SPfd_4.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Power spectral density"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The one sided periodogram estimate $S_x(f)$ of the ${\\it power}$ spectral density (PSD) is defined in terms of the Fourier transform of a finite sample of data\n",
    "as\n",
    "$$\n",
    "S_x[k] \\equiv 2\\times\\frac{1}{T}\\left|\\hat{x}[k]\\right|^2\\ .\n",
    "$$\n",
    "The division by $T$ is what turns an \"energy\" into a \"power\"; it also implies that the estimate at a given frequency will be independent on $T$.\n",
    "The factor $2$ takes into account that we use only positive frequencies.\n",
    "\n",
    "Note that the dimensions of the PSD:\n",
    "$$\n",
    "[ S_x[k] ] = [ dt^2 / T ] = \\frac{1}{Hz}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "### importing the library\n",
    "from __future__ import print_function\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "from scipy.fftpack import fft"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Let's simulate a noise time series"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f653b2b1f90>]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Il16qG4NvvLG6bIcdauvCO3WqvXi7GzfT7sug7nxx/fzn5uvef7+9eWXCCglB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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f653b2b1fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "sampling=1024\n",
    "nsample = 32768\n",
    "dt = 1.0 / sampling\n",
    "T = nsample * dt\n",
    " \n",
    "x = np.linspace(0.0, nsample*dt,nsample)\n",
    "nu=50.0# frequency in Hz of the sine function\n",
    "sigma = 1.0\n",
    "y = np.sin(nu * 2.0*np.pi*x) +sigma * np.random.normal(size=nsample)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, y, label=\"Data\")\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### ...and perform its Fourier Transform"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "yf = fft(y)\n",
    "xf = np.linspace(0.0, 1.0/(2.0*dt), nsample/2) \n",
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The simple Power Spectral Density (One Sided, since we are considering only the positive Frequencies)\n",
    "is "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7f6538c0d590>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f653b791fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Let's check the (one sided) periodogram\n",
    "oneSidedPeriodogram = 2/T*abs(yf[0:nsample/2])**2\n",
    "plt.plot(xf,oneSidedPeriodogram)\n",
    "plt.xlabel('Frequency')\n",
    "plt.ylabel('PSD')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Using standard scipy library with its module signal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We obtain the same results, apart the normalization factor for the amplitude"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7f652ee72890>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f65386044d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    " \n",
    "from scipy import signal\n",
    "\n",
    "freqs, P_xx = signal.periodogram(y, sampling, scaling = 'density')\n",
    "plt.plot(freqs, P_xx)\n",
    "plt.xlabel('Frequency')\n",
    "plt.ylabel('PSD') "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, in the plot we can clearly identify a Frequncy (the $50$ Hz) on a random noise. If in some way we can listen to these data, we will clearly hear a whistle at that frequency. What happen if we have a more featured surroinding noise?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "** Let's simulate data using the model we introduced in our first post of these series, which was based on ARMA serie**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f652c7a0f90>]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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AcZ6H9wZwX+PtCQB+Uut81ZqIrAvgB0qp2wBAKfWtUupLNNNyAaAl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\nEmOQJyIqMQZ5IqISY5AnIiqx/wfJc4bY0rWSTAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f652ee72ed0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import statsmodels.api as sm\n",
    "from statsmodels.tsa.arima_process import arma_generate_sample\n",
    "np.random.seed(12345)\n",
    "\n",
    "arparams = np.array([.75, -.25])\n",
    "maparams = np.array([.65, .35])\n",
    "arparams = np.r_[1, -arparams]\n",
    "maparam = np.r_[1, maparams]\n",
    "## simulate a simple sinusoidal function\n",
    "x = np.linspace(0, 100, nsample)\n",
    "y = arma_generate_sample(arparams, maparams, nsample)\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, y, label=\"Data\") "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Now, how does look like its PSD?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7f652b6f1c90>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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rO2d3N1x8cfIsSnHoK+sb36g84W/Wau55mNVg7Ni+Hkapm2+Gd76zfyBZtQpe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\nDh5mZlY3Bw8zM6ubg4eZmdXt/wNq0UhRB0bVYAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f652ee08bd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    " \n",
    "freqs, P_xx = signal.periodogram(y, sampling, scaling = 'density')\n",
    "plt.plot(freqs, P_xx)\n",
    "plt.xlabel('Frequency')\n",
    "plt.ylabel('PSD') "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We cannot recognize a single dominant contribution to a given frequency, but we found a forest of lines.\n",
    "Each of these lines identify the intensity of the signal at the corresponding frequency"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we plot the data in a semilogaritmic plot, we can see better what we call a **broad band** noise. That is a noise where the **spectrum** of frequency contributing to it is very wide"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7f652edaee50>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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WEz1ujDfKTfgoIiICXddMNgHx4EADtgH+Hm27uw+qeQ6rpJqJiEj10tZMuurN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\nv9YAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f652b5c0790>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.semilogy(freqs, P_xx)\n",
    "plt.xlabel('Frequency')\n",
    "plt.ylabel('PSD') "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
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