Welcome to EE472.ipynb

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   "source": [
    "# Welcome to EE472 Tele. 1 Lab.\n",
    "\n",
    "All students should be familiar with at least some basic features of MATLAB, you have used MATLAB extensively in ES302 and other courses. We will simulate some basic analog communication experiments similar to those carried out in course EE419 Tele.1.\n",
    "First step, you will learn how to define time, sampling frequency and data vectors, also how to plot signals in time and frequency domains, and how to create functions in Matlab. Remember to use ```Help``` whenever you need further explanation in MATLAB.\n",
    "\n",
    "## Introduction\n",
    "In computers, we cannot have a continuous signal since it means it is defined at every point, infinite number of points require unlimited memory! But instead, we will define the signal at different time intervals and this will convert the signal from continuous to discrete. The signal value at a time interval is called a sample and the number of samples in one second is called the sampling frequency. The sampling process will limit the bandwidth of the signal to the half of sampling frequency, Nyquist theorem.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 15,
     "metadata": {
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 15,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "% Define the sampling frequency\n",
    "Fs = 1e2;\n",
    "\n",
    "% Determine the duration of the signal in seconds\n",
    "du = 1; % one second\n",
    "\n",
    "% Create the time vector\n",
    "t  = 0: 1/Fs: du;\n",
    "\n",
    "% Create the frequency vector\n",
    "f  = -Fs/2: 1/du :Fs/2;\n",
    "\n",
    "% Specify the frequency of a signal\n",
    "fc = 10; % note that fc, in Hz, should be less than Fs/2\n",
    "\n",
    "% Create the signal time vector\n",
    "y  = cos(2*pi*fc*t);\n",
    "\n",
    "% Convert the signals to frequency domain\n",
    "Y  = abs( fftshift( fft(y)/length(y) ));\n",
    "\n",
    "% Plot the signal in each domain\n",
    "figure, plot(t,y); title('time domain')\n",
    "figure, plot(f,Y); title('frequency domain')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "The previous steps are very helpful as a starter. Yet they are very inefficient and difficult to make corrections in longer programs.\n",
    "In MATLAB, you can define a function for repeated operations so you do not have to write them more and more.\n",
    "Define a function for converting signals from time to frequency domain:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 17,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function X = sig_spec(x)\n",
    "% sig_spec, is just a name, you can use your own, determines the spectrum of x by calculating FFT\n",
    "% X is the signal in frequency domain\n",
    "% x is the signal\n",
    "X = abs( fftshift( fft(x)/length(x) ));\n",
    "\n",
    "end\n",
    "\n",
    "% test your function\n",
    "figure, plot(f,sig_spec(y));title('frequency domain')\n"
   ]
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