Signal_Processing_for_Communications_Vetterli/numex_module_4.ipynb

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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Module 4: Fourier Transforms\n",
      "============================\n",
      "\n",
      "DFS &amp; DFT\n",
      "-------------\n",
      "\n",
      "The discrete Fourier transform (DFT) of a signal is defined using the following formula:\n",
      "\n",
      "$$X(k)=\\sum_{n=0}^{N-1} x(n) e^{-j(2\\pi/N)(n)(k)},\n",
      "k=0,\\ldots,N-1.$$\n",
      "\n",
      "Similar formula is used for the discrete Fourier series (DFS) of a periodic signal.\n",
      "\n",
      "The DFS / DFT is a vector of complex numbers which can be\n",
      "represented via their magnitude $|X(n)|$ (absolute value) and\n",
      "phase $\\arctan \\frac{\\text{Im}(X(n))}{\\text{Re}(X(n))}$,\n",
      "$n=0,\\ldots,N-1$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Transformation matrix\n",
      "\n",
      "DFS / DFT can be easily implemented using the transform matrix ${W}_N$. We recall that for a signal of length $N$ the transform matrix has as $n$-th row and $k$-th column element ${W}_N(n,k)=e^{-j(2\\pi/N)(n)(k)}=W_N^{(n)(k)}$, with $n,k=0,\\ldots,N-1$, and there we denote $W_N=e^{-j(2\\pi/N)}$.\n",
      "\n",
      "In Python, a DFT matrix can be generated using the following code."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def dftmatrix(N):\n",
      "    '''construct DFT matrix'''\n",
      "    import numpy as np\n",
      "    \n",
      "    # create a 1xN matrix containing indices 0 to N-1\n",
      "    a = np.expand_dims(np.arange(N), 0)\n",
      "    \n",
      "    # take advantage of numpy broadcasting to create the matrix\n",
      "    WN = np.exp(-2j*np.pi*a.T*a/N)\n",
      "    \n",
      "    return WN"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The transformation $X$ of the signal $x$, i.e. $X(k)$; is then given by (here using a short constant signal), multiplying the (here) $3\\times 3$ DFT Matrix and the input signal (in its vector form)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "\n",
      "N = 6\n",
      "WN = dftmatrix(N)\n",
      "\n",
      "x = np.ones(N)\n",
      "X = np.dot(WN, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The inverse transformation $x(n)=\\frac{1}{N} \\sum_{k=1}^{N} X(k) e^{j(2\\pi/N)(n)(k)}$ is then computed using:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x2 = 1./N * np.dot(WN.T.conjugate(), X)\n",
      "print 'Error: ',(np.abs(x - x2).sum())\n",
      "print '(This small error is due to machine precision.)'"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Error:  5.38784573873e-15\n",
        "(This small error is due to machine precision.)\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Magnitude and Phase\n",
      "\n",
      "Let's start with a simple signal: unitary step function (periodic)\n",
      "of length N=128 (we shall see later what motivates us in choosing a\n",
      "power of 2)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.append(np.ones(64),np.zeros(64))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can see what this signal looks like in the next figure"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y = np.linspace(0,127,128)\n",
      "# The plotting is done by the pylab.stem function\n",
      "stem(y,x)\n",
      "xlabel('Samples')\n",
      "ylabel('x')\n",
      "ylim([0,2])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x1076c1990>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The transformation matrix and the transform is then"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = x.flatten();\n",
      "N = 128\n",
      "W128 = dftmatrix(N)\n",
      "X  = np.dot(W128,x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      " The stem plot of the magnitude (absolute value) of the DSF or DFT"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "magnitude = abs(X)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "is given by"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y = np.linspace(0,127,128)\n",
      "y[:] = y[:]/len(y) #we are normalizing the frequencies\n",
      "# Plotting is done by the pylab.stem function\n",
      "stem(y,magnitude)\n",
      "ylabel('Magnitude')\n",
      "xlabel('Normalized Frequencies')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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RETUumz2CtLQ09O/fH4WFhVi6dCmKiooQFxeHIUOG1Lnhc+fOITY2FpWVlaisrMS0adPw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\n/2wy7HBzdd7LAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10780e250>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "When it comes to compute the phase we can use the Python command "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "phase = np.angle(X)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Numerical intermezzo\n",
      "\n",
      "It is important to notice that, when the phase is close to $k\\pi$\n",
      "with $k=0,1,2,\\ldots$, its computation is extremely sensitive to\n",
      "numerical errors.\n",
      "\n",
      "Consider for instance the third coefficient $X(3)$ of the DFT /\n",
      "DFS of the step function.\n",
      "\n",
      "If we theoretically compute such value, by remarking that is a\n",
      "geometric series $X(3)=\\sum_{n=1}^{N/2} a^n$ with\n",
      "$a=e^{-j(2\\pi/N)(n-1)(3-1)}$, we obtain\n",
      "$X(3)=\\frac{1-e^{-j(2\\pi/N)2 N/2 }}{1-e^{-j(2\\pi/N)2}}= 0$, i.e.,\n",
      "a null complex number to which we assign by definition a phase of\n",
      "$0$ degrees. Such a result applies to all coefficients with odd\n",
      "index.\n",
      "\n",
      "If we numerically compute such value"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X[2] = sum(W128[2])\n",
      "X[2]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "(-3.4416913763379853e-15+3.8441472227646045e-15j)"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We obtain -3.44e-15 + 3.84e-15i which give us a phase of -0.7608\n",
      "radian! Such a result is nothing but the numerical error of the sum\n",
      "computation.\n",
      "\n",
      "Notice that the theoretical phase of $X$ is"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Theoretical_phase = np.linspace(-1.5,1.5,128)\n",
      "Theoretical_phase[0:127:2] = 0\n",
      "stem(y,Theoretical_phase)\n",
      "ylabel('Angle(X)')\n",
      "xlabel('Normalized Frequencies')\n",
      "xlim([0,1])\n",
      "ylim([-2,2])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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IiYmRrBcQEABPT090794dPXr0QGFhoWQ9qaDBXAsiovtzmjWNiIgI7NmzB8nJ\nyfetp1KpYDAYMHDgwDbbbJyGarwjKjU1AXfuSP1qzLUgIuoIRYJGcHCw7Lpyo2DTqCIvbzWMxhXw\n9KySrMcFbiKi9nPohXCVSoXHH38csbGxePvtt2V/r3Ed4y43EiQi6mQ2G2lotVqUlZWZla9duxZT\np06V1cbhw4cxdOhQVFRUQKvVIjg4GOPHj5f1XU9PP2RkTORGgkREnchmQePTTz/tcBtDhw4FAAwZ\nMgQ///nPUVhYeJ+goW/2swbu7g1ITIxjkCAi+pHBYIDBYOhQG4on91las7h9+zYaGhrQr18/VFdX\nIy8vD2lpafdpSW/6iVt+EBGZ02g00Gg0pvfp6elWt6FI0NizZw9SU1NRWVmJxMREqNVqHDhwAKWl\npVi8eDFycnJQVlaG6dOnAwDq6+sxd+5cJCQkWGxTp+M0FBGRrblUch8REcnXnmunQ989RUREjoVB\ng4iIZGPQICIi2Rg0iIhINgYNIiKSjUGDiIhkY9AgIiLZGDSIiEg2Bg0iIpKNQYOIiGRj0CAiItkY\nNIiISDYGDSIiko1Bg4iIZGPQICIi2Rg0iIhINgYNIiKSjUGDiIhkY9AgIiLZFAkav//97xESEoKo\nqChMnz4d33//vWS93NxcBAcHIygoCBs2bLBzL4mIqDVFgkZCQgLOnj2Lr7/+Gg8++CDWrVtnVqeh\noQFLly5Fbm4uzp07h+zsbJw/f16B3joXg8GgdBccBs/FT3gufsJz0TGKBA2tVotu3RoPPWbMGFy5\ncsWsTmFhIUaOHImAgAD06NEDs2fPxt69e+3dVafDvxA/4bn4Cc/FT3guOkbxNY13330XTz75pFn5\n1atX4e/vb3rv5+eHq1ev2rNrRETUiputGtZqtSgrKzMrX7t2LaZOnQoAWLNmDXr27Ilf/OIXZvVU\nKpWtukZERO0lFLJlyxYxduxYUVNTI/n5kSNHhE6nM71fu3atWL9+vWTdwMBAAYAvvvjiiy8rXoGB\ngVZfu1VCCAE7y83NxQsvvICDBw9i8ODBknXq6+vx0EMP4R//+AeGDRuG0aNHIzs7GyEhIXbuLRER\nNVFkTSMlJQW3bt2CVquFWq3G888/DwAoLS1FYmIiAMDNzQ2vvfYadDodQkNDMWvWLAYMIiKFKTLS\nICIi56T43VNyyUn0S01NRVBQEKKionDy5Ek799B+2joX27dvR1RUFCIjIzFu3DicPn1agV7ah9wE\n0K+++gruqzkDAAALVElEQVRubm746KOP7Ng7+5JzLgwGA9RqNcLDw6HRaOzbQTtq61xUVlbiiSee\nQHR0NMLDw7F161b7d9IOFi5cCG9vb0RERFisY/V10+pVEAXU19eLwMBA8d1334m7d++KqKgoce7c\nuRZ1cnJyxOTJk4UQQhQUFIgxY8Yo0VWbk3Mu8vPzxY0bN4QQQhw4cKBLn4umehMmTBCJiYli9+7d\nCvTU9uSci6qqKhEaGipKSkqEEEJUVFQo0VWbk3Mu0tLSxB//+EchRON5GDhwoKirq1OiuzZ16NAh\nceLECREeHi75eXuum04x0pCT6Ldv3z7Mnz8fQGPC4I0bN1BeXq5Ed21Kzrl49NFH0b9/fwCWkydd\ngdwE0KysLDz11FMYMmSIAr20Dznn4oMPPsCMGTPg5+cHABZvQnF2cs7F0KFDcfPmTQDAzZs3MWjQ\nILi52SwDQTHjx4/HAw88YPHz9lw3nSJoyEn0k6rjihdLa5Me33nnHcnkSVcg98/F3r17sWTJEgCu\nm/8j51wUFRXh+vXrmDBhAmJjY/H+++/bu5t2IedcLF68GGfPnsWwYcMQFRWFV1991d7ddAjtuW46\nRWiV+xddtFrTd8ULhDW/0+eff453330Xhw8ftmGPlCPnXCxbtgzr16+HSqWCEMLsz4irkHMu6urq\ncOLECfzjH//A7du38eijj+KRRx5BUFCQHXpoP3LOxdq1axEdHQ2DwQCj0QitVouvv/4a/fr1s0MP\nHYu1102nCBq+vr4oKSkxvS8pKTENsS3VuXLlCnx9fe3WR3uRcy4A4PTp01i8eDFyc3PvOzx1ZnLO\nxfHjxzF79mwAjYufBw4cQI8ePZCUlGTXvtqanHPh7++PwYMHw8PDAx4eHoiLi8PXX3/tckFDzrnI\nz8/HihUrAACBgYEYPnw4vvnmG8TGxtq1r0pr13Wz01ZcbKiurk6MGDFCfPfdd+LOnTttLoQfOXLE\nZRd/5ZyLy5cvi8DAQHHkyBGFemkfcs5Fc88++6z461//asce2o+cc3H+/HkxadIkUV9fL6qrq0V4\neLg4e/asQj22HTnn4re//a3Q6/VCCCHKysqEr6+v+M9//qNEd23uu+++k7UQLve66RQjjeaJfg0N\nDVi0aBFCQkLw5ptvAgCSk5Px5JNPYv/+/Rg5ciT69OmDLVu2KNxr25BzLl5++WVUVVWZ5vF79OiB\nwsJCJbttE3LORVch51wEBwfjiSeeQGRkJLp164bFixcjNDRU4Z53Pjnn4qWXXsKCBQsQFRWFe/fu\n4X/+538wcOBAhXve+ebMmYODBw+isrIS/v7+SE9PR11dHYD2XzeZ3EdERLI5xd1TRETkGBg0iIhI\nNgYNIiKSjUGDiIhkY9AgIiLZGDSIiEg2Bg2yu27duuF3v/ud6f3GjRuRnp5u1z5oNBqcOHECAJCY\nmGjavK69DAYDpk6dKlnev39/qNVqqNVqJCQkdOg49vLmm2+67N5U1DFOkdxHrqVnz57Ys2cPXnzx\nRQwaNMjqPcIaGhrQvXv3DvWh+TFzcnI61FZb4uPjsW/fPsnP6uvrHXJ31a6UGEnW4UiD7K5Hjx74\n5S9/iT/96U9mnxUXF2PixImIiorC448/btoX59lnn8WvfvUrPPLII1i+fDkWLFiAJUuW4NFHH0Vg\nYCAMBgPmz5+P0NBQLFiwwNTe888/j4cffhjh4eHQ6/WS/QkICMB//vMfvPHGG6YRwfDhwzFx4kQA\nQF5eHsaOHYtRo0Zh5syZqK6uBtD4oJ+QkBCMGjUKe/bssfj7ts6f3bp1K5KSkjBp0iRotVrcvn0b\nCxcuxJgxYxATE2MKMDU1NZg9ezZCQ0Mxffp0PPLII6bRUd++fU3t7d692/Q7V1RU4KmnnsLo0aMx\nevRo5OfnAwD0ej0WLlyICRMmIDAwEFlZWabvb9u2DVFRUYiOjjZtk63X65GZmQkAMBqNmDx5MmJj\nYxEXF4dvvvkGALBr1y5EREQgOjoa8fHxFn9/cjGdt8MJkTx9+/YVN2/eFAEBAeL7778XGzduNO0D\nNGXKFLFt2zYhhBDvvvuumDZtmhBCiPnz54upU6eKe/fuCSEa95GaM2eOEEKIvXv3in79+okzZ86I\ne/fuiVGjRolTp04JIYS4fv26EKLxwTwajUacPn1aCCGERqMRx48fF0IIERAQ0GLfobq6OjF+/Hjx\n8ccfi4qKChEXFydu374thBBi/fr14uWXXxY1NTXC399fXLp0SQghxMyZM8XUqVPNftfPP/9c9O/f\nX0RHR4vo6GixZs0asXXrVuHn5yeqqqqEEEK8+OKL4i9/+YsQovFBSQ8++KCorq4WmZmZYtGiRUII\nIU6fPi3c3NxMfe7bt6/pGLt37xbPPvusEEKIOXPmiC+//FII0bgHWUhIiBCi8aFD48aNE3fv3hWV\nlZVi0KBBor6+Xpw5c0Y8+OCDpt+/qU96vV5kZmYKIYSYOHGiKCoqEkI0Pqhn4sSJQgghIiIiRGlp\nqRBCiO+///6+/8/JdTjeuJi6hH79+mHevHnYtGkTPDw8TOUFBQX429/+BgB45plnsHz5cgCN00lP\nP/10i2mlpjWE8PBw+Pj4ICwsDAAQFhaG4uJiREVFYefOnXj77bdRX1+Pa9eu4fz58/d99CXQ+PjL\nSZMmITExER9//DHOnTuHsWPHAgDu3r2LsWPH4ptvvsHw4cMRGBho6utbb70l2d748ePx97//3fT+\nvffeg1arxYABAwA0jmT+/ve/Y+PGjQCAO3fu4F//+he++OIL/OY3vwEAREREIDIyss3z+tlnn+H8\n+fOm9z/88AOqq6uhUqmQmJiIHj16YNCgQfDy8kJZWRn++c9/YubMmaZ9l5r61KS6uhr5+fl4+umn\nTWV3794FAIwbNw7z58/HzJkzMX369Db7Rq6BQYMUs2zZMsTExLSYTgLMp3Oa9O7du8X7nj17Amhc\nWO/Vq5epvFu3bmhoaMB3332HzMxMHDt2DP3798eCBQtQW1t73z5t3boVJSUleP31101lWq0WH3zw\nQYt6X3/9taw+W9KnT58W7z/66CPJLcottds8eNbU1LSof/ToUdO5aa55Wffu3VFfX296zogl9+7d\nwwMPPCD57OjNmzejsLAQOTk5GDVqFI4fP+6Sm/5RS1zTIMU88MADmDlzJt555x3TRXDs2LHYsWMH\nAGD79u2Ii4trV9tCCPzwww/o06cPPD09UV5ejgMHDtz3O8ePH0dmZmaLu4YeeeQRHD58GEajEUDj\nv7yLiooQHByM4uJifPvttwCA7Oxsq/rWnE6nw6ZNm0zvmy7QcXFxpmB15swZnD592lTH29sbFy5c\nwL1797Bnzx7T+UtISGjRVuvg1pxKpcLEiROxa9cuXL9+HQBQVVXVop/9+vXD8OHDsXv3blNZUz+M\nRiNGjx6N9PR0DBkyxCWflEnmGDTI7pr/K/mFF15AZWWl6X1WVha2bNmCqKgobN++vcVjOFvfZdX8\nvdRnkZGRUKvVCA4Oxty5c/HYY49Z7I8QAn/+859RVVWFCRMmQK1W45e//CUGDx6MrVu3Ys6cOYiK\nijJNTfXq1QtvvfUWEhMTMWrUKHh7e0veBaZSqST71rxs1apVqKurQ2RkJMLDw5GWlgYAWLJkCW7d\nuoXQ0FCkpaVh1KhRpu+sX78eU6ZMwbhx4zBs2DBT+aZNm3Ds2DFERUUhLCzMtB241DkCgNDQUKxY\nsQLx8fGIjo7GCy+8YFZ/+/bteOeddxAdHY3w8HDTQv3y5csRGRmJiIgIjBs3Ttb0GTk/bo1O5CQm\nTJiAzMxMxMTEKN0V6sI40iAiItk40iAiItk40iAiItkYNIiISDYGDSIiko1Bg4iIZGPQICIi2Rg0\niIhItv8PXNz2Tink9xcAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1078defd0>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      " ... a good paper and pencil exercise!\n",
      "\n",
      "Back to our phase computation\n",
      "-----------------------------\n",
      "\n",
      "So if we numerically compute the phase of a DFT / DFS&nbsp; obtained\n",
      "by the matrix multiplication and plot the phase we obtain"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "stem(y,phase)\n",
      "ylabel('Angle(X)')\n",
      "xlabel('Normalized Frequencies')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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mXFo07x454inTlyYdKkI5engEKcXKXkbRXD+N8gyJl2TwMnI6a9HSUhv2fVFR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\nhBACgAqBEEJIL/8fyNaWgjYNPIEAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107843050>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "where we can remark the phase computation error where the phase is\n",
      "supposed to be zero.\n",
      "\n",
      "As we shall see in the following, there exists an optimized\n",
      "algorithm that besides speeding up the computation of the DFT / DFS; \n",
      "it avoids, under certain circumstances, numerical errors.\n",
      "\n",
      "The inverse transformation, and it plot, read"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xx = np.real(np.dot(np.conjugate(W128),X) *1/128)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notice that, once again, we are here affected by numerical errors that add negligible imaginary parts to the results that is on the contrary supposed to be real. We shall therefore force the result to be real using the command np.real(), obtaining "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "stem(np.arange(xx.size),xx)\n",
      "ylim([0,2])\n",
      "xlim([0,128])\n",
      "ylabel('x')\n",
      "xlabel('Samples')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x107821dd0>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The Role of the Fourier transform \n",
      "-----------------------------\n",
      "\n",
      "### Two birds with one stone\n",
      "\n",
      "As discussed in the course, the Fourier transformation has a twofold\n",
      "reason to exist: (a) it enables to move into a domain where certain\n",
      "computations are easier to implement (e.g., convolution) and (b) it\n",
      "enables to represent the energy of the signal on the frequency\n",
      "domain rather than the time domain.The second reason to be can be\n",
      "put in evidence with a very simple numerical example.\n",
      "\n",
      "Load the file `frequencyRepresentation.mat`.\n",
      "The signal $x$ contained in the file is a sum of sinusoids, that\n",
      "is, is a sound composed of different tones. We would like to\n",
      "understand how many tones or sinusoids compose the signal.\n",
      "\n",
      "To import a `.mat` file in Python, use the following commands"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import scipy.io\n",
      "mat = scipy.io.loadmat('frequencyRepresentation.mat')\n",
      "print mat"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "{'x': array([[  7.30443561e-01,   1.32608160e+00,   1.68257383e+00, ...,\n",
        "         -1.32608160e+00,  -7.30443561e-01,  -2.93915232e-14]]), '__version__': '1.0', '__header__': 'MATLAB 5.0 MAT-file, Platform: MACI64, Created on: Wed Dec 19 14:17:23 2012', '__globals__': []}\n"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And then,"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "signal = mat['x']\n",
      "print signal \n",
      "type(signal)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[  7.30443561e-01   1.32608160e+00   1.68257383e+00 ...,  -1.32608160e+00\n",
        "   -7.30443561e-01  -2.93915232e-14]]\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "numpy.ndarray"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In the time domain we can either visualize the signal using a basic plot function (the figure below only displays 500 samples out of the 4000),"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "signal = signal.reshape(signal.size,)\n",
      "plot(np.arange(signal.size),signal)\n",
      "xlim([0,500])\n",
      "xlabel('Samples')\n",
      "ylabel('x')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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D8cMho8F9c52MPblt0dFhJxITJwYLpIPdvwfDFpcu2bPqeDNrywqnX3C0Iygi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\nAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107874050>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "or play it!"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Actually, as you can remark, none of these approaches allow us to clearly understand how many sinusoids compose the signal.\n",
      "\n",
      "Let's now take a look at the DFS /DFT. Once created, the transformation matrix (here $N=4000$) `W4000`, we get"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "signal = signal.reshape(signal.size,1)\n",
      "W4000 = dftmatrix(4000)\n",
      "X = np.dot(W4000,signal)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The stem plot in normalized frequencies yields:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "normFrequ = np.arange(1,X.size+1,dtype=float)/float(X.size)\n",
      "plot(normFrequ,abs(X))\n",
      "ylabel('|X|')\n",
      "xlabel('Normalized Frequencies')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAZAAAAEPCAYAAABsj5JaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAMASAgQAYAkBAgCwhAABAFhCgAAALPl/qU0QJ4NBP04AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1078d9750>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### DFS and DFT as projections on a base of complex sinusoids\n",
      "\n",
      "In short, using the DFS or the DFT we are projecting a signal of\n",
      "length N on a basis of N complex sinusoids $W_N^{n,k}$ with\n",
      "$k=1,\\ldots,N$.\n",
      "\n",
      "Let's see what happens if we reduce the dimension of the base, that is the number of sinusoids used to represent the signal.\n",
      "\n",
      "One complex sinusoid (of order $0$, i.e. a constant), taken from the first 128 samples of the `frequencyRepresentation` signal yields:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.append(np.ones(64),np.zeros(64))\n",
      "W128 = dftmatrix(128)\n",
      "X1 = np.dot(W128[0],x)\n",
      "xx1 = np.real(np.conjugate(W128[:,0])*X1*1/128)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Two,three,... complex sinusoids will be:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X2 = np.dot(W128[0:2],x)\n",
      "xx2 = np.real(np.dot(np.conjugate(W128[:,0:2]),X2)*1/128)\n",
      "\n",
      "X3 = np.dot(W128[0:3],x)\n",
      "xx3 = np.real(np.dot(np.conjugate(W128[:,0:3]),X3)*1/128)\n",
      "\n",
      "X4 = np.dot(W128[0:4],x)\n",
      "xx4 = np.real(np.dot(np.conjugate(W128[:,0:4]),X4)*1/128)\n",
      "\n",
      "X5 = np.dot(W128[0:5],x)\n",
      "xx5 = np.real(np.dot(np.conjugate(W128[:,0:5]),X5)*1/128)\n",
      "\n",
      "X6 = np.dot(W128[0:6],x)\n",
      "xx6 = np.real(np.dot(np.conjugate(W128[:,0:6]),X6)*1/128)\n",
      "\n",
      "X7 = np.dot(W128[0:7],x)\n",
      "xx7 = np.real(np.dot(np.conjugate(W128[:,0:7]),X7)*1/128)\n",
      "\n",
      "X8 = np.dot(W128[0:8],x)\n",
      "xx8 = np.real(np.dot(np.conjugate(W128[:,0:8]),X8)*1/128)\n",
      "\n",
      "X9 = np.dot(W128[0:9],x)\n",
      "xx9 = np.real(np.dot(np.conjugate(W128[:,0:9]),X9)*1/128)\n",
      "\n",
      "X10 = np.dot(W128[0:10],x)\n",
      "xx10 = np.real(np.dot(np.conjugate(W128[:,0:10]),X10)*1/128)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "and so on, up to considering 10 complex sinusoids. If now we plot the signal representation using 1,2 and up to 10 sinusoids we obtain the following figure:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(np.arange(xx1.size),xx1)\n",
      "plot(np.arange(xx2.size),xx2)\n",
      "plot(np.arange(xx3.size),xx3)\n",
      "plot(np.arange(xx4.size),xx4)\n",
      "plot(np.arange(xx5.size),xx5)\n",
      "plot(np.arange(xx6.size),xx6)\n",
      "plot(np.arange(xx7.size),xx7)\n",
      "plot(np.arange(xx8.size),xx8)\n",
      "plot(np.arange(xx9.size),xx9)\n",
      "plot(np.arange(xx10.size),xx10)\n",
      "xlabel('Samples')\n",
      "ylabel('x')\n",
      "xlim([0,128])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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7hCv8T/HcTwuZYiBl4lXll+qxS7fpPO89TnzFjGXGJsq3Zv01v1JenvJBOVNT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\nEAShmhGJXxAEoZr5f82MYtcHsRVLAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107821090>"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Three tones\n",
      "    \n",
      "Let us consider another example. We create a simple signal $x$\n",
      "composed of 3 tones $x_1$, $x_2$, and $x_3$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy import constants as c\n",
      "y = np.linspace(0,999,1000)\n",
      "x1 = np.sin((y*2*c.pi*40)/1000)\n",
      "x2 = np.sin((y*2*c.pi*80)/1000)\n",
      "x3 = np.sin((y*2*c.pi*160)/1000)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can put together these signals to get a final signal composed of $N=5000$ samples, with silence in between the 3 different sine waves.\n",
      "\n",
      "This signal can be played by using Audacity, or the winsoud module."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.append(x1,[np.zeros(1000),x2,np.zeros(1000),x3])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's generate the Fourier matrix $W_N$ (with $N=5000$) and\n",
      "compute the DFS or DFT of the signal"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "W5000 = dftmatrix(5000)\n",
      "X = np.dot(W5000,x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The magnitude of the Fourier transformation is"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "normFrequ = np.arange(1,X.size+1,dtype=float)/float(X.size)\n",
      "plot(normFrequ,abs(X))\n",
      "ylabel('|X|')\n",
      "xlabel('Normalized Frequencies')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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YsWNHQC+LtufNmjVLzJ07V2zbts0HlapPybI4e/asMBgMoqKiQgghxJkzZ3xR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\nYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1078ad290>"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "where we can clearly see the three tones.\n",
      "\n",
      "Let's now reduce the base over which the signal is represented, and take:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "W5000reduced = np.append(W5000[0:600],W5000[4399:5000],axis=0)\n",
      "iW5000reduced = np.conj(np.append(W5000[:,0:600],W5000[:,4399:5000],axis=-1))\n",
      "\n",
      "Xapprox = np.dot(W5000reduced,x)  # DFT/DFS on a reduced base\n",
      "xapprox = np.dot(iW5000reduced,Xapprox)*1/5000 # inverse DFT/DFS of the DFT/DFS on a reduced base\n",
      "xapprox = np.real(xapprox) # real value of the result to get rid of imaginary numerical errors"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So `xapprox` is the signal $x$ projected on a reduced base\n",
      "\n",
      "Play the signal `xapprox` and take a look at its DFS / DFT:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = np.dot(W5000,xapprox)\n",
      "\n",
      "plot(np.arange(1,5001,dtype=float)/float(5000),abs(X))\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x107860f50>"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "You will notice the difference!"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Numerical optimization: Fast Fourier Transform - FFT\n",
      "----------------------------------------------------\n",
      "\n",
      "If we take a look at the matrix form of the DFS or DFT\n",
      "\n",
      "    X = dot(WN, x) # We assume here WN to have dimensions N times N and x to have dimensions N times 1\n",
      "\n",
      "we see that it requires $O(N^2)$ multiplications. As `N` increases, the computational burden is way too high. Already in the case of the previous example with $N=5000$ the number of multiplication is not negligible.\n",
      "\n",
      "The Fast Fourier Transform (FFT) is an algorithm that optimizes the computation of the DFS/DFT by reducing the number of multiplications to $2 N \\log_2 N$. Standard FFT algorithm requires $N$ to be a power of 2, and, if it is not the case, the algorithm adds zeros (zero padding) to the signal to reach a number of samples that is a power of 2.\n",
      "\n",
      "Python implements the Fast Fourier Transform using the following command: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy import fftpack as f\n",
      "X = f.fft(x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The `fft()` function is also optimized with respect to numerical\n",
      "errors.\n",
      "    \n",
      "In the first numerical example presented in Fourier Transform, we have indeed chosen a step function of length 128 so as to fall into the\n",
      "framework of an optimized algorithm w.r.t. the computational burden and the numerical errors.\n",
      "\n",
      "The FFT function also accepts an additional parameter `M` enabling to choose how many samples of `x` are used to compute the FFT\n",
      "\n",
      "    X = scipy.fftpack.fft(x,M)\n",
      "    \n",
      "### Getting familiar with the FFT: Truncating and Zero Padding\n",
      "\n",
      "Consider a signal $x$ composed of $N$ samples and let's use the function\n",
      "\n",
      "\n",
      "    X = fft(x,M)\n",
      "\n",
      "to compute the FFT. \n",
      "\n",
      "\n",
      "#### Truncating\n",
      "\n",
      "If $M<N$ then the FFT is computed using the first $M$ samples of $x$.\n",
      "\n",
      "Let's take another two tones signal, with N=4000,"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y = np.linspace(0,3999,4000)\n",
      "x1 = np.sin((y*2*c.pi*40)/1000)\n",
      "x2 = np.sin((y*2*c.pi*50)/1000)\n",
      "x = x1 + x2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "that we can of course play using the `scipy.io.wavfile.write` function.\n",
      " \n",
      "Let's plot the spectrum and represent the frequency axis relatively\n",
      "to the number of samples, i.e., between 0 and 1. We are going to\n",
      "compute the fft using all the samples, that is"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M = 4000\n",
      "X = f.fft(x,M)\n",
      "#Plot versus the normalized frequencies\n",
      "normFrequ = np.arange(1,M+1,dtype=float)/float(M)\n",
      "plot(normFrequ,abs(X))\n",
      "ylabel('|X|')\n",
      "xlabel('Normalized Frequencies')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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9erQmTJigN998UxUVFRo6dKgk6dSpUxo6dKg+//xz9evXT5GRkd5an3/++TaP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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107e05190>"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As expected, the spectrum shows the energy of the signal concentrated on two precise frequencies.\n",
      "\n",
      "Let us now reduce the number of samples used to compute the DFS / DFT, and take for instance $M=500$: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M = 500\n",
      "X = f.fft(x,500)\n",
      "\n",
      "#Plot versus the normalized frequencies\n",
      "normFrequ = np.arange(1,M+1,dtype=float)/float(M)\n",
      "plot(normFrequ,abs(X))\n",
      "ylabel('|X|')\n",
      "xlabel('Normalized Frequencies')\n",
      "ylim([0,300])\n",
      "show()\n",
      "     "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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YkZGRAIDIyEicPn0aZrMZW7ZswWuvvYaWlhbU1NTg+PHj3X6UI9D2cY5JSUmY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\niMit/wcrsLKsAAAABElEQVSxPdzcwbas1wAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107821c90>"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "and $M=50$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M = 50\n",
      "X = f.fft(x,50)\n",
      "\n",
      "#Plot versus the normalized frequencies\n",
      "normFrequ = np.arange(1,M+1,dtype=float)/float(M)\n",
      "plot(normFrequ,abs(X))\n",
      "ylabel('|X|')\n",
      "xlabel('Normalized Frequencies')\n",
      "ylim([0,35])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x107e08dd0>"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can see that, as the number of samples decreases, it becomes more difficult to distignuish the two spectral lines that are on the contrary clearly visible on the DSF / DFT computed using all the 4000 samples. So the number of samples used to compute the DFS / DFT affects the resolution of the latter.\n",
      "\n",
      "#### Zero Padding\n",
      "\n",
      "As mentioned earlier, if $M<N$ the number of samples of the signal is increased to $M$ by adding zeros.\n",
      "\n",
      "Since truncating decreases the resolution of the DFS / DFT, one might think that zero padding increases such a resolution. But we shall see that this is not the case.\n",
      "\n",
      "So let's take the DFS / DFT computed using only 50 samples and reconstruct the signal using such a DFS / DFT."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M = 50\n",
      "X = f.fft(x,M)   # X is of length 50\n",
      "x50 = f.ifft(X)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 34
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "where we have used here the inverse FFT function `scipy.fftpack.ifft()`.\n",
      " \n",
      "We have obtained a signal $x_{50}$ of $N=50$ samples and we compute the FFT with $M=4000$, that is, we zero pad the signal so as to have the initial number of samples before computing the FFT."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M = 4000\n",
      "X = f.fft(x50,M)\n",
      "\n",
      "#Plot versus the normalized Frequencies\n",
      "normFrequ = np.arange(1,M+1,dtype=float)/float(M)\n",
      "plot(normFrequ,abs(X))\n",
      "ylabel('|X|')\n",
      "xlabel('Normalized Frequencies')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Y+IrIyJbbRo4EPv1U/jEHBTUfc7xwQXZR9+6V8xNZWcC2bUZotQY8/7w8tU2j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\nERGBYUBERGAYEBERGAZERATg/wGmJSBqI6JiZQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10780ef10>"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As shown in the plot, the two spectral lines are still not distinguishable even if the FFT has now been computed using 4000 samples."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}