DSB1.ipynb

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   "cell_type": "code",
   "execution_count": 1,
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   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy as sp\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from ipywidgets import interactive\n",
    "#plt.style.use('ggplot')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## INHOUD:\n",
    "\n",
    "* [AD Conversie](#Analog_Digital_Conversion)\n",
    "    * [Sampling](#Sampling)\n",
    "    * [Quantization](#Quantization)\n",
    "    \n",
    "    \n",
    "----    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "------------\n",
    "## Analog Digital Conversion\n",
    "\n",
    "Two steps are mandatory in order to do digital processing on analog signals.\n",
    "![Analog, discrete and digital signals](figures/analog_discrete_digital.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "# Sampling, Aliasing en Nyquist:\n",
    "\n",
    "\n",
    "### Nyquist (Folding) frequency: $$\\dfrac{f_s}{2}$$\n",
    "### Nyquist rate: $$f_s=2 f_{max}$$\n",
    "### Nyquist interval: $$\\left[-\\dfrac{f_s}{2}, \\dfrac{f_s}{2}\\right]$$\n",
    "\n",
    "### Nyquist condition: $$f_s \\geq 2 f_{max}$$\n",
    "----"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "def aliasingExample(fs=2.1,f1=1):\n",
    "\n",
    "    T = 1/fs #sampling period\n",
    "\n",
    "    tstart = 0\n",
    "    tend = 100\n",
    "\n",
    "    dt = 1./100;\n",
    "\n",
    "    t = np.arange(tstart,tend,dt) #time axis\n",
    "    n = np.arange(tstart,tend*fs) #discrete time intervals \n",
    "\n",
    "    f2 = np.mod(f1,fs) #determine aliasing frequency\n",
    "\n",
    "    if f2>fs/2:\n",
    "        f2 = f2-fs\n",
    "\n",
    "    w1 = 2*np.pi*f1\n",
    "    w2 = 2*np.pi*f2\n",
    "\n",
    "    phi = -.1*np.pi\n",
    "\n",
    "    def genSig(arg,type='sin'):\n",
    "        if type == 'cos':\n",
    "            return np.cos(arg)\n",
    "        else:\n",
    "            return np.sin(arg)\n",
    "\n",
    "    y1 = genSig(w1*t+phi)\n",
    "    y2 = genSig(w2*t+phi)   \n",
    "    y2_discr = genSig(w1*n*T+phi)\n",
    "    t_discr = n*T\n",
    "\n",
    "    fig,ax = plt.subplots(ncols=1,nrows=2,num=None, figsize=(14,10), facecolor='w', edgecolor='k')\n",
    "    ax[0].plot(t,y1,'r', label='original: ' + str(f1) + ' Hz', linewidth=3)\n",
    "    ax[0].plot(t,y2,'b', label='reconstructed: ' + str(f2) + ' Hz')\n",
    "\n",
    "    ax[0].stem(t_discr,y2_discr,linefmt='k-', markerfmt='ko', basefmt='k.', label='samples')\n",
    "\n",
    "    ax[0].set_xlabel('Time [s]')\n",
    "    ax[0].set_ylabel('Amplitude [au]')\n",
    "    ax[0].legend(fontsize='large',loc='lower right')\n",
    "    ax[0].grid('on')\n",
    "    ax[0].set_xlim([0,4])\n",
    "\n",
    "    # determine frequency axis y1\n",
    "    n = y2_discr.size\n",
    "    fvec_2 = np.arange(0,fs,fs/n); # Frequency axis start at 0 ends at Fs, resolution is Fs/N\n",
    "    fvec_2 = fvec_2[range(int(n/2))]\n",
    "\n",
    "    Y2_discr = np.fft.fft(y2_discr)/n # fft computing and normalization\n",
    "    Y2_discr = Y2_discr[range(int(n/2))]\n",
    "\n",
    "    ax[1].stem(fvec_2,2*abs(Y2_discr))\n",
    "    ax[1].set_xlabel('Frequency [Hz]')\n",
    "    ax[1].set_ylabel('Amplitude [au]')\n",
    "    ax[1].set_xlim([0,2])\n",
    "    ax[1].grid('on')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>interactive</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in the Jupyter Notebook or JupyterLab Notebook, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
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      ],
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       "interactive(children=(FloatSlider(value=2.1, description='fs', max=3.0, step=0.05), FloatSlider(value=1.0, description='f1', max=2.0, step=0.5), Output()), _dom_classes=('widget-interact',))"
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   "source": [
    "interactive(aliasingExample, f1 = (0,2,0.5), fs=(0,3,0.05))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Quantization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Supporting functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {},
   "outputs": [],
   "source": [
    "def biquant(NoBits, Xmin, Xmax, x):\n",
    "    # function pq = biquant(NoBits, Xmin, Xmax, value)\n",
    "    # This routine is created for simulation of the bipolar quantizer.\n",
    "    #\n",
    "    # NoBits: number of bits used in quantization.\n",
    "    # Xmax: overload value.\n",
    "    # Xmin: minimum value\n",
    "    # value: input to be quantized.\n",
    "    # pq: output of the quantized value\n",
    "    # I: coded integer index\n",
    "    \n",
    "    L = 2**NoBits; #Levels\n",
    "    R = Xmax-Xmin; #Range\n",
    "    Q = R/L; #Resolution\n",
    "    I = np.round((x-Xmin)/Q);\n",
    "    \n",
    "    if ( I == L):\n",
    "        I = I-1\n",
    "\n",
    "    if I < 0:\n",
    "        I = 0\n",
    "\n",
    "    pq = Xmin + I*Q;\n",
    "    return pq"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Het samplen zorgt voor een opdeling in de tijd. Kwantiseren is nodig om het signaal op te delen in de amplitude. Het aantal niveaus (levels) wordt bepaald door het aantal bits van de AD converter. \n",
    "\n",
    "Stel we hebben een 3 bits analoog naar digitaal (AD) converter, het aantal levels wat we dan verkrijgen is:\n",
    "\n",
    "\\begin{equation}\n",
    "levels = 2^{n} = 2^3 = 8\n",
    "\\end{equation}\n",
    "\n",
    "De resolutie (vaak in Volt uitgedrukt) wordt bepaald door de verhouding in het bereik van de AD converter en het aantal levels:\n",
    "\n",
    "\\begin{equation}\n",
    "resolutie (Q) = \\frac{bereik (R)}{levels} = \\frac{2}{8} = 0.25~V\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f6edda0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = 1000  # length of signal\n",
    "\n",
    "# generate signal\n",
    "A = 1\n",
    "f = 4\n",
    "t = np.linspace(0,1,N)\n",
    "x = A*np.sin(f*2*t)\n",
    "t_discr = t[::20]\n",
    "x_discr = x[::20]\n",
    "\n",
    "nBits = 3\n",
    "xQ = [biquant(nBits,-A,A,xi) for xi in x_discr]\n",
    "\n",
    "# plot (quantized) signals\n",
    "fig, ax1 = plt.subplots(figsize=(10, 6), facecolor='w', edgecolor='k')\n",
    "\n",
    "ax1.plot(t,x, 'b', label=r'continous signal')\n",
    "ax1.stem(t_discr,x_discr,label=r'discrete signal',linefmt='r-', markerfmt='ro', basefmt='r.')\n",
    "ax1.stem(t_discr,xQ,label=r'digital signal',linefmt='g-', markerfmt='go', basefmt='g.')\n",
    "ax1.set_xlabel('t [s]')\n",
    "ax1.legend(fontsize='large',loc=0)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Impulse function:\n",
    "\\begin{equation}\n",
    "\\delta (t) = \\begin{cases}\n",
    "                  \\inf & \\text{for $t=0$}.\\\\\n",
    "                  0 & \\text{otherwise}\n",
    "                \\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "### The function is defined by the integral:\n",
    "$$ \\int\\limits_{-\\infty}^\\infty \\delta (t) dt = 1$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1109c0240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,2)\n",
    "ax[0].arrow(x=0, y=0, dx=0, dy=1.5, head_width=0.1, head_length=0.1, fc='k', ec='k')\n",
    "ax[0].axis((-1,1,0,2))\n",
    "ax[0].set_title('Continuous')\n",
    "ax[1].stem((0,),(1,))\n",
    "ax[1].axis((-1,1,0,2))\n",
    "ax[1].set_title('Discrete')\n",
    "plt.subplots_adjust(hspace=9)\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step function:\n",
    "\\begin{equation}\n",
    "u (t) = \\begin{cases}\n",
    "                  1 & \\text{for $t \\geq 0$}.\\\\\n",
    "                  0 & \\text{otherwise}\n",
    "                \\end{cases}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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jd2ddhWQ2duTuTW/fTW2radSqnj5gQsnj8cD6QcqtUZxKmslx3WJnn3AIu+88Yruy3XcewdknHNL09t3StohGJf6FwLuVOQZ4NCIeBG4Ajpc0StIo4Pi8zKwTOK5bbOaUcXz+zUewy4gsNY0buTuff/MRhVe41NO+G9oWVXQ55wKyC1qjJfWRrWjYGSAirgAWkS15W0227O09+bFNki4CluVPdWH/BTFrHFWcebBqHNfD08wp41hw+58B+N4HX9nS9p3c9g9/Wb28aJuiq3pOrXI8gA8NcGweMK/ogKw2nukZOse1pcrf3DUzS4wTfxeQZ3rMrAZO/B0um40wMyvOid/MLDFO/F3AMz1mVgsn/g7niR4zq5UTfxfwxV0zq4UTv5lZYpz4O5wX9ZhZrZz4u4A812NmNXDiNzNLjBN/hwuv6zGzGjnxdwFP9JhZLZz4O5wv7ppZrZz4zcwS48TfDTzXY2Y1cOLvcJ7pMbNaOfF3Af/0opnVolDilzRd0n2SVks6p8LxL0takd9+L2lzybFtJccWNnLwZvVybFuKqv7mrqQRwGXAG4A+YJmkhRFxT3+diPhfJfU/DEwpeYotETG5cUO27XiuZ8gc25aqImf8U4HVEbEmIp4CrgZmDFL/VGBBIwZnxXjHhiFzbFuSiiT+ccDaksd9edkOJE0CDgB+WVK8m6ReSUslzRyoE0mz83q9GzduLDAss7o1PbYd1zYcFUn8lc4nB5pgmAX8MCK2lZRNjIge4O3AVyQdWKlhRMyNiJ6I6BkzZkyBYRl4y4Y6NT22Hdc2HBVJ/H3AhJLH44H1A9SdRdlH4YhYn/+7BriJ7edIrQE80zNkjm1LUpHEvww4WNIBknYhewPssIJB0iHAKODWkrJRknbN748GpgH3lLc1axPHtiWp6qqeiNgq6QzgBmAEMC8iVkm6EOiNiP43yqnA1RHb7R7zUuBKSc+Q/ZG5uHTFhNXPe/UMnWPbUlU18QNExCJgUVnZeWWPL6jQ7hbgiDrGZwV4Vc/QObYtRf7mrplZYpz4O5xnesysVk78XcB79ZhZLZz4zcwS48Tf4SLCF3fNrCZO/GZmiXHi73C+uGtmtXLi7wKe6TGzWjjxm5klxom/w3nLBjOrlRN/N/CyHjOrgRO/mVlinPjNzBLjxN8FPNFjZrVw4jczS4wTfwfr/10QX9s1s1o48ZuZJcaJ38wsMYUSv6Tpku6TtFrSORWOny5po6QV+e39JcdOk/SH/HZaIwefuv4vb3k//qFzbFuKqv7mrqQRwGXAG4A+YJmkhRV+WPp7EXFGWdu9gfOBHrL9xJbnbR9pyOjN6uDYtlQVOeOfCqyOiDUR8RRwNTCj4POfACyOiE35G2IxMH1oQzVrOMe2JalI4h8HrC153JeXlftvku6S9ENJE2psi6TZknol9W7cuLHAsKx/mx6v6hmypse249qGoyKJv1JaKd8a7Hpg/4h4GfAL4Koa2maFEXMjoiciesaMGVNgWGZ1a3psO65tOCqS+PuACSWPxwPrSytExMMR8WT+8BvAUUXb2tCFt+asl2PbklQk8S8DDpZ0gKRdgFnAwtIKkvYreXgKcG9+/wbgeEmjJI0Cjs/LrIE80zNkjm1LUtVVPRGxVdIZZEE9ApgXEaskXQj0RsRC4H9KOgXYCmwCTs/bbpJ0EdkbDODCiNjUhNdhVjPHtqWqauIHiIhFwKKysvNK7s8B5gzQdh4wr44x2gB8cbd+jm1Lkb+5a2aWGCd+M7PEOPF3sGe3bPBcj5nVwInfzCwxTvxmZolx4u9gUflL0GZmg3LiNzNLjBO/mVlinPg72N9X9bR3HGbWWZz4zcwS48RvZpYYJ/4u4N/cNbNaOPGbmSXGib+D+eKumQ2FE7+ZWWKc+M3MEuPE38H6t2zwTI+Z1cKJ38wsMYUSv6Tpku6TtFrSORWOnynpHkl3SfoPSZNKjm2TtCK/LSxva9ZOjm1LUdXf3JU0ArgMeAPQByyTtDAi7impdifQExFPSPoX4IvA2/JjWyJicoPHbXhVT70c25aqImf8U4HVEbEmIp4CrgZmlFaIiF9FxBP5w6XA+MYO06wpHNuWpCKJfxywtuRxX142kPcBPyt5vJukXklLJc0cwhjNmsWxbUmqOtVD5UUjFX8BRNI7gR7gNSXFEyNivaQXAb+UtDIi/lih7WxgNsDEiRMLDMv6/yd4y4Yha3psO65tOCpyxt8HTCh5PB5YX15J0nHAucApEfFkf3lErM//XQPcBEyp1ElEzI2InojoGTNmTOEXYFaHpse249qGoyKJfxlwsKQDJO0CzAK2W8EgaQpwJdkbY0NJ+ShJu+b3RwPTgNILZ9YAvrg7ZI5tS1LVqZ6I2CrpDOAGYAQwLyJWSboQ6I2IhcAlwHOBHyjLQn+OiFOAlwJXSnqG7I/MxWUrJqwOEf7N3Xo4ti1VReb4iYhFwKKysvNK7h83QLtbgCPqGaBZMzm2LUX+5q6ZWWKc+DuYJ3rMbCic+M3MEuPE3wXkZT1mVgMn/g7mRT1mNhRO/GZmiXHi72T9u3O2dxRm1mGc+M3MEuPE3wV8bdfMauHE38HCK/nNbAic+M3MEuPE3wU802NmtXDi72Bex29mQ+HEb2aWGCf+LuAtG8ysFk78HcwzPWY2FE78ZmaJceLvAp7pMbNaOPF3MP/mrpkNRaHEL2m6pPskrZZ0ToXju0r6Xn78Nkn7lxybk5ffJ+mExg3drH6tjO2V6x5l2sW/5Lo71xUe33V3rmPaxb/kgHN+WnPbetu3q601X9UfW5c0ArgMeAPQByyTtDAi7imp9j7gkYg4SNIs4AvA2yQdCswCDgPGAr+Q9OKI2NboF5Iyz/QMTTtie93mLcy5ZiUAM6eMG3R81925jjnXrGTL09tqbltv+3a1tdaomviBqcDqiFgDIOlqYAZQ+uaYAVyQ3/8hcKmyNYYzgKsj4kngT5JW589362Adrlr/GIed9/NaXkeSnunfltmT/EPV0tge//gGPnfz5dmDJSN4YOLIwUf3581csLXC35Eibett3662udMffAyAB379vEL1G9m+U9t+v4b6RRL/OGBtyeM+4OiB6kTEVkmPAvvk5UvL2lb8ky9pNjA7f/jXey468b4CY6tkNPDQENvWo1398q4vMPpdib3mOvuelP/b9Ngujevn7P483r3m988ee2rJ6uWDDXKXfQ86aqBj1drW275dbct0any1q99J1atkiiT+SqeT5VcVB6pTpG1WGDEXmFtgPIOS1BsRPfU+T6f0286+u+A1Nz22Oz2u29m3X3PzFLm42wdMKHk8Hlg/UB1JOwHPBzYVbGvWLo5tS1KRxL8MOFjSAZJ2IbugtbCszkLgtPz+W4BfRrbWcCEwK18ZcQBwMHB7Y4ZuVjfHtiWp6lRPPq95BnADMAKYFxGrJF0I9EbEQuCbwL/nF7g2kb2ByOt9n+xi2VbgQy1Y0VP3x+oO67edfXf0a+6w2O7o/9Yd1m87+25Jv/KXgMzM0uJv7pqZJcaJ38wsMV2d+CV9TFJIGt2i/i6R9DtJd0m6VlKxb6sMvb9BtxtoYr8TJP1K0r2SVkn6SKv6zvsfIelOST9pZb/DRavjOu+z62M7pbju2sQvaQLZV/H/3MJuFwOHR8TLgN8Dc5rVUcl2AycChwKn5tsItMJW4KyIeClwDPChFvYN8BHg3hb2N2y0Ka4hjdhOJq67NvEDXwY+Tgt/ryQiboyIrfnDpWRru5vl2e0GIuIpoH+7gaaLiAcj4o78/uNkwdqSTVgkjQf+Cfi3VvQ3DLU8riGN2E4prrsy8Us6BVgXEb9t4zDeC/ysic9fabuBlu+Ale9WOQW4rUVdfoUs8T3Tov6GjWES15BAbHd7XBfZsmFYkvQLYN8Kh84FPgEc3+p+I+LHeZ1zyT42fqcZY+gfSoWylp4FSnou8CPgoxHxWAv6OwnYEBHLJR3b7P7aoV1xXa3vlGI7hbju2MQfEcdVKpd0BHAA8Nt818rxwB2SpkbEX5rVb0n/pwEnAa+P5n5Joq1bBkjamezN8Z2IuKZF3U4DTpH0RmA34HmSvh0R72xR/03XrrgerO+SMXR9bKcS113/BS5J9wM9EdH0nfYkTQe+BLwmIjY2ua+dyC6yvR5YR7b9wNsjYlUz+837FnAVsCkiPtrs/gYYw7HAxyLipHb0326tjOu8v66P7ZTiuivn+NvoUmAvYLGkFZKuaFZH+YW2/u0G7gW+34qkn5sGvAt4Xf46V+RnK9a9UojtZOK668/4zcxsez7jNzNLjBO/mVlinPjNzBLjxG9mlhgnfjOzxDjxt4mkkZL+R8njsZJ+2KS+Zko6r+T+oSXH/lXS65rRr1kjSLpJUk0/QC7pK5JeXaXOSZI+Xd/oOpMTf/uMBJ5N/BGxPiLe0qS+Pg5cnt+fSbbjYb+vAS3b0tms2STtDRwTEUuqVP0p2Tdm92jBsIYVJ/72uRg4MP+SyCWS9pd0N4Ck0yVdJ+l6SX+SdIakM/O9upfmgY2kAyX9XNJySTdLekl5J5JeDDwZEQ9J+gfgFOCSvN8DI+IBYB9JlfZoMduBpD0l/VTSbyXdLeltefl5kpblZXPzb8L2n7F/WdKSfK/7V0i6RtIfJH0mr7O/sv3+r1K25/8PKyVkScdLulXSHZJ+kO+rU+4twM9L2twv6dN5m5X975N824mbyLahSIoTf/ucA/wxIiZHxNkVjh8OvJ1si9rPAk9ExBTgVuDdeZ25wIcj4ijgY/z9rL7UNKB/q9lbgIXA2Xm/f8zr3JHXMytiOrA+Io6MiMP5e5K9NCJekZftzvYJ9amIeDVwBfBj4ENkMX66pH3yOocAc/M9/x+j5BMxgLIfnvkkcFxEvBzoBc6sML5pwPKysofyNl8ne6/06wVeVfyldwcn/uHrVxHxeL4vyqPA9Xn5SmD//EznH4AfSFoBXAnsV+F59gOq7a2yARjbmGFbAlYCx0n6gqRXRcSjeflrJd0maSXwOuCwkjYLS9quyve+fxJYw983ZFsbEb/J738b+Meyfo8hm6b8TR7zpwGTKoyvUsz3b7i2HNi/pDzJ2O/Y3TkT8GTJ/WdKHj9D9v/tOcDmiJhc5Xm2AM+vUme3vJ5ZVRHxe0lHAW8EPi/pRuCLZJ84eyJiraQLyOKqX2n8lsd2fx4q3z+m/LGAxRFxapUhbinru7T/bWyf95KMfZ/xt8/jZJteDUm+T/ifJL0Vsp0FJR1Zoeq9wEFV+n0xcPdQx2JpkTSWbOrx28C/Ai/n74n2ofzT6FAWKkyU9Mr8/qnAr8uOLwWmSTooH8ce+TWscuUxP5gkY9+Jv00i4mGyj6x3S7pkiE/zDuB9kn4LrKLyz9MtAab0X2gj+xm7s/MLxQcq23/8ILK5TrMijgBuz6dbzgU+ExGbgW+QTeVcR7aVcq3uBU6TdBewN9l8/LPyac/TgQV5naXADgsayFbrHFuwz9fm9ZPi3TkTIOmrwPUR8YsKx94EvDwiPtX6kZlllP3U4U/yC8ONeL5fAyflf5AGqvNC4LsR8fpG9NlJfMafhs8BA61V3gn4Py0ci1krnAVMrFJnYl4vOT7jNzNLjM/4zcwS48RvZpYYJ34zs8Q48ZuZJcaJ38wsMf8fuTuTtdhNPfwAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113c3e828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def step_func(t):\n",
    "    if t >= 0:\n",
    "        u = 1\n",
    "    else:\n",
    "        u = 0\n",
    "    return u    \n",
    "        \n",
    "fig, ax = plt.subplots(1,2)\n",
    "tc = np.arange(-6,6,1e-2)\n",
    "td = np.arange(-6,6,1)\n",
    "uc = np.array([step_func(ti) for ti in tc])\n",
    "ud = np.array([step_func(ti) for ti in td])\n",
    "ax[0].plot(tc,uc)\n",
    "ax[0].axis((-5,5,0,2))\n",
    "ax[0].set_title('Continuous')\n",
    "ax[0].set_xlabel('time (t)')\n",
    "ax[1].stem(td,ud)\n",
    "ax[1].axis((-5,5,0,2))\n",
    "ax[1].set_title('Discrete')\n",
    "ax[1].set_xlabel('sample (n)');"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***"
   ]
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