make the substitution $m = -n$ in the sum for negatave $n$.  Note: $c_{-m} = c_m^*$ because $x(t)$ is real. $$x(t) = x(t+T_0) = c_0 + \\sum_{m=1}^{\\infty}c_m^* e^{-j2\\pi mt/T_0} + \\sum_{n=1}^{\\infty}c_n e^{j2\\pi nt/T_0}$$ Then substitute $m = n$ in the sum over $m$, and combine the two sums.  Note that m is just a dummy index.  $$x(t) = c_0 + \\sum_{n=1}^{\\infty} \\left[c_n e^{j2\\pi nt/T_0} + c_n^* e^{-j2\\pi nt/T_0}\\right]$$  Now define $a_n \\equiv 2 \\mathscr {Re}[c_n]$ and $b_n \\equiv -2\\mathscr {Im} [c_n]$ so $c_n =  1/2(a_n - jb_n)$ into the Fourier series and use Euler's identity for the complex exponential.\n",
    "\n",
    "$$x(t) = a_0/2 + \\sum_{n=1}^{\\infty} \\left[1/2(a_n - jb_n)(cos(2\\pi nt/T_0) + j sin(2\\pi nt/T_0) + 1/2(a_n + jb_n)(cos(2\\pi nt/T_0) - j sin(2\\pi nt/T_0)\\right]$$ $$=  a_0/2 + \\sum_{n=1}^{\\infty} [a_n cos(2\\pi nt/T_0) +b_n sin(2\\pi nt/T_0)]$$  From this we can see the $c_0$, the DC component is the same as $a_0/2$ and $a_n = \\mathscr {Re}(2c_n)$ is the amplitude of the cosine components of the signal $x(t)$.  The coefficients of the sine components of $x(t)$ are $b_n = -\\mathscr {Im}(2c_n)$.  Note that this shows that the real part of $c_n$ corresponds to the even part of the function $x(t)$ and the imagaginary part of $c_n$ corresponds to the odd part of $x(t)$, because cosines are even functions and sines are odd functions.\n",
    "\n",
    "3. Sample the triangle wave starting at zero.  Make your samples $T$ seconds apart.  Take samples over exactly one period of the wave, and use $N = 8$ samples.  Write the vector of those numbers.  Call it $x(kT)$ where $k \\in \\{0, 1, 2, 3, 4, 5, 6, 7\\}$.\n",
    "4. Find the DFT of the sequence in the previous problem, and call it $X(n)$ where $n \\in \\{0, 1, 2, 3, 4, 5, 6, 7\\}$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 3 4 5 6 7]\n",
      "[0.    0.125 0.25  0.375 0.5   0.625 0.75  0.875]\n"
     ]
    },
    {
     "data": {
      "image/png": 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n6+50m0h6tqTbJD21eD/a6bq7pF02S3pL8foMSfs6XXc9P93WA398ouSIeETSYxMlT7dO0meK19dIOtf2XNO79YoF2yQibomIXxZvv63a7Ei9rp5jRZL+WtIHJbV2KpTuVE+bvFnSRyPiZ5IUEeNtrrET6mmXkLS0eH2Skswu1m0BXs9EyY9vExGTkh6U9LS2VNcZdU0ePc2lkm5a1Iq6w4LtYvt5kp4RETe0s7AOqudYOV3S6bb/2/a3bZ/Xtuo6p552+UtJF9s+oNocB29rT2nNaWpCh0VQz0TJdU2m3EPq/vvavlhSWdLLF7Wi7jBvu9gekPSPki5pV0FdoJ5jpaTaMMorVPuf2lbbZ0bEzxe5tk6qp11eL+nTEfEPtl8k6V+Kdpla/PIa12098HomSn58m2JezpMkHWpLdZ1R1+TRtl8l6c8lrY2IiTbV1kkLtcuwpDMl3Wp7n6QXSrq+x09k1vv78+WIeDQifixpj2qB3svqaZdLJX1JkiJim6Qh1R501dW6LcB3Snq27dNsH6/aScrrZ2xzvaQ3Fq/Pl/T1KM489KgF26QYKtikWnj3w5imtEC7RMSDEbEsIlZGxErVzg2sjYhenlW7nt+f61Q76S3by1QbUvlRW6tsv3raZb9qE7TL9nNUC/BqW6tsQFcFeDGm/dhEybslfSki7rL9V7bXFpt9UtLTbN8j6Z2Sjnr5WC+os00+JOnJkv7V9u22Zx6cPafOdukrdbbJVyQ9YPtuSbdI+rOIeKAzFbdHne3yLklvtn2HpM9LuiRDx5Bb6QEgqa7qgQMA6keAA0BSBDgAJEWAA0BSBDgAJEWAA0BSBDgAJEWAoy/YPsX2646y7gTb37B93FHWH2/7m8WjG4CuQYCjX5wr6flHWfcmSddGxP/NtbJ4BOkWSXP+AwB0CgGOnmf7pZI+LOn84lEDp83Y5CJJXy62fZLt/7R9h+07p/Xaryu2A7oG/yVEz4uIb9neKendEXHn9HXFw42eGRH7ikXnSbovIl5brD+pWH6npN9uU8lAXeiBo1+sUu3RqTMtkzT9Wdjfk/Qq2x+w/TsR8aAkFcMrj9geXvxSgfoQ4Oh5tp8m6cGIeHSO1b9S7dGhkqSI+IGkF6gW5O+3/RfTth1Uf0zNhiQYQkE/OE1HmeMwIn5m+zjbQxFxxPZySYci4irbD6uY0af4R6B6lH8EgI6gB45+8H1Jy4qTki+eY/1XJb20eH2WpB22b1dthqMNxfIx1eZKBLoGzwNH3ytmNHpnRLxhnm2ulfTeiJhrHB3oCHrg6HsRcZukW+a7kUfSdYQ3ug09cABIih44ACRFgANAUgQ4ACRFgANAUgQ4ACRFgANAUv8PbUAV9n8HEqAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(5+0j)\n",
      "(-1.5088834764831844-1.5088834764831844j)\n",
      "0j\n"
     ]
    },
    {
     "data": {
      "image/png": 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PkxQiom37S5KekzQo6fGIOFHxWAuyfVDS70naZHtG0t9FxGPVTrWgHZI+L+ln3b1nSfqbiHi2wpkWcoukJ7rf+R+Q9N2IWPfH+pL4iKRDnb/XVZP0nYj4QbUjLekvJR3o3txNS/qziudZkO0PqHPq7i+Kr53l+CEAYH6ZtlYAAPMg5ACQHCEHgOQIOQAkR8gBIDlCDgDJEXIASO7/AaEbQlZWhwB+AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "A = 10\n",
    "N = 8\n",
    "T_0 = 1\n",
    "T = T_0/N\n",
    "k = np.arange(0,N)\n",
    "print(k)\n",
    "t = k*T\n",
    "print(t)\n",
    "x = np.piecewise(t,[(t<3*T_0/8) & (t>=0), (t>=3*T_0/8) & (t<7*T_0/8), t>= (7*T_0/8)],\\\n",
    "                 [lambda t: 2*A/T_0*(t+T_0/8), lambda t: -2*A/T_0*(t-7/8*T_0),\\\n",
    "                 lambda t: 2*A/T_0*(t-7*T_0/8)])\n",
    "plt.plot(t,x,\"*\")\n",
    "plt.title('$x(t)$')\n",
    "plt.xlabel('$t$ (s)')\n",
    "plt.show()\n",
    "X = 1/N*np.fft.fft(x)  # Notice this is 1/N * what we were asked.\n",
    "print(X[0])\n",
    "print(X[1])\n",
    "print(X[2])\n",
    "f = np.arange(-1/2/T, 1/2/T,1/T_0) # np.fft.fftfreq(N,T)\n",
    "plt.plot(k, np.real(X), '.', k, np.imag(X), '.')\n",
    "plt.show()\n",
    "\n",
    "X = np.fft.fftshift(X)\n",
    "\n",
    "plt.plot(f,np.real(X),'o',f,np.imag(X),'*')\n",
    "plt.title('$X(f)$')\n",
    "plt.legend(['Real', 'Imaginary'])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5. Approximate the formula for the complex Fourier series coefficients, $c_n$ with the DFT.  (You did this in class a while back).  Compare the numerical results You got above with this approximation?\n",
    "6. When you compared the $c_n$ to the $a_n$ and $b_n$ you should have noted that negative $n$ terms paired with positive $n$ terms to make the sinusoidal waves you learned about in high school.  In previous work, you showed that if $x(t) \\in \\mathscr R$ then $c_{-n} = c_n^*$, and $X(-n) = X(n)^*$.  You also learned that $X(n+N) = X(n)$.  Use that to figure out from the $X(n)$ what an approximation for $c_{-1}$ is. What valid negative frequency components can you find using the DFT approximation, and what values of $n$ goes with them?\n",
    "7.  What is the highest frequency component you can resolve using the eight samples you took in $T_0$ seconds using the DFT?  What would this be in general in terms of $T$?\n",
    "8. The triangle wave has higher frequency components than you could resolve in the problem above.  What if the wave we sampled only contained those frequency components that we could resolve above?  To figure out how this would work, sample $sin(2\\pi t/T_0)$ using eight samples over one period as you did above.  Take the DFT of that, and compare it to the complex Fourier series coefficients for this wave.  What do you notice?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.5*I\n",
      "0\n"
     ]
    }
   ],
   "source": [
    "import sympy as sp\n",
    "sp.init_printing()\n",
    "t,n,T_0, A = sp.symbols(('t', 'n', 'T_0','A'))\n",
    "c_n  = 1/T_0*(sp.integrate(sp.sin(2*sp.pi*t/T_0)*sp.exp(-sp.I*2*sp.pi*n*t/T_0),\\\n",
    "                           (t, 0, T_0)))\n",
    "c_n\n",
    "c_1 = c_n.subs({n:1, T_0:1})\n",
    "c_2 = c_n.subs({n:2, T_0:1})\n",
    "print(c_1.evalf())\n",
    "print(c_2.evalf())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(-5.456039645643629e-17-0.5j)\n",
      "(1.5308084989341915e-17-1.3877787807814457e-17j)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = 10\n",
    "N = 8\n",
    "T_0 = 1\n",
    "T = T_0/N\n",
    "t = np.arange(0,T_0,T)\n",
    "x = np.sin(2*np.pi*t/T_0)\n",
    "plt.plot(t,x,\"*\")\n",
    "plt.title('$x(t)$')\n",
    "plt.xlabel('$t$ (s)')\n",
    "plt.show()\n",
    "X = 1/N*np.fft.fft(x)\n",
    "print(X[1])\n",
    "print(X[2])\n",
    "X = np.fft.fftshift(X)\n",
    "f = np.arange(0, 1/T,1/T_0) # np.fft.fftfreq(N,T)\n",
    "plt.plot(f,np.real(X),'o',f,np.imag(X),'*')\n",
    "plt.title('$X(f)$')\n",
    "plt.legend(['Real', 'Imaginary'])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this is exact to within machine precision!  This isn't a proof, but it appears the inaccuracies on the triangle wave are caused by aliasing, not the rectangular integration."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "9. Find the output voltage due to the input of the triange wave above in the circuit below.  First do this analytically, in terms of R and C, then with the DFT, and use $T_0 = 1$ second and use the values for R and C. ![Low Pass Filter](Low_Pass.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}