chapter10/astroml_chapter10_Modeling_Toolkit_for_Time_Series_Analysis.ipynb

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   "cell_type": "markdown",
   "metadata": {
    "id": "W_l-7O_Ed1MV"
   },
   "source": [
    "\n",
    "# Modeling Toolkit For Time Series Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Wz5lwLnld1Mb"
   },
   "source": [
    "## Introduction\n",
    "In this notebook, we are going to learn fundamental concepts and tools for analyzing time series data. From an astronomical perspective, this field has predominantly focused on unevenly sampled data, low signal-to-noise data, and heteroscedastic errors. The most common applications range from the detection of variability and periodicity to the treatment of nonperiodic\n",
    "variability and searches for localized events.  \n",
    "Despite being set in the context of time series, many tools and results are readily applicable in other domains, which we will see later in this notebook. We will extend this toolkit by introducing tools for analysis\n",
    "in the frequency domain, such as Fourier analysis, discrete Fourier transform, wavelet analysis and\n",
    "digital filtering.  \n",
    "The main data sets used in this chapter include light curves obtained by the [LINEAR survey](https://arxiv.org/abs/1505.02082) (Lincoln Near-Earth Asteroid Research)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "5MH0kypUd1Me"
   },
   "source": [
    "## Main concepts for time series analysis\n",
    "The time series discussed here will be limited to two-dimensional scalar data sets: pairs of random variables, ($t_1$, $y_1$),...,($t_N$; $y_N$), with no assumptions about the sampling of the time coordinate t. In many ways, analysis methods discussed here are closely related to the parameter estimation and model selection problems discussed in the context of regression. However, unlike regression problems where different y measurements are typically treated as independent random\n",
    "variables, in such models the value of $y_{i+1}$ directly depends on the preceding value $y_i$.  \n",
    "  \n",
    "The main tasks of time series analysis are:  \n",
    "1. To characterize the presumed temporal correlation between different values of y, including its significance.  \n",
    "2. To forecast (predict) future values of y."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "cjlARqU7d1Mh"
   },
   "source": [
    "### Import Functions and Packages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "id": "AdBdcKyud1Mj"
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "from astroML.datasets import fetch_rrlyrae_templates\n",
    "\n",
    "from scipy.signal import fftconvolve\n",
    "\n",
    "from scipy import fftpack\n",
    "from matplotlib import mlab\n",
    "\n",
    "from astroML.datasets import fetch_LIGO_large\n",
    "\n",
    "from scipy.fftpack import fft\n",
    "from scipy.stats import norm\n",
    "\n",
    "from astroML.fourier import PSD_continuous"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from astroML.plotting import setup_text_plots\n",
    "setup_text_plots(usetex=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "TY_9dGYfd1Mz"
   },
   "source": [
    "## 1. Fourier analysis\n",
    "Fourier analysis plays a major role in the analysis of time series data. In Fourier analysis, general\n",
    "functions are represented or approximated by integrals or sums of simpler trigonometric functions.   \n",
    "\n",
    "For periodic functions, such as periodic light curves\n",
    "in astronomy, it is often true that a relatively small number of terms (less than 10) suffices to reach\n",
    "an approximation precision level similar to the measurement precision."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "__8znqDsd1M0"
   },
   "source": [
    "## Some useful concepts\n",
    "The **Fourier transform** of function h(t) is defined as\n",
    "\n",
    "$$H(f)=\\int_{-\\infty}^{\\infty} h(t) exp(-i2\\pi ft)dt$$\n",
    "\n",
    "with **inverse transformation**\n",
    "\n",
    "$$h(t)=\\int_{-\\infty}^{\\infty} H(t) exp(-i2\\pi ft)df$$\n",
    "\n",
    "where t is time and f is frequency (for time in seconds, the unit for frequency is hertz, or Hz; the\n",
    "units for H(f) are the product of the units for h(t) and inverse hertz.  \n",
    "For a real function h(t), H(f) is in general a complex function.  \n",
    "In the special case when h(t) is an even function such that h(-t) = h(t), H(f) is real and even as well. For example, the\n",
    "Fourier transform of a pdf of a zero-mean Gaussian $N(0, \\sigma)$ in the time domain is a Gaussian\n",
    "$H(f) = exp(-2\\pi^2 \\sigma^2 f^2)$ in the frequency domain.  \n",
    "When the time axis of an arbitrary function\n",
    "h(t) is shifted by $\\Delta t$, then the Fourier transform of $h(t + \\Delta t)$ is\n",
    "\n",
    "$$\\int_{-\\infty}^{\\infty}h(t+\\Delta t)exp(-i2\\pi ft)dt = H(f)exp(i2\\pi f\\Delta t)$$\n",
    "\n",
    "Therefore, the **Fourier transform of a Gaussian $N(\\mu, \\sigma)$** is\n",
    "\n",
    "$$H_{Gauss}(f)=exp(-2\\pi^2\\sigma^2f^2)[cos(2\\pi f\\mu)+isin(2\\pi f \\mu)]$$\n",
    "\n",
    "This is known as **\"white noise\"** since there is no frequency\n",
    "dependence (also known as \"thermal noise\" or Johnson's noise).  \n",
    "An important quantity in time series analysis is the one-sided **power spectral density (PSD)** function\n",
    "(or power spectrum) defined for $0 \\leq f < \\infty$ as\n",
    "\n",
    "$$PSD(f) \\equiv |H(f)|^2+|H(-f)|^2$$\n",
    "\n",
    "The PSD gives the amount of power contained in the frequency interval between f and f + df.  \n",
    "The total power is the same whether computed in the frequency or the time domain:\n",
    "\n",
    "$$P_{tot} \\equiv \\int_0^{\\infty}PSD(f)df=\\int_{-\\infty}^{\\infty}|h(t)|^2dt$$\n",
    "\n",
    "This result is known as **Parseval's theorem**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "3ZOruSvtd1M2"
   },
   "source": [
    "### (1) Fourier Reconstruction of RR-Lyrae Templates\n",
    "Below is an example of a truncated Fourier representation of an RR Lyrae light curve. As we can see, the more terms that are included in the sum, the better is the resulting approximation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "uDhfCgond1M3"
   },
   "source": [
    "#### Import a RR Lyrae template\n",
    "From astroML.datasets, we take the RR Lyrae (variable star) light curve as an example. RR Lyrae has a periodical pulsation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 52
    },
    "executionInfo": {
     "elapsed": 959,
     "status": "ok",
     "timestamp": 1603475202530,
     "user": {
      "displayName": "Brigitta Sipocz",
      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GgSZ5lvx2m0gmkej0jid6to7H2aVpPwrDwknsPBVg=s64",
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   "outputs": [],
   "source": [
    "templates = fetch_rrlyrae_templates()\n",
    "x, y = templates['115r'].T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "jm4osB-Zd1ND"
   },
   "source": [
    "#### Show Fourier reconstruction results\n",
    "The thick dashed line shows the true curve; the gray lines show the approximation based on 1, 3, and 8 Fourier modes (sinusoids)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
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     "timestamp": 1603475206059,
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   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 360x360 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5, 5))\n",
    "fig.subplots_adjust(hspace=0)\n",
    "\n",
    "kvals = [1, 3, 8]\n",
    "subplots = [311, 312, 313]\n",
    "\n",
    "for (k, subplot) in zip(kvals, subplots):\n",
    "    ax = fig.add_subplot(subplot)\n",
    "\n",
    "    # Use FFT to fit a truncated Fourier series\n",
    "    y_fft = np.fft.fft(y)\n",
    "    y_fft[k + 1:-k] = 0\n",
    "    y_fit = np.fft.ifft(y_fft).real\n",
    "\n",
    "    # plot the true value and the k-term reconstruction\n",
    "    ax.plot(np.concatenate([x, 1 + x]),\n",
    "            np.concatenate([y, y]), '--k', lw=2)\n",
    "    ax.plot(np.concatenate([x, 1 + x]),\n",
    "            np.concatenate([y_fit, y_fit]), color='gray')\n",
    "\n",
    "    label = \"%i mode\" % k\n",
    "    if k > 1:\n",
    "        label += 's'\n",
    "\n",
    "    ax.text(0.02, 0.1, label, ha='left', va='bottom',\n",
    "            transform=ax.transAxes)\n",
    "\n",
    "    if subplot == subplots[-1]:\n",
    "        ax.set_xlabel('phase')\n",
    "    else:\n",
    "        ax.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "\n",
    "    if subplot == subplots[1]:\n",
    "        ax.set_ylabel('amplitude')\n",
    "    ax.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "\n",
    "    ax.set_xlim(0, 2)\n",
    "    ax.set_ylim(1.1, -0.1)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Ud72xBPPd1NP"
   },
   "source": [
    "### (2) Convolution Theorem\n",
    "A convolution of two functions $a(t)$ and $b(t)$ is given by\n",
    "\n",
    "$$(a*b)(t) \\equiv \\int_{-\\infty}^{\\infty} a(t')b(t-t')dt'$$\n",
    "\n",
    "In the above equation, the function a can\n",
    "be thought of as the \"convolving pattern\" of the measuring apparatus, and the function $b$ is the\n",
    "signal. In practice, we measure the convolved (or smoothed) version of our signal, $[a * b](t)$, and\n",
    "seek to uncover the original signal $b$ using the presumably known $a$.  \n",
    "The convolution theorem states that if $h = a * b$, then the Fourier transforms of $h$, $a$, and $b$ are\n",
    "related by their pointwise products:\n",
    "\n",
    "$$H(f)=A(f)B(f)$$\n",
    "\n",
    "Thus a convolution of two functions is transformed into a simple multiplication of the associated\n",
    "Fourier representations. Therefore, to obtain $b$, we can simply take the inverse Fourier transform\n",
    "of the ratio $H(f)=A(f)$. In the absence of noise, this operation is exact.  \n",
    "  \n",
    "For further explanation, we will see a schematic representation of the convolution theorem below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "-pn_fvkld1NQ"
   },
   "source": [
    "#### Generate data and define function\n",
    "Generate random x, y with a given covariance length. Then we define a normalized top-hat window function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "id": "pBOLaHxgd1NS"
   },
   "outputs": [],
   "source": [
    "# Generate data\n",
    "np.random.seed(1)\n",
    "x = np.linspace(0, 1, 500)\n",
    "h = 0.01\n",
    "C = np.exp(-0.5 * (x - x[:, None]) ** 2 / h ** 2)\n",
    "y = 0.8 + 0.3 * np.random.multivariate_normal(np.zeros(len(x)), C)\n",
    "\n",
    "# define function\n",
    "w = np.zeros_like(x)\n",
    "w[(x > 0.12) & (x < 0.28)] = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ItbkRO3Od1Nb"
   },
   "source": [
    "#### Show data and window function\n",
    "The graph below show simulated data $D(x)$ in black line, and top-hat window function $W(x)$ in gray boxes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 345
    },
    "executionInfo": {
     "elapsed": 707,
     "status": "ok",
     "timestamp": 1603475216373,
     "user": {
      "displayName": "Brigitta Sipocz",
      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GgSZ5lvx2m0gmkej0jid6to7H2aVpPwrDwknsPBVg=s64",
      "userId": "02384660603430700458"
     },
     "user_tz": 420
    },
    "id": "vJ0v8K_Id1Nc",
    "outputId": "b37c8854-d9db-4df3-e18c-baf423a3f571"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0, 2.0)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Set up the plots\n",
    "fig = plt.figure(figsize=(5, 5))\n",
    "ax1 = fig.add_subplot(111)\n",
    "\n",
    "# plot the data and window function\n",
    "ax1.plot(x, y, '-k', label=r'data $D(x)$')\n",
    "ax1.fill(x, w, color='gray', alpha=0.5,\n",
    "        label=r'window $W(x)$')\n",
    "ax1.fill(x, w[::-1], color='gray', alpha=0.5)\n",
    "\n",
    "ax1.legend()\n",
    "ax1.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "\n",
    "ax1.set_xlabel('$x$')\n",
    "ax1.set_ylabel('$D$')\n",
    "\n",
    "ax1.set_xlim(0.01, 0.99)\n",
    "ax1.set_ylim(0.0, 2.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "FVbNylWZd1Nk"
   },
   "source": [
    "#### Compute the Fourier transforms of the signal and window"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "id": "KtaTXE0Xd1Nl"
   },
   "outputs": [],
   "source": [
    "y_fft = np.fft.fft(y)\n",
    "w_fft = np.fft.fft(w)\n",
    "\n",
    "yw_fft = y_fft * w_fft\n",
    "yw_final = np.fft.ifft(yw_fft)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "fx5OjrM5d1Nq"
   },
   "source": [
    "#### Show Fourier Transform result of two functions\n",
    "The function on the top shows the Fourier transform of $D(x)$ and the bottom shows the transform of $W(x)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 351
    },
    "executionInfo": {
     "elapsed": 635,
     "status": "ok",
     "timestamp": 1603475233896,
     "user": {
      "displayName": "Brigitta Sipocz",
      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GgSZ5lvx2m0gmkej0jid6to7H2aVpPwrDwknsPBVg=s64",
      "userId": "02384660603430700458"
     },
     "user_tz": 420
    },
    "id": "59zTPJZsd1Ns",
    "outputId": "ef7cce16-23b4-4809-82a4-5b5563c6a06c"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, '$x$')"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Set up the plots\n",
    "fig = plt.figure(figsize=(5, 5))\n",
    "\n",
    "# plot the Fourier transforms\n",
    "N = len(x)\n",
    "k = - 0.5 * N + np.arange(N) * 1. / N / (x[1] - x[0])\n",
    "\n",
    "ax = fig.add_subplot(211)\n",
    "ax.plot(k, abs(np.fft.fftshift(y_fft)), '-k')\n",
    "\n",
    "ax.text(0.95, 0.95, r'$\\mathcal{F}(D)$',\n",
    "        ha='right', va='top', transform=ax.transAxes)\n",
    "\n",
    "ax.set_xlim(-100, 100)\n",
    "ax.set_ylim(-5, 85)\n",
    "\n",
    "ax.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "\n",
    "ax = fig.add_subplot(212)\n",
    "ax.plot(k, abs(np.fft.fftshift(w_fft)), '-k')\n",
    "\n",
    "ax.text(0.95, 0.95,  r'$\\mathcal{F}(W)$', ha='right', va='top',\n",
    "        transform=ax.transAxes)\n",
    "\n",
    "ax.set_xlim(-100, 100)\n",
    "ax.set_ylim(-5, 85)\n",
    "\n",
    "ax.set_xlabel('$x$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "5VXs79uLd1N7"
   },
   "source": [
    "#### Calculate the product of Fourier transforms\n",
    "We multiply the above two Fourier transform functions and get the product of them (shown below). The inverse transformation of this product give us the convolution of $[D*W](x)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 357
    },
    "executionInfo": {
     "elapsed": 590,
     "status": "ok",
     "timestamp": 1603475237115,
     "user": {
      "displayName": "Brigitta Sipocz",
      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GgSZ5lvx2m0gmkej0jid6to7H2aVpPwrDwknsPBVg=s64",
      "userId": "02384660603430700458"
     },
     "user_tz": 420
    },
    "id": "Zu8iOIO1d1N8",
    "outputId": "7a3c74a1-4b9c-49e1-db19-fa94e0763eb5"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, '$k$')"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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zACmbGICkVXwARrf6ioqKzBBkAFIuMABJq1QtwOgyA5BygQFIWiXqBIlmlNkJQrlQkmqmUsoOwAXAISItSikHgDYAXQCaAfgBuAH4AHTHl0XEl7uq03TAFiDplDIARSSglOoG4Iia/C0RCQCAUmozAE9kuWYA/XHlphzVm6aJVNcAo8sMQMqFyZwCO5VSjUqpOgANRhgiHJLx5XH6+vrgdDrNl8fjmUQVaLpgC5CmwuPxmFkCoCrT9VO2AONFTml9AKCUao2bbQcQiCuPU11dja6urkx2S9MYW4A0FW63G263GwCglDqZ6foZtQCVUu7IdUEg3MLrjCoHEpSJUjIeh2V0ciRrARrzEz0+i2iyJtICdAFoiHSAeBE+BXYgfH3PB2CNUsoPYGuCMlFKbAGSTmkDUEQ8AKIv1MX37MZfxONFPZqwYDCIoqIis6XHa4CUTxwHSFoFg0GUlpZixowZABK3AIuKiszpDEDKpow6QYiyzQjAefPm4dFHH8Udd9wRM3/9+vW47LLLUFpaai5PlC0MQNLKCMCioiI888wz4+YvW7YMy5Yt45ejU07wFJi0MgIwHbYAKRcYgKTVRAOwpKTEXJ4oWxiApFUwGITNZku7nFIKNpuNAUhZxQAkrSbaAgTCp8EMQMomBiBpxQAknRiApBUDkHRiAJJWDEDSiQFIWjEASScGIGnFACSdGICk1cjISEYByMdhUTYxAEkrtgBJJwYgacUAJJ0YgKQVA5B0YgCSVgxA0okBSFoxAEknBiBpxQAknRiApBUDkHRiAJI2IpLxOEAGIGUTA5C0MQY1MwBJFwYgaWOEGQOQdGEAkjYMQNKNAUjaGGE2kUfiG8sxACmbGICkDVuApBsDkLSZTCeIiCAUCuWyWmQhDEDSZjItwOj1iKaKAUjaMABJNwYgacMAJN1KUs1UStkBuAA4RKQlUnYD8AHoBuBPVRYRX85qThc8BiDpljIARSSglOoG4IhMcgPwRKY3A+hPU27KZeXpwsYAJN0yPQVuEJFA5GfHBMpESTEASbepXAO0Z1gGAPT19cHpdJovj8czhSrQhYwBSFPl8XjMLAFQlen6KU+BE+hUStkjrbzABMrjVFdXo6urK9N60jTEAKSpcrvdcLvdAACl1MlM159IALoANCilHAA8ANYopfwAtiLc2ZGqTJQUA5B0SxuAIuJBOPgM8ees6cpECTEASTeOAyRtGICkGwOQtGEAkm4MQNKGAUi6MQBJGwYg6cYAJG0m8zis6PWIpooBSNoEg0EUFRWhuLh4QsuzBUjZxgAkbYLB4IQfhw+cf3Q+A5CyhQFI2mTypegAW4CUfQxA0oYBSLoxAEkbBiDpxgAkbTINQF4DpGxjAJI2mQagUorfDUxZxQAkbTINQIDfDUzZxQAkbRiApBsDkLRhAJJuDEDShgFIujEASRsGIOnGACRtGICkGwOQtGEAkm4MQNJmZGRkUgHIx2FRtjAASRu2AEk3BiBpwwAk3RiApA0DkHRjAJI2DEDSjQFI2mT6RGgAfBgCZRUDkLRhC5B0YwCSFiLCACTtGICkxejoKICJfyWmgQFI2cQAJC0y/VJ0AwOQsokBSFowAKkQMABJCwYgFYKSTBZWSjkAtAHoAtAMwA/ADcAHoDu+LCK+rNaWpg0GIBWCjAIw4lsiEgAApdRmAB4RCSilmgH0x5WbsldVmk6mEoBjY2MIhUIoLi7ORdXIQiZzCuxUSjUqpeoANBhhCMCRoEyUkPFEl8kEYPT6RFORUQswckrrAwClVGvcbDuAQFx5nL6+PjidTrPsdrvhdrszqQZNA1NpARrrz5gxI+v1oguLx+OBx+MxilWZrp/pNUA3gFcjrTwHgA6llD1SDgDojCuPU11dja6urkzrSdNMNgKQKLoBpZQ6men6mV4D9CJ8CuxA+PqeD8AapZQfwNYEZaKEGIBUCCZ9ChzFk6ZMNA4DkAoBxwGSFkNDQwAyD8CysrKY9YmmggFIWpw+fRoAMHv27IzWmzVrVsz6RFPBACQtTp06BSDzADSWN9YnmgoGIGkxODgIAJgzZ05G6xnLG+sTTQUDkLQwAuyiiy7KaD2jBcgApGxgAJIWg4ODmDVrFoqKMvsIMgApmxiAlDNHjhxBbW0t3n333XHzTp06lfH1P+B8J0iia4BtbW24+uqr2UFCE8YApJzZvXs3fD4fNmzYMK7FNjg4mPH1PyD8pUjl5eXjtnf06FH86Ec/wsGDB7Fv374p1ZusgwFIOXPw4EEopXDs2DE89NBDMfMGBwcn1QIEwqfB0QEoIrjvvvtw9uxZc79EE8EApJw5ePAgrrjiCjz88MP43e9+h127dpnzshmAr7zyCrxeL5599lnYbDYGIE0YA5By5tChQ7jqqqvwxBNPYNGiRfjxj3+MsbExAJO/BgiEA9C4Bvjll19i06ZNaGhowIMPPoja2locOnQoa++BpjcGIOWEiODQoUNYsmQJKioq8PDDD+Ojjz7C559/DmDy1wCB8FhAowW4d+9e9Pb24rHHHkNxcTGWLFnCFiBNGAOQcuLEiRM4c+YMlixZAgBwOMLPxz1+/DiA7J0CG9urra0FACxZsgSffvopQqHQlOpP1sAApJwwWmFXXXUVAGDBggUAwoEVCoVw+vTprJwCGwFobP+qq65CMBjEkSNHplJ9sggGIOWEcR3OaAFGB+CXX34JIPP7gA3xLcCKigrzdNrYH68D0kQwACknDh48iPLycsyfPx8AMHfuXJSVleH48eOTvg/YYFwDFBEcP34cCxYsgFIKwPkA5HVAmggGIOXEsWPHsHjxYjOYlFJYsGBBTABOpQU4NjaGs2fPmgFoqKqqQnl5OY4dOzb1N0HTHgOQcqKvrw81NTUx04wAnOyjsAzRj8SKD0ClFKqrq9HX1zfJmpOVMAApJ3p7e5MGYDZagAAwMDCAzz77LCYAAaCmpga9vb2T2jZZCwOQciJZAJ44cQKBQADA1K4BAsDhw4cRCoUYgDRpDEDKumAwiIGBgYQBGAwG4fOFv1drqi3Ajz/+2NxuNAYgTRQDkJJ699138dlnn2W83smT4a9nTRSAALB//34AUw9AYzvJAlBEMt724cOH0d3dPal60YWHAUgJHTp0CCtWrMCmTZsyXtdofcUHoDEk5uOPP4ZSKuOnQRviW4DGdg01NTUYGRmZ1PeGbNiwAbfeeiv8fv+k6kYXFgYgJfT4448jFArh9ddfNwcuJ9LT04NbbrkFH374oTnN6IFN1gI8cODApJ4GbTCuAR44cABlZWWoqqqKmW/sN7on2Ov1wuVypQw2n8+H999/H6dOnUJLS8uk6kYXFgYgjbN3715s374dLpcL586dw+uvv55wORHBAw88gLfffhtNTU3mdKMFWF1dHbP8pZdeijvvvBPDw8OTPv0Fzj8Venh4GPfff7851tBg7NeoRygUws9+9jPs3LkTW7ZsSbrd7du3AwBuvfVWPPfcc7yOaAEMQBrnH//4BwDgT3/6E+bPn49t27YlXO7FF1/Em2++CafTiTfeeAP/+te/ACQ/BVZK4Y9//CMaGhqwaNGiSdevpKQE8+fPxx133IFnn3123Hxjv0Y92tracODAATidTng8HuzcuTPhdrdt24bly5ejpaUF586dS7ocTSMiktdXfX29UGFbtWqVXH311SIi8tBDD4nNZhO/3x+zzK9//WtRSsk3v/lNOXXqlFRXV8sPfvADERF55JFHpKSkRMbGxhJuf3h4WE6fPj2lOg4MDEgoFEo47/jx4wJAnn/+eRERqaurk+uuu05OnTolS5YskbKyMvnrX/8as87+/fsFgDz33HMyMjIiFRUV8pOf/GRKdaT8AtAlGeYRW4AUQ0Swe/duLF++HACwbt06jIyM4G9/+5u5THd3N376059i1apV+Oc//4nZs2fjzjvvhNfrxejoqDkGMP7U1FBaWjrpDhCD3W5Peg0x+hS4t7cX3d3duOeeezB79mzs2rULS5cuxfr163HixAlznW3btqGoqAirV69GSUkJbrjhBuzevXtKdaTCxwCcpkQEwWAw4/UOHToEv99vBqDT6URtba15Giwi2LRpE+bOnYuXX34Z5eXlAACXy4XBwUHs2bMn4SDofCotLYXdbkdvby/eeusts35A+F7hP//5zxgdHcVjjz0GIPyetm3bhhUrVuDSSy8FACxfvhwffvhhyg6gZILB4KSG4FD+MQCnoU8//RROpxPXXnstvvjii4TLhEIh7NixAwcPHjQfUw/AbPUYAaiUwrp167Bz50709vbi73//O9555x08+eSTMXdyrFixAkC4tzXRfcD5VlNTg76+PuzcuRNz5sxBfX29Oc/hcODnP/85XnzxRXzwwQf44IMPcPjwYaxbt85cZvny5QiFQujs7DSnjY6OYt++fXjrrbeSBtzBgwcxf/58fPvb30567KmAZHrOPNUXrwHm1uHDh8Vut0tlZaXMnDlTbrjhBjl79uy45f7v//5PAAgAmTdvnvzyl7+UsbExue++++Tiiy+Oub5mXB976KGH5Otf/7pceeWVMjIyMm6b119/vaxYsUKuuOIKWb9+fU7fZzo33XSTrFixQi6//HK58847x80PBAJit9vljjvukB/+8IdSVlYm/f395ny/3y8A5Mknn5Th4WF54IEHpLKy0jxmr7322rht9vX1SW1trXnsFy1aJCdPnszp+6TzMIlrgFkPOAB2AJsBNAJwxM9nAObG8PCwHD58WJYtWyaVlZXS09Mjf/nLXwSAuN1uEREZGhqSl156ST7++GO55JJL5KabbpIXXnhBbr/9dgEgv/71r6W8vFzuueeecdvfuHGj+Z//5ZdfTliHhx9+WGw2m9hsNtm0aVNO3286d911l1x00UXm+0rkqaeeMt/T1q1bx81fvny5zJ8/X55++mkBIOvWrZOXX35ZrrzySrn++utl3759sn37dhkdHZVQKCS33XablJaWyu7du+X999+X0tJS+e53vys+n09GR0dz/ZYtbzIBqCTL1yqUUpsBeEQkoJRqFpGm6PnXXnutvPTSS2brM7olGi1VebLzprqdkZERFBUVoaysDKWlpRgZGcG5c+cwPDwcPphRz75TSqG0tBRlZWX4/PPPzcfA//vf/8ZFF12EhQsXYtGiRVi4cCHOnTuH/v5+DA8P48CBAxARXHbZZXjvvffgcDhw8803IxAIQCmFTz75BPv370dlZSXefvttDA4O4q677sIrr7xiDvt4/fXXsWrVKgDAI488gubmZjQ2NmLv3r0xT0p+9913cdNNN2F0dBRLly7F/v37UVpaigMHDpjf4WE4deoUvvrVr2LWrFnYt28fiouLEe/o0aO47777sGvXLng8HmzYsGHcMvnyzDPPYOvWrbjtttvwwgsv4OKLLx63zODgIGpra7Fw4UK8//77sNlsMfN37dqFm2++GQBw2223YceOHQDCw4PWr19vLldXV4eamhrs2LEDzz//PDZu3AgA+M1vfoMHH3wQAHDFFVdg1apVePXVV7F48WJ84xvfgN/vR319vfncxFmzZuHNN9+E3++H0+nEkSNHUFFRgSVLlsBms6GmpgZFRUU4evQojh07hqNHj2J0dBQ33HADysrKUFZWhpqaGgwODmJgYACjo6Oorq6GUgpjY2MQEcyYMQMzZ85EaWnpuM+r8XO6f42fx8bGEAqFEAqFzJ+NyynGctGvZJLNy3S60+ncIyLOpDtKJNPETPcC0Jbo56hpYuVXZWWlVFRUJJ1fXFwsxcXFAkCqq6ulqKho3DKLFi2SiooKWb58udxyyy0CQG655Rb5wx/+IF1dXTF/FUdGRuTuu++WhQsXyrJly+SVV16Ru+++22wVGnbs2CEA5Be/+EXSv7C9vb3S19eX4m9wWLLhL/k2kXocO3ZMBgcHk85fvXq1FBcXy0cffWROGx0dle9///ty//33y+9//3u55pprZOHChbJp06aYfY6Njck777wjra2tsmzZMgEg3/ve92Tp0qUya9YsWbBgwbjfrc1mM0+1bTabKKWSflbmzJkjs2fP1v6ZLqBXQbQA20RkdeTnDhFZGT2/pqZGou8C+M53voPbb7/dWDd+W0nLk503le3YbDaMjY0hGAxieHgYNpsNM2fORFlZmbmsnA96DA8P49y5c7jkkktgs9lQWlpqfntZIBDAsWPHcOzYMcyYMQNVVVWw2WxwOBwQEXzxxRdYvHgx/ve//+GTTz5BVVUVRATz5s0bd4fFwMAA7HZ7yr+yE7F3715cd911CVt3VjU0NIT//Oc/uOaaa6a0nbGxMZw+fXrcI8BOnDgBv9+PUCgEv9+PpUuXYs6cOTh69CgWLFiA4eFhHDlyBKOjo/jiiy8gIli4cCEWLlyI2bNnIxQKma36oaEh9PX1Yfbs2bj44otRXFyM/v5+iIj5Ox0aGsK5c+dieqon86+IoKioCMXFxearqKjIHJoUHzTJJJs30ek7duzAG2+8AQDo6en5r4hcnnRnCeT6FNgMQ4PT6ZSurq6s7pOISCmV8SlwSQ7q4QGwRinlB7A1B9snIsqKrAegiAQQDkEiooKW94HQhfRlNR5PYeR0odQDYF2SYV0SK6S6AKhKv0isvAeg8bTgQlAov7xCqQfAuiTDuiRWSHUBUJ1+kVi8FY6ILCvrvcBpd6jUaQAH87rT5KoAFEKTtFDqAbAuybAuiRVSXZaIyKxMVsh7ABIRFQqeAhORZTEAiciycjEQmmhKlFJ2AG4APgDdIuLTVAcnAEekHj4AbQC6ADTns05KKUf0vgH4oen4KKUaAWwEEIjUoxl5PC6R34sL4SdNtcR/VpDpscn05uHJvBB+RFYjgM1RZfORWfHlPNWpEUAHwr+81kg99hg/56MOUXWJ2beO4xH3u3JFPkQuHccl8t7tkZ+b8/n+o+rgjqpDh/F70VSXmH3rPD4A6qJ+duk4LpF9GlkScywyPTZ5aQFK+L7g7kjFgfCHy3xkFoD+uHJTsm1lkU8iD2pQSrki074l4TtZdDD3Hf9IMeTneBjWAHhVRLxKqQ6E/9rn+7g0iIjxxbyOlEvmiIh4AEApVYdwywIAnJEWh09EupOtmyPmvqHx+BjvWynlinxGHNB7XOKPhSOTY6PrFHhKlc6GAvxFFsoHvND+49vzvL94a+X8My19AKCUakX4D0NeSPg0Lnrf0ez5qoch8lmwAwnrlrfjkoAd4VPz6HJKhdAJYk9Tzpn4X6SIeEWkHXn+JabZtz2fdYmyVkSaNB2XzsjvBoj9QOdV5HrXVqWUQynljqpTXv8oJdi37uPjMvar87hExB+LjI5NVluAkQ9MtICIeBMs2qmUskdOqwIJyvmqU8wvEuFTvwBy9ItMVpcE++7I1fFIV5eoeVsjrWIXcnxcEtD+RKHIZZGNANYi3MJpRbgl7EB+L0kAgDdu3z7oPT6VkTokqls+uAA0RPYZ/1nJ6NjkbSB05D/5SoQPkh/ha01+nO9hM8v5Os2K1KlLRLojB9N4deXzVC9+39B0PCJ1cSH8Owrg/H98LceFKNd4JwgRWVYhXAMkItKCAUhElsUAJCLLYgBSQVJKNUfGIhLlDAOQCpWdPc6UawxAKlSVSim7UqoxamArUVYxAKlQ2REeexjQWw2azhiAVHAig7F9CN+O59X4gAqa5hiAVIiM26r6o57UQ5R1vBOEiCyLLUAisiwGIBFZFgOQiCyLAUhElsUAJCLLYgASkWUxAInIsv4fZ+GK1xlaG+8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5, 5))\n",
    "ax = fig.add_subplot(111)\n",
    "ax.plot(k, abs(np.fft.fftshift(yw_fft)), '-k')\n",
    "\n",
    "ax.text(0.95, 0.95, ('Pointwise\\nproduct:\\n' +\n",
    "                     r'$\\mathcal{F}(D) \\cdot \\mathcal{F}(W)$'),\n",
    "        ha='right', va='top', transform=ax.transAxes,\n",
    "        bbox=dict(fc='w', ec='k'), zorder=2)\n",
    "\n",
    "ax.set_xlim(-100, 100)\n",
    "ax.set_ylim(-100, 3500)\n",
    "\n",
    "ax.set_xlabel('$k$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "HagyS8qjd1OD"
   },
   "source": [
    "#### Perform the convolution\n",
    "Use equation $(a*b)(t) \\equiv \\int_{-\\infty}^{\\infty} a(t')b(t-t')dt'$ we take the convolution of D(x) and W(x)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "id": "B2MDOl4hd1OE"
   },
   "outputs": [],
   "source": [
    "y_norm = np.convolve(np.ones_like(y), w, mode='full')\n",
    "valid_indices = (y_norm != 0)\n",
    "y_norm = y_norm[valid_indices]\n",
    "\n",
    "y_w = np.convolve(y, w, mode='full')[valid_indices] / y_norm\n",
    "\n",
    "# trick: convolve with x-coordinate to find the center of the window at\n",
    "#        each point.\n",
    "x_w = np.convolve(x, w, mode='full')[valid_indices] / y_norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "uuBZGDyvd1OJ"
   },
   "source": [
    "#### Show convolution result\n",
    "The result of $[D*W](x)$ can be viewed as the data signal smoothed with the window (top-hat) function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 351
    },
    "executionInfo": {
     "elapsed": 581,
     "status": "ok",
     "timestamp": 1603475244686,
     "user": {
      "displayName": "Brigitta Sipocz",
      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GgSZ5lvx2m0gmkej0jid6to7H2aVpPwrDwknsPBVg=s64",
      "userId": "02384660603430700458"
     },
     "user_tz": 420
    },
    "id": "fMzeGpgDd1OL",
    "outputId": "bd99ecbe-2418-4167-bd03-4c682a1b67e1"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0, 1.99)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the convolution\n",
    "fig = plt.figure(figsize=(5, 5))\n",
    "ax = fig.add_subplot(111)\n",
    "ax.plot(x_w, y_w, '-k')\n",
    "\n",
    "ax.text(0.5, 0.95, \"Convolution:\\n\" + r\"$[D \\ast W](x)$\",\n",
    "        ha='center', va='top', transform=ax.transAxes,\n",
    "        bbox=dict(fc='w', ec='k'), zorder=2)\n",
    "\n",
    "ax.text(0.5, 0.05,\n",
    "        (r'$[D \\ast W](x)$' +\n",
    "         r'$= \\mathcal{F}^{-1}\\{\\mathcal{F}[D] \\cdot \\mathcal{F}[W]\\}$'),\n",
    "        ha='center', va='bottom', transform=ax.transAxes)\n",
    "\n",
    "ax.set_xlabel('$x$')\n",
    "ax.set_ylabel('$D_W$')\n",
    "\n",
    "ax.set_xlim(0.01, 0.99)\n",
    "ax.set_ylim(0, 1.99)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "P8CKAQQ_d1OQ"
   },
   "source": [
    "#### Show Convolution Diagram\n",
    "Here we take all the process together and express them in a clockwise order. \n",
    "* Top-left are the original data signal $D(x)$ (solid line) and top-hat window functions $W(x)$ (gray box). \n",
    "* Moving on to the top-right panel, we have the Fourier transform of two functions, with $F(D)$ as the Fourier of $D(x)$ at the top, and $F(W)$ as the Fourier of $W(x)$ at the bottom. \n",
    "* Next we multiply $F(D)$ and $F(W)$ to get a pointwise function in the bottom-right panel. \n",
    "* At last, we can take the inverse Fourier of $F(D) \\cdot F(W)$ and get the convolution of function $D(x)$ and $W(x)$, as shown in the bottom-left panel."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 232
    },
    "executionInfo": {
     "elapsed": 580,
     "status": "error",
     "timestamp": 1603476937455,
     "user": {
      "displayName": "Zhouyangguang Zhao",
      "photoUrl": "",
      "userId": "12804106343770884783"
     },
     "user_tz": 420
    },
    "id": "PTo1FfJ0d1OR",
    "outputId": "4046c59f-d30a-4b14-c421-92110d502c16"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 6 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Set up the plots\n",
    "fig = plt.figure(figsize=(5, 5))\n",
    "fig.subplots_adjust(left=0.09, bottom=0.09, right=0.95, top=0.95,\n",
    "                    hspace=0.05, wspace=0.05)\n",
    "\n",
    "#----------------------------------------\n",
    "# plot the data and window function\n",
    "ax = fig.add_subplot(221)\n",
    "ax.plot(x, y, '-k', label=r'data $D(x)$')\n",
    "ax.fill(x, w, color='gray', alpha=0.5,\n",
    "        label=r'window $W(x)$')\n",
    "ax.fill(x, w[::-1], color='gray', alpha=0.5)\n",
    "\n",
    "ax.legend()\n",
    "ax.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "\n",
    "ax.set_ylabel('$D$')\n",
    "\n",
    "ax.set_xlim(0.01, 0.99)\n",
    "ax.set_ylim(0, 2.0)\n",
    "\n",
    "#----------------------------------------\n",
    "# plot the convolution\n",
    "ax = fig.add_subplot(223)\n",
    "ax.plot(x_w, y_w, '-k')\n",
    "\n",
    "ax.text(0.5, 0.95, \"Convolution:\\n\" + r\"$[D \\ast W](x)$\",\n",
    "        ha='center', va='top', transform=ax.transAxes,\n",
    "        bbox=dict(fc='w', ec='k'), zorder=2)\n",
    "\n",
    "ax.text(0.5, 0.05,\n",
    "        (r'$[D \\ast W](x)$' +\n",
    "         r'$= \\mathcal{F}^{-1}\\{\\mathcal{F}[D] \\cdot \\mathcal{F}[W]\\}$'),\n",
    "        ha='center', va='bottom', transform=ax.transAxes)\n",
    "\n",
    "for x_loc in (0.2, 0.8):\n",
    "    y_loc = y_w[x_w <= x_loc][-1]\n",
    "    ax.annotate('', (x_loc, y_loc), (x_loc, 2.0), zorder=1,\n",
    "                arrowprops=dict(arrowstyle='->', color='gray', lw=2))\n",
    "\n",
    "ax.set_xlabel('$x$')\n",
    "ax.set_ylabel('$D_W$')\n",
    "\n",
    "ax.set_xlim(0.01, 0.99)\n",
    "ax.set_ylim(0, 1.99)\n",
    "\n",
    "#----------------------------------------\n",
    "# plot the Fourier transforms\n",
    "N = len(x)\n",
    "k = - 0.5 * N + np.arange(N) * 1. / N / (x[1] - x[0])\n",
    "\n",
    "ax = fig.add_subplot(422)\n",
    "ax.plot(k, abs(np.fft.fftshift(y_fft)), '-k')\n",
    "\n",
    "ax.text(0.95, 0.95, r'$\\mathcal{F}(D)$',\n",
    "        ha='right', va='top', transform=ax.transAxes)\n",
    "\n",
    "ax.set_xlim(-100, 100)\n",
    "ax.set_ylim(-5, 85)\n",
    "\n",
    "ax.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "ax.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "\n",
    "ax = fig.add_subplot(424)\n",
    "ax.plot(k, abs(np.fft.fftshift(w_fft)), '-k')\n",
    "\n",
    "ax.text(0.95, 0.95,  r'$\\mathcal{F}(W)$', ha='right', va='top',\n",
    "        transform=ax.transAxes)\n",
    "\n",
    "ax.set_xlim(-100, 100)\n",
    "ax.set_ylim(-5, 85)\n",
    "\n",
    "ax.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "ax.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "\n",
    "#----------------------------------------\n",
    "# plot the product of Fourier transforms\n",
    "ax = fig.add_subplot(224)\n",
    "ax.plot(k, abs(np.fft.fftshift(yw_fft)), '-k')\n",
    "\n",
    "ax.text(0.95, 0.95, ('Pointwise\\nproduct:\\n' +\n",
    "                     r'$\\mathcal{F}(D) \\cdot \\mathcal{F}(W)$'),\n",
    "        ha='right', va='top', transform=ax.transAxes,\n",
    "        bbox=dict(fc='w', ec='k'), zorder=2)\n",
    "\n",
    "ax.set_xlim(-100, 100)\n",
    "ax.set_ylim(-100, 3500)\n",
    "\n",
    "ax.set_xlabel('$k$')\n",
    "\n",
    "ax.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# Plot flow arrows\n",
    "ax = fig.add_axes([0, 0, 1, 1], xticks=[], yticks=[], frameon=False)\n",
    "\n",
    "arrowprops = dict(arrowstyle=\"simple\",\n",
    "                  color=\"gray\", alpha=0.5,\n",
    "                  shrinkA=5, shrinkB=5,\n",
    "                  patchA=None,\n",
    "                  patchB=None,\n",
    "                  connectionstyle=\"arc3,rad=-0.35\")\n",
    "\n",
    "ax.annotate('', [0.57, 0.57], [0.47, 0.57],\n",
    "            arrowprops=arrowprops,\n",
    "            transform=ax.transAxes)\n",
    "ax.annotate('', [0.57, 0.47], [0.57, 0.57],\n",
    "            arrowprops=arrowprops,\n",
    "            transform=ax.transAxes)\n",
    "ax.annotate('', [0.47, 0.47], [0.57, 0.47],\n",
    "            arrowprops=arrowprops,\n",
    "            transform=ax.transAxes)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "cugjz8ifd1OW"
   },
   "source": [
    "## 2. Discrete Fourier Transform\n",
    "In practice, data are always discretely sampled. When the spacing of the time interval is constant,\n",
    "the discrete Fourier transform is a powerful tool.\n",
    "Although in astronomy, temporal data are rarely sampled with uniform spacing (LIGO data, however, as a counterexample, will be discussed in the later notebook), uniformly sampled data is a good place to start because we have fast algorithm for this situation.  \n",
    "When computing the Fourier transform for discretely and uniformly sampled data, the Fourier\n",
    "integrals from \n",
    "\n",
    "$$H(f)=\\int_{-\\infty}^{\\infty} h(t) exp(-i2\\pi ft)dt$$\n",
    "$$h(t)=\\int_{-\\infty}^{\\infty} H(t) exp(-i2\\pi ft)df$$\n",
    "\n",
    "are translated to sums. Here we have t for time and f for frequency.\n",
    "  \n",
    "The discrete Fourier transform of the vector of values hj is a complex vector of length N defined\n",
    "by\n",
    "\n",
    "$$H_k=\\sum_{j=0}^{N-1}h_j exp[-i2\\pi jk/N]$$\n",
    "\n",
    "where k=0,...,(N-1).  \n",
    "The corresponding inverse discrete Fourier transform is defined by\n",
    "\n",
    "$$h_j=\\sum_{j=0}^{N-1}H_k exp[-i2\\pi jk/N]$$\n",
    "\n",
    "where j=0,...,(N-1). Unlike the continuous transforms, here the units for $H_k$ are the same as\n",
    "the units for $h_j$ . Given $H_k$, we can represent the function described by $h_j$ as a sum of sinusoids,\n",
    "as was done in the first figure."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Boa3Dfngd1OX"
   },
   "source": [
    "### (1) The Nyquist sampling theorem\n",
    "Let us define $h(t)$ to be band limited if $H(f) = 0$ for $|f| > f_c$, where $f_c$ is the band limit, or the\n",
    "Nyquist critical frequency. If $h(t)$ is band limited, then there is some \"resolution\" limit in $t$ space,\n",
    "$t_c = 1/(2f_c)$ below which h(t) appears \"smooth.\" When $h(t)$ is band limited, then according to the Nyquist sampling theorem we can exactly reconstruct h(t) from evenly sampled data when $\\Delta t < t_c$,\n",
    "as\n",
    "\n",
    "$$h(t)=\\frac{\\Delta t}{t_c} \\sum_{k=-\\infty}^{k=\\infty} h_k \\frac{sin[2\\pi f_c (t-k\\Delta t)]}{2\\pi f_c (t-k\\Delta t)}$$\n",
    "\n",
    "This result is known as the Whittaker-Shannon, or often just Shannon, interpolation formula\n",
    "(or \"sinc-shifting\" formula).  \n",
    "Note that the summation goes to infinity, but also that the term\n",
    "multiplying $h_k$ vanishes for large values of $|t - k\\Delta t|$.  \n",
    "For example $h(t) = sin(2 \\pi t/P )$ has a period\n",
    "P and is band limited with $f_c = 1/P$. If it is sampled with $\\Delta t$ not larger than $P/2$, it can be fully\n",
    "reconstructed at any $t$ (it is important to note that this entire discussion assumes that there is no\n",
    "noise associated with sampled values $h_j$ ). On the other hand, when the sampled function h(t) is not\n",
    "band limited, or when the sampling rate is not sufficient (i.e., $\\Delta t > t_c$), an effect called **\"aliasing\"**\n",
    "prevents us from exactly reconstructing h(t). In such a case, all of the power\n",
    "spectral density from frequencies $|f| > f_c$ is aliased (falsely transferred) into the $-f_c < f < f_c$\n",
    "range. The aliasing can be thought of as inability to resolve details in a time series at a finer detail\n",
    "than that set by $f_c$. The figure below will show us the effect of aliasing."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "mF11YtRbd1Oa"
   },
   "source": [
    "#### Define functions and terms\n",
    "Here we defind a Gaussian shaped data and its Fourier transform as an example."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "id": "1_b_9Cybd1Ob"
   },
   "outputs": [],
   "source": [
    "def gaussian(x, a=1.0):\n",
    "    return np.exp(-0.5 * (x / a) ** 2)\n",
    "\n",
    "\n",
    "def gaussian_FT(f, a=1.0):\n",
    "    return np.sqrt(2 * np.pi * a ** 2) * np.exp(-2 * (np.pi * a * f) ** 2)\n",
    "\n",
    "a = 1.0\n",
    "t = np.linspace(-5, 5, 1000)\n",
    "h = gaussian(t, a)\n",
    "\n",
    "f = np.linspace(-2, 2, 1000)\n",
    "H = gaussian_FT(f, a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "WmZ9o3iad1Og"
   },
   "source": [
    "#### Show a comparison of well-sampled and over-sampled results\n",
    "A visualization of aliasing in the Fourier transform. \n",
    "We have two sets of four panels, and in each set:\n",
    "* the top-left panel shows a signal and a regular sampling function\n",
    "* the top-right panel shows the Fourier transform of the signal and sampling function\n",
    "* the bottom-left panel shows the sampled data\n",
    "* the bottom-right panel shows the convolution of the Fourier-space representations\n",
    "  \n",
    "In the top four panels, the data is well sampled, and there is little to no aliasing.   \n",
    "In the bottom panels, the data is not well sampled (the spacing between two data points is larger) which leads to aliasing, as seen in the overlap of the convolved Fourier transforms. The aliasing effect can be recognized if the Fourier transform is nonzero at $|f| = 1/(2 \\Delta t)$ in this example."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "id": "BmBMhJeLd1Oi"
   },
   "outputs": [],
   "source": [
    "# Two plots: one well-sampled, one over-sampled\n",
    "N = 12\n",
    "\n",
    "def plot_aliasing(dt):\n",
    "    \"\"\"\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "\n",
    "    dt : float\n",
    "       \n",
    "\n",
    "    \"\"\"\n",
    "    # define time-space sampling\n",
    "    t_sample = dt * (np.arange(N) - N / 2)\n",
    "    h_sample = gaussian(t_sample, a)\n",
    "\n",
    "    # Fourier transform of time-space sampling\n",
    "    df = 1. / dt\n",
    "    f_sample = df * (np.arange(N) - N / 2)\n",
    "\n",
    "    # Plot the results\n",
    "    fig = plt.figure(figsize=(7, 3.5))\n",
    "    fig.subplots_adjust(left=0.07, right=0.95, wspace=0.16,\n",
    "                        bottom=0.1, top=0.85, hspace=0.05)\n",
    "\n",
    "    # First plot: sampled time-series\n",
    "    ax = fig.add_subplot(221)\n",
    "    ax.plot(t, h, '-k')\n",
    "\n",
    "    for ts in t_sample:\n",
    "        ax.annotate('', (ts, 0.5), (ts, 0), ha='center', va='center',\n",
    "                    arrowprops=dict(arrowstyle='->'))\n",
    "    ax.text(0.03, 0.95,\n",
    "            (\"Signal and Sampling Window\\n\" +\n",
    "             r\"Sampling Rate $\\Delta t$\"),\n",
    "            ha='left', va='top', transform=ax.transAxes)\n",
    "    ax.set_ylabel('$h(t)$')\n",
    "    ax.set_xlim(-5, 5)\n",
    "    ax.set_ylim(0, 1.4)\n",
    "    ax.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "    ax.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "    ax.set_title('Time Domain: Multiplication')\n",
    "\n",
    "    # second plot: frequency space\n",
    "    ax = fig.add_subplot(222)\n",
    "    ax.plot(f, H, '-k')\n",
    "    for fs in f_sample:\n",
    "        ax.annotate('', (fs, 1.5), (fs, 0), ha='center', va='center',\n",
    "                    arrowprops=dict(arrowstyle='->'))\n",
    "    ax.text(0.03, 0.95,\n",
    "            (\"FT of Signal and Sampling Window\\n\" +\n",
    "             r\"$\\Delta f = 1 / \\Delta t$\"),\n",
    "            ha='left', va='top', transform=ax.transAxes)\n",
    "    ax.set_ylabel('$H(f)$')\n",
    "    ax.set_xlim(-1.5, 1.5)\n",
    "    ax.set_ylim(0, 3.8)\n",
    "    ax.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "    ax.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "    ax.set_title('Frequency Domain: Convolution')\n",
    "\n",
    "    # third plot: windowed function\n",
    "    ax = fig.add_subplot(223)\n",
    "    for (ts, hs) in zip(t_sample, h_sample):\n",
    "        if hs < 0.1:\n",
    "            continue\n",
    "        ax.annotate('', (ts, hs), (ts, 0), ha='center', va='center',\n",
    "                    arrowprops=dict(arrowstyle='->'))\n",
    "    ax.plot(t, h, ':k')\n",
    "    ax.text(0.03, 0.95, \"Sampled signal: pointwise\\nmultiplication\",\n",
    "            ha='left', va='top', transform=ax.transAxes)\n",
    "    ax.set_xlabel('$t$')\n",
    "    ax.set_ylabel('$h(t)$')\n",
    "    ax.set_xlim(-5, 5)\n",
    "    ax.set_ylim(0, 1.4)\n",
    "    ax.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "    ax.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "\n",
    "    # fourth plot: convolved PSD\n",
    "    ax = fig.add_subplot(224)\n",
    "    window = np.array([gaussian_FT(f - fs, a) for fs in f_sample])\n",
    "    ax.plot(f, window.sum(0), '-k')\n",
    "    if dt > 1:\n",
    "        ax.plot(f, window.T, ':k')\n",
    "    ax.text(0.03, 0.95, \"Convolution of signal FT\\nand window FT\",\n",
    "            ha='left', va='top', transform=ax.transAxes)\n",
    "    ax.set_xlabel('$f$')\n",
    "    ax.set_ylabel('$H(f)$')\n",
    "    ax.set_xlim(-1.5, 1.5)\n",
    "    ax.set_ylim(0, 3.8)\n",
    "    ax.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "    ax.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "\n",
    "    if dt > 1:\n",
    "        fig.suptitle(r\"Undersampled data: $\\Delta t > t_c$\")\n",
    "    else:\n",
    "        fig.suptitle(r\"Well-sampled data: $\\Delta t < t_c$\")\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 269
    },
    "executionInfo": {
     "elapsed": 1235,
     "status": "ok",
     "timestamp": 1603476277340,
     "user": {
      "displayName": "Brigitta Sipocz",
      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GgSZ5lvx2m0gmkej0jid6to7H2aVpPwrDwknsPBVg=s64",
      "userId": "02384660603430700458"
     },
     "user_tz": 420
    },
    "id": "LThinIzQn7L7",
    "outputId": "c899453b-3f53-4e0f-9dc8-a04453af577a"
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 504x252 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_aliasing(0.9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 269
    },
    "executionInfo": {
     "elapsed": 1469,
     "status": "ok",
     "timestamp": 1603476278010,
     "user": {
      "displayName": "Brigitta Sipocz",
      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GgSZ5lvx2m0gmkej0jid6to7H2aVpPwrDwknsPBVg=s64",
      "userId": "02384660603430700458"
     },
     "user_tz": 420
    },
    "id": "1AZiZtcen9CE",
    "outputId": "cdaa3fd8-bbe8-409e-cd32-029491e86c6d"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x252 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_aliasing(1.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "WOMT-ozOd1Op"
   },
   "source": [
    "### (2) The window function\n",
    "The figure above shows the relationship between sampling and the window function: the sampling window\n",
    "function in the time domain can be expressed as the sum of delta functions placed at sampled\n",
    "observation times. The Fourier transform of a set of delta functions with spacing $\\Delta t$ is another set of delta functions with spacing $1/\\Delta t$; this result is at the core of the Nyquist sampling theorem. By the convolution theorem, pointwise multiplication of this sampling window with the data is equivalent to the convolution of their Fourier\n",
    "representations, as seen in the right-hand panels. When data are nonuniformly sampled, the impact of sampling can be understood using the same framework.  \n",
    "The sampling window is the sum of delta functions, but because the delta functions\n",
    "are not regularly spaced, the Fourier transform is a more complicated, and in general complex,\n",
    "function of f. The PSD can be computed using the discrete Fourier transform by constructing a\n",
    "fine grid of times and setting the window function to one at the sampled times and zero otherwise.\n",
    "The resulting PSD is called the **spectral window function**, and models how the Fourier-space signal\n",
    "is affected by the sampling. As discussed in detail in [Fourier analysis with unequally-spaced data](http://adsabs.harvard.edu/full/1975Ap%26SS..36..137D), the observed **power spectral density (PSD)** is a convolution of\n",
    "the true underlying PSD and this spectral window function.  \n",
    "We will see an example of an irregular sampling window in the figure below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "iIz1qxLdd1Oq"
   },
   "source": [
    "#### Generate the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "id": "6F7LlasQd1Or"
   },
   "outputs": [],
   "source": [
    "Nbins = 2 ** 15\n",
    "Nobs = 40\n",
    "f = lambda t: np.sin(np.pi * t / 3)\n",
    "\n",
    "t = np.linspace(-100, 200, Nbins)\n",
    "dt = t[1] - t[0]\n",
    "y = f(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "VCl1Y5m7d1Ov"
   },
   "source": [
    "#### Select observations and compute PSD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "id": "s1-8TqU_d1Ow"
   },
   "outputs": [],
   "source": [
    "# select observations\n",
    "np.random.seed(42)\n",
    "t_obs = 100 * np.random.random(40)\n",
    "\n",
    "D = abs(t_obs[:, np.newaxis] - t)\n",
    "i = np.argmin(D, 1)\n",
    "\n",
    "t_obs = t[i]\n",
    "y_obs = y[i]\n",
    "window = np.zeros(Nbins)\n",
    "window[i] = 1\n",
    "\n",
    "# Compute PSDs\n",
    "Nfreq = int(Nbins / 2)\n",
    "\n",
    "dt = t[1] - t[0]\n",
    "df = 1. / (Nbins * dt)\n",
    "f = df * np.arange(Nfreq)\n",
    "\n",
    "PSD_window = abs(np.fft.fft(window)[:Nfreq]) ** 2\n",
    "PSD_y = abs(np.fft.fft(y)[:Nfreq]) ** 2\n",
    "PSD_obs = abs(np.fft.fft(y * window)[:Nfreq]) ** 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "zuKDXgj0d1O3"
   },
   "source": [
    "#### Scale PSDs for plotting\n",
    "Normalize the true PSD so it can be shown in the plot: in theory it's a delta function, so normalization is arbitrary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "id": "y34Wrsc5d1O3"
   },
   "outputs": [],
   "source": [
    "PSD_window /= 500\n",
    "PSD_y /= PSD_y.max()\n",
    "PSD_obs /= 500"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "RKwFwO9ad1O8"
   },
   "source": [
    "#### Show figure\n",
    "An illustration of the impact of a sampling window function of resulting PSD. \n",
    "* The top-left panel shows a simulated data set with 40 points drawn from the function $y(t|P) = sin(t)$ (i.e., $f = 1/(2\\pi) ~ 0.16$). \n",
    "* The bottom-left panel shows radom sampling illustracted by the vertical lines. \n",
    "* The bottom-right panel shows the PSD of sampling times, or spectral window. \n",
    "* The top-right panel shows the PSD computed for the data set from the top-left panel; it is equal to a convolution of the single peak (shaded in gray) with the window PSD shown in the bottom-right panel (e.g., the peak at f ~ 0.42 in the top-right panel can be traced to a peak at f ~ 0.26 in the bottom-right panel)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "id": "DKI0Q-5fd1O9",
    "outputId": "f50a807e-2b99-4b09-f1ec-e985f37bee7a"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x288 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Prepare the figures\n",
    "fig = plt.figure(figsize=(8, 4))\n",
    "fig.subplots_adjust(bottom=0.15, hspace=0.2, wspace=0.25,\n",
    "                    left=0.12, right=0.95)\n",
    "\n",
    "# First panel: data vs time\n",
    "ax = fig.add_subplot(221)\n",
    "ax.plot(t, y, '-', c='gray')\n",
    "ax.plot(t_obs, y_obs, '.k', ms=4)\n",
    "ax.text(0.95, 0.93, \"Data\", ha='right', va='top', transform=ax.transAxes)\n",
    "ax.set_ylabel('$y(t)$')\n",
    "ax.set_xlim(0, 100)\n",
    "ax.set_ylim(-1.5, 1.8)\n",
    "\n",
    "# Second panel: PSD of data\n",
    "ax = fig.add_subplot(222)\n",
    "ax.fill(f, PSD_y, fc='gray', ec='gray')\n",
    "ax.plot(f, PSD_obs, '-', c='black')\n",
    "ax.text(0.95, 0.93, \"Data PSD\", ha='right', va='top', transform=ax.transAxes)\n",
    "ax.set_ylabel('$P(f)$')\n",
    "ax.set_xlim(0, 1.0)\n",
    "ax.set_ylim(-0.1, 1.1)\n",
    "\n",
    "# Third panel: window vs time\n",
    "ax = fig.add_subplot(223)\n",
    "ax.plot(t, window, '-', c='black')\n",
    "ax.text(0.95, 0.93, \"Window\", ha='right', va='top', transform=ax.transAxes)\n",
    "ax.set_xlabel('$t$')\n",
    "ax.set_ylabel('$y(t)$')\n",
    "ax.set_xlim(0, 100)\n",
    "ax.set_ylim(-0.2, 1.5)\n",
    "\n",
    "# Fourth panel: PSD of window\n",
    "ax = fig.add_subplot(224)\n",
    "ax.plot(f, PSD_window, '-', c='black')\n",
    "ax.text(0.95, 0.93, \"Window PSD\", ha='right', va='top', transform=ax.transAxes)\n",
    "ax.set_xlabel('$f$')\n",
    "ax.set_ylabel('$P(f)$')\n",
    "ax.set_xlim(0, 1.0)\n",
    "ax.set_ylim(-0.1, 1.1)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "LQnha-UWd1PE"
   },
   "source": [
    "### (3) The Fast Fourier Transform (FFT)\n",
    "The **Fast Fourier transform (FFT)** is an algorithm for computing discrete Fourier transforms in\n",
    "O(N logN) time, rather than O(N2) using a naive implementation. The algorithmic details for\n",
    "the FFT can be found in NumRec. The speed of FFT makes it a widespread tool in the analysis\n",
    "of evenly sampled, high signal-to-noise ratio, time series data.  \n",
    "An example below shows such analysis for a function with a single dominant frequency: a sine wave whose amplitude is modulated by a Gaussian."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "G0QsDAord1PF"
   },
   "source": [
    "#### Generate the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "id": "2y0B-e3Bd1PG"
   },
   "outputs": [],
   "source": [
    "np.random.seed(1)\n",
    "\n",
    "tj = np.linspace(-25, 25, 512)\n",
    "hj = np.sin(tj)\n",
    "hj *= norm(0, 10).pdf(tj)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "pgCSQNhUd1PK"
   },
   "source": [
    "#### Show result with data and calculated PSD\n",
    "The discrete Fourier transform (bottom panel) for two noisy data sets shown in the top panel. For 512 evenly sampled times t (dt = 0.977), points are drawn from h(t) = a + sin(t)G(t), where G(t) is a Gaussian N(mu = 0,sigma = 10). Gaussian noise with sigma = 0.05 (top data set) and 0.005 (bottom data set) is added to signal h(t). The value of the offset a is 0.15 and 0, respectively. The discrete Fourier transform is computed as described in this section. For both noise realizations, the correct frequency f = (2pi)-1 ~ 0.159 is easily discernible in the bottom panel. Note that the height of peaks is the same for both noise realizations. The large value of abs(H(f = 0)) for data with larger noise is due to the vertical offset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "id": "ePznsDVSd1PL",
    "outputId": "983c15ef-8510-46ba-9220-b52f0f0e13cb"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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lV8tstbwA7R6TeoRYtrDU62TK6dXKH61wNNGWREym2ULNdqeomr0s5JAnj8cT9Xz1s8SMK2HF68WRasX/9vf3d2smSIe0U5Z6byyRcdu3b0dnZydKSkqirBwtR7k6ljGekUGjRmLVRSDOB6Jj1ESD7erqUhS8eI5IuxVLzkrAt54VoBfTZyWg3Aj1IJKZNWl0f/W1otGL/LMyl1hYNMA3ClI9GKFlGelZS1p5YjQKrkZrIFE9mCUsLDnqQUshaeWP+KtuS+qYTL2RbHWcrEiXVt6I55i9LNS/yy97LUtYbvt6caTqdiVehnawpCwZYzUIb/3AAezgnO+2/SSHMAruFseA0aOiepXWajiOmZWlV8DAN4HtwOiRczV6MWqccyVMyWy01y5mAd/yrAj1mzue0VMAmDdvXtQsD6P4UrvpEI0IgKVRVNmiEYr2xIkTuPbaa1FSUqKrLAB70xPFs6wq7wULFkTJro5FBEYv/6aVb1a7yXrX641ky3+10mU0VdnuHHijiRlGdUVuV5xzdHd3O9sNZ4yVI7yM2kGEVwY6BaCCMbYCQCvn/JCdhzmBWcFoVSQnniWjp7D1nidbOFaVnLrCqBd4sDswEeu1WvktK3w7ysxOwzCSOZa0WLF61ZaWiN0Uc7q1XDlbtmzRjbW1+iw9ZOVtty7HaggYXa9liVqxGo3Saudla9QD0pNZ65y8vDx0dnYiFAqdMTxZDedc9wOgPJbfnPzMmTOHBwIBvnbtWh4IBLjMwMAA37ZtGw8EAnzbtm18YGBA83f18XiR77tt2zbe3NzMt23b5ugztIjnWfFcOzAwwDdv3sw3bNjAt2zZopvfTsowMDDA165dq5zvRp6r64/6fyGHLIsTdc6sXuuh106MnjEwMGAos1l64i2LWNMaD+KZWVlZ7dyGLjK0LDnnXQDAGJvJOT/EGJsAYDLn/JD4zSomuzv6AFTwb/boiUK93L3AbEZCvF1EPcxm48hYsY6ErGaWkvpZemFQVq61g+w38/l86OjoiClfjWTQGq2Uu1xqa97ukm2xoNWtV7sI1D0GJ+pcrPfQaydaqN1EZoMxeltFx1OvhBw1NTWOTRKw+syFCxdiZGRk2M51pj5LxthTAMojOy/uQnizsWdikFFzd8fI/+K+mpgtd69XYFYKMt6YO7NupF4lkBsEYD74IJ4ld/vFPfSW39IafFGvfmQFPTeC3QZixTcq0qDucqn/t7tkmxmxxifOmzfPlvKw8pxYFZBeO7Hrn9QK35IHRmSlGcugnho9X2QsbdPOYKpdTJUl5/zHEd/lKYQtQFsWpYTm7o5WMAvZ0Sswq6E6dt/iVgvRyCFt5NuxIq+8oIDeoI+V0BaraVFbFVqDa7GiFVtqxaqT/8aL3XogwrlCoVDU4Eq8vkIr99BDr53Y9U9qRSCIATKra7sK7L4c5HNiaZta0QNOYTbAczPnfLPU5X5F/VuMzy22euLJkyfh9Xrh8XiQlZUVteGQE8TS6KwWopFDWm0lWi1U+U0vFhQoKSmJKbjZalqsKF0gditdHVdn5T6xKhS9e9utB+pAbas4peSNgvnVeWb3mVrn60VpmBHPyyGWvDIbTI1nd0dTpyaAhwH8FMCKyKch8neCHecogEYAxZHvL6l+qwDQqHXdlVde6Zgz36kBH6fuE++gS7LSonWO1gDBli1bYkqP3sCJ+j6xpjmewSF5AGLz5s18y5YtowZFEjWQaIQ6HfL/bsijh54sbskoPxfhMEhnBngiyvR3AMAYm4jw4E6s3fAWAPcwxk4hsu0tAB/nvAXh7v1cra1wPR7PqIVRY8WpAR8n/DSAtYEKoyB8J2SwGm5hFKMqz+ONpaysdrFjLT/14JDWvc2ulf3CcqA2YD/G0gnU6dAK8UqmPGrkeqs12SOZAzoy6rECO1iewcM5P8MY42Jk3O6DuMHujhGFqf4NwDd78DiB034uq+iNWlsZqEiFim+GWfyb3e653gwX8XLRWv3djnyxjOBrzZLRur+TGOWbkU9Xa8AmUYMeephNd7Sz6n882J09ZITVGTwNACoR3tubA/i/tp+UAthpKE6uTGM0am1WeMlU8FYXkdBbHV0Pqwpfvq9eyIpYiBmA4juL1b9p1zeqtbhKIon1Rak1YJOoQQ89jOqt2cvVSeSBuHnz5sXVps0GeK7j4amNOznnj0eOTYxJahfQa+hW3rJOWnSydaIetTZTNE51t61gdSDHbt5YVfha3WX16KvswLe7iZvR8+LJ40RZ/068KPWmTCYCq+sEJLJOq2WQB+LiLSczy/I+xtiPEZ7ieDeAjQhPe0wLtKwTq29ZJy06J6dgqnHSAtbyoVoZUTfDauPQ6i5rjb7KZRlPd04vHXaWaTO6T7w4FcMY65RJI+yuNJQs1DJorZgUazmZzeD5Z/l/xtgtAO5BbEHpSUO9AoxsnRiFFsS7go7TWFGETlZQLR+q2eCOk1h5lt5gjZMTCuxa06lQV/RIlCLXyg+3xgRk1DKoyyaecrK1RBvnfFPMT0oiWoHX4u3i8Xh0fU+p8GaUsSJPLBU0kVZSolchj2ewJpZnGB1LB5KZR6nw0kikDBmzrYRMrA02GdsN2CFR8ljZtiLWZ9vdEoMgkoWGP9PWHjxZiRTOLcTbxa6CifW6RJEoeWpqakzjIYVV297e7vi9zRDz14PBoKPn9/b24vHHH0dvb2/MsrmJ3Xwhoom1TgsyUlmmC+rKn6zGYEUJx6r0xL3Pnz+vKCa7SspupRbnt7W16eZfb28vfvCDH6CxsRE33HAD9u/fbytdqUC8jX2sE++LPO22lYiFVOteC9Q+yVTymcbr+3n22WfR2Nio/N/Y2IiBgQEUFRXhoYce0t3PHIh9LrPWIg+9vb1YvXq1EmJUVlaGjo4O/OM//iNuvfVWU1kSQW9vL5599lnbz05Xv6lVEt1O4/Zn2pkb6cZnzpw5cc8HTfRisU7MWY7nPqlGIBDgTU1NvKmpiQcCAeV/n8/HAfA1a9Yk5Lla+dfU1MQRnkjBy8rK+IYNG3hdXR2vqKjgAPiyZcssLZbrFIFAgC9btiyh+ZCuxDpvP9b2Aptzw8dEN9wJP5oRsXaP1N3hRPgok+3n6u3txX333Yef//znyrEnn3wSW7ZsQWtreO/yo0ePorm52XHfoZx/vb29aG5uxtatWwGEly/r6enB7t27MW7cOHR2dsLj8eDtt9/Gk08+6agceuzfvx833HAD3n77bXi9XnznO98BkP6+VKew206T7pawo1nd+DhhWSaaVLMI/X4/r6ur4wsWLOANDQ28ubmZb9iwgQcCAb5mzZqEWVKBQECxHgHwuro6vmjRIuV/9SeRVt2aNWuU5/h8Pu73+5W0+/1+7vV6o35PtHUZCASUZxYVFUWlX1i/yZBDT7ZE1otEEe/KT7BpWbquDM0+6aAs3UY0uIaGBr5y5UpeUlKiKILCwkJFaSay+6dWlFVVVVGKsa6ujjc0NPC6ujoOgJeXlydUQcjyNDU1af4uuwaqq6t5Y2NjwmURZSMUp9fr5atWrUq64l6zZg33+/1RedDU1OS60ozV8IjF1UbKMsGk2ltYbSWJz6RJk/iMGTNGKS+v18v9fr/jcmhZckKBC98l59/kX0NDg3K+ljKLB1kZmJWVbPElShb5JdLQ0KDIJp5bUlLCFyxYkDSFKcpKTrfcCxDlpVV+iUYovQ0bNthqZ2RZSspSVlJuKizRZWpqauJ+v58vW7aMt7W18YaGBu7z+ZQBhEWLFkUdT4SC4pwr1uKkSZP4ggUL+IoVK5Tnqa0n0Tic7P76/X7u8/n4qlWrbDUsWYlUVVU5kj/C/WB38Mbv9ysvEycVlTyYo3Vvv9/Py8rKlN9kpZpIS1dW1HK65RfrypUro16+ieqNyO04EAjw1atX8w0bNvDVq1crZWjlpRcLKassEd5KohHAcoR3cjQ8Lj5Tp05VlJIoNPFmFN0GkZltbW2KotJrfFpK16gw1OeLStXQ0DCqwgHgubm5yne5O+z1eh0pbCFPW1sb9/l8fMWKFVHP16rUspUgusFOKQVZGdhtULJVd9ttt/HHHnuMHz161BFZhBKyKpNZtz0WxIu1rq5O9yUi6rb8cpPL06kXmxyhIOSqrq5WXnLiI54t6m55ebmitOM1UuR6KIwIuR3Lcsn1WtQRdVnGY/0GAgEO4ChPUWUpbyux2uy49LvSCHw+n5LRK1euVDJT/VdkuFYmypahULpavjxhMYmuifp7Y2Oj8kzhc/J4PIqVJ6wb8b9Zo7FSuE1NTYqyk9Mq0ml0b5FW2YKIVynIA0mrVq2KKW0inx988EHe3NzMH3vsMdtyyC8Q2aq327CdHGiRrVW7+RyrhWxFFlEP5Xog13/5XFl+2XUiuxKM8ljPNyp/fD6fUqenT5/OJ0+eHPW7UNrTp09XylXUddEG5XuJtqAlm5bRwG3osKTNDWeMvcQ5v9vqd+m65AhIEMSYg3Nuecc5t+Isi60enzBhgvLd5/MBAOrq6rBy5UpUVVUpv4nva9asQVNTk+HDy8vLo66Rqa6uxqJFiwAAkyZNijq/rq4OTU1Nihxer1d51po1a3TfSIFAQLlGTVVVFdra2rBmzRoEAoFR1/r9fixbtgyrVq1SZBD3ampq0r3O6BMIBJTl66qrq9HQ0ICmpiZb9/H7/airq8OiRYvg9/sRCARikkVOo9/vj6nXIudvU1NTXD0gv9+P6urqmO61Zk14p+eKigrMnTtXyZtY07Rs2TKljBobG03zVr6mqakJ1157LRYsWIDCwkJUV1dj1apVlspaXZZ+v19pK+Kv1+vFhg0b4PP5sGrVqlFtqaioSPku2pNcdwXi//nz56Ouri4qnYFAAE1NTWhoaMDKlStRV1eHFStWKO1RbpNtbW1KuYnjDQ0Nuu2urq4OAI5p/qhHPBXLzgc6uzvqHRefa665ZtQAgtyVFh/Z7Fb7RlauXMnnz5+vjA7LsX/V1dUcAJ8/f74SziI+criN3BVS+5msdPe0ugDiI/xr8u+iWyuHnIhjTji8tUbRfT6fpfvLAxPLli2LWQaB7CKI1U0h1wknZEEMbhM5X5zIG60BIqOykUe5Gxsb+YIFC3hzczN/8MEH4xpAk/Nk5cqVUWFPcv0R7ibRdS4pKeErVqzgdXV1vLGxMcrNYdRdtiKHaLeivOU6JNq3cCOUlpYq/lfZXYQUH+CpR3ggZzbC29/Wq4+rrxOj4SIzRMbGoizUPhR1YWk55BMx8i6etXLlSkVBl5eXawZwNzQ0KA1Qz9EdjxxCGYuKLhq51nNEXohrysrKHBnFVisFK+lTD845NWKqjiCwIo+QRdSfiooKw0FGu/LIssij53L9VfsFxe9NTU28sLAwrjqjNTgk6qJQWqKuCt+xMCbUxoYTA0RaSla+t5BRllPL75uyyjLWj1boUKJwIyRJLljR0IQFLEYiReWLZdDCDLWiEsHjWta6OM9JZSDLIV4gsmVtdcAqXotSSx5h5RuNBsv5IgZCnK4/6lFyMQNIy3qqq6tTei4i/5yUSeslZTS4KPfCkoWRUpXJWGWZqcjdc7nhxxOSE6sM8gijaPjirSyOiy6WE91vLeR0GylB0WjFCG0iArnlbl9VVZWSB3LvRrzsEhkXKZBdJ7JFJ3f9E/kCyTRIWaYx6phOozd2op4vGpqw8GQrVxxPpKWgDnPRavDqQPhErWak7gJXV1dHWWtCMdnpsjshk+yPb2pqUmJunfRpjwVIWRIxI1uSejFxyWiAWsHG8otEy/pMlHIQ3XHxshDyyMdk10Uy8kc9KYOUYmyQsiRiRlZIsvUmzxlOJurRU2HhihHWWAPh45FFtmTlTzLXplS7AGhdzNiwqyzHxErphDVKSkrQ0NCgfN+4cSOA6JW93aC1tRVDQ0MAgK6uLjzzzDNoampCc3Nz0mQQca5iQzYgHKs3b948eDyepOaNKKfe3l5l5Xki8ZCyJEyRlWgyWbVqlaKcOOdYuXIl3nrrLXR1dSVdlpKSEhQVFUXtXLlq1aqkb0mhlsmNchmrkLIkUpaSkhK88MIL+MEPfoC3334bd9xxBz755BPXrFzxTDf27SHcJyP3DScyi1g3+CIII+zuG06WJZHyUHeTSAXGxIZlBEEQ8ULKkiAIwgKkLAmCICxAypIgCMICpCwJgiAsQMpSRUtLi9siJBRKX/qSyWkDUj99SVOWjLFixlgjY2w5Y6xCdXw5Y6wxWbIYkeoFFi+UvvQlk9MGpH76kmlZ1gNo4Zy/DOARcZBz3gdgVxLlIAiCsE0yg9Lncs7XRL5XGJ4psXPnzn7GmKzUewAEHJUsmlLGWCLv7zaUvvQlk9MGJCd9pQDKIt+vtHOhWzN4iq2eyDkfn0A5CIIgLOG4smSMLVcd6uOctwL4hDFWHOl29zn9XIIgiESStIU0GGPFAO4BcApAJ8IK08c5b2GM1QNYAuBRznlnUgQiCIKwQcqvOkQQZkRexPUIv4R3ZcoLN5IuH4AKzvmaTEtnJD21CI9hdALYgRROH8VZSkTCmHyMsfrIX81wp3RFHaaVQenTjLRIdzQiRTItnfcgvLVDC4BHkeLpI2UZTVoVnl0yuPHNjaQNsBFpkYZkVDo55y2c8z7G2GyE62VKp4+UpUS6FZ4DZGL6it0WIEkUuy2Ag9zLOX9UdazYDUGMIGWpTVoUnsMUuy1AHHwS8X8BmR1pkXHpjETP/CriBkrp9I3JldINwps0Cy/dwp2M0qciLdOnQQuAexhjpwD8ym1hHMYHYG6kPmZUOhljPoTdP/ciPKjzK6Rw+mg0XCJSeI8irDiUwkMk3IlznvbTMuUwLYTTlVHpI4hEQcqSIAjCAuSzJAiCsAApS4IgCAuQsiQIgrAAKUuCIAgLkLIkMgbG2NNuy0BkLqQsiYwgEodYkeZz3IkUhpQlkSmcArAx1VaqITIHUpZEpuADsEuaLkcQjkLKksgU+hBeDGSyy3IQGQrN4CEIgrAAWZYEQRAWIGVJEARhAVKWBEEQFiBlSRAEYQFSlgRBEBYgZUkQBGGBpG0rYbTncWSFcgCAzvYHBEEQrpJMy1Jz29XIfjGnIkqS5vUSBJGSJHPDsrmc8zWR77JSbAWwiTG2A+F9YaIoKCjg2dnZyv+lpaUoKyvDsWPHwBjD5ZdfnlChZXp6elBWVuboPUdGRpCVZf+dlQhZYoVkSV05AJJF/fxAIAAACAaDg5zzAssXc86T8gHwkvR9o/R9NsLzelcDWK2+zuPxcDXDw8O8ubmZ//znPx/1WyKZM2eO4/d84403YrouEbLECskymlSRg3OSRQ8AA9yGDktmN1xvT2Af57yVj96ne0zQ0tLitggEQVggmd3wqD2PI+sO+gC0RvyWnQA2JlGelGDdunVui0AQhAWSpiw5530IK0wZ8b/uftWlpaWJEsk29fX1jt/z/vvvx/PPP58SssQKyTKaVJEDIFkM6LFzcsqvOlRbW8t37NgRdWxkZAS/+MUvAABNTU1uiOUYzz//PO6//363xSCIMQdjbCfnvNbq+WkZlM4Yc1sExyBFSRDpQVoqy0wikxQ/QWQypCxdJtXdIARBhCFl6TKxDO4QBJF8SFm6zJtvvum2CARBWICUpcuQZUkQ6QEpS5e588473RaBIAgLkLJ0mRQL0iUIQgdSli6zePFit0UgCMICpCxdZsqUKW6LQBCEBUhZusy5c+fcFoEgCAskTVkyxooZY42MseWRFYfk3+oZY7Mjqw+NKWiJNoJID1JlW4lOzvkuGKw+JJNJUwTVi4QQBJGaJFNZzo0s0wZEbyuxBEBFRGmOuT14yLIkiPTALZ9lser/HRGLc9Rq6T09PaitrVU+maZcaDScIJJHS0uLoksA2FosN5krpX/CGCuOWJd90vGDRheVlZVldFe1ubnZbREIYsxQX1+vxDYzxgJ2rk2FbSXE8QqENy0bU8yaNcttEQiCsECqbCsRc9+ac57WAz61tbU4duyY22IQBGECxVm6DClKgkgPSFm6zK9//Wu3RSAIwgKkLF2GLEuCSA9IWboMWZYEkR6QsnSZOXPmuC0CQRAWiHs0nDFWA6AWAEc4uHx3vPccS2RakD1BZCoxW5aMsXLG2MMAJgFoBfBS+DBbwRib6ZB8Gc1HH32U1mFPBDGWiMuy5Jz/TnWoHUA7Y6w8nvuOFVasWIHe3l4cP37cbVEIgjAhZmXJOe+K5TciTH9/PyZOnIjbb7/dbVEIgrBAPN3wFU4KMtbYtGkTbr31Vrz//vtui0IQhAXiGQ2fxBi7jjF2nVPCxALn3M3Hx0xHRwcAICcnB8PDwy5LQxCEGfH4LFsA3APgbsaYWDloJ4BWzvmheAXLdM6cOYP8/Hzccsst6OjowLe+9S23RSIIwoB4fJZnAPyOMfaXyHcwxiYC8DHGZnPOX5XPZ4wVI7xaeieAXZzzTtXvjQivpN5n5fmMsbS1KgcHB5GXlwcA+O1vf4vy8nJSlgSR4sQdlC4Upfq7BprbSgBAZHm2ynhlSRf279+vhAzdf//96Oqi8TCCSHXiGeC5Wes45/wVAFpKU29bCfG/4SLAmcTOnTsBABMmTEAoFMLp06ddloggCDPi6YZvjgSlFwMQrX1S5PtfTC4vFl8YYz7OeStjbLbWiWJbCYG80nG60t3dDSC8luXjjz+Oe+65x2WJCGJs0NLSIs+aS962EiIoPeKrnGwSX6m3rcQpxpgPwFyE/Zkvyxdl2rYSg4ODGBoaAmMMs2fPxiOPPILh4WH09PSgrKzMbfEIIqOJZ1uJeLrhExljTzHGXgTwsIVAdLF9xHJEtpVgjNVHtsAV2nByrPKkC0eOHAFjDMXFxSgqKsLWrVuRnZ2NtrY2t0UjCMKAeAZ4/hnAo5zzexGe4vhfjLEJeidzzvs45y2c85c557s4552c8xbpt7vF/5nMgQMHAABerxcAMHXqVADAF1984ZpMBEGYE4+y3CFGvznnmzjn/wfhuEvCgH379gEAKivDg/8//OEPAdAiwASR6sSjLLWCHCkGxoDBwUGcOXMGnHNMmzYNAPD9738fAHD+/Hk3RSMIwoR4lOXPGGPvMMZejCzLdh20FSgR4ejRowCA3Nxc5OfnAwA2btwIACgsLCSFSRApTMzKknNeyzlfGvFZdgFYAhf2/U6nWTyHDh0CAEyfPl05FggEwBjDpEmT8Omnn7okGUEQZjiyrUTEZ/k4yGdpiJipM3v2NyGlv/jFL1BSUgLGGHbt2uWWaARBmBBP6NB16mPJXMcy3VYYD4VCOH78OEZGRlBVVaUc37p1Ky655BIA34yUEwSResRjWS5hjH0XABhjDYyxnzokU0Zy5MgRcM4xNDSk+CuBcJCsCEY/deqUW+IRBGFCPMpyJ+f81YjCLEF4BSJaEFgH4a8sLi6OOl5bW6tYliMjI7S2JUGkKPEoy8pIEPqPAfwyEnNJoUM6HD58GABw1VVXRR2vr69XlOWll16K/fv3J102giDMiWc0/HcIL7W2mnN+NjI/fKJjkmUQoVAIx44dw8jICG688cao38aPH4/JkycjOzsb48aNU1YkIggitYh3NHwj53xT5HsFgPQadUkSR48excjICPr6+kZ1w48dO4asrCyUloYXQPn8889dkJAgCDPiGQ3/fwDuY4z9FwBwztsja1kSKoS/Mjs7e9RvW7duBQClKy4C1wmCSC3isSw3cs7/GcAaxthMh+TJSIS/8oorrhj1m1hbT4yIDwwMpFWgPUGMFeJRlhMBJbay2OxkxlgxY6yRMbY8so2EfNzHGKuPrGuZUYRCIXz11VfgnEctYixYt24dgG8sy5KSElpUgyBSEEfmhgO418K2uHp78NyD8ApGLQAetStEqlthwl/59ddfayrL+++/HwCiRsTb29uTKiNBEOY4NTe8FeG54b8zuERzD57IGpd9kW0lRs33E9tKiI+0JHxaILrgZ8+ehcfjGfX7HXfcASAcf5mbm4u8vDxSlgSRIFpaWhRdgmRuKyGIjIhvYoy9bHpymGKNY/dyzkdZlum+rYQY3Jk4UTuqSliWjDGUlZXh2LFjFGtJEAnClW0ltLCyB0/ke5/8g7zVhNVnpcPccNlfeeWVV2qeI6dDdMV7enqSIh9BENZxxLK0iNiD5xS+UYw+hDcpewTAvZHvtv2WqcqXX36JkZERnDlzBkuWLNE8R/a5CmWZn5+vGZNJEIR7OGpZGqG3Bw/nvJVzviSyB0/GKMqRkRFs27YNALB9+3Zcd911muc9//zzyncRPjRjxgxa25IgUoykKcuxRkdHB3p7e1FYWIhgMIiioiLN8958803lu7Asi4uLaW1LgkgxSFkmAM65YlXm5+dHLfarRrYsx48fj4KCAmRlZdG0R4JIMUhZJoCDBw/i+PHjKCoqwr59+zBnzhzdc++8807lO2NMsS6PHDmScDkJgrAOKcsEIKzKa665Bi+99BJuu+023XNFGINA+C0vXLiAwcHBxAlJEDHQ19eHp556Cq+++uqYq5+kLB3myJEjOHz4MPLz87F27VqsWLFCN2wIABYvXhz1v7AsKysrFaWbDC5evJi0ZyWC/fv344033kB/f7/bomQsw8PDePnll3HixAns2bMHv//978fU6v6kLB1GKLjS0lJ88MEH+Nd//VfD86dMmRL1v1CWV199Nerr63H27NnECBohFArhrbfewi9/+Uts2LAh5aePanHkyBG8+OKLaG9vxwsvvIChoSG3RcpIWltb8dVXX2HChAkoLS1FT08PnnnmGXR2drotWlJIe2WZSo3766+/xv79+5Gbm4vf/OY3+M///E/NKY4y586di/pfdMOHh4exZMkS/PjHP05YGg8fPoynnnoKO3bsAOccH374IbZv3+7IvTnnOHPmjCP3MqKvrw8vvvgihoeHkZ2dja+++gqvv/56StWLTKCjowMfffQRsrKysHz5cqxYsQLV1dU4f/481q5di7a2tozP87RXlqmEsCoHBwdx9dVXY9myZabXqOe6FxUVoaioCKFQCE1NTfjss8/whz/8wVE5h4aG8M477+B//ud/cPr0aVx66aW45ZZbAACbNm2Ke2764OAg/vznP+M//uM/8MYbb2BkZMQJsXWfEwwGUVlZiYcffhj5+fn48ssvsXnz5oQ8cyzS19eH//3f/wUA3HLLLZg2bRry8/Nx33334YYbbgDnHOvXr8cbb7yR9u4cI9JWWabadMfe3l588cUXyMrKwnPPPYcnnnjC0nVa895FV/zcuXN44YUX8NOf/hR+v98RObu7u/H000/jo48+AmMMN954Ix5++GHccMMNykDUunXrYn7e6dOn8fvf/16Z397e3o6XXnrJ8UY0MjKCV199FSdPnkRpaSmWL1+OSy+9FHfffTcYY9i2bRt2797t6DPTjcHBQXz00Uf4/PPPY7b6hJ/ywoULmDVrFubPn6/8lpWVhVtuuQXf+973kJOTg927d+O5555LuOvILdJWWeoRCoXwySef4JNPPknqaJ3ovnZ3d+MnP/nJKF+kHlqrKImu+MmTJ3HNNdfg3/7t33DffffFlZ5z586htbUVzz77LHp7e1FaWoof/ehHuPnmm5UV3OfNm4cbb7wRnHO89NJLtldtP3LkCJ555hn09PSgtLQU3/3ud1FQUICOjg786U9/crQ8Nm3aBL/fj8LCQnz/+99HQUEBgPDAmLDo161bpyxkMpYIhULYtm0bnnjiCbzzzjt45ZVX8Pzzz8ekxISfcuLEifjOd76jaaRcc801eOihhzBhwgR0d3fjt7/9LV5//XUcP3487rScPXsWn376KQ4fPuz6zqcs1f0MtbW1XMv6euyxx3Dx4kX87Gc/Q25uLkKhED7++GN8+OGHCAaDAICCggLMmzcP8+bNQ2Fhoe4zRkZGcOjQIezduxfZ2dm46qqrMG3aNGRlWXuXnD17Fk888QRGRkawefNmbN68WXMLCS0WL16sbC0h2LlzJ958802UlJSgpqYGM2bMwKpVq1BeXo5///d/t3RfAAgEAujo6MC+ffvQ3d2tHJ8/fz5uvvlm5OSMXhqAc45169ahvb0dhYWF+OEPf6gobyN2796NdevWYWRkBJWVlVi+fDkKCgpw4sQJrF27Fv39/bj88svxwAMP6M5mskp7ezveeOMNZGVl4e///u8xc+bMUee88847+Oijj1BQUIAVK1agpKTE9nMGBwcxMDCASZMmxd2TGRgYgN/vx/79+3Hx4kVUV1fD6/Vi/Pjxcd1XRhgKH3zwgdIGpk6dit7eXly4cAH5+flYunQprrvuOkvp2bdvH1544QVkZWXhH/7hHzBt2jTD8/v7+/H2229j7969yrHp06dj3rx58Hq9ttrTl19+iS+//DLqhZ2bm4sZM2agvLwc5eXluOyyy2yXC+ccQ0NDuHjxIoqKinZyzkcvMqtD2ivLn/70p9i1axc+/PBDnD9/HkB4+wbGmJLReXl5qK2txfz58zFu3DgA4Uw7fvw49uzZg88//3xUyElRURG8Xi+uuuoqzJw501D5rV+/Hm1tbdi/fz/+6Z/+yTAIXc3WrVtHhQ8FAgE89dRTUW/S3NxcdHZ2Yv78+bj11ltRWFgIzjlGRkai/g4ODuLgwYPYt28fent7leuzs7NRWVmJhQsXYvr06YYyjYyM4MUXX4Tf78eECRPwox/9CBMmTNA9d9OmTfjggw8AANdffz2WLl0a1TBOnz6NP/7xjzh9+jRKSkrw4IMPjlokZGhoCIcPH8aBAwcUy/Tyyy/H5ZdfjtLSUiX/Dx8+jD/84Q8YGRnBnXfeqTs7Sk7D5MmTsWLFCsMXpiAYDGLfvn3Yu3cvOjs7MTw8jLy8PEyZMkX5TJ06FRMnTjRsqJxznDx5Evv27YPf78dXX32led7UqVPh9Xrh9XqVTevsMjQ0hB07dmD79u0YGBgAEG4DixcvRkVFBfr7+/Hmm28qrpWqqirccccdussGAmE/5dNPP40LFy5gyZIlWLBggWV5Tp8+jY8//hjt7e1Kb2LixImYO3cupk6diuzs7FGfkZERHDhwAF988UWUgszJycHMmTNx5syZUatxFRYWYubMmcjPzwfnfFR7GBkZweDgIAYHB3HhwgXlI3Rec3NzairLyPJs9QivLLSLc95pdFxgpiwLCgpw4cIFAMC0adOwaNEiVFSEV3o7fPgw3n//fSW0IScnBzU1NSgqKsKePXuilMmkSZNwzTXXYHh4GHv37sXp06eV3woLC3HllVdi4sSJGBgYQDAYHPUXCFew3/zmN7by5dixY5pd9v7+fnR1deHQoUM4fPhwlKxWKSwsxKxZs3DllVeisrISeXl5lq8dGhrCH//4Rxw9ehRlZWW4/vrro34XisLv98Pv94MxhmXLlmmuBi/Ss3btWpw4cQLjx4/Hgw8+iOzsbOzfvx8HDx7EoUOHdP2a2dnZuPTSS3H55Zdj7969CAaD+Nu//VssXbrUMA2hUAj//d//ja+//hpTp07F1VdfjaKiIowbN04ZSPN4PDh//jz27t2Ljo4OdHV1Rfn3ioqKFAUk4/F4UFJSorwUGGNRn0AgEBUNkJ2djfLycsyaNQs5OTnYt28fDh48GJXm0tJSTJs2DXl5eZqfoaEhnD17FufOnYv6K7/op06disWLF6OysjJKmXPOsWfPHvz1r39VrMxbb70Vf/M3f4P+/n6cPXs26p779+9HIBDArFmzcN9998VkWQ8ODuLTTz9FW1ubrXjMnJwcVFVV4Vvf+haqq6uRn58PIOxK6urqQldXFzo7O2P2jebk5IAxhn/5l39JWWXZiPC2En2MsdVihSG944Jrr72Wv/XWW6Pu99xzzymW16WXXoqamhpMmTJFKVS5cE+ePIndu3ePmkJYUFCAyspKVFZW4pJLLlEqOuccvb29SsH09fWZpq+jowNPP/20rgWmx5QpUyztuXPu3DkcOnQIr7zyCoLBIMrKypCVlQXGGLKyspRPdnY2LrnkEsycOROXXXaZco5AXemNGsGFCxfw+uuvR704tMjLy8PSpUs1N2STGRwcxF//+lccP35cyWeZ0tJSTJ8+HWVlZYol0dPTM6pRTJ8+HX/3d39nqVvX39+v5JkVsrKyMGXKFFRUVGDmzJnweDwYGBhQZDl58iR6enqUF7QRhYWFmDFjBmbMmIErrrgCubm5Ub8PDQ2hu7sbXV1dOHz4cFw+3UsuuQRz5szB9OnTDct0YGAA77//viVf7rhx4/C9731P8QfHCuccR48exb59+3D+/HmMjIxgeHhYsf7E97KyMlRWVmL69OmmL3bOOc6ePYsTJ05geHhYaQfyCysrKwu5ubnIz8+PeulkZ2djeHgYV1xxRcoqy5c453db/S7IycnhcqMQVsH8+fMxYcIEfPbZZ6OUjV6ahPXIGENnZye6u7ujzjW6bsaMGYpv9MKFCwiFQhgaGkIoFMLw8DBWr16N22+/PYacscfFixfx0EMP4csvv8TFixcV/4v4rg7INkqf+jetRubxeHDttdciPz9f8/eLFy9iz549lmMqs7OzsXTpUpSXl+P8+fM4evQoDh8+jKNHjypuFDV5eXkoLS1FWVkZCgoKsGvXLluB5+PGjcOsWbNQWFgIj8cT9begoAAjIyM4evQoOjs7cejQIUtKa/z48Rg3blzUy1l8gPCLxs4izllZWbjsssuU7UXEJycnR/k+PDyMgYEBDAwMoL+/X/keDAZth2dVV1dj4cKFyqpY8v3EJ14FnoqcP39eeXFevHjxMOd8ptVrk7n4r0yx1ePXXXddWm8rYcavf/1r/OQnP7F8fk5OjuNxl8mGc47Tp0+juLjYstM/UQgfl9UBuUxiLKcdcHlbCRP0tpXQ3W5iLDAWt71ljGHy5MmuK0oAiutiLDKW0x4LyR7guQfAKYQHc/oQ3lbiL/JxznnUqrd6AzwEQRDxwBiz5bNMWjc8sg2uOgK7RfV3zDFnzhzs3LnTbTEIgjDB/X7QGCfd9kEniLEKKUuXcXIGB0EQiSPllWUq7aGdCCvwjjvuiOm6VLJISZbRpIocAMligK0pUyk/3bGoqIhrzaBwg9ra2pQJYyJZtEkVWVJFDoBk0YMxFuScW16oIOUty0ynubnZbREIgrBAyluWjLFzAPa5LUeEUgC2AlkTCMmiTarIkipyACSLHldyzi0PGqS8siQIgkgFqBtOEARhAVKWBEEQFnBrIQ1NYl3z0gVZfAAqOOdrXJajFkAFwtNEW1NEllY3y0f6XVn6zy1ZGGMVAF4CsAPAarfzhTFWH5GlgnP+sluyMMaWA3gE4SnPpzjnj7glS+Q3n/hu2I7ECsOp8AHQCKA48n212XE3ZIn8XwGg0eU8qZeOb3RZFh+A2ZFPqpTP0+J3F/OlIlkyWJBlOQCfkMtlWWbLdScF8mV25Hu90T1SrRs+l39jCVRYOO6GLMlGUw7OuVgweTaAXZpXJk+WVoSthHsRVlKuySL9fzBJcpjJUssYWx4pJzdlWQKgImLVudqGeGSxHMaYjyehR2QkC4BWAL9jjD2N8KI+uqSaspQptnk8kbjxTC2KNY7dy1WryyeJYvkfHu7WbET4Te2aLElugIaycM47OeetPNzlTXhX00iWCDsisrheXyLd4mKtE5OA/NwKhPOjD8D/Nboo1ZRlKq15mSrrbOrKEbESfhXxjbkmC2OskTFWHFFSS9yUBcCpiA9qLsLuAddkYYzVS8ddLSMk19IWGLUhn8YxN2TxRV5opi+QlIqzjHXNy2TKwjlviTjKlwB4lCfYaW+QJ5345o3YmQzr0kAWMX/Nh7DzPFmDTXrlUwzgdwj7chM+GdkgX1oRVpIVCFt1rtVbRLehvhQoo3okKU+MZEFkwCtybLJRvqSUsiQIgkhVUq0bThAEkZKQsiQIgrAAKUuCIAgLpNQMHoKIF2mmRl8yBneIsQMpSyLTuAfhKZfJCtQnxgjUDScyhohVebfbchCZCSlLImOITGdLShwuMfYgZUkQBGEBUpZExhDphu90Ww4iM6EBHiIjiMyTB8JTDAnCcciyJDKKRM/VJ8YuNDecIAjCAmRZEgRBWICUJUEQhAVIWRIEQViAlCVBEIQFSFkSBEFYgJQlQRCEBf4/j4bcra26rC0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 360x270 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the results\n",
    "fig = plt.figure(figsize=(5, 3.75))\n",
    "fig.subplots_adjust(hspace=0.35)\n",
    "ax1 = fig.add_subplot(211)\n",
    "ax2 = fig.add_subplot(212)\n",
    "\n",
    "offsets = (0, 0.15)\n",
    "colors = ('black', 'gray')\n",
    "linewidths = (1, 2)\n",
    "errors = (0.005, 0.05)\n",
    "\n",
    "for (offset, color, error, linewidth) in zip(offsets, colors,\n",
    "                                             errors, linewidths):\n",
    "    # compute the PSD\n",
    "    err = np.random.normal(0, error, size=hj.shape)\n",
    "    hj_N = hj + err + offset\n",
    "    fk, PSD = PSD_continuous(tj, hj_N)\n",
    "\n",
    "    # plot the data and PSD\n",
    "    ax1.scatter(tj, hj_N, s=4, c=color, lw=0)\n",
    "    ax1.plot(tj, 0 * tj + offset, '-', c=color, lw=1)\n",
    "    ax2.plot(fk, PSD, '-', c=color, lw=linewidth)\n",
    "\n",
    "# vertical line marking the expected peak location\n",
    "ax2.plot([0.5 / np.pi, 0.5 / np.pi], [-0.1, 1], ':k', lw=1)\n",
    "\n",
    "ax1.set_xlim(-25, 25)\n",
    "ax1.set_ylim(-0.1, 0.3001)\n",
    "\n",
    "ax1.set_xlabel('$t$')\n",
    "ax1.set_ylabel('$h(t)$')\n",
    "\n",
    "ax1.yaxis.set_major_locator(plt.MultipleLocator(0.1))\n",
    "\n",
    "ax2.set_xlim(0, 0.8)\n",
    "ax2.set_ylim(-0.101, 0.801)\n",
    "\n",
    "ax2.set_xlabel('$f$')\n",
    "ax2.set_ylabel('$PSD(f)$')\n",
    "\n",
    "plt.show()"
   ]
  },
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