first-order-lpf.ipynb

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    "# How to Implement a First-Order Low-Pass Filter in Discrete Time\n",
    "\n",
    "Copyright &copy; 2021 [Joshua Marshall](https://www.ece.queensu.ca/people/j-marshall/)\n",
    "\n",
    "We often teach or learn about filters in continuous time, but then need to implement them in discrete time (e.g., in code) on data acquired at discrete sample times.  This notebook shows one way to design and implement a simple first-order low-pass filter in discrete time."
   ]
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    "## Filter Model Derivation\n",
    "\n",
    "Let's start with an integrator, which can be approximated in discrete time by any number of methods.  For this example, we use a [trapezoidal approximation](https://en.wikipedia.org/wiki/Trapezoidal_rule)\n",
    "\n",
    "$$\\int_{(k-1)T}^{kT}f(\\tau)d\\tau \\approx \\frac{T}{2}\\left(f_k+f_{k-1}\\right),$$\n",
    "\n",
    "where $T>0$ be the step tiem (i.e., sample period), $k=1,2,\\ldots$ is the time index, and $f_k:=f(kT)$.  Thus, the integral from $t=0$ to the current time $kT$ is \n",
    "\n",
    "$$i_k=\\int_{0}^{kT}f(\\tau)d\\tau \\approx i_{k-1} + \\frac{T}{2}\\left(f_k+f_{k-1}\\right).$$\n",
    "\n",
    "Now, let the [$z$-transform](https://en.wikipedia.org/wiki/Z-transform) of $i(t)$ and $f(t)$ be $I(z)$ and $F(z)$, respectively, where $z^{-1}$ denotes a unit time delay of period $T$.  Therefore, taking the $z$-transform of the integral approximation above yields\n",
    "\n",
    "$$G_I(z) = \\frac{T}{2}\\left(\\frac{1+z^{-1}}{1-z^{-1}}\\right) =\\frac{I(z)}{F(z)}.$$\n",
    "\n",
    "This formula is often called [Tustin's method](https://en.wikipedia.org/wiki/Bilinear_transform).  For those more familar with the Laplace domain, Tustin's method uses a first-order approximation for the integral $1/s$ such that\n",
    "\n",
    "$$s\\approx \\frac{2}{T}\\left(\\frac{1-z^{-1}}{1+z^{-1}}\\right).$$\n",
    "\n",
    "We now have the machinery to build a first-order low-pass filter.  For simplicity, let $Y(z)$ be the out put of our filter with input $X(z)$.  In continuious time, the [frequency response of a low-pass filter](https://en.wikipedia.org/wiki/Low-pass_filter#Frequency_response) is given by \n",
    "\n",
    "$$G_{\\rm LPF}(s) = \\frac{\\omega_c}{s+\\omega_c},$$\n",
    "\n",
    "where $\\omega_c~\\textup{[rad/s]}$ is the filter's [cut-off frequency](https://en.wikipedia.org/wiki/Cutoff_frequency).  Substituting our above approximtion for $s$ into $G_{\\rm LPF}(s)$ we obtain\n",
    "\n",
    "$$G_{\\rm LPF}(z) = \\frac{T\\omega_c(1+z^{-1})}{(T\\omega_c+2)+(T\\omega_c-2)z^{-1}} = \\frac{Y(z)}{X(z)},$$\n",
    "\n",
    "where $Y(z)$ is the filtered version of $X(z)$.\n",
    "\n",
    "Therefore, using our expression for the filter transfer function $G_{\\rm LPF}(z)$ we can write\n",
    "\n",
    "$$ Y(z)\\left(T\\omega_c+2 + (T\\omega_c-2)z^{-1}\\right) = T\\omega_c(1+z^{-1})X(z).$$ \n",
    "\n",
    "Thus, in the time domain, we obtain\n",
    "\n",
    "$$(T\\omega_c + 2)y_k + (T\\omega_c-2)y_{k-1} = T\\omega_c\\left(x_k+x_{k-1}\\right),$$\n",
    "\n",
    "which can be rearranged to obtain\n",
    "\n",
    "$$y_k = \\left(\\frac{2-T\\omega_c}{2+T\\omega_c}\\right)y_{k-1} + \\left(\\frac{T\\omega_c}{2+T\\omega_c}\\right)\\left(x_k+x_{k-1}\\right),$$ \n",
    "\n",
    "where $x_k$ is the input at time $t=kT$ and $y_k$ is the filtered output at time $t=kT$."
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   "source": [
    "## Example Implementation\n",
    "\n",
    "Let's try out our design on some contrived data.\n",
    "\n",
    "### Create Test Input Data\n",
    "\n",
    "Let's create some input data with two frequencies that are sufficiently far apart."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "306f534c",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n"
   ]
  },
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   "source": [
    "Set the simulation time as `SIM_TIME` \\[s\\] and the sample period $T$ as `T` \\[s\\]."
   ]
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  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "d3621dfa",
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   "outputs": [],
   "source": [
    "SIM_TIME = 15\n",
    "T = 0.01\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "5ea46ebc",
   "metadata": {},
   "outputs": [],
   "source": [
    "t = np.arange(0.0, SIM_TIME, T)\n",
    "N = np.size(t)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa5ed675",
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   "source": [
    "Create two contrived periodic signals $x_1$ and $x_2$ with frequencies $f_1$ and $f_2$ \\[Hz\\], respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "a9dbd990",
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   "outputs": [],
   "source": [
    "f_1 = 0.25\n",
    "f_2 = 5.0\n",
    "x_1 = np.sin(2.0 * np.pi * f_1 * t)\n",
    "x_2 = np.sin(2.0 * np.pi * f_2 * t)\n"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "### Plot the Input Signal\n",
    "\n",
    "Now plow the signals."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "e1fcbdb0",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "75462c8f",
   "metadata": {},
   "outputs": [],
   "source": [
    "%config InlineBackend.figure_format = 'svg'\n",
    "plt.rc('text', usetex=True)\n",
    "plt.rc('text.latex', preamble=r'\\usepackage{amsmath}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "3279cb7b",
   "metadata": {},
   "outputs": [
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169.037969 \nL 164.116337 174.091234 \nL 164.332918 175.832471 \nL 164.549499 174.091234 \nL 164.766079 169.037969 \nL 165.199241 151.249734 \nL 166.065564 111.474031 \nL 166.282145 106.420766 \nL 166.498726 104.679529 \nL 166.715306 106.420766 \nL 166.931887 111.474031 \nL 167.365049 129.262266 \nL 168.231372 169.037969 \nL 168.447952 174.091234 \nL 168.664533 175.832471 \nL 168.881114 174.091234 \nL 169.097695 169.037969 \nL 169.530856 151.249734 \nL 170.397179 111.474031 \nL 170.61376 106.420766 \nL 170.830341 104.679529 \nL 171.046921 106.420766 \nL 171.263502 111.474031 \nL 171.696664 129.262266 \nL 172.562987 169.037969 \nL 172.779567 174.091234 \nL 172.996148 175.832471 \nL 173.212729 174.091234 \nL 173.42931 169.037969 \nL 173.862471 151.249734 \nL 174.728794 111.474031 \nL 174.945375 106.420766 \nL 175.161956 104.679529 \nL 175.378536 106.420766 \nL 175.595117 111.474031 \nL 176.028279 129.262266 \nL 176.894602 169.037969 \nL 177.111182 174.091234 \nL 177.327763 175.832471 \nL 177.544344 174.091234 \nL 177.760925 169.037969 \nL 178.194086 151.249734 \nL 179.060409 111.474031 \nL 179.27699 106.420766 \nL 179.493571 104.679529 \nL 179.710151 106.420766 \nL 179.926732 111.474031 \nL 180.359894 129.262266 \nL 181.226217 169.037969 \nL 181.442797 174.091234 \nL 181.659378 175.832471 \nL 181.875959 174.091234 \nL 182.09254 169.037969 \nL 182.525701 151.249734 \nL 183.392024 111.474031 \nL 183.608605 106.420766 \nL 183.825186 104.679529 \nL 184.041766 106.420766 \nL 184.258347 111.474031 \nL 184.691509 129.262266 \nL 185.557832 169.037969 \nL 185.774412 174.091234 \nL 185.990993 175.832471 \nL 186.207574 174.091234 \nL 186.424155 169.037969 \nL 186.857316 151.249734 \nL 187.723639 111.474031 \nL 187.94022 106.420766 \nL 188.156801 104.679529 \nL 188.373381 106.420766 \nL 188.589962 111.474031 \nL 189.023124 129.262266 \nL 189.889447 169.037969 \nL 190.106027 174.091234 \nL 190.322608 175.832471 \nL 190.539189 174.091234 \nL 190.75577 169.037969 \nL 191.188931 151.249734 \nL 192.055254 111.474031 \nL 192.271835 106.420766 \nL 192.488416 104.679529 \nL 192.704996 106.420766 \nL 192.921577 111.474031 \nL 193.354739 129.262266 \nL 194.221062 169.037969 \nL 194.437642 174.091234 \nL 194.654223 175.832471 \nL 194.870804 174.091234 \nL 195.087385 169.037969 \nL 195.520546 151.249734 \nL 196.386869 111.474031 \nL 196.60345 106.420766 \nL 196.820031 104.679529 \nL 197.036611 106.420766 \nL 197.253192 111.474031 \nL 197.686354 129.262266 \nL 198.552677 169.037969 \nL 198.769257 174.091234 \nL 198.985838 175.832471 \nL 199.202419 174.091234 \nL 199.419 169.037969 \nL 199.852161 151.249734 \nL 200.718484 111.474031 \nL 200.935065 106.420766 \nL 201.151646 104.679529 \nL 201.368226 106.420766 \nL 201.584807 111.474031 \nL 202.017969 129.262266 \nL 202.884292 169.037969 \nL 203.100872 174.091234 \nL 203.317453 175.832471 \nL 203.534034 174.091234 \nL 203.750615 169.037969 \nL 204.183776 151.249734 \nL 205.050099 111.474031 \nL 205.26668 106.420766 \nL 205.483261 104.679529 \nL 205.699841 106.420766 \nL 205.916422 111.474031 \nL 206.349584 129.262266 \nL 207.215907 169.037969 \nL 207.432487 174.091234 \nL 207.649068 175.832471 \nL 207.865649 174.091234 \nL 208.08223 169.037969 \nL 208.515391 151.249734 \nL 209.381714 111.474031 \nL 209.598295 106.420766 \nL 209.814876 104.679529 \nL 210.031456 106.420766 \nL 210.248037 111.474031 \nL 210.681199 129.262266 \nL 211.547522 169.037969 \nL 211.764102 174.091234 \nL 211.980683 175.832471 \nL 212.197264 174.091234 \nL 212.413845 169.037969 \nL 212.847006 151.249734 \nL 213.713329 111.474031 \nL 213.92991 106.420766 \nL 214.146491 104.679529 \nL 214.363071 106.420766 \nL 214.579652 111.474031 \nL 215.012814 129.262266 \nL 215.879137 169.037969 \nL 216.095717 174.091234 \nL 216.312298 175.832471 \nL 216.528879 174.091234 \nL 216.74546 169.037969 \nL 217.178621 151.249734 \nL 218.044944 111.474031 \nL 218.261525 106.420766 \nL 218.478106 104.679529 \nL 218.694686 106.420766 \nL 218.911267 111.474031 \nL 219.344429 129.262266 \nL 220.210752 169.037969 \nL 220.427332 174.091234 \nL 220.643913 175.832471 \nL 220.860494 174.091234 \nL 221.077075 169.037969 \nL 221.510236 151.249734 \nL 222.376559 111.474031 \nL 222.59314 106.420766 \nL 222.809721 104.679529 \nL 223.026301 106.420766 \nL 223.242882 111.474031 \nL 223.676044 129.262266 \nL 224.542367 169.037969 \nL 224.758947 174.091234 \nL 224.975528 175.832471 \nL 225.192109 174.091234 \nL 225.40869 169.037969 \nL 225.841851 151.249734 \nL 226.708174 111.474031 \nL 226.924755 106.420766 \nL 227.141336 104.679529 \nL 227.357916 106.420766 \nL 227.574497 111.474031 \nL 228.007659 129.262266 \nL 228.873982 169.037969 \nL 229.090562 174.091234 \nL 229.307143 175.832471 \nL 229.523724 174.091234 \nL 229.740305 169.037969 \nL 230.173466 151.249734 \nL 231.039789 111.474031 \nL 231.25637 106.420766 \nL 231.472951 104.679529 \nL 231.689531 106.420766 \nL 231.906112 111.474031 \nL 232.339274 129.262266 \nL 233.205597 169.037969 \nL 233.422178 174.091234 \nL 233.638758 175.832471 \nL 233.855339 174.091234 \nL 234.07192 169.037969 \nL 234.505081 151.249734 \nL 235.371404 111.474031 \nL 235.587985 106.420766 \nL 235.804566 104.679529 \nL 236.021147 106.420766 \nL 236.237727 111.474031 \nL 236.670889 129.262266 \nL 237.537212 169.037969 \nL 237.753793 174.091234 \nL 237.970373 175.832471 \nL 238.186954 174.091234 \nL 238.403535 169.037969 \nL 238.836696 151.249734 \nL 239.703019 111.474031 \nL 239.9196 106.420766 \nL 240.136181 104.679529 \nL 240.352762 106.420766 \nL 240.569342 111.474031 \nL 241.002504 129.262266 \nL 241.868827 169.037969 \nL 242.085408 174.091234 \nL 242.301988 175.832471 \nL 242.518569 174.091234 \nL 242.73515 169.037969 \nL 243.168311 151.249734 \nL 244.034634 111.474031 \nL 244.251215 106.420766 \nL 244.467796 104.679529 \nL 244.684377 106.420766 \nL 244.900957 111.474031 \nL 245.334119 129.262266 \nL 246.200442 169.037969 \nL 246.417023 174.091234 \nL 246.633603 175.832471 \nL 246.850184 174.091234 \nL 247.066765 169.037969 \nL 247.499926 151.249734 \nL 248.366249 111.474031 \nL 248.58283 106.420766 \nL 248.799411 104.679529 \nL 249.015992 106.420766 \nL 249.232572 111.474031 \nL 249.665734 129.262266 \nL 250.532057 169.037969 \nL 250.748638 174.091234 \nL 250.965218 175.832471 \nL 251.181799 174.091234 \nL 251.39838 169.037969 \nL 251.831541 151.249734 \nL 252.697864 111.474031 \nL 252.914445 106.420766 \nL 253.131026 104.679529 \nL 253.347607 106.420766 \nL 253.564187 111.474031 \nL 253.997349 129.262266 \nL 254.863672 169.037969 \nL 255.080253 174.091234 \nL 255.296833 175.832471 \nL 255.513414 174.091234 \nL 255.729995 169.037969 \nL 256.163156 151.249734 \nL 257.029479 111.474031 \nL 257.24606 106.420766 \nL 257.462641 104.679529 \nL 257.679222 106.420766 \nL 257.895802 111.474031 \nL 258.328964 129.262266 \nL 259.195287 169.037969 \nL 259.411868 174.091234 \nL 259.628448 175.832471 \nL 259.845029 174.091234 \nL 260.06161 169.037969 \nL 260.494771 151.249734 \nL 261.361094 111.474031 \nL 261.577675 106.420766 \nL 261.794256 104.679529 \nL 262.010837 106.420766 \nL 262.227417 111.474031 \nL 262.660579 129.262266 \nL 263.526902 169.037969 \nL 263.743483 174.091234 \nL 263.960063 175.832471 \nL 264.176644 174.091234 \nL 264.393225 169.037969 \nL 264.826386 151.249734 \nL 265.692709 111.474031 \nL 265.90929 106.420766 \nL 266.125871 104.679529 \nL 266.342452 106.420766 \nL 266.559032 111.474031 \nL 266.992194 129.262266 \nL 267.858517 169.037969 \nL 268.075098 174.091234 \nL 268.291678 175.832471 \nL 268.508259 174.091234 \nL 268.72484 169.037969 \nL 269.158001 151.249734 \nL 270.024324 111.474031 \nL 270.240905 106.420766 \nL 270.457486 104.679529 \nL 270.674067 106.420766 \nL 270.890647 111.474031 \nL 271.323809 129.262266 \nL 272.190132 169.037969 \nL 272.406713 174.091234 \nL 272.623293 175.832471 \nL 272.839874 174.091234 \nL 273.056455 169.037969 \nL 273.489616 151.249734 \nL 274.355939 111.474031 \nL 274.57252 106.420766 \nL 274.789101 104.679529 \nL 275.005682 106.420766 \nL 275.222262 111.474031 \nL 275.655424 129.262266 \nL 276.521747 169.037969 \nL 276.738328 174.091234 \nL 276.954908 175.832471 \nL 277.171489 174.091234 \nL 277.38807 169.037969 \nL 277.821231 151.249734 \nL 278.687554 111.474031 \nL 278.904135 106.420766 \nL 279.120716 104.679529 \nL 279.337297 106.420766 \nL 279.553877 111.474031 \nL 279.987039 129.262266 \nL 280.853362 169.037969 \nL 281.069943 174.091234 \nL 281.286523 175.832471 \nL 281.503104 174.091234 \nL 281.719685 169.037969 \nL 282.152846 151.249734 \nL 283.019169 111.474031 \nL 283.23575 106.420766 \nL 283.452331 104.679529 \nL 283.668912 106.420766 \nL 283.885492 111.474031 \nL 284.318654 129.262266 \nL 285.184977 169.037969 \nL 285.401558 174.091234 \nL 285.618138 175.832471 \nL 285.834719 174.091234 \nL 286.0513 169.037969 \nL 286.484461 151.249734 \nL 287.350784 111.474031 \nL 287.567365 106.420766 \nL 287.783946 104.679529 \nL 288.000527 106.420766 \nL 288.217107 111.474031 \nL 288.650269 129.262266 \nL 289.516592 169.037969 \nL 289.733173 174.091234 \nL 289.949753 175.832471 \nL 290.166334 174.091234 \nL 290.382915 169.037969 \nL 290.816076 151.249734 \nL 291.682399 111.474031 \nL 291.89898 106.420766 \nL 292.115561 104.679529 \nL 292.332142 106.420766 \nL 292.548722 111.474031 \nL 292.981884 129.262266 \nL 293.848207 169.037969 \nL 294.064788 174.091234 \nL 294.281368 175.832471 \nL 294.497949 174.091234 \nL 294.71453 169.037969 \nL 295.147691 151.249734 \nL 296.014014 111.474031 \nL 296.230595 106.420766 \nL 296.447176 104.679529 \nL 296.663757 106.420766 \nL 296.880337 111.474031 \nL 297.313499 129.262266 \nL 298.179822 169.037969 \nL 298.396403 174.091234 \nL 298.612983 175.832471 \nL 298.829564 174.091234 \nL 299.046145 169.037969 \nL 299.479306 151.249734 \nL 300.34563 111.474031 \nL 300.56221 106.420766 \nL 300.778791 104.679529 \nL 300.995372 106.420766 \nL 301.211953 111.474031 \nL 301.645114 129.262266 \nL 302.511437 169.037969 \nL 302.728018 174.091234 \nL 302.944599 175.832471 \nL 303.161179 174.091234 \nL 303.37776 169.037969 \nL 303.810922 151.249734 \nL 304.677245 111.474031 \nL 304.893825 106.420766 \nL 305.110406 104.679529 \nL 305.326987 106.420766 \nL 305.543568 111.474031 \nL 305.976729 129.262266 \nL 306.843052 169.037969 \nL 307.059633 174.091234 \nL 307.276214 175.832471 \nL 307.492794 174.091234 \nL 307.709375 169.037969 \nL 308.142537 151.249734 \nL 309.00886 111.474031 \nL 309.22544 106.420766 \nL 309.442021 104.679529 \nL 309.658602 106.420766 \nL 309.875183 111.474031 \nL 310.308344 129.262266 \nL 311.174667 169.037969 \nL 311.391248 174.091234 \nL 311.607829 175.832471 \nL 311.824409 174.091234 \nL 312.04099 169.037969 \nL 312.474152 151.249734 \nL 313.340475 111.474031 \nL 313.557055 106.420766 \nL 313.773636 104.679529 \nL 313.990217 106.420766 \nL 314.206798 111.474031 \nL 314.639959 129.262266 \nL 315.506282 169.037969 \nL 315.722863 174.091234 \nL 315.939444 175.832471 \nL 316.156024 174.091234 \nL 316.372605 169.037969 \nL 316.805767 151.249734 \nL 317.67209 111.474031 \nL 317.88867 106.420766 \nL 318.105251 104.679529 \nL 318.321832 106.420766 \nL 318.538413 111.474031 \nL 318.971574 129.262266 \nL 319.837897 169.037969 \nL 320.054478 174.091234 \nL 320.271059 175.832471 \nL 320.487639 174.091234 \nL 320.70422 169.037969 \nL 321.137382 151.249734 \nL 322.003705 111.474031 \nL 322.220285 106.420766 \nL 322.436866 104.679529 \nL 322.653447 106.420766 \nL 322.870028 111.474031 \nL 323.303189 129.262266 \nL 324.169512 169.037969 \nL 324.386093 174.091234 \nL 324.602674 175.832471 \nL 324.819254 174.091234 \nL 325.035835 169.037969 \nL 325.468997 151.249734 \nL 326.33532 111.474031 \nL 326.5519 106.420766 \nL 326.768481 104.679529 \nL 326.985062 106.420766 \nL 327.201643 111.474031 \nL 327.634804 129.262266 \nL 328.501127 169.037969 \nL 328.717708 174.091234 \nL 328.934289 175.832471 \nL 329.150869 174.091234 \nL 329.36745 169.037969 \nL 329.800612 151.249734 \nL 330.666935 111.474031 \nL 330.883515 106.420766 \nL 331.100096 104.679529 \nL 331.316677 106.420766 \nL 331.533258 111.474031 \nL 331.966419 129.262266 \nL 332.832742 169.037969 \nL 333.049323 174.091234 \nL 333.265904 175.832471 \nL 333.482484 174.091234 \nL 333.699065 169.037969 \nL 334.132227 151.249734 \nL 334.99855 111.474031 \nL 335.21513 106.420766 \nL 335.431711 104.679529 \nL 335.648292 106.420766 \nL 335.864873 111.474031 \nL 336.298034 129.262266 \nL 337.164357 169.037969 \nL 337.380938 174.091234 \nL 337.597519 175.832471 \nL 337.814099 174.091234 \nL 338.03068 169.037969 \nL 338.463842 151.249734 \nL 339.330165 111.474031 \nL 339.546745 106.420766 \nL 339.763326 104.679529 \nL 339.979907 106.420766 \nL 340.196488 111.474031 \nL 340.629649 129.262266 \nL 341.495972 169.037969 \nL 341.712553 174.091234 \nL 341.929134 175.832471 \nL 342.145714 174.091234 \nL 342.362295 169.037969 \nL 342.795457 151.249734 \nL 343.66178 111.474031 \nL 343.87836 106.420766 \nL 344.094941 104.679529 \nL 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129.896579 266.163904 \nL 130.113159 263.481572 \nL 130.546321 254.259141 \nL 131.412644 233.668592 \nL 131.629225 230.961378 \nL 131.845805 229.909348 \nL 132.062386 230.5979 \nL 132.278967 232.941724 \nL 132.712128 241.466449 \nL 133.578451 260.581268 \nL 133.795032 262.903346 \nL 134.011613 263.563946 \nL 134.228194 262.477767 \nL 134.661355 255.569051 \nL 135.96084 226.780276 \nL 136.17742 225.670862 \nL 136.394001 226.303076 \nL 136.610582 228.591623 \nL 137.043743 237.009025 \nL 137.910066 255.922497 \nL 138.126647 258.197081 \nL 138.343228 258.811349 \nL 138.559809 257.68001 \nL 138.99297 250.684537 \nL 140.292455 221.664733 \nL 140.509035 220.521179 \nL 140.725616 221.120523 \nL 140.942197 223.377479 \nL 141.375358 231.735566 \nL 142.241681 250.546076 \nL 142.458262 252.79822 \nL 142.674843 253.391381 \nL 142.891424 252.240272 \nL 143.324585 245.20929 \nL 144.62407 216.115493 \nL 144.84065 214.964384 \nL 145.057231 215.557544 \nL 145.273812 217.809689 \nL 145.706973 226.162276 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202.580421 \nL 161.95053 199.978167 \nL 162.16711 199.038741 \nL 162.383691 199.847514 \nL 162.600272 202.319147 \nL 163.033433 211.122122 \nL 163.899756 230.882977 \nL 164.116337 233.384987 \nL 164.332918 234.232802 \nL 164.549499 233.341075 \nL 164.766079 230.795165 \nL 165.199241 221.869738 \nL 166.065564 201.968765 \nL 166.282145 199.453609 \nL 166.498726 198.601412 \nL 166.715306 199.497521 \nL 166.931887 202.056577 \nL 167.365049 211.03457 \nL 168.231372 231.145547 \nL 168.447952 233.73498 \nL 168.664533 234.670131 \nL 168.881114 233.865633 \nL 169.097695 231.406822 \nL 169.530856 222.655118 \nL 170.397179 203.099096 \nL 170.61376 200.669535 \nL 170.830341 199.902631 \nL 171.046921 200.883713 \nL 171.263502 203.527399 \nL 171.696664 212.673544 \nL 172.562987 233.115808 \nL 172.779567 235.786898 \nL 172.996148 236.8032 \nL 173.212729 236.079326 \nL 173.42931 233.700593 \nL 173.862471 225.107329 \nL 174.728794 205.860772 \nL 174.945375 203.50692 \nL 175.161956 202.815026 \nL 175.378536 203.870399 \nL 175.595117 206.58764 \nL 176.028279 215.878611 \nL 176.894602 236.600897 \nL 177.111182 239.339885 \nL 177.327763 240.423209 \nL 177.544344 239.765464 \nL 177.760925 237.45195 \nL 178.194086 228.986331 \nL 179.060409 209.983459 \nL 179.27699 207.688022 \nL 179.493571 207.053512 \nL 179.710151 208.165223 \nL 180.143313 215.123303 \nL 181.442797 244.04615 \nL 181.659378 245.175806 \nL 181.875959 244.563221 \nL 182.09254 242.293682 \nL 182.525701 233.912421 \nL 183.392024 215.0636 \nL 183.608605 212.803566 \nL 183.825186 212.203196 \nL 184.041766 213.347776 \nL 184.474928 220.367751 \nL 185.774412 249.44501 \nL 185.990993 250.595774 \nL 186.207574 250.002959 \nL 186.424155 247.751848 \nL 186.857316 239.403398 \nL 187.723639 220.603916 \nL 187.94022 218.352806 \nL 188.156801 217.75999 \nL 188.373381 218.910754 \nL 188.806543 225.938978 \nL 190.106027 255.007989 \nL 190.322608 256.152569 \nL 190.539189 255.552199 \nL 190.75577 253.292164 \nL 191.188931 244.921768 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266.299188 \nL 207.432487 268.858244 \nL 207.649068 269.754353 \nL 207.865649 268.902155 \nL 208.08223 266.387 \nL 208.515391 257.496732 \nL 209.381714 237.5606 \nL 209.598295 235.014689 \nL 209.814876 234.122963 \nL 210.031456 234.970778 \nL 210.248037 237.472788 \nL 210.681199 246.31049 \nL 211.547522 266.036618 \nL 211.764102 268.508251 \nL 211.980683 269.317024 \nL 212.197264 268.377598 \nL 212.413845 265.775344 \nL 212.847006 256.711352 \nL 213.713329 236.430268 \nL 213.92991 233.798764 \nL 214.146491 232.821743 \nL 214.363071 233.584586 \nL 214.579652 236.001965 \nL 215.012814 244.671515 \nL 215.879137 264.066357 \nL 216.095717 266.456333 \nL 216.312298 267.183955 \nL 216.528879 266.163904 \nL 216.74546 263.481572 \nL 217.178621 254.259141 \nL 218.044944 233.668592 \nL 218.261525 230.961378 \nL 218.478106 229.909348 \nL 218.694686 230.5979 \nL 218.911267 232.941724 \nL 219.344429 241.466449 \nL 220.210752 260.581268 \nL 220.427332 262.903346 \nL 220.643913 263.563946 \nL 220.860494 262.477767 \nL 221.293655 255.569051 \nL 222.59314 226.780276 \nL 222.809721 225.670862 \nL 223.026301 226.303076 \nL 223.242882 228.591623 \nL 223.676044 237.009025 \nL 224.542367 255.922497 \nL 224.758947 258.197081 \nL 224.975528 258.811349 \nL 225.192109 257.68001 \nL 225.62527 250.684537 \nL 226.924755 221.664733 \nL 227.141336 220.521179 \nL 227.357916 221.120523 \nL 227.574497 223.377479 \nL 228.007659 231.735566 \nL 228.873982 250.546076 \nL 229.090562 252.79822 \nL 229.307143 253.391381 \nL 229.523724 252.240272 \nL 229.956885 245.20929 \nL 231.25637 216.115493 \nL 231.472951 214.964384 \nL 231.689531 215.557544 \nL 231.906112 217.809689 \nL 232.339274 226.162276 \nL 233.205597 244.978286 \nL 233.422178 247.235242 \nL 233.638758 247.834586 \nL 233.855339 246.691032 \nL 234.288501 239.679265 \nL 235.587985 210.675754 \nL 235.804566 209.544416 \nL 236.021147 210.158684 \nL 236.237727 212.433268 \nL 236.670889 220.834708 \nL 237.537212 239.764142 \nL 237.753793 242.052689 \nL 237.970373 242.684902 \nL 238.186954 241.575489 \nL 238.620116 234.635779 \nL 239.9196 205.877998 \nL 240.136181 204.791819 \nL 240.352762 205.452419 \nL 240.569342 207.774496 \nL 241.002504 216.274359 \nL 241.868827 235.41404 \nL 242.085408 237.757865 \nL 242.301988 238.446417 \nL 242.518569 237.394386 \nL 242.73515 234.687172 \nL 243.168311 225.436297 \nL 244.034634 204.874193 \nL 244.251215 202.19186 \nL 244.467796 201.17181 \nL 244.684377 201.899432 \nL 244.900957 204.289408 \nL 245.334119 212.927631 \nL 246.200442 232.3538 \nL 246.417023 234.771179 \nL 246.633603 235.534021 \nL 246.850184 234.557001 \nL 247.066765 231.925497 \nL 247.499926 222.828067 \nL 248.366249 202.580421 \nL 248.58283 199.978167 \nL 248.799411 199.038741 \nL 249.015992 199.847514 \nL 249.232572 202.319147 \nL 249.665734 211.122122 \nL 250.532057 230.882977 \nL 250.748638 233.384987 \nL 250.965218 234.232802 \nL 251.181799 233.341075 \nL 251.39838 230.795165 \nL 251.831541 221.869738 \nL 252.697864 201.968765 \nL 252.914445 199.453609 \nL 253.131026 198.601412 \nL 253.347607 199.497521 \nL 253.564187 202.056577 \nL 253.997349 211.03457 \nL 254.863672 231.145547 \nL 255.080253 233.73498 \nL 255.296833 234.670131 \nL 255.513414 233.865633 \nL 255.729995 231.406822 \nL 256.163156 222.655118 \nL 257.029479 203.099096 \nL 257.24606 200.669535 \nL 257.462641 199.902631 \nL 257.679222 200.883713 \nL 257.895802 203.527399 \nL 258.328964 212.673544 \nL 259.195287 233.115808 \nL 259.411868 235.786898 \nL 259.628448 236.8032 \nL 259.845029 236.079326 \nL 260.06161 233.700593 \nL 260.494771 225.107329 \nL 261.361094 205.860772 \nL 261.577675 203.50692 \nL 261.794256 202.815026 \nL 262.010837 203.870399 \nL 262.227417 206.58764 \nL 262.660579 215.878611 \nL 263.526902 236.600897 \nL 263.743483 239.339885 \nL 263.960063 240.423209 \nL 264.176644 239.765464 \nL 264.393225 237.45195 \nL 264.826386 228.986331 \nL 265.692709 209.983459 \nL 265.90929 207.688022 \nL 266.125871 207.053512 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228.590301 \nL 283.452331 227.932556 \nL 283.668912 229.01588 \nL 284.102073 235.901762 \nL 285.401558 264.485366 \nL 285.618138 265.540738 \nL 285.834719 264.848844 \nL 286.0513 262.494993 \nL 286.484461 253.930173 \nL 287.350784 234.655171 \nL 287.567365 232.276438 \nL 287.783946 231.552564 \nL 288.000527 232.568867 \nL 288.217107 235.239956 \nL 288.650269 244.417429 \nL 289.516592 264.828366 \nL 289.733173 267.472052 \nL 289.949753 268.453134 \nL 290.166334 267.68623 \nL 290.382915 265.256668 \nL 290.816076 256.538403 \nL 291.682399 236.948943 \nL 291.89898 234.490132 \nL 292.115561 233.685633 \nL 292.332142 234.620785 \nL 292.548722 237.210217 \nL 292.981884 246.222937 \nL 293.848207 266.299188 \nL 294.064788 268.858244 \nL 294.281368 269.754353 \nL 294.497949 268.902155 \nL 294.71453 266.387 \nL 295.147691 257.496732 \nL 296.014014 237.5606 \nL 296.230595 235.014689 \nL 296.447176 234.122963 \nL 296.663757 234.970778 \nL 296.880337 237.472788 \nL 297.313499 246.31049 \nL 298.179822 266.036618 \nL 298.396403 268.508251 \nL 298.612983 269.317024 \nL 298.829564 268.377598 \nL 299.046145 265.775344 \nL 299.479306 256.711352 \nL 300.34563 236.430268 \nL 300.56221 233.798764 \nL 300.778791 232.821743 \nL 300.995372 233.584586 \nL 301.211953 236.001965 \nL 301.645114 244.671515 \nL 302.511437 264.066357 \nL 302.728018 266.456333 \nL 302.944599 267.183955 \nL 303.161179 266.163904 \nL 303.37776 263.481572 \nL 303.810922 254.259141 \nL 304.677245 233.668592 \nL 304.893825 230.961378 \nL 305.110406 229.909348 \nL 305.326987 230.5979 \nL 305.543568 232.941724 \nL 305.976729 241.466449 \nL 306.843052 260.581268 \nL 307.059633 262.903346 \nL 307.276214 263.563946 \nL 307.492794 262.477767 \nL 307.925956 255.569051 \nL 309.22544 226.780276 \nL 309.442021 225.670862 \nL 309.658602 226.303076 \nL 309.875183 228.591623 \nL 310.308344 237.009025 \nL 311.174667 255.922497 \nL 311.391248 258.197081 \nL 311.607829 258.811349 \nL 311.824409 257.68001 \nL 312.257571 250.684537 \nL 313.557055 221.664733 \nL 313.773636 220.521179 \nL 313.990217 221.120523 \nL 314.206798 223.377479 \nL 314.639959 231.735566 \nL 315.506282 250.546076 \nL 315.722863 252.79822 \nL 315.939444 253.391381 \nL 316.156024 252.240272 \nL 316.589186 245.20929 \nL 317.88867 216.115493 \nL 318.105251 214.964384 \nL 318.321832 215.557544 \nL 318.538413 217.809689 \nL 318.971574 226.162276 \nL 319.837897 244.978286 \nL 320.054478 247.235242 \nL 320.271059 247.834586 \nL 320.487639 246.691032 \nL 320.920801 239.679265 \nL 322.220285 210.675754 \nL 322.436866 209.544416 \nL 322.653447 210.158684 \nL 322.870028 212.433268 \nL 323.303189 220.834708 \nL 324.169512 239.764142 \nL 324.386093 242.052689 \nL 324.602674 242.684902 \nL 324.819254 241.575489 \nL 325.252416 234.635779 \nL 326.5519 205.877998 \nL 326.768481 204.791819 \nL 326.985062 205.452419 \nL 327.201643 207.774496 \nL 327.634804 216.274359 \nL 328.501127 235.41404 \nL 328.717708 237.757865 \nL 328.934289 238.446417 \nL 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      "text/plain": [
       "<Figure size 640x480 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.subplot(311)\n",
    "plt.plot(t, x_1)\n",
    "plt.ylabel(\"$x_1(t)$\")\n",
    "plt.setp(plt.gca(), xticklabels=[])\n",
    "plt.subplot(312)\n",
    "plt.plot(t, x_2)\n",
    "plt.ylabel(\"$x_2(t)$\")\n",
    "plt.setp(plt.gca(), xticklabels=[])\n",
    "plt.subplot(313)\n",
    "plt.plot(t, x_1 + x_2, alpha=0.5, label=\"$x_1+x_2$\")\n",
    "plt.plot(t, x_1, label=\"$x_1$ (low-frequency signal)\")\n",
    "plt.xlabel(\"$t$ [s]\")\n",
    "plt.ylabel(\"$x(t)$\")\n",
    "plt.legend()\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb7af82a",
   "metadata": {},
   "source": [
    "### Filter Implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "6ccb6ca3",
   "metadata": {},
   "outputs": [],
   "source": [
    "def lpf(x, omega_c, T):\n",
    "    \"\"\"Implement a first-order low-pass filter.\n",
    "\n",
    "    The input data is x, the filter's cutoff frequency is omega_c\n",
    "    [rad/s] and the sample time is T [s].  The output is y.\n",
    "    \"\"\"\n",
    "    y = x\n",
    "    alpha = (2 - T * omega_c) / (2 + T * omega_c)\n",
    "    beta = T * omega_c / (2 + T * omega_c)\n",
    "    for k in range(1, N):\n",
    "        y[k] = alpha * y[k - 1] + beta * (x[k] + x[k - 1])\n",
    "    return y\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d94aa64f",
   "metadata": {},
   "source": [
    "Pick a cutoff frequency $\\omega_c$ somewhere between the low and high frequency components in the signal, and then run the filter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "834c63a5",
   "metadata": {},
   "outputs": [],
   "source": [
    "omega_c = 2.0 * np.pi * (f_1 + f_2) / 2.0\n",
    "y = lpf(x_1 + x_2, omega_c, T)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eec23804",
   "metadata": {},
   "source": [
    "### Plot the Output Signal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "5d9ff70b",
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 640x480 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.subplot(211)\n",
    "plt.plot(t, x_1 + x_2, alpha=0.5, label=\"$x_1+x_2$ (input)\")\n",
    "plt.plot(t, x_1, label=\"$x_1$ (low-frequency signal)\")\n",
    "plt.ylabel(\"$x(t)$\")\n",
    "plt.setp(plt.gca(), xticklabels=[])\n",
    "plt.legend()\n",
    "plt.subplot(212)\n",
    "plt.plot(t, y, alpha=0.5, label=\"$y$ (filtered output)\")\n",
    "plt.plot(t, x_1, label=\"$x_1$ (low-frequency signal)\")\n",
    "plt.xlabel(\"$t$ [rad/s]\")\n",
    "plt.ylabel(\"$y(t)$\")\n",
    "plt.legend()\n",
    "plt.savefig('sample-lpf.svg', bbox_inches='tight')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "67603e9d",
   "metadata": {},
   "source": [
    "Notice how the filtered signal $y_k$ does not perfectly match $x_{1,k}$ but that the higher-frequency component $x_{2,k}$ is significantly attenuated.  Play around with the filter cut-off frequency $\\omega_c$ to get a feeling for how the filter behaves and what to expect should you use this for your particular application."
   ]
  }
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