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   "source": [
    "# Homework \\#6 - Python exercise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%pylab inline\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Filtering a Sequence of Independent Random Variables\n",
    "  \n",
    "Let $x[n]$ be a real-valued white Gaussian random process, with zero mean and variance $\\sigma_x^2=3$. We filter the process with the FIR filter $h[n]$ where\n",
    "$$\n",
    "h[1]=1/2, \\quad h[2]=1/4, \\quad h[3]=1/4, \\quad\n",
    "h[n]=0~\\forall~n\\neq 1,2,3\n",
    "$$\n",
    "Moreover, at the output of the filter, we add white Gaussian noise $z[n]$ with unit variance. \n",
    "\n",
    "* Write a routine in Python to generate $N$ samples of the input process, $N$ samples of the additive Gaussian noise and compute the output of the system.\n",
    "* write a routine to estimate the power spectral density of the output\n",
    "* compare the numerical estimation of the PSD with its theoretical value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solution\n",
    "\n",
    "For a comparison with the theoretical value we need the exact PSD of the output process, which is ...\n",
    "\n",
    "The following function computes a single $N$-point realization of the output process\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_y(N):\n",
    "    h = np.array([0, 0.5, 0.25, 0.25])\n",
    "    # generate x[n]\n",
    "    x = ...\n",
    "    # filter it with h[n]; you may want to consult the Jupyter notebook FIRImplementation for\n",
    "    #  details about border effects when filtering a finite-length data set\n",
    "    y1 = ...\n",
    "    # generate z[n]\n",
    "    z = ...\n",
    "    # generate y[n]\n",
    "    y = y1 + z\n",
    "    return y    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following function estimates the PSD using $M$ independent realizations of the process:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def estimate_psd(N, M):\n",
    "    # estimate the PSD using M realizations\n",
    "    ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function computes the values of the theoretical PSD for the same frequency bins as the estimated PSD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def theoretical_PSD(N):\n",
    "    # define range for omega\n",
    "    omega = np.arange(0, N) * 2 * np.pi / N \n",
    "    p = ...\n",
    "    return p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# compare the experimental PSD to the theoretical PSD for different values of M\n",
    "N = 50\n",
    "for M in [50, 500, 5000]:\n",
    "    plt.plot(estimate_psd(N, M), 'r--', label='estimate')\n",
    "    plt.plot(theoretical_PSD(N), 'b-', label='theoretical')\n",
    "    plt.legend()\n",
    "    plt.show()  "
   ]
  }
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