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    "## Lecture 26: Time Series Analysis\n",
    "J.R. Gladden, University of Mississippi, Dept. of Physics\n",
    "Physics 503"
   ]
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   "source": [
    "#### Imports and Initialization\n",
    "Set up the details of sampling rates and signal duration."
   ]
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   "source": [
    "%matplotlib qt4\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
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     "text": [
      "Nyquist frequency: 200.00 Hz\n"
     ]
    }
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   "source": [
    "plt.close('all')\n",
    "\n",
    "Fs = 400.0          #Sampling rate (samples per second)\n",
    "Ts = 1.0/float(Fs)  # Sampling interval (s)\n",
    "Td = 20.0           # Duration of signal\n",
    "\n",
    "print(\"Nyquist frequency: %3.2f Hz\" % (Fs/2.0))\n",
    "# Time vector\n",
    "t = np.arange(0,Td,Ts)\n"
   ]
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   "source": [
    "### Sample signal types\n",
    "Generate some signals to analyze such as multitone, chirp, gaussian, and gaussian shaped wavepacket."
   ]
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   "source": [
    "def chirp(t, A=1.0, w0 = 10.0, ramp = 0.2):\n",
    "    return np.sin(2.0*np.pi*ramp*w0*t*t)\n",
    "    \n",
    "def multitone(t, ws, A=1.0):\n",
    "    y = np.zeros(len(t))\n",
    "    for w in ws:\n",
    "        y+=A*np.sin(2.0*np.pi*w*t)\n",
    "    return y\n",
    "\n",
    "def gaussian(t,sigma=5.0, A=1.0):\n",
    "    return A*np.exp(-( (t-t.max()/2.0)/sigma)**2)\n",
    "    \n",
    "def wavepacket(t,sigma=0.5, A=1.0, omega=None):\n",
    "    if not omega: omega=20*sigma\n",
    "    return A*np.exp(-( (t-t.max()/2.0)/sigma)**2)*np.sin(2*np.pi*omega*t)"
   ]
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   "source": [
    "### Perform FFT and Visualize\n"
   ]
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   "source": [
    "y=multitone(t,(5,20,80))\n",
    "#y=chirp(t, ramp=0.1)\n",
    "#y=gaussian(t,sigma=0.2)\n",
    "#y=wavepacket(t,sigma=4,omega=50)\n",
    "\n",
    "N=len(t)\n",
    "\n",
    "#Compute the normalized FFT of a real signal\n",
    "Y=np.fft.rfft(y)/N\n",
    "freqs = np.fft.rfftfreq(len(y),d=Ts)\n",
    "\n",
    "#Visualize results\n",
    "plt.close('all')\n",
    "plt.subplot(211)\n",
    "plt.plot(t,y)\n",
    "plt.xlabel(\"Time (sec)\")\n",
    "plt.ylabel(\"Signal (A.U.)\")\n",
    "\n",
    "plt.subplot(212)\n",
    "plt.plot(freqs,abs(Y))\n",
    "plt.xlabel(\"Frequency (Hz)\")\n",
    "plt.ylabel(\"FFT Amplitude\")\n",
    "\n",
    "plt.figure()\n",
    "plt.specgram(y,Fs=Fs)\n",
    "plt.xlabel(\"Time (s)\")\n",
    "plt.ylabel(\"Frequency (Hz)\")\n",
    "\n",
    "plt.show()"
   ]
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