{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Difference equation:\n",
    "\n",
    "$$y[n] = b_0x[n] + b_1x[n-1] + ... + b_Mx[n-M]$$\n",
    "\n",
    "for this simple filter:\n",
    "\n",
    "$$y[n] = x[n] + x[n-1]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can estimate the effect of frequency from the difference equation!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "A6sjaIP/2PV/AAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xe449110>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "from scipy.signal import freqz\n",
    "import  numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "frq, resp = freqz([1,-1])\n",
    "plt.semilogy(frq, np.abs(resp))\n",
    "plt.title('Frequency Response')\n",
    "plt.grid()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "iBhCKe4REUMoxT0iYgj9f7xm9XnqIjORAAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xe282bd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(frq, np.angle(resp))\n",
    "plt.title('Phase Response')\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$y[n] = x[n] + 0.5 \\cdot x[n-1]$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xe9e9190>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "frq, resp = freqz([1,1])\n",
    "plt.semilogy(frq, abs(resp))\n",
    "\n",
    "frq, resp = freqz([1,-1])\n",
    "plt.semilogy(frq, abs(resp))\n",
    "plt.title('Frequency Response')\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Transfer function\n",
    "\n",
    "$$ Y(z) = H(z) \\cdot X(z) $$\n",
    "\n",
    "$$ H(z) = \\frac{Y(z)}{X(z)} $$\n",
    "\n",
    "\n",
    "# z-transform\n",
    "\n",
    "$$\\mathcal{Z}[x[n-M]] = z^{-M}X(z)$$\n",
    "\n",
    "z-transform on the difference equation:\n",
    "\n",
    "$$Y(z) = b_0z^{0}X(z) + b_1z^{-1}X(z) + ... + b_Mz^{-M}X(z) - a_1z^{-1}Y(z) - a_2z^{-2}Y(z) - ... - a_Mz^{-M}Y(z)$$\n",
    "\n",
    "$$ [1 + a_1z^{-1} + a_2z^{-2} - ... + a_Mz^{-M}] \\cdot Y(z) = [b_0z^{0} + b_1z^{-1} + ... + b_Mz^{-M}]\\cdot X(z) $$\n",
    "\n",
    "$$H(z) = \\frac{Y(z)}{X(z)} = \\frac{b_0z^{0} + b_1z^{-1} + ... + b_Mz^{-M}}{ 1 + a_1z^{-1} - a_2z^{-2} - ... - a_Mz^{-M}}$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Pole-zero analysis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-1., -1.]), array([], dtype=float64), 1.0)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.signal import tf2zpk\n",
    "\n",
    "tf2zpk([1,2,1],[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([], dtype=float64), array([-1.]), 1.0)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tf2zpk([1],[1, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-1.]), array([], dtype=float64), 2.0)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tf2zpk([2, 2],[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def PoleZeroPlot(b, a):\n",
    "    (zeros,poles,gain) = tf2zpk(b, a)\n",
    "    angle = np.linspace(-np.pi,np.pi,50)\n",
    "    cirx = np.sin(angle)\n",
    "    ciry = np.cos(angle)\n",
    "    plt.figure()\n",
    "    plt.plot(poles.real, poles.imag, 'x', zeros.real, zeros.imag, 'o', cirx,ciry, 'k-')\n",
    "    plt.grid()\n",
    "    \n",
    "    plt.xlim((-2, 2))\n",
    "    plt.xlabel('Real')\n",
    "    plt.ylim((-1.5, 1.5))\n",
    "    plt.ylabel('Imag')\n",
    "    plt.gcf().set_figwidth(5)\n",
    "    return (zeros,poles,gain)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 1.,  1.]), array([], dtype=float64), 1.0)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xe449970>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "PoleZeroPlot([1,-2,1],[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([], dtype=float64), array([-1.]), 1.0)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xe514b10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "PoleZeroPlot([1],[1,1])"
   ]
  },
  {
   "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": "2.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}