random_signals_LTI_systems/acoustic_impulse_response_measurement.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Random Signals and LTI-Systems\n",
    "\n",
    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Measurement of Acoustic Impulse Responses\n",
    "\n",
    "The propagation of sound from one position (e.g. transmitter) to another (e.g. receiver) conforms reasonable well to the properties of a linear time-invariant (LTI) system. Consequently, the impulse response $h[k]$ characterizes the propagation of sound between theses two positions. Impulse responses have various applications in acoustics. For instance as [head-related impulse responses](https://en.wikipedia.org/wiki/Head-related_transfer_function) (HRIRs) or room impulse responses (RIRs) for the characterization of room acoustics. \n",
    "\n",
    "The following example demonstrates how an acoustic impulse response can be estimated with [correlation-based system identification techniques](correlation_functions.ipynb#System-Identification) using the soundcard of a computer. The module [`sounddevice`](http://python-sounddevice.readthedocs.org/) provides access to the soundcard via [`portaudio`](http://www.portaudio.com/)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.signal as sig\n",
    "import sounddevice as sd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Generation of the Measurement Signal\n",
    "\n",
    "We generate white noise with a uniform distribution between $\\pm 0.5$ as the excitation signal $x[k]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "fs = 44100  # sampling rate\n",
    "T = 5  # length of the measurement signal in sec\n",
    "Tr = 2  # length of the expected system response in sec\n",
    "\n",
    "x = np.random.uniform(-.5, .5, size=T*fs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Playback of Measurement Signal and Recording of Room Response\n",
    "\n",
    "The measurement signal $x[k]$ is played through the output of the soundcard and the response $y[k]$ is captured synchronously by the input of the soundcard. The length of the played/captured signal has to be of equal length when using the soundcard. The measurement signal $x[k]$ is zero-padded so that the captured signal $y[k]$ includes the complete system response.\n",
    "\n",
    "Be sure not to overdrive the speaker and the microphone by keeping the input level well below 0 dB."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Playback level:  -6.02060087394  dB\n",
      "Input level:  -2.23183822753  dB\n"
     ]
    }
   ],
   "source": [
    "x = np.concatenate((x, np.zeros(Tr*fs)))\n",
    "y = sd.playrec(x, fs, channels=1)\n",
    "sd.wait()\n",
    "y = np.squeeze(y)\n",
    "\n",
    "print('Playback level: ', 20*np.log10(max(x)), ' dB')\n",
    "print('Input level: ', 20*np.log10(max(y)), ' dB')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Estimation of the Acoustic Impulse Response\n",
    "\n",
    "The acoustic impulse response is estimated by cross-correlation $\\varphi_{yx}[\\kappa]$ of the output with the input signal. Since the cross-correlation function (CCF) for finite-length signals is given as $\\varphi_{yx}[\\kappa] = \\frac{1}{K} \\cdot y[\\kappa] * x[-\\kappa]$, the computation of the CCF can be speeded up with the fast convolution method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = 1/len(y) * sig.fftconvolve(y, x[::-1], mode='full')\n",
    "h = h[fs*(T+Tr):fs*(T+2*Tr)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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AKIXBM7E770wC47hxyS+gSOFaiJL0hz+EAChJ/x19Kv7iF9JHPypdf33u9uvrc0czr10b\n5otdUuf735fWrSv5cDul2D5/7z1phx2S+ebm4vUmT5YWLWp7/QsXSkuXdq2NaNuUKX3nvfv66z3/\neomvSFCr4isldCt379RN0lclmaTTJP1O0q87u642ttFP0jxJu0vqL+lFSXvn1Tlb0vXR9MmS7o6m\n94nq94+Wnxe1t911Rst7rLU1+btxoxc1fXpSr6nJ/cADk/saG91bWtwXLy6+bOyRR9zvvTdMv/CC\n+7XXuq9Ykdzf0uL+3nttr6Ncra3uS5cmbY5t2FBYtn5957YxcmQ4mbZgQW75woXuP/hB0oaVK92b\nm8N9gweH+9N+9Sv3a64J02vWuL/8cnJfc3PYxjPPJGVr15bXvi9/OazrxRfDOuLHOW2a+1tvhW25\nh3bGz3t8gjBNcv/kJ8P0e++FZWPx/mttTZ6HDRsK1xH7wAfcH3ssTC9Z4v6nP4Xnp6UllL35ZnmP\nrVzr17u/9lpo29NPJ4/vkkuSaW7Jba+9wt/3388t33vv3Pn5891/+9vC5d1z5xctcv/HP8J+iMsa\nG90PPjiZ/8Y3kv3v7n7CCaF8333dv/td91/8Iswfemjuvm1tDe+l+fPD/K67ut96a5iePdv9lVeS\nup/7XFjH6tVh/qKLwuv/e98L5a2tyWMePz53O2vX5j6+1tZkPn4P5T9u97CtGTPC6y/2+uvugwaF\nOvFnwiuvhPfCNtvkbnfVqtz5998Pz01XtLS4z5sXHsPKle7r1oXy//gP9wkTcutu2BD+Pv108XWt\nXx/2bTELFrjPmROmzzrL/cILc+9fvNj94x93v/zyMP/226E96e+fGTMKv48k91Gjim+ztdV96FD3\n5cuT59Y9fF499VQy/8Yb7nPnhs+uhQvDduPHW+pzy939Jz9J2vPuu7n3nXFG28vGz8Xy5e6HHx4+\nm4t91i5cmKx79eqwv/K/rzZuDK+j++8v3M64ce5PPBGmP/IR90MOSfaje9jeSy/lLjNxovvo0cn8\nO+8kn4/pz3r38J09a1bY/po14fst3t6MGcVfnxs2uC9bVli+enVu25YsCducPTspW7nS/aqr0u2X\nu3cwZ3WosvSspBuj4Le3pP8j6WJJH5a0VUc3Xsb2PiVpcmp+rKSxeXUmSzokmq6T9E40fYmki/Pq\nHVrOOqPynA+tG290/9KXCj/Uf/7z3Plf/artL5H99gt/GxrC3wEDwt9rr03qXH557jI//an77rsX\nrmv77ZPpj37U/fTTk/nVq91/9KPCZRYtSoKT5P6tb4W/O+3k/qlPJeV//7v7Djsk8/GLd8GC3PUt\nXuy+3Xa5Zffe6z5pUjJ/7LHhgyX+oorLzz47d7mTTkqmX3st/P3xj5OybbfNrf/b34YPz3j+ttvc\n77gjt87ZZ7v/8peFz8NNNxWW/fjH7gcckFt23XXuX/lKYd0RI9xvuMH9iCOK7+c//jGZfvzxZHrm\nzNx9/Y1vhA/4L3wheU2Uul1xRe78Ndck0/ffn7uvxoxx79ev9LoeeqjtbV18cdv3Z/32b/+WO5//\n2ix1i98Hlbql31/l3HbaKXyppstWrAh/hwxJyhobc+s8+aT7s8+G6fQX9llnuV92WW7dm28O4TNd\n9o9/FLblnXcKy15/vXC76X9W0p8zxfbR+PG5ZZtvngTs738/KR850n3YsGT+rruS6QceSKbvv9/9\n299O5j/xifC3f//wGR2/5+6+O3e7P/hB+PvLX4YwXqy9f/pTCEPpEC25n3xy2/tw4MCkHfFtypRk\n+je/CZ9h+a/RPfYo3NbMmcW3sXx58o/OTjuF/flf/1VYL/25Uewz9uabw+M58cTyX6O//rX7Bz9Y\n+vGlX3vDh+eWbdzofsopYTr9unzhhcLHt2xZ+22ZMcN9662T+Z/9rLzHkP58njw5ec/tuWduvfif\nsfjxfehD7v/5n+E7YvTopF5LS+E2wj9j6nAI7NB1As1spKQXJB0WBar9Ja2X9Jykv7r7X8peWXnb\n+5qko9399Gj+mwqB79xUnRlRnSXR/DxJh0hqkDTF3f8Yld8s6ZFosZFtrTMqd5U9DgboWy66SLr6\n6mq3AgBQvsoODJG7T44m749uMrM6SfspnH7tblVOYQ2p6froBvR9HfjfEABQFU3RrfPKHhhiZgcV\nK3f3Fnd/zt0ntFe3ExZLGpaaHyYpv/vxYkm7RtutkzTQ3ZcVWXZotGw564w0pG71HWr4Kad0qDpq\n3HXXdW35zXrZZdnjELj99tVtR1916aXVbkGhyy9vv84jj7RfJ/a5z3W+LUBf8KEPde/6Cq9AUK/c\nnNIJHeifd5mk0WXeLuumPoF1kl5TGMSxuUoPDLkhmh6lwoEhmyv0WXxNYWBIu+tM9wmcPTv0kZkw\nIfRta252//3vQ/+Pxx8P5+tbW8PtpZeS8/PuoVOxFPpu/fSn7lOnhv4s06eH/i9PPx36lc2fnwyG\nmDo1bDPuhD10aChvaUk6hsa3118PfR7iPoRr1iQdc+M6W24Z+hcsWBAGQcR9NkaPDnXjeo2N4f7W\n1tAvZcIE91dfTQa1XHBB6ER79NGhr4176IM3YUJYf9yBOr3t99/P7bf29tuhb8of/hCeA8n91FND\nJ2QprLe11f3OO8P8IYeE7ab7Uc2Z4/7Pf4bnZbfdkuc67rQshX3iHjrkxmWHHZZ0hk73g2ltDR3P\nr7wy97l98MFQN+7jk35cUuiDOXVq6Dif7u/kHvbT9Onh8cZtmDs33Bf3cYoHf7z5ZuiHOH58qN/c\nHF4bLS1hX221Vaj/mc8k60+3Jd7e+PHhsUydGsrefz/0Zdq4MXQePuCAsD//9rdQb5ttwvJf+lLY\n1vnnu9fXF/Yzye/vmdVb3Je3vdvmm4eO2vnlRx+dO5/ftza+5fc7zu+LF9/Wrs3tV/XWW8n0+vWh\nHVLoE5t+vdTXJ/0Z0+Vnnpk7bxb6oB5+eG7/vXffDa+pdD85Kbzn4j6C8brXrQuDrOKywYNDefy+\nlUKf4vT7Me6PGA8OkcI64kFkcdmyZeE9v2BBWK8U+tTF99fVJdO/+114P8ycGd5T55+fPNbHHsv9\nnMjvI9fSEt7r8WOaMCGZvuuu0I93xYrcz6j0Otaudb/vvtDf0D30SzML/bfTr4H0c//yy+G9mu4v\ntnRpeN7j74B168Jzt2JF+Lz46EfdL700t4/600+H5e65JwxsSQ/geeaZpL9kPMDmuuvcr746WceI\nEaFvc7rfW3Oz+/PPhz51d9wRvguPOy70eUv3qYtfjw89FF4/d92V7Pd9900GIMWf2+nXcvp76YYb\nwt8zzgiv0cbGpA/qqFHu554bXitvvhm+Y9evD89N+rX4xS+GNkuh79+aNbnPezxo5YILwmvtYx8L\n5Y8+Gsqvuio8V+PGJcutWxe+t5uawvzPfha+C+6+O/f94u7+3HPu++8fHvODD4byf//33H01aVLu\ne+WMM8Jrb/783P6PL7+cTG+2We7rJvTvl7tXcGBIyZVIB0rq3x3rKrLuYyS9qjCi95KobJyk46Lp\nAZImSporaYqk3VPLXhotN1uh32DJdRbZrnfG178eBlR0h/QooNjo0eFFldbSEjqb5otHBfa0JUty\nR2y1tuaOboz99Kfhw7mYWbNy5ydOzB0BXEz8RZHvpptyR8O5h4Ev6eAaj35syw9/mLyx8/3lL0n4\nzBd/4NW6V17J/QKUCgej9LVbsUE/xW4PPJAEq/Tt1Vdz5xcvzv2Sib+wVq/OrdfSEr5sLr7Yfeed\nk/dLPFI9/SXy9tu5wTIeHRl/0cbv86VLk9GEa9aEYNKe9Jeae/jMKTXCNn/UpHt4z6RH90run/98\nbp3ly3NHyba2hi/r9FUA0sE0/qd63brCEcDtaW0NwejKK91PO638Zf7612S+uTk8Dx29EsPq1aWv\nHlFKY2PyPLz2Wu5oUPewvvzRr13R3Bz+iW7PqFEhNKeVsy/iUb6ltPcZG8v/vO5ua9aE8Nye444L\n7/FYa2vhiOD33ivc7wsXFv/Oyzd/fu7+2LChvNdQqeexMyGwQwND0szsG5IOVhgx/FdJR7n7rZ1a\nWQ3iF0OQNS0tyW/1xq6+OgwS6a2GDAk/5RZ77TXpIx9J5t3Dacsnn8xdbvbscC2/2OrV4SLTixeH\nCz4/+KB02GHhV0TS1/Zrbg6nbNK/HLJokTR0aPiJuyOPDD+F1173gJaWcK3Bjv5sXUe9/bY0eHDh\nfu/K+gYMCD+5B6BndeYXQ9r8KDKzzaKwV8xGST+RtFLhMis7dmTDAGpfb+vLmC+/X9oee4RgltbY\nmPt7wz/6kbTjjsm0JG29tXT00dK3vy3ddJM0bVoIgFLSX/Tee4v/asjQoeHvEUdIb7xR3nNaV1f5\nACiFPkvdFQDj9REAgd6jvY+j8yRNi2fM7H/NbHwUDIdIanX3h939HHf/RSUbCqDn1XIIrK9v+/6T\nT5Yuu6z0/WedFf4OGCBtsUVS3r9/Mn/FFYWhcccdpYMPTub/9V+lr3xF+upXk7Jly3J/qi62225t\ntxkAelKbp4PNrJ+kk9z9rmj+JEmvKFxw+RBJwyW9rdAXr8ndn694i3sIp4ORNcVOB//mN7m/l1tt\nm20mtbaGaXfpvvukr32teN347Rufmt1yS+mf/yx+X+zJJ6UDDwx1BwwIp3cBoDfo9tPB7r4xDoDR\n/ER3f8Xdb3b30929XtK/SXpJ0qc702gAtSs/JFXbww9LRx0V+uRJ0uGHS1//epg+5ZQQ7tJH6aTQ\nJ+/JJ3P7BpoVf2yf/az0gQ+EsEkABNDXdXpgSF/HkUBkTbEjgb/7nXTuucXrV9Kee0rz5uWWXXKJ\n9POfF6+/bl0IbwCQVd1+JBBAtn3yk9XZ7jbbFJZtvXXp+gRAAOg4QiCAkgYN6rltfec7yXT6VO3R\nR0uvvCJ973s91xYAyAJCIICSKtUnsNhgkzFjCsvGjpUmT5b22ScM1AAAdJ8iV7UCgKBSITAe4Rtb\nt07aaqtwGnj1aumWW6T99qvMtgEAAUcCAZTU3SGw1HX7ttoqd3sEQACoPEIggJIGD67MeksNvK/l\ni1MDQF/D6WAAJW2/fdfXscce0uuvh+n0b+oW8+ij4dQwAKDy+L8bQEXcFV1m/kMfCn/33jtc3PmO\nO6SLLpLWry9c5uCD2/85OABA9+Bi0SVwsWhkTbGLRbt3vl+gu/TMM+Gn2o44ovjRv/aODAIAysPF\nogFUnXsS6j71KWmLLarbHgBAcYRAAB0yblz3revCC7kINABUCwNDAHTIHnt0rP6IEaEfYDFXX931\n9gAAOocjgQCKGj26eHlH++/17y+demrX2wMA6F6EQACSCgeA7LRTYZ2//rX4sg0N3d4cAECFEQIB\nlGXYMOmzny1+39ln92xbAABdRwgEUFT+kcF+/dpf5utfr0xbAADdjxAIoKg4BC5fXrw8bdttw19+\n9g0Aeg9GBwMoKr50y+DB0rXXSkOHhvliA0PKOUoIAKgthEAARaV/N/i888pbhl/+AIDeg5M3ADqk\nVNDr10866KCebQsAoPM4EgigW7S0VLsFAICO4EggAABABtVsCDSzbc2s0czmmNljZjaoRL3RUZ05\nZnZaqvwgM5thZnPN7NpU+dVmNsvM/m5m/2VmA3vi8QB9Bf3+AKBvqNkQKGmspEZ330vSE9F8DjPb\nVtLlkkZEtytSoe4GSWPcfbik4WY2Mip/TNLH3H0/SXMkXVLZhwEAAFB7ajkEHi/p9mj6dkknFqlz\ntKTH3H2lu6+U1CjpGDPbWdLW7j4tqndHvLy7N7p7a1Q+VdLQSj0AoC/KPxL45z9Xpx0AgK6p5RC4\no7u/FU2/JWnHInV2kbQoNb9I0pAi5Yuj8nzflvRw15sKZMfAVAeK88+XvvSl6rUFANB5VR0dbGaN\nkor8TL1+mJ5xdzezbu2JZGY/lLTB3f/UnesF+rr99692CwAA3aGqIdDdjyx1n5m9ZWY7ufvS6PTu\n20WqLZZUn5ofJul/ovL0ad6hUVm87v8r6VhJX2yrfQ0NDZum6+vrVV9fX7IuAABAT2lqalJTU1OX\n1mFeo0P9zOyXkpa5+1VmNlbSIHcfm1dnsKTnJB0oyeJpd19pZlMlnSdpmqSHJP3W3SdHA0R+Lenz\n7v5uG9sIoJM/AAAMyElEQVT3Wn1ugErYuFGqS/1bWOrlP3++tMceYfqCC6Tx4yvfNgBA28xM7l7k\n191Lq+U+gVdKOtLM5kg6PJqPL/1ykyS5+wpJP5H0rELYGxcNEJGksyXdLGmupHnuPjkq/52kD0pq\nNLMXzOz6nnpAQF/A/0YA0DfU7C+GuPtySUcUKX9O0ump+dsk3Vai3ieKlA/v3pYCAAD0PrV8JBBA\nDTvsMOnYY6vdCgBAZ9XskUAAtSk+HfzUU9VtBwCgazgSCAAAkEGEQAAdwsAQAOgbCIEAAAAZRAgE\nAADIIEIggA7hdDAA9A2EQAAAgAwiBALokP79q90CAEB3IAQC6JDdd5dmzKh2KwAAXWVOB5+izMx5\nbpAlGzdKdanLx/PyB4Dew8zk7taRZTgSCAAAkEGEQAAAgAwiBAIAAGQQIRAAACCDCIEAAAAZRAgE\nAADIIEIgAABABhECAQAAMogQCAAAkEGEQAAAgAwiBAIAAGQQIRAAACCDCIEAAAAZRAgEkOMjH5Ge\neabarQAAVBohEECOrbaSDj202q0AAFQaIRAAACCDajYEmtm2ZtZoZnPM7DEzG1Si3uiozhwzOy1V\nfpCZzTCzuWZ2bZHlvm9mrWa2bSUfBwAAQC2q2RAoaaykRnffS9IT0XyOKMBdLmlEdLvCzAZGd98g\naYy7D5c03MxGppYbJulISQsq+xCA3ses2i0AAPSEWg6Bx0u6PZq+XdKJReocLekxd1/p7islNUo6\nxsx2lrS1u0+L6t2Rt/w1kn5QmWYDAADUvloOgTu6+1vR9FuSdixSZxdJi1LziyQNKVK+OCqXmZ0g\naZG7v9TtLQb6AI4EAkA21FVz42bWKGmnInf9MD3j7m5m3g3b21LSpQqngjcVd3W9AAAAvU1VQ6C7\nH1nqPjN7y8x2cvel0endt4tUWyypPjU/TNL/ROVDU+VDFY4MfkTS7pL+buFwx1BJz5nZCHcvWH9D\nQ8Om6fr6etXX1+dXAfocjgQCQO1rampSU1NTl9Zh7l0+wFYRZvZLScvc/SozGytpkLuPzaszWNJz\nkg5UOKL3nKQD3X2lmU2VdJ6kaZIekvRbd5+ct/x8SQe5+/Ii2/dafW6ASti4Uaqrkw44QHr++Wq3\nBgDQEWYmd+/Qv/G13CfwSklHmtkcSYdH8/GlX26SJHdfIeknkp5VCHvjogEiknS2pJslzZU0Lz8A\nRkh5QCQ+Ati/f3XbAQDoGTV7JLDaOBKIrGltlfr1kxYtkoYMqXZrAAAd0ZkjgYTAEgiByJo4BLa2\n0i8QAHqbvnY6GAAAABVCCAQAAMggQiAAAEAGEQIBAAAyiBAIAACQQYRAAACADCIEAgAAZBAhEAAA\nIIMIgQAAABlECAQAAMggQiAAAEAGEQIBAAAyiBAIAACQQYRAADnMqt0CAEBPIAQCAABkECEQAAAg\ngwiBAAAAGUQIBAAAyCBCIAAAQAYRAgEAADKIEAgAAJBBhEAAAIAMIgQCAABkECEQAAAggwiBAAAA\nGUQIBAAAyCBCIAAAQAbVbAg0s23NrNHM5pjZY2Y2qES90VGdOWZ2Wqr8IDObYWZzzezavGXONbNZ\nZvaymV1V6ccCAABQa2o2BEoaK6nR3feS9EQ0n8PMtpV0uaQR0e0KMxsY3X2DpDHuPlzScDMbGS3z\nBUnHS9rX3T8u6VcVfyQAAAA1ppZD4PGSbo+mb5d0YpE6R0t6zN1XuvtKSY2SjjGznSVt7e7Tonp3\npJY/S9Iv3L1Zktz9nUo9AAAAgFpVyyFwR3d/K5p+S9KORersImlRan6RpCFFyhdH5ZI0XNLnzGyK\nmTWZ2Se7t9lA7+Re7RYAAHpSXTU3bmaNknYqctcP0zPu7mbWXV9RdZIGu/uhZnawpImS9ihWsaGh\nYdN0fX296uvru6kJAAAAndfU1KSmpqYurcO8Rv/9N7PZkurdfWl0evcv7v7RvDqjojpnRvM3Svof\nSf8b1d87Kj9F0ufc/Swze0TSle7+v9F98yQd4u7L8tbttfrcAJWwcaNUV8cRQQDojcxM7m4dWaaW\nTwdPkjQ6mh4t6YEidR6VdJSZDTKzwZKOlPSouy+VtNrMDjEzk3SqpP+OlnlA0uGSZGZ7Sdo8PwAC\nAAD0dbUcAq+UdKSZzVEIbVdKmy79cpMkufsKST+R9KykaZLGRQNEJOlsSTdLmitpnrtPjspvlbSH\nmc2QdJekTZeVAQAAyIqaPR1cbZwORtZwOhgAeq++djoYAAAAFUIIBAAAyCBCIAAAQAYRAgEAADKI\nEAgAAJBBhEAAAIAMIgQCAABkECEQAAAggwiBAAAAGUQIBAAAyCBCIAAAQAYRAgEAADKIEAgAAJBB\nhEAAkqTNNpMuuKDarQAA9BRz92q3oSaZmfPcAACA3sDM5O7WkWU4EggAAJBBhEAAAIAMIgQCAABk\nECEQAAAggwiBAAAAGUQIBAAAyCBCIAAAQAYRAtHnNDU1VbsJ6AL2X+/Fvuvd2H/ZQwhEn8MHWe/G\n/uu92He9G/svewiBAAAAGUQIBAAAyCB+O7gEM+OJAQAAvUZHfzuYEAgAAJBBnA4GAADIIEIgAABA\nBmU6BJrZSDObbWZzzeziIvcPMLN7ovunmNlu1Wgniitj/33PzF4xs7+b2eNmtms12oni2tt/qXpf\nNbNWMzuwJ9uH0srZd2Z2UvT+e9nM/tjTbURpZXx27mpmfzGz56PPz2Oq0U4UMrNbzewtM5vRRp3f\nRvv272Z2QFvry2wINLN+kq6TNFLSPpJOMbO986qNkbTM3YdLGi/pqp5tJUopc/89L+kgd99P0r2S\nftmzrUQpZe4/mdnWks6XNKVnW4hSytl3ZjZc0lhJh7n7xxX2IWpAme+9yyTd7e4HShol6fqebSXa\ncJvCvivKzI6VtGeUW74j6Ya2VpbZEChphKR57v6GuzdLulvSCXl1jpd0ezR9n6Qv9mD70LZ295+7\nN7n7+9HsVElDe7iNKK2c958k/UTSlZLWS+rQqDdUTDn77nRJ17n7Kkly93d7uI0orZz91yppYDQ9\nSNLiHmwf2uDuT0pa0UaVTbnF3adKGmRmO5aqnOUQOETSwtT8oqisaB13b5G0ysy27ZnmoR3l7L+0\nMZIermiL0BHt7r/o9O8Qd4/3G5cyqA3lvPeGS/oXM/ubmT1jZkf3WOvQnnL2X4Okb5rZQkkPSTq3\nZ5qGblBs/5Y8AJLlEMgXSu9W9v4zs29KOlDS1ZVrDjqozf1nZptJukbSheniirYI5SrnvVcnaU9J\nn5d0iqSbzGxg24ugh5Sz/74h6TZ3HybpWEl3VrZJ6Gb5n5Ul93mWQ+BiScNS88MUEnN+nV0lyczq\nJA109+U90zy0o5z9JzM7QtKlko6PTn2gNrS3/7aW9DFJTWY2X9KhkiYxOKQmlPPeWyTpQXff6O5v\nSJqjEApRfeXsv29LmihJ7j5F0hZmtn3PNA9dlL9/h6qN0/lZDoHTJQ03s93NbHNJJ0ualFdnkqTR\n0fTXJD3Rg+1D29rdf9GoqH+XdBx9kmpOm/vP3Ve5+w7u/mF3/7DCwJDj3P35KrUXiXI+Ox+QVC9J\nUXjYS9LrPdlIlFTO/ntT0hGSFA0a2YLP0F5jkqTTJMnMDpW00t3fKlW5rqdaVWvcvcXMzpH0qKR+\nkm5x91lmNk7SdHd/UNItkiaY2VxJyxRGSaEGtLP/nnX3PyuMBv6ApHvNTJIWuPuJVWs0Ninz/Yca\nVM6+c/dHzewoM3tF0kZJF7p7W53Z0UPKfO99X+EU/ncVTiWOLr1G9CQzu0uhm8X2UZ/NKyT1lyR3\nv9HdHzazY81snqR1kr7V5vr42TgAAIDsyfLpYAAAgMwiBAIAAGQQIRAAACCDCIEAAAAZRAgEAADI\nIEIgAABABhECAQAAMogQCAAAkEGEQACoEDN7qtptAIBSCIEA0EVmtreZXZJf7u6frkZ7AKAchEAA\n6LovSHoxv9DM1kZ/dzezWWb2BzN72cweNbMtitT/gJk9ZGYvmtkMMzupB9oOIKMIgQDQBWZ2jKQx\nkoaa2U55d6d/nH1PSde5+8clrZT01SKrGylpsbvv7+6fkDS5Em0GAIkQCABd4u6PSFri7je5+9I2\nqs5395ei6eck7V6kzkuSjjSzK83sM+6+upubCwCbEAIBoAuio39thb/Y+tT0Rkl1+RXcfa6kAyTN\nkPRTM/tRtzQSAIogBAJA1xwsaZqZHWxmW3VlRWa2s6T33f2Pkn4l6cDuaCAAFFPwnygAoEOWSDpI\n0jx3/2fefV5iuti8JH1C0tVm1ippg6Szuq2VAJDH3It9DgEAAKAv43QwAABABhECAQAAMogQCAAA\nkEGEQAAAgAwiBAIAAGQQIRAAACCDCIEAAAAZRAgEAADIoP8PyTfnwDJ5+SUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106b89240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10, 5))\n",
    "t = 1/fs * np.arange(len(h))\n",
    "plt.plot(t, h)\n",
    "plt.axis([0.0, 1.0, -1.1*np.max(np.abs(h)), 1.1*np.max(np.abs(h))])\n",
    "plt.xlabel(r'$t$ in s')\n",
    "plt.ylabel(r'$\\hat{h}[k]$');"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "\n",
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples, 2016-2017*."
   ]
  }
 ],
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