E3-Fourier_properties.ipynb

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    "# Exercise 3: Fourier properties\n",
    "\n",
    "With this exercise you will get a better understanding of some of the Fourier theorems and of some useful properties of the DFT. You will write code to implement and verify several properties of the DFT that are discussed in the lectures. You will also learn to use the `dftModel.py` module of sms-tools, which contains the basic python functions implementing the DFT. There are five parts in the exercise: 1) Minimize energy spread in DFT of sinusoids, 2) Optimal zero-padding, 3) Symmetry properties of the DFT, 4) Suppressing frequency components using DFT model, and 5) FFT size and zero-padding. \n",
    "\n",
    "### Relevant Concepts\n",
    "\n",
    "__DFT of sinusoids:__ When a real sinusoid has an integer number of cycles in $N$ samples, the frequency of the sinusoid exactly matches one of the bin frequencies in an $N$ point DFT. Hence the DFT spectrum of the sinusoid has a value of zero at every DFT bin except at the two bins that match the frequency of the sinusoid. Otherwise, the energy of the sinusoid is spread over all the bins. When there are multiple sinusoids, the equations extend to each sinusoid. \n",
    "\n",
    "\\begin{eqnarray}\n",
    " x[n]&=&A_{0}\\cos\\left(2\\pi k_{0}n/N\\right)=\\frac{A_{0}}{2}{\\textstyle e}^{j2\\pi k_{0}n/N}+\\frac{A_{0}}{2}{\\textstyle e}^{-j2\\pi k_{0}n/N}\\\\\n",
    " X[k] &=& \\frac{A_0}{2} \\,\\,\\, \\mathrm{for} \\,\\,\\, k = k_0, -k_0; \\,\\,\\,\\, 0 \\,\\,\\,\\mathrm{otherwise}\n",
    "\\end{eqnarray}\n",
    "\n",
    "__Zero-padding:__ Zero-padding a signal is done by adding zeros at the end of the signal. If we perform zero-padding to a signal before computing its DFT, the resulting spectrum will be an interpolated version of the spectrum of the original signal.  In most implementations of the DFT (including the FFT algorithms) when the DFT size is larger than the length of the signal, zero-padding is implicitly done. \n",
    "\n",
    "__Zero phase windowing:__ Zero phase windowing of a frame of a signal puts the centre of the signal at the zero time index for DFT computation. By moving the centre of the frame to zero index by a circular shift, the computed DFT will not have the phase offset which would have otherwise been introduced (recall that a shift of the signal causes the DFT to be multiplied by a complex exponential, which keeps the magnitude spectrum intact but changes the phase spectrum). When used in conjunction with zero-padding, zero phase windowing is also useful for the creation of a frame of length of power of 2 for FFT computation (`fftbuffer`). \n",
    "\n",
    "If the length of the signal $x$ is $M$ and the required DFT size is $N$, the zero phase windowed version of the signal, `dftbuffer`, for DFT computation can be obtained by (works for both even and odd $M$): \n",
    "\n",
    "    hM1 = floor((M+1)/2)\n",
    "    hM2 = floor(M/2)\n",
    "    dftbuffer = zeros(N)\n",
    "    dftbuffer[:hM1] = x[hM2:]                              \n",
    "    dftbuffer[-hM2:] = x[:hM2]\n",
    "\n",
    "__Real, even and odd signals:__ A signal is real when it does not have any imaginary component, and all sounds are real signals. A signal $x$ is even if $x[n] = x[-n]$, and odd if $x[n] = -x[-n]$. For a signal of length $M$ (and $M$ is odd), in the context of a zero phase windowed signal and its DFT, the signal is even if $x[n] = x[M-n]$ and odd if $x[n] = -x[M-n]$, $1 \\leq n \\leq M-1$. The DFT properties show that for real input signals, the magnitude spectrum is even and the phase spectrum is odd. Furthermore, when the input signal is both real and even, the DFT is real valued, with an even magnitude spectrum and imaginary component equal to zero. In summary, if $x$ is an input signal of length $M$ ($M$ is odd) and $X = \\mathrm{DFT}(x,M)$, then for $1 \\leq k \\leq M-1$\n",
    "\n",
    "If $x$ is real, $\\left|X[k]\\right| = \\left|X[M-k]\\right|$ and $\\boldsymbol{<}\\!X[k] = -\\boldsymbol{<}\\!X[M-k]$\n",
    "\n",
    "If $x$ is real and even, $\\left|X[k]\\right| = \\left|X[M-k]\\right|$ and $\\mathrm{imag}(X[k]) = 0$\n",
    "\n",
    "\n",
    "__Positive half of the DFT spectrum:__ Audio signals are real signals. Due to the symmetry properties of the DFT of a real signal, it is sufficient to store only one half of the magnitude and phase spectra. To save on both storage and computation, we will just store just the half spectrum when possible. \n",
    "\n",
    "From an $N$ point DFT ($N$ even), we can obtain the positive half of the spectrum by considering only the first $(N/2)+1$ samples of the DFT. We can compute the magnitude spectrum of the positive half (in dB) as $m_X = 20\\log_{10}\\left|X[0:(N/2)+1]\\right|$, where $X$ is the DFT of the input. \n",
    "\n",
    "__Filtering:__ Filtering involves selectively suppressing certain frequencies present in the signal. Filtering is often performed in the time domain by the convolution of the input signal with the impulse response of a filter. The same operation can also be done in the DFT domain using the properties of DFT, by multiplying the DFT of the input signal by the DFT of the impulse response of the filter. In this assignment, we will consider a very simple illustrative filter that suppresses some frequency components by setting some DFT coefficients to zero. It is to be noted that the convolution operation here is circular convolution with a period $N$, the size of the DFT. \n",
    "\n",
    "If $x_1[n] \\Leftrightarrow X_1[k]$ and $x_2[n] \\Leftrightarrow X_2[k]$, $x_1[n] * x_2[n] \\Longleftrightarrow X_1[k]\\,X_2[k]$"
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    "\n",
    "## Part 1 - Minimize energy spread in DFT of sinusoids\n",
    "\n",
    "Given an input signal consisting of two sinusoids, complete the function `minimize_energy_spread_dft()` to select the first `M` samples from the signal and return the positive half of the DFT magnitude spectrum (in dB), such that it has only two non-zero values. \n",
    "\n",
    "`M` is to be calculated as the smallest positive integer for which the positive half of the DFT magnitude spectrum has only two non-zero values. To get the positive half of the spectrum, first compute the `M` point DFT of the input signal (for this you can use the `fft()` function of `scipy.fftpack`). Consider only the first `(M/2)+1` samples of the DFT, computing the magnitude spectrum of the positive half (in dB) as `mX = 20*log10(abs(X[:M/2+1]))`, where `X` is the DFT of the input signal.\n",
    "\n",
    "The input arguments to this function are the input signal `x` (of length W >= M) consisting of two sinusoids of frequency `f1` and `f2`, the sampling frequency `fs` and the value of frequencies `f1` and `f2`. The function should return the positive half of the magnitude spectrum `mX`. For this question, you can assume the input frequencies `f1` and `f2` to be positive integers and factors of `fs`, and that `M` is even. \n",
    "\n",
    "Due to the precision of the FFT computation, the zero values of the DFT are not zero but very small values < 1e-12 (or -240 dB) in magnitude. For practical purposes, all values with absolute value less than 1e-6 (or -120 dB) can be considered to be zero. \n",
    "\n",
    "HINT: The DFT magnitude spectrum of a sinusoid has only one non-zero value (in the positive half of the DFT spectrum) when its frequency coincides with one of the DFT bin frequencies. This happens when the DFT size (`M` in this question) contains exactly an integer number of periods of the sinusoid. Since the signal in this question consists of two sinusoids, this condition should hold true for each of the sinusoids, so that the DFT magnitude spectrum has only two non-zero values, one per sinusoid. \n",
    "\n",
    "`M` can be computed as the Least Common Multiple (LCM) of the sinusoid periods (in samples). The LCM of two numbers `x`, `y` can be computed as: `x*y/gcd(x,y)`, where gcd denotes the greatest common divisor."
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.fftpack import fft, fftshift\n",
    "import numpy as np\n",
    "from math import gcd, ceil, floor\n",
    "import sys\n",
    "sys.path.append('../software/models/')\n",
    "from dftModel import dftAnal, dftSynth\n",
    "from scipy.signal import get_window\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def minimize_energy_spread_dft(x, fs, f1, f2):\n",
    "    \"\"\"\n",
    "    Inputs:\n",
    "        x (numpy array) = input signal \n",
    "        fs (float) = sampling frequency in Hz\n",
    "        f1 (float) = frequency of first sinusoid component in Hz\n",
    "        f2 (float) = frequency of second sinusoid component in Hz\n",
    "    Output:\n",
    "        positive half of DFT spectrum (in dB) (numpy array) \n",
    "    \"\"\"\n",
    "    \n",
    "    hN = int(floor(x.size+1/2))   \n",
    "    hN1 = int(floor((hN+1)/2))\n",
    "    hN2 = int(floor(hN/2))\n",
    "    \n",
    "    M = fs/gcd(f1,f2)\n",
    "    \n",
    "    fftbuffer = np.zeros(int(floor(M)))\n",
    "    fftbuffer[:hN1] = x[hN2:]\n",
    "    fftbuffer[hN-hN2:] = x[:hN2]\n",
    "    \n",
    "    # Frequency spectrum signal\n",
    "    X = fft(fftbuffer)\n",
    "    \n",
    "    # Positive half frequency spectrum signal [dB]\n",
    "    XpH_dB = 20*np.log10(abs(X[:int(floor(M/2))]))\n",
    "    \n",
    "    return XpH_dB  \n",
    "    "
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run and test the function `minimize_energy_spread_dft()`. Plot the output spectrum to validate the result. Here are two possible examples to use.\n",
    "\n",
    "_Test case 1:_ For an input signal `x` sampled at `fs = 10000`Hz that consists of sinusoids of frequencies `f1 = 80`Hz and `f2 = 200`Hz, you need to select `M = 250` samples of the signal to meet the required condition. In this case, output `mX` is 126 samples in length and has non-zero values at bin indices 2 and 5 (corresponding to the frequency values of 80 and 200 Hz, respectively). You can create a test signal `x` by generating and adding two sinusoids of the given frequencies.\n",
    "\n",
    "_Test case 2:_ For an input signal `x` sampled at `fs = 48000` Hz that consists of sinusoids of frequencies `f1 = 300`Hz and `f2 = 800`Hz, you need to select `M = 480` samples of the signal to meet the required condition. In this case, output `mX` is 241 samples in length and has non-zero values at bin indices 3 and 8 (corresponding to the frequency values of 300 and 800 Hz, respectively). You can create a test signal `x` by generating and adding two sinusoids of the given frequencies."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3ddd5b3ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Non-zero value samples:\n",
      "2\n",
      "5\n"
     ]
    }
   ],
   "source": [
    "## Test case 1:\n",
    "fs = 10000\n",
    "f1 = 80\n",
    "f2 = 200\n",
    "M = 250\n",
    "\n",
    "x = np.sin(2*np.pi*f1*np.arange(M)/fs)+np.sin(2*np.pi*f2*np.arange(M)/fs)\n",
    "\n",
    "XpH_dB = minimize_energy_spread_dft(x,fs,f1,f2)\n",
    "\n",
    "# Plotting\n",
    "plt.plot(XpH_dB)\n",
    "ymax = np.max(XpH_dB)                                  # Maximum amplitude value in dBs for plotting\n",
    "plt.ylim(-120,ymax+3)                                  # limiting y axis to maximum value plus 3 dB\n",
    "plt.ylabel(\"Amplitude [dB]\")\n",
    "plt.xlabel(\"Samples\")\n",
    "plt.show()\n",
    "\n",
    "# RESULT VERIFICATION\n",
    "# Print non-zero values samples\n",
    "print(\"Non-zero value samples:\")\n",
    "for i in range (XpH_dB.size):\n",
    "    if(XpH_dB[i]>0):\n",
    "        print(i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3db58a4940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Non-zero value samples:\n",
      "3\n",
      "8\n"
     ]
    }
   ],
   "source": [
    "## Test case 2:\n",
    "fs = 48000\n",
    "f1 = 300\n",
    "f2 = 800\n",
    "M = 480\n",
    "\n",
    "x = np.sin(2*np.pi*f1*np.arange(M)/fs)+np.sin(2*np.pi*f2*np.arange(M)/fs)\n",
    "\n",
    "\n",
    "XpH_dB = minimize_energy_spread_dft(x,fs,f1,f2)\n",
    "\n",
    "# Plotting\n",
    "plt.plot(XpH_dB)\n",
    "ymax = np.max(XpH_dB)                                  # Maximum amplitude value in dBs for plotting\n",
    "plt.ylim(-120,ymax+3)                                  # limiting y axis to maximum value plus 3 dB\n",
    "plt.ylabel(\"Amplitude [dB]\")\n",
    "plt.xlabel(\"Samples\")\n",
    "plt.show()\n",
    "\n",
    "# RESULT VERIFICATION\n",
    "# Print non-zero values samples\n",
    "print(\"Non-zero value samples:\")\n",
    "for i in range (XpH_dB.size):\n",
    "    if(XpH_dB[i]>0):\n",
    "        print(i)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 2 - Optimal zero-padding\n",
    "\n",
    "Given a sinusoid, complete the function `optimal_zeropad()` to compute the DFT of the sinusoid after zero-padding and return the positive half of the magnitude spectrum (in dB). Zero-padding needs to be done such that one of the bin frequencies of the DFT coincides with the frequency of the sinusoid. Choose the minimum zero-padding length for which this condition is satisfied. \n",
    "\n",
    "The input arguments are the sinusoid `x` of length `M`, sampling frequency `fs` and the frequency of the sinusoid `f`. The output is the positive half of the magnitude spectrum `mX` computed using the `N` point DFT `(N >= M)` of `x` after zero-padding `x` to length `N` appropriately as required. \n",
    "\n",
    "To get the positive half of the spectrum, first compute the `N` point DFT of the zero-padded input signal (for this you can use the `fft()` function of `scipy.fftpack`, which is already imported in this script). Consider only the first `(N/2)+1` samples of the DFT and compute the magnitude spectrum of the positive half (in dB) as `mX = 20*log10(abs(X[:(N/2)+1]))`, where `X` is the `N` point DFT of the zero-padded input.\n",
    "\n",
    "For this exercise, you can assume that the frequency of the sinusoid `f` is a positive integer and a factor of the sampling rate `fs`. The input parameters will be given in such a way that `N` will be even.  Note that the numerical value of `f` is an integer but the data type is float, for example 1.0, 2.0, 55.0 etc. This is to avoid issues in python related with division by a integer.\n",
    "\n",
    "Due to the precision of the FFT computation, the zero values of the DFT are not zero but very small values < 1e-12 (or -240 dB) in magnitude. For practical purposes, all values with absolute value less than 1e-6 (or -120 dB) can be considered to be zero. \n",
    "\n",
    "HINT: One of the DFT bin frequencies coincides with the frequency `f` of a sinusoid when the DFT size (`N` in this question) contains exactly an integer number of periods of the sinusoid. For example, if `f = 100` Hz and `fs = 1000` Hz, one period of the sinusoid has 10 samples. Then given a signal of length `M = 25` samples, there are 2.5 periods in it. The minimum zero-padding length here would be 5 samples (0.5 period), so that the DFT size `N = 30` corresponds to 3 periods of a sinusoid of frequency `f=100` Hz."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def optimal_zeropad(x, fs, f):\n",
    "    \"\"\"\n",
    "    Inputs:\n",
    "        x (numpy array) = input signal of length M\n",
    "        fs (float) = sampling frequency in Hz\n",
    "        f (float) = frequency of sinusoid in Hz\n",
    "    Output:\n",
    "        positive half of DFT spectrum of N point DFT after zero-padding (numpy array)\n",
    "    \"\"\"\n",
    "    ## Your code here\n",
    "    k = f*x.size/fs\n",
    "    k = ceil(k)\n",
    "    \n",
    "    N = int(fs*k/f)\n",
    "    M = len(x)\n",
    "    \n",
    "    xw = np.concatenate((x,np.zeros(N-M)),axis = 0)\n",
    "\n",
    "    X = fft(xw)\n",
    "    \n",
    "    #Positive part of the spectrum in decibels\n",
    "    Xp_dB = 20*np.log10(abs(X[:int(N/2)+1])) \n",
    "    \n",
    "    return Xp_dB\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run and test the function `optimal_zeropad()`, validating the result by ploting the output array. Here are two possible examples.\n",
    "\n",
    "_Test case 1:_ For a sinusoid `x` with `f = 100.0` Hz, `M = 25` samples and `fs = 1000.0` Hz, you will need to zero-pad by 5 samples and compute an `N = 30` point DFT. In the magnitude spectrum, you can see a maximum value at bin index 3 corresponding to the frequency of 100 Hz. The output array you return is 16 samples in length. \n",
    "\n",
    "_Test case 2:_ For a sinusoid `x` with `f = 250.0` Hz, `M = 210` samples and `fs = 10000.0` Hz, you will need to zero-pad by 30 samples and compute an `N = 240` point DFT. In the magnitude spectrum, you can see a maximum value at bin index 6 corresponding to the frequency of 250 Hz. The output array you return is 121 samples in length. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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hk5k1MT9i91g5YvpYjpg+livOOZxfP7+eny5q49rfVmbZHd0yjg8cN5P3vW06Y/PJTJyDoSQiEmPV2UEdxVIsr+Df2tHF3/3iWWZPKvCn75zNqXMmc2zr+H2uYnE2k+rZDXPD1k7uWPIaty5u429//gz/3y+f5ay5U7lwfgunHDyJ9AibUqwkIhJjPS2RzqFNeQ3Lth2V8ZoF75jNRSe0RhxNPEzeL8enTp3NpaccyLI1b3Hr4jZ+vuQ1frl0LVPH5Ljg2Jl84LiZI2YNkJKISIwValoicVSNK5/TR0lvZsaRM8Zy5IyxXHHuYfzP8vXcuriNhQ++yDW/WcWxreP4wHEtnHf0/oxJ8C6Y+s2LxFg+aIlsi+kMrW1BC6k5p90bB5LLpDnnqP0556j9Wf/WDn6+5DV+uqiNr9z+NF//xTLOPnIanz99DgdPSV7rRElEJMaagyv8jph2Z3UEyW1fXdk9HFPGjGLBOw7i06fOZmlbZXbXTx5fTTad4lsXHh11eEOm37xIjFU/nNtj2p1VXeFdUBIZMjPj6JZxHN0yjkdfeiO2rc092bemUIgkTNwH1qsLIQvqztor+WxGSaRezOxbZvacmS01s9vNbFzNuSvMbKWZPW9m74kyTpEw5GM+sF5tIRU0sL5XklyZIHZJBLgXONLd3wa8AFwBYGZzgYuAI4CzgR+YmS5/ZEQrZJPSElES2RuFXJqOYjx/x3sSuyTi7ve4ezUlPwLMDG6fD9zs7p3u/hKwEjghihhFwpJJp8hlUvFtiQTJLd+k67m9UVB3VsN8Eviv4PYMYHXNubbgmMiIVsjF9wOmvbNEPpvWxk57qZDg7qxI2qBmdh8wrY9TV7r7HcFjrgRKwE3Vp/Xx+D43QzGzBcACgNZWraKVZMtn49vV0V4sa3pvHeRz6d32MkmKSH777n7mQOfN7BPAecAZvnPXrDagpeZhM4E1/fz7C4GFUNmUaq8DFolQnAddO4olzcyqg+ZshmKpm65y96A32IqL2EVrZmcDfw28z907ak7dCVxkZjkzOxCYAzwWRYwiYYp1S6SzpDUidVCI+aLSgcTxt/99IAfcG5SPfsTd/9Tdl5nZLcCzVLq5PufuyfuJiwxRIZdh6454tkTaO8tqidRB9We4rVhKXNn42CURdz94gHNXAVeFGI5I5ArZDOve2hF1GH1qL5aYUNh3t4atl2pLJK7dlgPpN4mY2Z2DeP4md7+4fuGISG/5XDrW60Raxjd2X/R9QbVLcEQlEeBw4FMDnDfgn+sbjoj0Vshm4ls7q7NMPqvurL21syUSz4uFgQyURK509wcGerKZfb3O8YhIL/lcOrYDru3Fklar10Eh5iX/B9Lv7Cx3v2VPTx7MY0Rk7zRnMxTL3RRL3VGHsgt3p6OogfV6iPvmYwPpN4mY2SQz+5qZfcHMms3sGjN7xszuMLN+B79FpL6quwZuj9k0385SN+VuV0ukDpI8sD7QOpH/pDLVtroe40XgA8AvgWsbH5qIwM4ijNtidpXaU3xR60T2WnXzsSSuWh/otz/V3b9ilcUar7j7t4Ljz5nZ50KITUSoXYgWtyQSbEillsheG9WUImUjryVSBgjKjmzsdS5enbMiI1jPxlQxu0rt2UtEs7P2mpkltpLvQJcQs4O1IlZzm+D+gQ2PTESAmi1yY/YBUx0EzqslUhdxnoU3kIF+++fX3P52r3O974tIg8R1Idq24AOvWbOz6qKQy8Ru3Gsw+k0ie1ojIiLhqHZnxa0IY3WMRqXg66M5l4nduNdgDFT25Gn62a8DINi+VkQarGf6Z8yuUqv9983qzqqLfDa+5W0GMtBv/7zge3Um1r8H3z8CdOz+cBFphHzPPuvxSiLVlpHKntRHcy7Dms3xLLQ5kIG6s14BMLOT3f3kmlNfNrOHgf/d6OBEpHZgPV5XqdWWiKb41kc+mxlZK9ZrFMzslOodM3s7UGhcSCJSK50yRjelY/cB01EskU4ZuUzs9rZLpEIu0zNZIUkGcwlxKXCdmY2lMkayBfhkQ6MSkV0UYrgHd7WCb7B5nOyl5lw6dl2Wg7HHJOLui4GjzWwMYO6+pfFhiUitfDZ++6y3d5Y0qF5H+WyG7V1lyt1OOpWcxDxQAcbzau+7+1u9E0jvx4hIYxRymdiNiXQUtZdIPVUTcty6LfdkoMuIb5nZa1RWqPfn/1ApyCgiDVTIxm9MZJtaInXVUyOtWGa/UcnZZ32gd8A64B/28PwVdYxFRPqRz2XY0lGMOoxddBRLWmhYR7UbU02NOJahGGiK77tCjENEBlDIplmzOV7dWe2dZaaPS84Vc9zFtbzNnsR2bp6Z/S8zczObFNw3M7vazFaa2VIzOzbqGEXCUohhSQxtjVtf+Wq15piNfe1JLJOImbUA7wZerTl8DpUNsuYAC4BrIghNJBKFbFyn+CqJ1EtzQnc3jGUSAb4L/BW71u46H/iRVzwCjDOz/SOJTiRk+Vxlim9le594qEzx1eyseolrjbQ92WMSMbO8mf2tmf1bcH9OI6f2mtn7gNfc/alep2YAq2vutwXHREa85lyGUrdTLMdjP7hyt7O9Sy2ReirEtLzNngzmHXA9sBg4KbjfBvyUvZjaa2b3AdP6OHUl8BXgrL6e1sexPi/LzGwBlS4vWltbhxmlSHxU12N0dJbJZaK/+q9ONy6oJVI3PTtYJqw7azBJ5CB3/6CZfQjA3bfbXtY5cPcz+zpuZkdR2TXxqeAlZgJPmNkJVJJXS83DZwJr+vn3FwILAebPnx+f9r/IMFWvUrd1lhhfyEYczc4KvhpYr5+eQpsjrTsLKJrZaIKrfjM7COhsRDDu/rS7T3H3We4+i0riONbdXwfuBD4ezNI6Edji7msbEYdI3ORjtjFV9Wq5oO6suqkW2hyJLZGvAf8NtJjZTcDJwMWNDKofvwLOBVZS2c/kkghiEIlE3AZdq/32aonUVxIr+Q6mAOO9ZvYEcCKVcYnL3X1jwyOrvPasmtvOzg2yRPYp1Sv+jph8wFSTWUG1s+qqkItfeZs9GWh73N6L+apdR61m1uruTzQuLBGpVR1Y3xaTro52bUjVEIUYVmvek4HeAd8Jvo8C5gNPUWmJvA14FDiln+eJSJ3FrcJre8/Auloi9dScy8TmQmGw+h1Yd/fT3P004BUqg9vz3f04YB6VcQkRCUlPSYyYDaxrnUh95XPp2EyeGKzBzM46zN2frt5x92eAYxoXkoj0FrfifOrOaoxCAlsig3kHLDeza4H/oDLN96PA8oZGJSK7GN1UXWwYjw+YnnUiGlivq0I2HZvJE4M1mCRyCfBZ4PLg/oOo+KFIqFIpi1URxvbOErlMikw6ruX3kqmQG1kD6wC4+w4qBRG/2/hwRKQ/+VwmRgPrKgPfCM25DO3FSqHNvSwMEpo9vgvM7CX6qFHl7rMbEpGI9KmQTcdmIVqlDLy6suotn83Q7bCjq5vRCfn5DuZSYn7N7VHAhcCExoQjIv2J08ZU7dpfvSGaa7bITUoS2WOHpru/UfP1mrt/Dzg9hNhEpEYhm4lN2ZOOoloijVCdMh2XbsvBGEx3Vu3K9RSVlsl+DYtIRPqUz6XZ1F6MOgygcqW83yi1ROqtOs6UpGm+g3kXfKfmdgl4CfiTxoQjIv0pZDO8uqkj6jCAypXytDGjog5jxNm5RW48xr4GYzBJ5FJ3f7H2gJkd2KB4RKQfhVx81hC0d5Y1O6sBdlYmSE5LZDCTvG8d5DERaaB8jMZEKlN8NSZSbztbIvH4PQ/GQFV8DwOOAMaa2QU1p8ZQmaUlIiEq5CobFsVhDUF7Z0l1sxqgOllhRCQR4FDgPGAc8N6a41uBTzcyKBHZXXUNQWepm1FN0bUCiqVuusreMx1V6mdEjYm4+x3AHWZ2krv/PsSYRKQPtV0dUSaR6vRTtUTqLx+zQpuDMVB31l+5+98DHzazD/U+7+5faGhkIrKLaldHR7HMxAjjqE4/1WLD+stmUmTTqdjUSBuMgd4F1Uq9i8IIREQGFpc1BNUKvnl1ZzVEdewrKQbqzvpF8P3G8MIRkf4UYrK7YTWJFdSd1RBJq+Q7UHfWL+ij8GKVu7+vIRGJSJ8KPTN3ou3qqK5V0TqRxohTeZvBGOhd8O3QoujFzD4PXEZlhfxd7v5XwfErgEuBMvAFd787qhhFwhaXQdf2noF1dWc1QqU7awSMibj7A9XbZpYFDqPSMnne3RtWwMfMTgPOB97m7p1mNiU4Phe4iMralenAfWZ2iLsn56ctshcKMdlnvV0D6w1VyGXYuiM5LZE9rlg3sz8EVgFXA98HVprZOQ2M6bPAN929E8Dd1wfHzwdudvdOd38JWAmc0MA4RGIlLmMi7RpYb6hCNj6bjw3GYMqefAc4zd3f5e7vBE6jsbscHgKcamaPmtkDZnZ8cHwGsLrmcW3BMZF9QiEbj4Voaok0VmVgPTkdLIN5F6x395U1918E1vf34MEws/uAaX2cujKIaTxwInA8cIuZzQb6qvPQ58C/mS0AFgC0trbuTagisTGqKUXKoh8T6egsYQajMmqJNEIhlx4xA+tVy8zsV8AtVD60LwQer9bTcvfbhvqi7n5mf+fM7LPAbe7uwGNm1g1MotLyaKl56ExgTT///kJgIcD8+fP7nWEmkiRmFouZO9s6y+Sb0qRSydgDPGmSNsV3MN1Zo4B1wDuBdwEbqGyP+14qtbXq7ecEOyea2SFAFtgI3AlcZGa5oBT9HOCxBry+SGzlY1AOvqNY0vTeBipk03SVnc5SMrq09vhOcPdLwgikxnXAdWb2DFAEPhG0SpaZ2S3As1Sm/n5OM7NkXxOHlkh7UXuJNFLPBIrOMrkEdBkOZnvcA4HPA7NqH9+oxYbB9OGP9nPuKuCqRryuSBLkY1ASo71Te4k0Um15m/GFbMTR7NlgLid+DvwQ+AXQ3dhwRGQglZZI9LOzVMG3caqz8DoSUoRxMO+EHe5+dcMjEZE9KuQyrN+6I9IY2oslJjfnIo1hJKu28qIutDlYg0ki/2hmXwPuATqrB939iYZFJSJ9ymejL4nR0VmmMFEtkUZJ2ha5g3knHAV8jMqMqWp3lgf3RSREzTGY/rmts6QKvg2Uz8ajMsFgDead8MfA7EbWyxKRwclnM5H3lXdodlZDNfcMrCdjTGQw60SeorLPuohErLqauTLrPXzuTntRs7MaqVqTLOoW52AN5nJiKvCcmT3OrmMi2k9EJGT5bAZ32N5VjmSG1PauMu7aS6SResZERlB31tcaHoWIDEpzbufGVFEkkeqgfkF7iTRMLpMinbKR0xKp3VcEwMxOBj4MPND3M0SkUXYddA1/mm31g03rRBrHzGIxC2+wBvVOMLNjqCSOPwFeAn7WyKBEpG9RryGodrGoO6ux4jALb7AG2mP9ECo7CX4IeAP4CWDuflpIsYlIL/mIVzNXX1cD642VzyanHPxAlxPPAQ8B763uJ2JmXwolKhHpUyHihWjVFpBaIo3VnKCNqQaa4vt+4HXg12b2b2Z2Bn1vDCUiIam2ACJrifQMrCuJNFKS9hTpN4m4++3u/kHgMOA3wJeAqWZ2jZmdFVJ8IlKj+uEd2ZhIz8C6urMaKZ/NJKZ21h4XG7p7u7vf5O7nUdlNcAnw5YZHJiK72bnXRLQD69pfvbGac+nIKxMM1mBWrPdw903u/q/urrpZIhGotgCiKgff0xLRwHpDjYjuLBGJn6gXorUXyzSlLRE77iVZIRf9DpaDpSQikiDVhWjRDaxrQ6owFLIZdnR1UyrHfx9AJRGRhIlyIdq2zrLGQ0JQnYUX9S6Wg6EkIpIwkbZEiiXNzApBzwSKBHRpKYmIJEwhF930z22dJfJqiTRc1ItKhyJ2ScTMjjGzR8xsiZktMrMTguNmZleb2UozW2pmx0Ydq0gUCtlMZFeoHcVyTyVhaZxqleQkrFqPXRIB/h74ursfA3w1uA9wDjAn+FoAXBNNeCLRKuSiq/DaroH1UKglsnccGBPcHgusCW6fD/zIKx4BxpnZ/lEEKBKlfIQtkfZiSQPrIYi6MsFQxPHd8EXgbjP7NpUk9/bg+Axgdc3j2oJja8MNTyRahVw6sv23OzrLGlgPQdQ10oYikiRiZvcB0/o4dSVwBvBYzZ20AAAN2UlEQVQld/+Zmf0J8EPgTPou/tjnRtNmtoBKlxetra11iVkkLqIcE9nWWVIF3xBUW3tqifTD3c/s75yZ/Qi4PLj7U+Da4HYb0FLz0Jns7Orq/e8vBBYCzJ8/v89EI5JU+VyGjmKZ7m4nlQqvsHap3E1nqVsVfEOQ15jIXlkDvDO4fTqwIrh9J/DxYJbWicAWd1dXluxzqjN3OrrC7epo14ZUock3JWexYRwvKT4N/KOZZYAdBN1SwK+Ac4GVQAdwSTThiUSrtpJvmIPcHdoaNzSplFHIphPREondu8Hdfwsc18dxBz4XfkQi8RJVSYzqtGINrIej0m0Z/yQSx+4sERlAdZ1G2Fep1dfTFN9wNOcykc3CGwolEZGEKUSVRIrVXQ2VRMKQT0h3lpKISMJEtYag2p2lgfVwJGVjKiURkYTpKYkRcn+5BtbD1ZyQjamUREQSpmeL3JCvUqsL37ROJByV7iyNiYhIne0cEwn3A6ZD3VmhinLzsaFQEhFJmKg2LNLAerjyWSUREWmAbCZFU9oiWCdSYlRTinSIpVb2Zc25NB1dlfI2caYkIpJAUVylthe1v3qYCrkM7rA95PI2Q6UkIpJAlf7y8Fsi6soKT1KKMCqJiCRQPpsOf0yks6zpvSFqjqi8zVApiYgkUD6XCf3DpaNY6qkgLI0XVXmboVISEUmgKCq8tmtDqlA1qztLRBolipIY7cWy1oiEKKrKBEOlJCKSQIVsOoLaWRpYD1O16zDulXyVREQSKB9FSyTkTbD2dbWbj8WZkohIAoVdnM/daS+WtSFViKpJZJuSiIjUWz6bZkdXN+WQVjN3liqvpYH18BR6Cm2qO0tE6qxahDGstSLV8RdN8Q1PJp0il0nFfotcJRGRBMrnwr1KrY6/5NUSCVUhl1F3lojUX3PI0z+rr6OB9XAVcuHPwhuqSJKImV1oZsvMrNvM5vc6d4WZrTSz583sPTXHzw6OrTSzL4cftUh8VKfadoTdElF3VqgKWbVE+vMMcAHwYO1BM5sLXAQcAZwN/MDM0maWBv4ZOAeYC3woeKzIPmnnGoKQWiJBslJLJFxJ2Gc9kneEuy8HMNttX4LzgZvdvRN4ycxWAicE51a6+4vB824OHvtsOBGLxEvYG1N1aEOqSBRyGbZs74o6jAHFbUxkBrC65n5bcKy/4yL7pELIFV63qSUSiShqpA1Vw94RZnYfMK2PU1e6+x39Pa2PY07fya7fCfJmtgBYANDa2rqHSEWSZ+eYSMgtEdXOCtU+3Z3l7mcO42ltQEvN/ZnAmuB2f8f7eu2FwEKA+fPnx3tvSZFhqK4TCWtMpPo6BXVnhao5AUkkbt1ZdwIXmVnOzA4E5gCPAY8Dc8zsQDPLUhl8vzPCOEUiVW0RhDX9s6OzTMpgVFPcPjJGtnw2TXuxjHt8r4WjmuL7x2bWBpwE3GVmdwO4+zLgFioD5v8NfM7dy+5eAi4D7gaWA7cEjxXZJzWlU2QzqdDWiWzrLFHIZvqaDCMNVMhlKHc7naXuqEPpV1Szs24Hbu/n3FXAVX0c/xXwqwaHJpIYYQ66dhS1IVUUajemGtUUz/EotU1FEiqfzYS32LBY1qB6BPIJKMKoJCKSUGGWg28PurMkXGGXtxkOJRGRhMqHWFepo1Nb40Yhn4B91pVERBIqzLpK29QSiURzyItKh0NJRCShCrl0aGMiGliPRkEtERFplEI2vDGRberOikTYi0qHQ0lEJKFCHRMpqjsrCj2FNpVERKTewhoT6e52Oopl7WoYgZ4pvhoTEZF6K+QyFEvddJUbu5q5o0v7q0cll0mRSZm6s0Sk/qpXqY3u0qp2pWhgPXxmRiGXUXeWiNRfWDN3eir4amA9EoVsumc/lzhSEhFJqLB2N6y2dDSwHo1CLhPaDpbDoSQiklCFkOoqtas7K1KFXHiLSodDSUQkoaq7GzZ6rUh7z/7q6s6KQiEX7y1ylUREEqpnn/WGt0S0v3qUCtlMaOuBhkNJRCShwhoTqV4Fa51INNSdJSINUR3obnhLJLgKbtbAeiTUnSUiDZHv6c4KqyWiMZEoFHIZrVgXkforhDiwns2kaErr4yIKhWw4lQmGS+8KkYRKp4xRTakQVqyXVfIkQjuLMMazNaIkIpJghWwmlO6svMZDIlNN4NtiuuAwkiRiZhea2TIz6zaz+TXH321mi83s6eD76TXnjguOrzSzq83MoohdJE7yIQy6thdLmt4bobhvTBVVS+QZ4ALgwV7HNwLvdfejgE8A/15z7hpgATAn+Do7hDhFYq2yMVXj14loUD06zTFPIpFcXrj7cqhUqOx1/Mmau8uAUWaWAyYAY9z998HzfgT8EfBfoQQsElNh1FVSSyRa+ZDK2wxXnMdE3g886e6dwAygreZcW3BMZJ+WD6HCa0dnWSVPIlTtzorrgsOGXV6Y2X3AtD5OXenud+zhuUcA/xc4q3qoj4f5AM9fQKXrC2CbmT2/54j7NIlKF1tcxT0+UIz1sMf47LLGB7Hw4wOeTvzPMAYGjPGc/xtiJHDAYB/YsCTi7mcO53lmNhO4Hfi4u68KDrcBM2seNhNYM8BrLwQWDuf1e8WyyN3n7/mR0Yh7fKAY6yHu8UH8Y4x7fJCMGPsSq+4sMxsH3AVc4e4PV4+7+1pgq5mdGMzK+jgwYGtGREQaL6opvn9sZm3AScBdZnZ3cOoy4GDgb81sSfA1JTj3WeBaYCWwCg2qi4hELqrZWbdT6bLqffwbwDf6ec4i4MgGh9bbXneJNVjc4wPFWA9xjw/iH2Pc44NkxLgbc+93fFpERGRAsRoTERGRZFES6YOZnW1mzwclVr4cdTy9mVmLmf3azJYH5WMujzqmvphZ2syeNLNfRh1LX8xsnJndambPBT/Lk6KOqTcz+1LwO37GzH5sZqNiENN1ZrbezJ6pOTbBzO41sxXB9/Exi+9bwe95qZndHkziiUxfMdac+19m5mY2KYrYhkpJpBczSwP/DJwDzAU+ZGZzo41qNyXgL9z9cOBE4HMxjBHgcmB51EEM4B+B/3b3w4CjiVmsZjYD+AIw392PBNLARdFGBcAN7F526MvA/e4+B7g/uB+VG9g9vnuBI939bcALwBVhB9XLDfRRusnMWoB3A6+GHdBwKYns7gRgpbu/6O5F4Gbg/Ihj2oW7r3X3J4LbW6l8+MVqBX+w3ucPqcyoix0zGwO8A/ghgLsX3X1ztFH1KQOMNrMMkGeA9VFhcfcHgU29Dp8P3BjcvpFKWaJI9BWfu9/j7tUl34+w67qz0PXzMwT4LvBXDLCYOm6URHY3A1hdcz/WJVbMbBYwD3g02kh28z0qfwzx3EkHZgMbgOuDLrdrzawQdVC13P014NtUrkrXAlvc/Z5oo+rX1GA9V3Vd15Q9PD5KnySGSwTM7H3Aa+7+VNSxDIWSyO6GVGIlSmbWDPwM+KK7vxV1PFVmdh6w3t0XRx3LADLAscA17j4PaCfaLpjdBOMK5wMHAtOBgpl9NNqoks3MrqTSHXxT1LHUMrM8cCXw1ahjGSolkd21AS019wcssRIVM2uikkBucvfboo6nl5OB95nZy1S6A083s/+INqTdtAFt7l5twd1KJanEyZnAS+6+wd27gNuAt0ccU3/Wmdn+AMH39RHHsxsz+wRwHvARj9/ahoOoXCw8FfzdzASeMLO+6g/GipLI7h4H5pjZgWaWpTKQeWfEMe0iKP3yQ2C5u/9D1PH05u5XuPtMd59F5ef3P+4eqytod38dWG1mhwaHzgCejTCkvrwKnGhm+eB3fgYxG/yvcSeVPYAIvseqLJGZnQ38NfA+d++IOp7e3P1pd5/i7rOCv5s24NjgfRprSiK9BINvlwF3U/mDvcXdl0Ub1W5OBj5G5Qq/Wh7m3KiDSqDPAzeZ2VLgGOD/RBzPLoJW0q3AE8DTVP5eI1/VbGY/Bn4PHGpmbWZ2KfBN4N1mtoLK7KJvxiy+7wP7AfcGfy//ElV8A8SYSFqxLiIiw6aWiIiIDJuSiIiIDJuSiIiIDJuSiIiIDJuSiIiIDJuSiMgwmNmVQXXdpcGU0T9o4Gv9xswSt/e27Bsi2dlQJMmCkvHnUVkM1hmU7M5GHJZIJNQSERm6/YGN7t4J4O4b3X2NmX3VzB4P9v5YGKwyr7YkvmtmDwb7lhxvZrcFe298I3jMrGC/ixuD1s2tQT2lXZjZWWb2ezN7wsx+GtRPE4mMkojI0N0DtJjZC2b2AzN7Z3D8++5+fLD3x2gqrZWqoru/A/gXKiVBPgccCVxsZhODxxwKLAz2vHgL+LPaFw1aPH8DnOnuxwKLgD9vzH9RZHCURESGyN23AccBC6iUk/+JmV0MnGZmj5rZ08DpwBE1T6vWX3saWBbsCdMJvMjOgp+r3f3h4PZ/AKf0eukTqWyU9rCZLaFSo+qAuv7nRIZIYyIiw+DuZeA3wG+CpPEZ4G1UdiFcbWZ/B9RuZdsZfO+uuV29X/077F2DqPd9A+519w/t9X9ApE7UEhEZIjM71Mzm1Bw6Bng+uL0xGKf4wDD+6Vbbuc/7h4Df9jr/CHCymR0cxJE3s0OG8ToidaOWiMjQNQP/ZGbjqGxwtJJK19ZmKt1VL1PZUmColgOfMLN/BVYA19SedPcNQbfZj80sFxz+Gyp7hotEQlV8RWIg2Ob4l8GgvEhiqDtLRESGTS0REREZNrVERERk2JRERERk2JRERERk2JRERERk2JRERERk2JRERERk2P4fnrdCWIj+IaUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3db5737c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Array length: 16\n",
      "Maximum value of array in sample: [3]\n"
     ]
    }
   ],
   "source": [
    "#Test case 1:\n",
    "f = 100\n",
    "fs = 1000\n",
    "M = 25\n",
    "\n",
    "x = np.sin(2*np.pi*f*np.arange(M)/fs)\n",
    "\n",
    "X = optimal_zeropad(x,fs,f)\n",
    "\n",
    "plt.plot(X)\n",
    "plt.ylim(-120,np.max(X)+3)\n",
    "\n",
    "plt.xlabel('Sample')\n",
    "plt.ylabel('Amplitude [dB]')\n",
    "plt.show()\n",
    "\n",
    "# RESULT VERIFICATION\n",
    "print(f\"Array length: {X.size}\")\n",
    "print(f\"Maximum value of array in sample: {np.where(X == np.max(X))[0]}\") # Returns maximum value of the array\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3db571b358>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Array length: 121\n",
      "Maximum value of array in sample: [6]\n"
     ]
    }
   ],
   "source": [
    "#Test case 2:\n",
    "f = 250\n",
    "fs = 10000\n",
    "M = 210\n",
    "\n",
    "x = np.sin(2*np.pi*f*np.arange(M)/fs)\n",
    "\n",
    "X = optimal_zeropad(x,fs,f)\n",
    "\n",
    "plt.plot(X)\n",
    "\n",
    "plt.xlabel('Sample')\n",
    "plt.ylabel('Amplitude [dB]')\n",
    "plt.show()\n",
    "\n",
    "# RESULT VERIFICATION\n",
    "print(f\"Array length: {X.size}\")\n",
    "print(f\"Maximum value of array in sample: {np.where(X == np.max(X))[0]}\") # Returns maximum value of the array"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 3 - Symmetry properties of the DFT\n",
    "\n",
    "Complete the function `test_real_even()` a function to check if the input signal is real and even using the symmetry properties of its DFT. The function will return the result of this test, the zerophase windowed version of the input signal (`dftbuffer`), and its DFT. \n",
    "\n",
    "Given an input signal `x` of length `M`, do a zero phase windowing of `x` without any zero-padding. Then compute the `M` point DFT of the zero phase windowed signal and use the symmetry of the computed DFT to test if the input signal `x` is real and even. Return the result of the test, the `dftbuffer` computed, and the DFT of the `dftbuffer`. \n",
    "\n",
    "The input argument is a signal `x` of length `M`. The output is a tuple with three elements \n",
    "`(isRealEven, dftbuffer, X)`, where `isRealEven` is a boolean variable which is `True` if `x` is real and even, else `False`. `dftbuffer` is the `M` length zero phase windowed version of `x`. `X` is the `M` point DFT of the `dftbuffer`. \n",
    "\n",
    "To make the problem easier, we will use odd length input sequence in this question (`M` is odd). \n",
    "\n",
    "Due to the precision of the FFT computation, the zero values of the DFT are not zero but very small values < 1e-12 in magnitude. For practical purposes, all values with absolute value less than 1e-6 can be considered to be zero. Use an error tolerance of 1e-6 to compare if two floating point arrays are equal. \n",
    "\n",
    "Caveat: Use the imaginary part of the spectrum instead of the phase to check if the input signal is real and even."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def test_real_even(x):\n",
    "    \"\"\"\n",
    "    Inputs:\n",
    "        x (numpy array)= input signal of length M (M is odd)\n",
    "    Output:\n",
    "        tuple including:\n",
    "        isRealEven (boolean) = True if input x is real and even, and False otherwise\n",
    "        dftbuffer (numpy array, possibly complex) = M point zero phase windowed version of x \n",
    "        X (numpy array, possibly complex) = M point DFT of dftbuffer \n",
    "    \"\"\"        \n",
    "        \n",
    "    # Center x signal in the origin\n",
    "    dftbuffer = np.zeros(x.size)\n",
    "    dftbuffer[:int(x.size/2+1)] = x[int(x.size/2):]\n",
    "    dftbuffer[int(x.size/2+1):] = x[:int(x.size/2)]\n",
    "    \n",
    "    X = fft(dftbuffer)\n",
    "    \n",
    "    # Verify if x is real and even (phase = 0)\n",
    "    cont = 0\n",
    "    for i in range (X.size):    \n",
    "        if(np.angle(X[i])>1e-6):\n",
    "            cont=cont+1\n",
    "    if(cont == 0):\n",
    "        isRealEven = True\n",
    "    else:\n",
    "        isRealEven = False\n",
    "        \n",
    "    \n",
    "    return isRealEven, dftbuffer, X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run and test the function `test_real_evenI ()`. Here are two possible examples.\n",
    "\n",
    "_Test case 1:_ If `x = np.array([ 2, 3, 4, 3, 2 ])`, which is a real and even signal (after zero phase windowing), the function returns \n",
    "\n",
    "```\n",
    "(True, array([ 4., 3., 2., 2., 3.]), array([14.0000+0.j, 2.6180+0.j, \n",
    "0.3820+0.j, 0.3820+0.j, 2.6180+0.j])) (values are approximate)\n",
    "```\n",
    "\n",
    "_Test case 2:_ If `x = np.array([1, 2, 3, 4, 1, 2, 3])`, which is not an even signal (after zero phase windowing), the function returns \n",
    "\n",
    "```\n",
    "(False,  array([ 4.,  1.,  2.,  3.,  1.,  2.,  3.]), array([ 16.+0.j, \n",
    "2.+0.69j, 2.+3.51j, 2.-1.08j, 2.+1.08j, 2.-3.51j, 2.-0.69j])) (values are approximate)\n",
    "```\n",
    "\n",
    "To get a more realistic example use a longer input signal and plot the real and imaginary parts of the output spectrum `X`. For example, use `x = get_window('hanning', 51, fftbins=False)`, which is real an even, and plot `x`and the real and imaginary part of the spectrum `X`.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "[ 4.  3.  2.  2.  3.]\n",
      "[ 14.00000000+0.j   2.61803399+0.j   0.38196601+0.j   0.38196601-0.j\n",
      "   2.61803399-0.j]\n"
     ]
    }
   ],
   "source": [
    "## TEST CASE 1:\n",
    "x = np.array([2, 3, 4, 3, 2])\n",
    "\n",
    "(isRealEven,dftbuffer,X) = test_real_even(x)\n",
    "\n",
    "print(isRealEven)\n",
    "print(dftbuffer)\n",
    "print(X)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "False\n",
      "[ 4.  1.  2.  3.  1.  2.  3.]\n",
      "[ 16.+0.j           2.+0.69589549j   2.+3.51351879j   2.-1.08208835j\n",
      "   2.+1.08208835j   2.-3.51351879j   2.-0.69589549j]\n"
     ]
    }
   ],
   "source": [
    "## TEST CASE 2:\n",
    "x = np.array([1,2, 3, 4, 1, 2, 3])\n",
    "\n",
    "(isRealEven,dftbuffer,X) = test_real_even(x)\n",
    "\n",
    "print(isRealEven)\n",
    "print(dftbuffer)\n",
    "print(X)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3db5686128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3db57db390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3db55c0400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = get_window('hanning', 51, fftbins=False)\n",
    "plt.plot(x)\n",
    "plt.show()\n",
    "\n",
    "(isRealEven,dftbuffer,X) = test_real_even(x)\n",
    "plt.plot(X.real)\n",
    "plt.show()\n",
    "plt.plot(X.imag)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 4 - Suppressing frequency components using DFT model\n",
    "\n",
    "Given a signal as input, complete the function `supress_freq_dft_model()` to suppress the frequency components <= 70Hz using the DFT. Return the filtered signal in the time domain.\n",
    "\n",
    "Use the DFT to implement a very basic form of frequency domain filtering. Use the functions `dftAnal()` and `dftSynth()` provided in the `dftModel.py` module. \n",
    "\n",
    "Use `dftAnal()` to obtain the magnitude spectrum (in dB) and phase spectrum of the audio signal. Set the values of the magnitude spectrum that correspond to frequencies <= 70 Hz to -120dB (there may not be a bin corresponding exactly to 70Hz, choose the nearest bin of equal or higher frequency, e.g., using `np.ceil()`).\n",
    "\n",
    "Use `dftSynth()` to synthesize the filtered output signal. Then return the filtered signal. \n",
    "\n",
    "Use a hamming window to smooth the signal. Hence, do not forget to scale the output signals by the sum of the window values (as done in `software/models_interface/dftModel_function.py`).  \n",
    "\n",
    "Please note that this question is just for illustrative purposes and filtering is not usually done this way - such sharp cutoffs introduce artifacts in the output. \n",
    "\n",
    "The input is a `M` length signal `x`, sampling frequency is `fs` and the FFT size `N`. The output is the filtered signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "def suppress_freq_dft_model(x, fs, N):\n",
    "    \"\"\"\n",
    "    Inputs:\n",
    "        x (numpy array) = input signal of length N (odd)\n",
    "        fs (float) = sampling frequency (Hz)\n",
    "        N (positive integer) = FFT size\n",
    "    Outputs:\n",
    "       output signal with filtering (N samples long) (numpy array) \n",
    "    \"\"\"\n",
    "    N = len(x)\n",
    "    w = get_window('hamming', N)\n",
    "    mX, pX = dftAnal(x,w,N)\n",
    "        \n",
    "    k=int(np.ceil(N/fs*70))                  #Frequency index that correspond aprox. 70Hz   \n",
    "\n",
    "    mX_filt = mX[:].copy()\n",
    "    for i in range(k):\n",
    "        mX_filt[i] = -120\n",
    "\n",
    "    outputScaleFactor = sum(w)\n",
    "\n",
    "    x_synth_filt = dftSynth(mX_filt,pX,N)*outputScaleFactor\n",
    "    \n",
    "    return x_synth_filt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run and test the function `suppress_freq_dft_model()`. Plot the spectrum of the output over the spectrum of the input signals to validate the result. Here are two possible examples to try.\n",
    "\n",
    "_Test case 1:_ For an input signal with 40Hz, 100Hz, 200Hz, 1000Hz components, the output should only contain 100Hz, 200Hz and 1000Hz components. \n",
    "\n",
    "_Test case 2:_ For an input signal with 23Hz, 36Hz, 230Hz, 900Hz, 2300Hz components, the output should only contain 230Hz, 900Hz and 2300Hz components. \n",
    "\n",
    "To understand the effect of filtering, you can plot the magnitude spectra of the input and output signals superposed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3db55a5e48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## TEST CASE 1:\n",
    "fs = 44100     \n",
    "f1 = 40\n",
    "f2 = 100\n",
    "f3 = 200\n",
    "f4 = 1000\n",
    "N = 2**13 #With N=2**13 frequency resolution = fs/N = 5,38Hz, to have a better resolution increase nº of samples\n",
    "\n",
    "x = np.sin(2*np.pi*f1*np.arange(N)/fs) + np.sin(2*np.pi*f2*np.arange(N)/fs) +\\\n",
    "    np.sin(2*np.pi*f3*np.arange(N)/fs) + np.sin(2*np.pi*f4*np.arange(N)/fs)\n",
    "\n",
    "x_filt = suppress_freq_dft_model(x,fs,N)\n",
    "\n",
    "w = get_window('hamming', N)\n",
    "\n",
    "mX, pX = dftAnal(x,w,N)\n",
    "mX_filt, pX_filt = dftAnal(x_filt,w,N)\n",
    "\n",
    "# Plotting original magnitude vs filter magnitude\n",
    "f = fs*np.arange(N/2+1)/(N)\n",
    "plt.plot(f,mX)\n",
    "plt.plot(f,mX_filt)\n",
    "plt.ylim(-120,3)\n",
    "plt.xlim(0,1500)\n",
    "plt.ylabel('Amplitude [dB]')\n",
    "plt.xlabel('Frequency [Hz]')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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i8Lz1XCXBLBqHIfDZ0Y8mwHVmCL9Fo4Um7PiLajqcOB1WqR8tNJo4xcBXasnEnUUD3C+l/KGqDwsh/hHm8fh5ArgL+CRg2yTgDanSyBYKIZoKIdpKKfdUd7CKMRqnYdjTyrlUaJwI4S8XoYUm7Fjfs8PhwGEYmAjtOtPELYb04fMLTWlbjNi63iuN0Ugp363uw6G8p6YIIc4DdkkpV1bY1R7YGfA6x9oW7BhThRBLhRBLi4qKg1s0NrrOsC4Kn3Dg0DGasCMDFmw6rZpyOhlAE5eYJgKJxBKYeHOdCSHSgVuAw8B0VLxkLLAZuFNKmV3bkwohvgXaBNl1P3AfcHqwjwXZFlQtpJTTgGkAiW27S3/FZj9Ou2I0vjKLBqykAHwqFVEEG76mNpg+Lw7KYjQmhhYaTXxiXdc+o6JFE1vXe1Wus5nAUlRMZDHwKioYPxZ4GTi5tieVUp4abLsQoj+QCay01r10AJYJIYajLJiMgLd3AHaHcr4id/k1M7ZVBghYsAlgEmDmOqr6qjU1QVqVmoXDUWrR6KQLTVwSkGAExJ9FA7SWUt5nZXttl1I+am1fL4S4xY7BSClXAa38r4UQ24ChVtbZp8DvhRCzUEkAR0OJzwAcyC8p99phCLw2JgNgqNXqPiPgotBCEzak6cMnBYbhr8Bg4IixH55GExIVJq+lFk2MxX6ruvv5AKSUUghxoMK+SHTemYNKbc5GpTdfU9sD2R6jseqqmcKhnHv6JhhWpOm1FrD5EzsMHNqi0cQjfteZ3zsSows2qxKaLpYVIQL+xnqdafvIACll54C/JSpmVGdsi9GUBqmt+ltaaGxB+qyeP/4CqRgx98PTaEIioIBsuecYu6dUJTSTAv5+rMK+iq9jCochMG1Mb/bPOsrFaDRhQ/q8pSU5/As2dXqzJi6p4DrDiLMYTXVraGIZp2HYHKPx+1OdyskYYxdFtCOlr5zQeDH0d6yJT6zr2hRxatEIIVZRSfowgJRygC0jCjOZ6cm8fF35AgL2x2gsiyZGV/FGO36LxhACpyEowYi54KhGExIVs84st3yseUmqcp2dYz374yJvWs+Xo4LxMUFKopMx3dPLbXM6BCVeG/6jAhYSQuyWi4h2pKlqPzmNwBiN/o41cYjVTNGfbSbizaKRUm4HEEKMllKODth1jxDiZ+Cfdg/OLgxhs0Xjb8YVoznv0U6pRWMInA6rwZwd66I0mkhznEUTm/eUUNoEJAshxvhfCCFGAcn2Dcl+7Ms6K1+CJlbrEkU9/kZQgVlnutSPJh45LhkgNu8poawivA6YLoRogorZHAWutXVUNpPgNHB77asMIBx+11lszj6iHenzYkolMv52DLH2w9NoQiKgIjyU3Vti7Z5SrdBIKbOAgUKINEBIKY/aPyx7aeRyUOSxI0bj72WvYzS2YvpUCwZ/jEbqTqaaOKVS11lsXe+Vus6EEOcEvpZSHqsoMhXfEyskJTgotkNofG717EgAyopraqEJL/5kAIeVdaZqnenvWBOHVEgwMhz+rLPYut6rsmgeFULsInjVZD//Aj4P75Dsp7HLQZHbPqGRTr/QxKY/NeoJKEHjr96sYzSauKTSBZueCA2odlQlNPuA/1bz+U1hHEu90SjBQaHHh5QSEc7y/VZVYb9FE6uVVqOe0h7qfovG0BaNJj6pLEbjixOhkVKeXI/jqFeSXA6khBKvSZLLEbbjmp5i5Yv0u850jMYegpSgEVKnN2vikIpxX0ei2u5308cIoaQ3xx2NLHEJd5xGVojRxKqZG+2oEjQqGUAIgSkMhHadaeIRy+3usCwZ4Y/RaKGJfhonKKEJZ+bZ8h2H2X3wGADCqS6G5MZJADz1zTo27M0L27kaOv4Fm41dZQ3mtEWjiUssoWmS0li9dqp7iukpqewTUUmDFJpGltAUhikh4GB+CRc8t4BPsrbilQaGZcm0b54CwK87DnLuMz/Zk+nWAPF6ldC0SPG7KLVFo4k/vD6TtbsOAdAspRFQNolds/MA89bvj9jYakq1QiOEaCyE+KsQ4iXrdfdYTWv244/LhCvz7IUfNgPgwosH1cceICM9DQAnJm6vyazFO8JyvoaOz+vBJw2aNlZC4xPaotHEH9e8toSnv10PQIs0JTRYMZovVmznmteWsPNQbJSdDMWieRUoAUZar3OAB20bUT0QzhhNQYmXWUt2cnb/tiTgxR0gNB0toXHgY3hmc57/YbO2asKAz6dK0DRtZNWUw4GhLRpNHLF02yF+3HQAp2p0TPNU5ToT1tIJF16chuCJbzZGbIw1IRSh6SqlfATwAEgpi6h6bU3UE84YzYfLcsgr9nLtmMxSoXFaQtOueSoA5w9sze2ndGffsRLey8qp8zkbMl+t2cvh/CIMhwOnQ12+HpGAUydcaOKIF37YTLPGLhwoS71FmhIah8OBRzpIEB6uGtWZT1buZsfB6LdqQhEatxCiEVZvGiFEV5SFE7P4XWd1jdGYpuTVBdsY2KEJgzs2DXCdqa/V5VIz7tN7pTOqawuGdGrG8/Oy7amz1gD4cFkON76ZhfR5y1ZIAz7hxCG10Gjigw178/h23X6uGtUZp1D3qJZpqo6x0xB4cOLCx9RxXXAIwfOW6z6aCUVo/gZ8CWQIIWYA3wF32Toqm/EnA9TVjTV/Uy5bcgu4ZnQmQghcwotbunD4v9WAEjRCCG47pTu7jxbzwTJt1dQUt9fk8a+Vm8CBqQppWnhFAobpZtr8zRwt0oKjiW1enL+ZRi4HV43szNUjOgCQamWwOgwDN05ceGmdlsTFwzrwftZO9hwtiuSQq6VaoZFSfgNcCFwNvA0MlVJ+b++w7MUfo1m2/TBS1r5dwCs/baVlaiIT+7cFOM6iqdg7Ylz3dAZmNOXZedl4fNqqqQnvLt3JriNF3DA2Ewc+igJCMj7DRQIe/jVnHY9/vSFyg9Ro6kjO4UI+XbGbKcMzaJacQL82ypIRVmdNv0XTSKgfwI3jumJKmDZ/S8TGHApVFdUc7H8AnYA9wG6go7UtZvHHaF7/ZTsfLd9Vq2Os3nWUHzcd4NrRmSQ41deYJPxCY72pgtAIIbj9lG7kHC7io2W1O29DpNjj45m52Qzt1IwbxnbBgUmJWRYm9Ar1I3ThY+aiHew+Et2zO42mMp6dtxlDCKaO66I2VCiq6TAEJbhINNT2jOaNuWBQe2Yu2kFuXvRGNKqyaB63Hs8Ci4BpwEvW3/+zf2j2EVh25tOVu2t1jOe/30xqopPLR3Qs3ZYofLhxBLFoylx043u2ol/7NJ6Zl41XWzUhMXPRDvYeK+aPp/egZWoiDkwyW6WV7vdZQnPhwJaYUvLGL9sjNVSNptbkHC7kvaU7mTI8g7ZNrHRmf/kq657idAg80kGSKHMR/+7krnhNydNzo7f0ZKVCI6UcL6UcD2wHBksph0ophwCDgOz6GqAdJDrL/tkLsg/WOFaz9UABc1bv4YqRnUhLKgtKJwgvblw4/IU6g/T3FkJw24Tu7DhUyIxFel1NdRwt9PC/uZsY0y2dUV3TEULQuXki/Tq0KH1PiVSC3jM9kTP7teHtxTsodOt0Z01s8dS3mzCE4OaTu5ZtLBWaMovGjYsEUXbP6tIyhctP7MiMRTuitgJJKMkAvaSUq/wvpJSrgRPsG5L9CCG4dHhHLhmagdtnsmb3sRp9ftr8zbgcBteM7lxueyJePLJsHU1l/b1P69OaMd3SeeyrDew/Vlzbf0aD4Jl5mzha5OG+ib3LNpq+0nbZAIWm+jvV6ePa0ZkcLfLwgXZNamKIVTlHeX9ZDteM7lxmzQDI8q4zf4wmSZS/p/zh1B6kJDr5+6dr6hR3totQhGadEOJlIcTJQoiTrAoB6+wemN38+8L+3Hl6D0DVKQuVfceK+SBrFxcP7UCr1KRy+1yifGWAyoRGCMED5/ejxGfyf59E54URDWzcl8drC7bxm8Ed6NOuzFWG6SuzFoEin/qek50mQzo1Y2CHJrz601ZMU3+vmuhHSskDn6+leeMEbpnQrfzO4ywalXWWUEFomiUn8OczevLLloPMWrKzPoZdI0IRmmuANcDtwB3AWmubbQghbhVCbBBCrBFCPBKw/V4hRLa174y6nqdVWhLtmzZi+Y4jIX/mlZ+24jVNpo7tety+igs2sTJF8B5faTUzPZk/ntaDL9fs5Z0ovDAijc+U3PX+r6QkOrnnrF7ld5reckLjt2iSHT6EEFw7JpMtBwr4YWNufQ5Zo6kVMxbtYPG2Q9x1Zs9yrnigLL5rWfBOh7JoEjneNXzZ8I6M6NKch2avI+dwdC3iDCW9uVhK+YSU8gLr8YSU0jZ/jxBiPDAJGCCl7As8Zm3vA0wB+gJnAs8JIercTKZf+zTW7Q3NdXa00MOMhds5Z0A7OrZofNz+iiVoMAxVm8gbPAtq6tgujOmWzt8/W0P2/uj0rUaKafO3sGLnEf5+Xl9apCSW32l6y6xFoNCnLuPGVibOxP5taZOWxMs/RXfKp0az81Ah/56zjrHd07l4aMbxb/AWq7Yj/mQAQ1AiXbjE8UJjGIJHJg9EALfMWEaJN3rKXYVSVHOrEGJLxYeNY7oZeFhKWQIgpfSXKJ0EzJJSlkgpt6ISEobX9WQ926Sx7UBBSAkBby3aToHbx00nHW/NQBDXGYCrEXiCC41hCP578UAaJzj53Yxl5BXrxYYAi7ce4rGvN3B2/7acN7Dd8W+QZoUYjd91pv4P/fGzn7MPsnTboXoZs0ZTU0xTcvcHvyKE4OHJA4J3+/UUqXuIhcOK0SQEsWgAOrZozKMXDWRlzlEe/Dx6IhyhuM6GAsOsx1hUavNbNo6pBzBWCLFICPGDEGKYtb09EOhjyrG21YlebVIxJWzal1/l+0q8Pl5bsI2x3dPLxwsCcOGvDBAoNI3BU7kZ2yotif9NGcSW3AJumbm8wS/kzDlcyC0zl5HRrBEPT+5fxY+vLD7mRrkbGhllP74rR3aiZWoij3y1QcfANFHJs/OyWbD5IPef3Zv2TRsFf5OnUN1D/EisEjSVZ1We2a8NU8d14c2F23k3StzyobjODgY8dkkpnwQm1OWkQohvhRCrgzwmodpLNwNGAH8G3hXqbhOskGfQO4gQYqoQYqkQYmlubtV++l5tVOHL6txnn6zYTW5eSdlCqiC4pBcPjvJCk9C4UovGz5ju6Tx0QT/mb8zl/z5Z3WBvjEcK3Vz96hKKPT6m/XYoqRX91QA+r+pY6iz7YXqs9OZGRplIN05wcuuEbizeqqrgajTRxLwN+/nvtxs5/4R2TBkWxGXmx1NU2uwMwGNKqwRN1d6Pu87oydju6dz70Sq+XbsvXMOuNaG4zgYHPIYKIW4CUutyUinlqVLKfkEen6AslQ+lYjFgAunW9sD/kQ6oSgXBjj/NWvcztGXLllWOpVOLZJJcRpX556YpeWn+Fnq3TWNMt/RK3+fCgwcn5XSiCtdZIJcM68gt47vy9uKdUZuiaCcH80u47KVF7DhYyItXDqFH60ouMX+8K8Cd4EYJTcWUzynDOtK+aSMe/WqDzkDTRA1bDxRw+9vL6dUmjX9fWInLzI+nqJxF4/aa1Vo0AE6HwQtXDKFfuzRumbmMRVsOhmv4tSIU19njAY9/A4OBi20c08dYFpMQogeQABwAPgWmCCEShRCZQHdgcV1P5jAE3VulVik0P2zKZdP+fKaOy6zyonBayQDlVvxX4zoL5E+n9+SGsZm8/st2/vLx6gZzc9yfV8ylLy1kc24+L101lFFdKxdzPFYeSjmhUZZPolH+x5fgNPjjaT1YtUutUdBoIs2+Y8Vc+coinA6DF68YUlrgt1I8heWudY/PpEQ6cYVQrTw50cmr1wynQ7NGXP3qEn7cFLkszFCE5jp/lQAp5WlSyqnA8fm64WM60EUIsRqYBVxlWTdrgHdR6dVfArdIKcOSVtGrTSrrq3Cdvb1oB+kpiZzdP0hgOgCn9ODGhbuc0IRm0YBaX3PfxN7cdFJXZizawc0zsuJ+hfuvOUeY9MzP7DxUxKtXD+OkHlVboKWiHfDjK7EsmmApnxcMas/QTs3495x1HC6w87LVaKrmaJGHq6Yv5lCBm1evHhY0c/U4PMXlrvXOLZLVgk0jtFtf8+QEZk0dSacWjblVsb6lAAAgAElEQVTutaV8uXpvbYdfJ0IRmvdD3BYWpJRuKeUVlittsJRybsC+h6SUXaWUPaWUX4TrnD3bpHIg3x20KF1uXglz1+9n8uD2pcUzg2L6cGDikU68vgBLxNUY3KHntAshuPvMnvzfOX34Zu0+Ln7xF3bFYZFIKSXvLNnBRS/8giEE7988klFVuCVL8R5v0XgsoTGCND8zDMGDF/TjWLGXh79YH5axazQ15UihmytfWcTm3HxevHIIAzOahvbBCskAfdqlcdGJXSvNOgtGy9RE3pk6kr7t07h5RhbPzsuud9d8VdWbewkhJgNNhBAXBjyuBpIq+1ws0tNKCNgUZC3Lx8t34TUlFw3tUPVBfGq27MFJq7SAdR+uRiG7zvz4Fx2+fNVQth0oZOJTP0ZsJmIHB/JLmPpmFnd/sIrBHZvx6e9H07ddk9A+7P8uA5IB3NK/MDZ49dpebdK4fkwm7yzdyYLNOjFAU78cyC9hyrSFrN+bx4tXDmFs92qs9kAqpDcDNG7UCOGrmXXepLGLt28YwXkD2/HoVxu49e3lFNWx8WNNqMqi6QmcAzQFzg14DAZusH9o9Ue3VikAbN5/fIrz+1k5DO7YlG6tqsl/sG5yV43twYAOAbMVV/VZZ5UxoVdrZt82hk4tGnPTW1n88d0VHMiP3lLg1eHxmUz/aSsTHvueHzbm8pezezPj+hOPX5BZ5UGOt2juOKOv+qOKH9/tp3YnMz2ZO99dydFCvV5JUz9szs1n8vML2HawgOlXDWNCr9Y1O0CFZABALQI3PWDWbClEksvBk5ecwD1n9WL2qj1MevYn1tawzmNtqap68ydSymuAc6SU1wQ8bpNSLqiX0dUTbdKSSEl0sqmC0Gw7UMCGfXmcG2zRYEVKlDXUplWF2UoNkgGC0alFMu/fNIrfj+/GZyt3c8rjPzBz0Y6YShQwTcmXq/dyxpPz+efnaxmY0ZQ5t43h+rFdMIwqMm6CESRGc8lIqz5UJRYNqHTnJy85gdy8Eu7/eFWDy+rT1D8LNh/ggmd/Jr/Yy8wbRjCmewiu4Yp4CsutGQPA38Y8iKu4OoQQ3HRSV16/ZjiHCz2c/+zPTJu/2fb7SVWuM3+75suEEP+r+LB1VPWMEIKurVLIriA0365T+een9g5hFlJ8VD0nVljMWYNkgMpIcBr86YyefHH7WHq1SeW+j1ZxxpPz+fzX3VEtOB6fyftZOZz+5HxueisLJLxy1VDeuHZ49RZiZQSJ0eBIUM/VuBMGZjTljlO78/mve3hvqc5C09iDlJI3F27nt68spnVaEh/fMprBHZvV7mDBLBr/tV+HCey4Hi356o5xnNyzJf+as57fvLCAdXvss26cVezz1y9YatvZo4jurVKOK8L43br99GidQkbzELJDSqz/pKSKQtMYfCXHVRyuDd1apTJr6ghmr9rDk99u4vczl9Oj9SauGZ3JeQPbkZxY1X9n/ZG9P593l+7kw2U5HMh306tNKk9NOYGz+7fF6Qgl/6QK/KLtDBQay/UWgt/65pO7sWDzQf7yyWp6tU0t7+bUaOpIXrGHez5cxexf93Byz5b879JBxxfKDBUp1bqxCjEakqx4ZvExaFRLAUNlpL145RA+XLaLh+as45ynf+K6MZncfkr3Wh+zMiq9M0kpP7OeXw/7WaOQbq1SeD8rh6OFHpo0dnG00MPibYe4sYpKAOUotoQmsUJQu3T2UQSJKXUepxCCcwa046x+bZm9ag/Pzcvm3g9X8dDsdUw6oR3nDGjHsM7N6n5DrwFSStbsPsY3a/fxzdp9rN1zDKchmNCrFZee2JGTe7SselFaTfAcv2ATw1BFNqtwnflxGIJnLhvMuU//xE1vZvHprWNIr0mMSKOphBU7j3D7rOXkHC7irjN7ctO4rjV3DQfic6u6fpUKzdHaH9tCCMHkIR2Y0KsV//lyPdPmb+FDG3o5VSo0QojPqKTEC4CU8rywjyaCdLcSArJz8xjSqTk/ZufiMyWn9G4V2gFKLZrKhKYwLELjx2EIzhvYjnMHtGXZjsPMWLiD97NymLFoB82TE5jQqxUju7RgeGbz0CyyGuAzJZv257Fk22GWbjvE4q2H2HO0GCFgSMdm3D+xN+cPak/LVBtu4MGEBpRVE2Imjn8mN/n5Bdz4ZhYzrj+xXHtvjaYmFHt8PPntJqbN30zrtCTevmEEwzOb1/3ApfHICr/fMAqNn2bJCTw8eQCXDMvgodnryArbkRVV+VoeC/O5opruVswge38+Qzo1Z+GWgyQnOBgYqmvF/58ezHUGdfKnVoUQgiGdmjOkU3MeOL8fP2zM5as1e/lm7T7ez1JxiNZpifRsk0avNqlkpifTOi2RVqlJtLaSIBKcRml9NtOUlHhNijw+DhW42X+smH15xew+UsymfXls2p9P9v58Srwq46VVaiLDOjfnpB4tmdC7lf3WQZASNIAKkIZg0fjp174JT1xyArfMXMbts5bz3OVDyteo02hCYPmOw/z5/V/J3p/PJUMzuP+c3rV3lVWkskmVDULjZ1DHZrx300iM34X3uFW5zn7w/y2ESAB6oSycDVLKuFti3b5ZIxKdRmkV50VbDjGkc/PQXVCVJQMk+IXG/kWXyYlOJvZvy8T+bTFNycb9eSzacoiVO4+wYV8ery04iNsbPCXSaQgchigVkGC0bZJE99apjOzSgt5t0xjWuTkZzRuFzy0WCsFiNKAKD3pr1iZpYv+2/PXsPvzz87X887M1/P28vvX7b9HELIcL3Dz69QbeXryDNmlJvH7t8OqrWtSUUqGx36IJxI7fQLXRYyHE2cALwGZUBeVMIcSN4VyZHw04DEHXlilk5+ZzML+ETfvzOX9QDboQlBxT2U8VUxFdydb+qtsQhBvDEPRqk0avNmXC5/WZ7M8rYe+xYmWpHCuhyOOjxGNS4vXhMyVJLgeNEhwkOQ2aJSdYlk8irdOSoiPZwFOk3GRGhQlAYiq4a/4dXzsmkz1Hi3jpx62kpyRyqw2BUE384DMlby/ewWNfbyCv2Ms1ozK547Tu4bNiAgmSyg/YLjR2EMqd43FgvJQyG0AI0RWYDcSH0Jg+mH0njLiZbq1SyNp+mMVbVbOsEV1ahH6c4mPHx2cAUqxZTsH+4/fVM06HQbumjWhXWe+LWKBCL5pSkprU+od371m9OVjg5vFvNuJyGpU2ttM0XKSUzNuwn0e+3MD6vXmM6NKcf5zXr7SqiC24C9RzRaHxe03iTGj2+0XGYgsQ+btmuNi9ArJehX2r6Z75PJ+u3M28Dftp5HIwoEOIZVFA/adXdJsBpLZVz3l7wjPeho43yLoCUEJTdLhWhzQMwaO/GYjXJ3n4i/U4DcH1Y0PMNtTEPVnbD/OfL9azeNshOrVozDOXDeLs/m3td7PmW7fZ5AouOcMBCallCUgxQChCs0YIMQdVOVkCFwFLhBAXAkgpP7RxfPZTZLX6TUwtLUXz7tIcxvVoiasmKcIlx45PBAB1kQgD8uKnVllECVL7CYBGTeHwtlof1mG11faZkgdnr8OUkqnjtGXTkFm58whPz93Et+v2k56SyAPn92PKsIya3RfqQr7VsCylzfH76mDBR4JQhCYJ2AecZL3OBZqj6p5JILaFptBqCNS4RbmKquefEELZmUCKjgR3nRkOSGmtLZpwUXQkuOUYhh+e02Hw5JQTEAL+NWc9x4q83Hl6D50g0MBYsu0QT8/NZv7GXNKSnNx5Wg+uHZNZ/zHKvL1qkpocpHRNvAmNVe8sfimwqvk2bkG7po24f2JvXluwjTP7BZlFVMWR7dDjzOD7UlpDXuTbqcYF+XshLUiShv+HJyXUQRhcDoOnpgwiNcnJM/OyOVbs4e/n9q3bwjtN1COl5KfsAzwzN5tFWw/RIjmBu8/sxRUjOgZvKV4f5O+F5FbBK4okNVGTrhghlKyzTOBWoHPg++NmwWa+5dKy3DE3jOvCDaFWA/BTfBQKcqFFJa6W1LZwVNfWCgt5+6Dd4OO3JzVRRQY9RWUp5bXEYQj+dUF/UpNcTJu/hWNFHh75zcCq+xFpYpJij49PVuxi+k/b2LAvjzZpSfzt3D5MGdax+u6XdpO3D1IrqbPYpD3sWFS/46kDodiCHwOvAJ8BNatLHQv4LQ1fHUrHH9ysnptXJjRtYOfCOs+2Gzw+rxL0lCA/vsCUzzoKDai1BPee1YsmjVw8+tUG9h0r4YUrh9CkUYRmt5qwsv9YMW8u3M6MRTs4VOCmd9s0HrtoIOcObEuiM0qqROTvhdRKXPgtusOq91RTxTBc73YTitAUSynjqlpzOfyxkxo2EirHoS3quUW34Ptb91WZbUd2QLNOtT9PQ6fwACCDz/IChSatbVhOJ4TglvHdaNskibs/+JXfPL+A6VcPC3tJH039IKVk+c4jvPnLdj7/dTdeU3Jq79ZcOzqTEV2aR18sLm8ftBsUfF+6da85tBna9K+/MdWSUITmKSHE34CvgdIaH1LKZbaNqj4ptvycdRGa3A2AgOaZwfd3GKaec5ZooakL/sy9oFk4ViKHDQHSCwd3oE1aEje+lcUFzy1g+tVDddXnGOJQgZsPl+XwzpKdbNqfT3KCg8tP7MQ1ozvTqUVypIcXnIIDau1dZV6SFtbC4gOb4kZo+gNXAhMoc51J63Xs42+AVRfX2bafoN0JwdNuQVk0zkZKaPr/pvbnaej40z1TqxIaewKko7ql8+HNo7j61SVc8uJC/nfpIE7rU8NuiZp6wzRVcP+dJTv5eu1ePD7JCRlNefjC/pwzsB0p0VDloip2WfP4DkOD72/RTWWk7VsD/S6sv3HVklC+7QuALvFY3wwIEJpa/vNK8iFnMYy6tfL3OFzQaRRs+ALOfFjHaWrLkR3qOZjQ+N1pNiZddG+dyke3jOL615dy45tLuf/sPlw7unP0uVwaMLuPFPHe0hzeXbqTXUeKaNrYxZUjOnPJsAx7V/GHm11LlZC0PSH4/oTG0H4IbJ4Lp/y1fsdWC0IRmpVAU+KpGkAg0jLSalD5txwbvwTTC12rMfD6XQif3AK7siqfpWiqZtcyaJwePL05tZ2qGOBPzLCJVqlJzJo6gj+8s4IHPl/L5tx8/nFe3/pbxKc5DrfX5Lt1+5i1ZCfzN+UiJYztns49Z/Xi9L6toye4XxO2zleekKpai3Q/Heb9C/Jzy0pdRSmhCE1rYL0QYgnlYzTxkd5MHV1ny16Hph2h05iq39frHJhzFyx+SQtNbclZAhnDg1uEhqH82Qc32T6MxglOnr98CI98tYEXftjMzkOFPHPZYJ2RVs9k78/jnSU7+XDZLg4WuGmTlsSt47tx0dCM2E7YOLYbdiyE8fdX/b5e58C8h2DFDBhzR/2MrZaEIjR/s30UkcRv0dTGdXZoi5p5jP/L8dWEK9KoKQy5ChZPgwl/gaYZNT9fQ6bggBKRgVMqf096N1W7rh4wDME9Z/WiS8tk7vtwFRc+9zPTrx4WvcHlOKHQ7eXzX/fwzpKdZG0/jNMQnNq7NZcMz2Bc95bx0VPo13cACX3Pr/p9rftA57Fq8jry9+CI3rhTtfa+lPKHwAfgBS62f2j1RF1iNFmvKz/qoMtDe/+I36nzLXyu5udq6Kz5SD33OKPy97Torio01EPvHz8XD83gzetO5GCBm/Of/bm08rcmfEgpWbHzCPd+uIrhD33HXe//yuFCN/ee1Ytf7j2FF64cwviereJDZHxeWPKKEpD0EFpWjLgZjuXA+s/tH1sdCEkChRAnAJehBGYr8IGdg6pXSi2aGrrOio/C0leh19mQFmJdtKYZMOBiWDpdiY62akJDSuUeaNW36lTO9kPU/+fORdDl5PoaHSO7tuCj343muteWcMXLi/j3hf2ZPKRDvZ0/XjlS6Oaj5bt4Z8lO1u/NI8llcHb/dkwZnsHQTs3iMwlj1btwdCec9Z/Q3t/jTGiWCfMfg97nBi9XEwVUatEIIXoIIf5PCLEOeAbYCQgp5Xgp5TN2DUgIcYIQYqEQYoUQYqkQYri1XQgh/ieEyBZC/CqECFKHpDbU0qJZNA1KjsK4P9fsc36/67d/r9nnGjLZ38Lu5TD8hqrf13kMGC6ViVPPZKYn8+HvRjGkUzPufG8lj321AdOU9T6OWMc0JQuyD3Dr28sZ/q/v+Mdna0lwGjx0QT8W338qj188kGGdo3BxZTjweeD7h6HtQOg5MbTPGA7lit+3ClbOsnd8daAqi2Y98CNwbkDTsz/Uw5geAf4hpfxCCDHRen0ycBbQ3XqcCDxvPdeN2sRoSvJg4bNqNtF2YM3O1zRDpULPfxSGXafSnjWVI6XKrGnSEU6oxkWZmAIdR8D6OXDqP+o9jbxp4wRev3Y4f/14Nc/My2brgQIeu2hg5GtmxQC5eSW8n5XDrCU72H6wkLQkJ5cOy+DiYRn0bVeDvlCxzLI3lOt34qM1u3b7TVbu+LkPQJ/zVLfZKKOqGM1kYC8wTwjxkhDiFFQrZ7uRgL8OfBNgt/X3JOANqVgINBVC1L3WSG1iNEteVk22xt1Vu3OOvgOadlLpzu7C2h2jobDuU9i9DMb9CZwJ1b9/4BSVNLDjF/vHFoQEp8HDk/tz38RezFm9hynTfmF/XnFExhLtmKbkh4253PxWFiP//R3/+XI9rdOSeOKSgSy+/1T+MalfwxGZwkMw90HoOFKlLdcEIeDM/6jKGd89YM/46kilQiOl/EhKeQnQC/ge+APQWgjxvBCiht9EjbgDeFQIsRN4DLjX2t4e5b7zk2Ntqxs1FRp3ASx4GrqdCh2G1O6ciSkw6VmVtfbdP2t3jIZA8VGVEt66P5xwWWif6XuB6leT9ZqtQ6sKIQRTx3XlhSuGsHFfPuc/8zNrd8dON0S72Z9XzNPfbWLco/O4avpiFm09xDWjO/PtH0/i3RtHcsGgDiS5GpgVOPcBdb1PfKx2lnjGMDjxRpXVuj0yk6yqCCXrrEBKOUNKeQ7QAVgB3FOXkwohvhVCrA7ymATcDPxBSpmBErdX/B8LNrxKjj/Viu8szc3NrWY0NVxHs/RV1SytttaMn8yxMHwqLHoetvxQt2PFK1//VdV7Ou8pVV0hFBKSVcLFmo/VLDGCnNG3De/dNBJTwkUvLOC7dQ27J9GKnUe4Y9ZyRj88l8e/2UinFo15+tJB/HLvBO4/u09ph9sGx9Yf1X1l+FRo06/2x5nwV1VL8YPrI37tV0RIGV0BSyHEUaCplFIKFfE7KqVME0K8CHwvpXzbet8G4GQpZZWtK4cOHSqXLl1a+Rv+2weO7VItl/+cXfXgPEXw5ABo1Ruu+rRm/7BguAvgxZOUG27q9zoLLZA1H8F7V8Oo2+D0GroD9q2F50fCSXfD+PtsGV6NhnOsmOtfX8rq3Ue5f2JvrhuTGZ/B7CC4vSZzVu3htQXbWLHzCCmJTi4a2oHfjuxMZrpec0TREXh+NDgT4aYf1USpLuxaBtPPUFmXl75T/fq+ShBCZEkpw7ayPBrrZuymrG30BMC/1PtT4LdW9tkIlADVvT9yaQmaEFxny95UM+yT6mjN+ElIhikzVfmbd66o1/UfUU3uRvj0NlX1+pT/q/nnW/dRq6YXvhAVXQhbpyXxzo0jOKNPGx6cvY77P16N1xd/rZ0COVbs4fnvNzP6P3O5450VHCv28M9JfVl43yn87dy+WmT8zPmzalVy4Ut1FxmA9oPhjH/Bpq/hx8frfrwwEY1LSW9AtSZwAsXAVGv7HGAikA0UAuFpMR1qjMbngQX/g4wTodPosJwagJY94MJpMOtS+Ogm+M30qM2FrxcKDsDMi9QM7zfTQ3eZVeSku9UitoXPRYVV0zjByXOXDy4tW3Ok0M2TlwyKu66dB/JLePXnrbzxy3byir2M7Z7OYxcNZGy3dN0OuyJZr6l1MyffW/t4bzCGXQ87F8O8B5WXpKpqGvVE1AmNlPIn4LhvXSof3y3hP2GI6c2r3lcLqWobrKuKXhPh9Afh67/AZ6lw3tMNs8Jz4SF460KVPXP1bFVDrra0HQB9JqnEjcFXqda3EcZftiY9JYEHZ6+j2JPFc5cPjovA975jxTw3L5tZS3bi9pmc1a8NN5/Ujf4dGkjWWE3ZvgBm36mSimq6Fq86hIBJzyhL6ZNbVFig2ynhPUcNia/pVK2wLBrpA9MX/C2mCT89Aa37VV0CpS6MulUlGCx/E764W52zIVF4CN6YBPvXwcVvhqfw6Gn/VP+n30ZXub7rx3bhoQv6MW/Dfq59bQkFJd5ID6nWHC308PAX6znp0XnMWLSD8wa249s/nsRzlw/RIlMZh7fBO1dCs84w+RV7PBjORJgyA1r2glmXw+Z54T9HDdBCIwNu6JVlnm38Ag5sgDF/sNfSGH+fKo63+EX4+Ka6NWOLJQ5vh9fOVp1Kp8yEHmHKnm/WWQn4qvdUNdwo4vITO/H4RQNZuOUg17y6hCJ3JZOcKKXI7eP57zcz9pG5vDh/M2f1a8vcO0/m0YsG0rVlA80eC4W8vfDG+WB6YMrbqtiuXSQ1gSs/huZdYOYlsOkb+85VDVpoArPuKnOfLX4J0jpAn2qqqdYVIZQLbcJfVQXXmZeojLR4ZsdCeGmCyvy7/F3oflp4jz/2j6pXzZw/RZ1wXzi4A09NGcTS7YeY+uZSSrzRLzZSSr5es5dT//sD//lyPUM7N2fObWN54pIT6Ngihkvz1weFh+DNCyB/P1z+gYrP2k1KS7j6c2jVC96eoqoPRAAtNNKkdIlOMKE5tAW2zIPBv62fMtxCqFXw5z2tWhBMG6/atcYbpk+V4XntbDXzun6uPYUwE5JVgcK9q5T7M8o4d2A7Hp48gB83HeC2t5dHdTbajoOFXPf6Uqa+mUVyooO3bxjB9KuH0bttWvUfbujk74fXz4WD2XDpTLXAsr5o3Byu+gwyT4JPb4Uv76s8TGATWmiQ4ExSfwYTmmVvgHDA4Cvrd1iDf6sC4p4ieOkUWPRi/MRt9q9TAjP3QRWwv2Gu6iVjF33Og36/gR8swYkyLh6awd/P7cNXa/Zxz4eriLa1baYpeeWnrZz2xA8s2nKQv5zdm9m3jWVk1xaRHlpscDQHXj1LTVove6deK4uXktQELnsXTrxZ1WmccZHqzFlPaKGRUgXOALwValKZJqx4WyUAhNoKIJx0PBFunK+qEn9xF7w5CQ5Us6g0mik+Cl/eqxao5a6HC6apYKidfmo/Ex+FRs3ho5vBE321x64encntp3Tn/awcnp0XPf/HOYcLuezlhTzw+VrGdk/nuztP5vqxXXTr6lDZtUxNFPP3w5UfVd/y3U4cTjjrYTj3Kdj2Ezw/SlVGrwf01SJlmYgc3VV+X85iyN+rqqNGitTWcPl7cO7/YNdyeO5E+Or+qFiIGDLFx9Tisf8NgoXPq06jty6DgZfUXxp34+Yq5XPfKvgq8utqgnHHqd25YFB7Hvt6I5+u3F39B2zm6zV7OevJH1mVc5RHJg/gpd8OpU2TpEgPK3ZY/aGyZJwJcO1XqrJ4NDDkapg6Dxq3gLcmqyzXknxbT6mFRpqQbgXlKvabX/cZOBJqXk013Aihbs63LVPFJX95Fp4erGIc0Sw4R3aqarJP9lPFQ9sNVqV2znlC3fjrmx5nwOjbYekr8Ot79X/+ahBC8PDk/gzv3Jy73l/J+r2RKcTpMyWPf72BqW9mkdkymS/vGMfFwzIaTNmcOuPzquv9/Wug7Qkq/ti6T6RHVZ7WfZXYDJ8Ki16A50bAhi9tO50WGiQ06QCuxnAgQGikVCXqu4yHpCgJdqa0UkkCU79XN+25D8IT/ZSFk7sx0qNTuAth7Scw42J4aoCyZDqPhRvmwRXvQ7sTIju+CX9Vpdg/ux32ro7sWIKQ6HTwzOWDSEtycfNby8grrt9MuYISL9e/voSn52Zz8dAOvHvjSDKa62yykDm6C14/R133g65UNRFTWkZ6VMFxNVIu5Wu/Ukkzb1+i1twc3Bz2U0VdUc1wU21RzQdbK1XfMg9S2qibIcCelfDiODjvmfpPBAgVfybVmo/VgtMOw2DAJdDzLCWe9UV+Lmz5HtZ9Apu+BW+R+i4HX6mSGuqywt8Oju2Bl8arJI8b5ir3ZJSxaMtBLnt5EWf2bcMzlw2qF2viUIGba15dzOrdx/jHeX25/MSO2oqpCWs/gc/uUElF5zyhqojHCl43/PKMEkhvCeJvB8NaVFMLzQOtYMTNcGSHarB1+0q1/acnVLvlP21SlkQ0k7dPrbtZMUMF2UH1cOk6HjKGQ4fh4buZet3Kxbh3NexcqIKKByxrKqWN6lve+1xVD64+0sFry56VMP1MtXL66tmQEH2z9ue+z+aRLzfw5CUncP4ge0vo7D9WzKUvLSTncBHPXDaY0/pEn/hGLfm5MOdOJTRtB8Lk6fZmUdpJ3j6Y9xBi0tNaaGpCtULzz3QY9XtwJKr01/v3gisJ3rwQju2GW6JrRXmVSKlu+hu/hI1fQc6SspTtlDaQ3l090tor8UxuqQKCDhcYTvXwecCdr4KDJcdUvaSju+BYDhzaqlbvm5Y7JyFVBTg7j4bO46DdoFqXJY8I6+fArMug9zlw0etRV8zUZ0ouemEBm3ML+OYP42iVZk8g/nCBm0um/ULO4SJevXoYJ3bRacshIaWqgfjFXeo3c/I9MOr26J5ghUi42wTE/jdSZyQIQ92AkSrXvUU31Qp40BWRHlzNEAJa9lSP0berNN49K1X23P51SoRWf6DSjGtCQooSp2ad1Mr9Vn1VcDO9Z2z/qHpNVCXVv7oXPr9DZfZFkavIYQgeu2ggZz31I/d9tIqXfjs07K6s/BIvV726mG0HC3lNi0zo7FujsrW2/Qjth6qOua16RXpUUUsM3yXChL8yQAvL1D24Sc3kPYUqiB3LuJLUWpyOJ5bf7imCglz1KDysLBTTqx7CoVpNJ6RCYiqktlGLvaLoBhxWRv5OdUz98TElqGf8K6r+rSqLwm8AABbWSURBVF1apvCn03vy0Jx1fL12H2f0bRO2Y5um5I/vrGDN7mO8eMUQRnVLD9ux45aiw/D9w6osVVIanP04DLkm6qzhaEMLjbQsGr/QHNhoZXAJtVAyHnE1UgH6aAvSR4oJf1Guj4XPKbEZf19Uic01ozvzXtZOHpy9lpN6tAxbW4Hnvs/m67X7+MvZvTlVx2SqxlOkqnP89ITyCAy9RmUwRiJNPwbRQoNUN5XEFGjaCfb8qmYtrfvpi6ihIASc8W/VWnv+I8rCO+VvUSM2TofB387ty+UvL2L6z1v53cl1DzTP35jL499sZNIJ7bhuTGYYRhmn+Lyw4i1lxeTtgW6nqa6vbQdEemQxRcMWGn8ihLAC2B1HqMwR06vK9WsaDoahYjQOl5q1eorgzIejRmxGd0tnfM+WvPjDFq4c0YnUpFp2HkX1kPnz+yvp1jKFhy8coFOYg+F1q0zOn/6r4rYdhsPkl+PXy2EzMZQiZAOlvWisH1rHkarememFHmdGbFiaCGEYcPZ/YcQtarX0Z7fXe5XbqvjDaT04WuThtZ+31ek4D8xey4F8N49fPJBGCTq2UA5PkYq/PD0YPv29cqVOmQnXfa1Fpg5oiwbKLJpup5alAXeoxzLemuhBCDjjIRXH+vExlRhywYtlhVcjyIAOTTm1d2te+nEL147JJDmx5j/frO2HeD8rh5tP7sqADvVQzDRWKDoCy15X5Z3y90HGiWrS0f20qLFqY5kGLjSWReO/jppmwJ82RGw4mihBCDjlryqr6Jv/UwvyprwFjZpFemTcfHJXvl23jw+X7+LKEZ1q9FnTlPzzs7W0Tkvk9+NjdEFhuDm4WVmvy2eAp0D1bJn8irJetMCEDS00UGbRaDSBjL5ddef8+GaYfpYqT1SfpX2CMLhjU/q3b8LrC7ZxRQ1LxHy+ag8rc47y+EUDa2UNxQ1SwtYfVCXxjV+puFy/38CIm9TKfk3YaeB32AquM42mIgMugis/VK2mXz4t4oU4hRBcPaoz2fvz+Tn7YMifk1Ly3LxsurVK4QKby9lELUVHYMnLqg/LG5NgVxacdDfcsRoueF6LjI007DtsxWQAjSYYmePgWquE+vQz1Sw4gpwzsC1NGrn4YFlOyJ/5fkMu6/fmcdNJXTGMBnS9S6nq8X14IzzeE2bfqRZXTnpOCcz4e6OyqGq80YDtZ45PBtBoKqN1X7j+W5h1Kcy8BE5/EEbeEhE/fqLTwVn92vDZyt0UuX0hZY69OH8z7Zokcd7ACHSKjQR5+2DlTFj2JhzaDIlpqpfT4N+qHjE6/lKvNHCh8cdo9EWnCYEm7eGaL1TM5uv7VaXss/+rOijWM+cNbMesJTuZu34/Zw9oW+V7tx8sYOGWQ/z5jJ4kOON4UuXzQvY3Slw2fqlaZ3QaDSfdBb3Pi8oK3Q2Fhi00OkajqSkJyfCb1+D7f6sqAoe2wMVvQnL9FqM8sUsLWqUm8unKXdUKzQfLdiEEXDg4TmMzBzfD8rdgxUzVej25FYy6VTUei9Vy/XFGwxYaHaPR1AbDgAn3qxbgn9wCL0+AS9+p1+q9DkMwsX9b3l68g2KPr9L6Z6Yp+SArhzHd0mnbpFG9jc92PEWq1fqyN1QFZWGoluuDf6ueHbWvnKAJPxGZygshLhJCrBFCmEKIoRX23SuEyBZCbBBCnBGw/UxrW7YQ4p6wDETHaDR1YcBFcM0cddN75TTY9E29nn58r1aUeE1+2VJ59lnWjsPsOlLE5MGRTcsOG3t+hdl/UoH9D2+AoztVccs/rIHL3oFeZ2uRiUIiZdGsBi4EXgzcKIToA0wB+gLtgG+FED2s3c8CpwE5wBIhxKdSyrV1GkWp0GiLRlNLOgxV7aDfngIzL1ZtBk68qV6uqRMzm9PI5eCHDbmM7xm8C+z3G/bjMAQTekd5l9iqKDoCq99X1suelapJYZ/zlPXSaUxsNdtroEREaKSU64Bgi80mAbOklCXAViFENjDc2pctpdxifW6W9d66CY2O0WjCQZMOcO1X8OFU+PIe1YV04qO2z6yTXA6Gdm7GL5srt2jmbzzA4I5NSatDEc6IICVsX6DEZe0n4C1S7cnPelRZklFQpUETOtEWo2kPBPZOzrG2AeyssL1CN68yhBBTgakAHTtW0XOlNEaj0dSRhGSVFDDvQfjxcZVSe9HrtreaGNGlBY9+tYFDBW6aJ5fPfjuQX8KqXUe587QelXw6Csnfr4L6y94ISEu+VKclxzi2CY0Q4lsgWDvA+6WUn1T2sSDbJMFjSbKyc0sppwHTAIYOHVrp+3SMRhNWDEP1KknvAZ/eCi+fquIG6d1tO+WJmUrIFm89xJn9yv/cftp0AICTera07fxhwTRhyzzIeg02zFHV0zuOgnF/hj6TdFpyHGCb0EgpT63Fx3KAjIDXHYDd1t+Vba89eh2Nxg4GToFmnWHW5fDyKcqy6TrellP1a98EpyFYmXPkOKGZvymX5skJ9GvXxJZz15lje1Ra8vI34MgOaNRcxbcGXwUtY8gK01RLtLnOPgVmCiH+i0oG6A4sRlk63YUQmcAuVMLAZXU/nbZoNDbRcURZksBbk2HiIzDs+rCfJsnloFfbVH7NOXLcvmXbDzOsc7PoKjljmrBlLix5RZXykT5VMfnUv0Ovc6KiHYMm/EREaIQQFwBPAy2B2UKIFVLKM6SUa4QQ76KC/F7gFimlz/rM74GvAAcwXUq5ps4D0etoNHbSrJNqmPXB9arG1uFtcOo/w54lNaBDUz5buRvTlKWicrjAzbaDhVw8LKOaT9cTJXmw4m1Y/CIczIbkljD6NrWoskXXSI9OYzORyjr7CPiokn0PAQ8F2T4HmBPmgahnbdFo7CIxVXVo/OJuWPC0CnZPejasGWn92zdh5qId7DpSREZzFc9YYVk4J2REuLnZwc2qY+Xyt8CdB+2HwoUvq9hLBEr3aCJDtLnO6hcdo9HUB4ZDpTuntoG5D0DBAbj4DUhMCcvhe7ROBWDD3rxSoVm7+xigYjgRYVcW/PhfWD8bDCf0uxCG3wgdhkRmPJqI0rCFRsdoNPWFEDDuT5DSGj67HV4/By57D1LqnhHWo7USrA378ji1jyp5vzk3nzb/397dR1lVnXcc//4CCGR4kVcFJIAgVgwgbwbjS1EjL0aipKK2KqCuxqYh1dUVWxOaFWxiXNXENraaRKmixiaKCQuiTRRQCyEqb46ARWQGyHJ4G15SZAbCy/D0j73Huc6aGe7MvXfuzD3PZ61Z99xzz9n37L3OPXv22fs8u0uH5n1+xgy2LYffPQxb34AOXUOex/21h+JPuGRXNN5H45rb6FtD/8SCWTD/izDz1xlfhDt3aEe/0zvywZ5DH68rLa9gcO+iDA82TWZQ+hq8/n3YsSZUplf9M4y5LUyH7RIv2f/Kex+Ny4dzJ4dpoQ9+GFo2h/ZknOQ5Z3Riy54KIMymWbq3kiG9snNrrkFla+DpqfCzL4f+py8+DHetD9NgeyXjomRfYT3WmcuXgZfAzS/CwR2xstmdWXI9iti+vxIzY/dHf6Li6AmG9M5hRbOvpOY5ofJNMOVB+PoaGHcHtOuQu+91rVKyKxrvo3H5NPDi2LLZAfOvCS2CJhrUs4jDx6rYe+gopeWVAAzORYvmWCUsvQ8eGw9b/wcmfAvuKobP3enPwLh6JfsK67HOXL4N+Dzc8ks4WAbPTYejFU1KZmDP0B+zbV8lJeWhrybrLZpNL8Gjnwud/cOvh79bBxP+MQzhdq4BXtGAt2hcfg24CKbPh90b4IUZUHW80UkM6hEqmu37KynZW0HnDm3p1TlLLYzDB2DBbfD8zaFSue03MO0n0KkVTz3gmlWyr7A+GMC1FOdOhqk/gtJlsGh2zbmZpr6nd6BdG7Ft32FKyysZ3KtTXdNwNN6WJfDYRbBpMVzxT3Dn8tAKc64RfHgz+GAA1zKMvhUO7YLX74fe58Eld6e9a9s2n6J/90+zfV9o0fz50AyfzzlZFY5jxQ+h9zC4+QXoMzKzNF1iJbui8cEArqW57B4o/19Ydh+cORyGXJn2rgN7FLFhx0H2HjqaWf/M4QMhPlvpshBJecqDPpLMZSTZV1h/YNO1NFKIhdbrPHjx9hA+P00DexSx4/+OABmMONtfCk9cDttXhFt5X3rEKxmXsYRXNN6icS3QaUVw08/C7auFfxNe0zCoZ80EYU1q0ewshicnhUjLs/4bxsxqfBrO1SHZV1jvo3EtVfezwxw2f1gJv38krV3GDKiZNrp/t46N+76yNeFZnrYd4fZXoP+4xu3vXAO8jwa8ReNappF/CR/8Fl67HwZfCX1GNLj5sL5dWPi3n2d/xTHatmnEOb17YwghU9QTZr0MXftleODOfVKyr7AfDyH1Fo1rgSS45t+gY7cQ8TmNW2ijPtPt4wjOaTmwFZ69DtoVwYxFXsm4nPCKBrxF41quT3eHSffDznWw9qnspn20IsQrqzoeKpluA7KbvnNRsq+wH/fR5PcwnGvQ8Okw8FJY9l048sfspGkGi2fD3vdh+lPQa2h20nWuDsmuaLyPxrUGEkz6PvzpIPzuX7OT5ts/hfcWwpXfgcFXZCdN5+qR7CusP0fjWos+I2DEjfDWT0K050zsK4Gl34FzJoV5Y5zLsYRXNN6ica3I5d8Cq4I3/6PpaZw8CYu+Bm07hAcyfWi/awbJvsL6czSuNek2AD57Pax9OoSJaYoNC+DDt8IAgy59snt8ztUj2RWN99G41ubiu+B4Jax6ovH7HquEpXOh7ygY+VdZPzTn6pPsK6z30bjW5oxhMOSqMNS56kTj9l31OBzaCZMegE8l+6fvmleyzzbvo3Gt0ZiZYTqB0mXp73P8CLz5GJx9eZhozblmlOwrrPfRuNZo6GQo6gXrnkl/n+L/gspyuPTvc3dcztUjLxWNpOmS3pN0UtLYlPVXSVoraUN8vSLlszFxfYmkR5SV6QO9ReNaoTbtYORNIQ5axd5Tb28Gbz0G/caEBz+da2b5usJuBL4MLK+1fh8w1cyGAzOBZ1M++zHwFeCc+Dc546PwWGeutRp+A5w8AR/85tTblq2G/SUw9g5vvbu8yEtFY2abzGxzHevfMbOd8e17QAdJ7SX1AbqY2ZtmZsAzwHVZOJDw6i0a19qcORy69ofNaVQ0xc+FoJnDrs39cTlXh5Z8hf0L4B0zOwr0A8pSPiuL69JT/j48Oh4+XAU/Ghnm3gCPdeZaLwnOnQKlr8Oxw/Vvd/wIbPwVDPsStM9gemfnMiD7+PZRlhOWlgJn1vHRHDNbFLd5A/iGma2pte/5wGJgopmVShoHPGBmX4ifXwr8g5lNree7v0K4zQbwWcKtOgc9Cbcnk87LoYaXRQ0vixrnmlnnbCWWs4nPqiuFxpJ0FrAQmGFmpXF1GXBWymZnATtr75vy3Y8Dj8f01pjZ2Pq2TRIvi8DLoYaXRQ0vixqS1px6q/S1qFtnkk4HXga+aWYrq9eb2S7gkKTxcbTZDGBRng7TOedcI+RrePM0SWXARcDLkl6JH80GhgDfllQc/3rHz74KzANKgFIgjV5Q55xz+ZazW2cNMbOFhNtjtdd/D/hePfusIfS3NNbjTdinUHlZBF4ONbwsanhZ1MhqWeRsMIBzzjkHLayPxjnnXOEp2IpG0mRJm2PImnvzfTzNQdL2GKanuHrUiKTukpZI2hJfu8X1iqF8SiStlzQ6v0efGUlPSiqXtDFlXaPzLmlm3H6LpJn5yEum6imLuZJ2pPR9Xp3y2TdjWWyWNCllfav/DUnqL+l1SZti2Ku74vpEnRsNlEPznBdmVnB/QBvCgIGzgdOAd4Fh+T6uZsj3dqBnrXUPAvfG5XuBf4nLVxMGVAgYD7yd7+PPMO+XAaOBjU3NO9Ad2Bpfu8XlbvnOW5bKYi7hmbXa2w6Lv4/2wKD4u2lTKL8hoA8wOi53Bj6IeU7UudFAOTTLeVGoLZoLgRIz22pmx4BfAEmNv3Et8HRcfpqa0D3XAs9Y8BZwegz10yqZ2XKg9rSTjc37JGCJmR0wsz8CS8hGTL1mVk9Z1Oda4BdmdtTMthFGdV5IgfyGzGyXma2Ly4eATYSoIok6Nxooh/pk9bwo1IqmH/BhyvvGhaxpvQx4VSHydXVkhDMsPIdEfK0eLp6EMmps3gu9TGbH20FPVt8qIkFlIWkgMAp4mwSfG7XKAZrhvCjUiqau6GVJGF53sZmNBqYAX5N0WQPbJrWMoP68F3KZ/BgYDFwA7AJ+GNcnoiwkdQJ+CdxtZh81tGkd6wqmPOooh2Y5Lwq1oikD+qe8bzBkTaGwGPnazMoJzyldCOypviUWX8vj5kkoo8bmvWDLxMz2mFmVmZ0EniCcG5CAspDUjnBxfc7MfhVXJ+7cqKscmuu8KNSKZjVwjqRBkk4DbiIE6SxYkookda5eBiYSgokuJsztQ3ytDt2zGJgRR9mMBw5W30ooII3N+yvAREnd4i2EiXFdq1er/20aNYFmFwM3KUzHMYgw19MqCuQ3JEnAfwKbzOzhlI8SdW7UVw7Ndl7kezREDkdZXE0YWVFKiBid92PKcX7PJowAeZcwl8+cuL4HsAzYEl+7x/UCHo3lswEYm+88ZJj/nxOa/scJ/3Xd0ZS8A7cTOj5LgNvyna8slsWzMa/r44WhT8r2c2JZbAampKxv9b8h4BLCrZ31QHH8uzpp50YD5dAs54VHBnDOOZdThXrrzDnnXAvhFY1zzrmc8orGOedcTnlF45xzLqe8onHOOZdTXtG4RJBUlRKhtjiG4SgIkmZJ2itpXnw/QdJLtbaZL+n6BtJ4SNJuSd/I9fG65MnLDJvO5cERM7ugvg8ltTWzE815QFn2vJnNburOZnaPpMpsHpBz1bxF4xIrtgQWSPo18Gpcd4+k1THI4H0p286Jc3AslfTz6v/8Jb0haWxc7ilpe1xuE1sJ1WndGddPiPu8KOl9Sc/Fp7aRNE7S7yW9K2mVpM6SVki6IOU4VkoakUGex6a06jZI8gfpXM55i8YlRUdJxXF5m5lNi8sXASPM7ICkiYRQGxcSnhBfHAOTVhJCbYwi/GbWAWtP8X13EMKXjJPUHlgp6dX42SjgfEKMqJXAxZJWAc8DN5rZakldgCPAPGAWcLekoUB7M1ufRn4vTckvwGeAl8xsDSGAIpIeAn6bRlrOZcQrGpcU9d06W2Jm1XO3TIx/78T3nQgVT2dgoZkdBpCUTsyvicCIlH6RrjGtY8AqMyuLaRUDA4GDwC4zWw1gMcKwpAXAtyXdQwiBMj/N/K4ws2uq30j6xH6SbiBMjjYxzfScazKvaFzSpfZLCHjAzH6auoGku6k/FPoJam5Bd6iV1tfN7BOBFyVNAI6mrKoi/A5V13eY2WFJSwiTS90AjD1Ffk5J0vnAfcBlZlaVaXrOnYr30ThX4xXgdoU5O5DUT1JvYDkwTVLHGCF7aso+24Excfn6Wml9NYZmR9LQGFW7Pu8DfSWNi9t3llT9j+A84BFgdUrrq0kkdSXMijjDzPZmkpZz6fIWjXORmb0q6Tzgzdg/XwHcYmbrJD1PiHj7B2BFym4/AF6QdCvwWsr6eYRbYutiZ/9eaqYLruu7j0m6Efh3SR0J/TNfACrMbK2kj4CnspDN64ABwBMxjzQ0Gs+5bPDozc41kqS5hArgB830fX2BN4A/szBBVe3PZxHC2Td5eHNMZy7NmC+XHH7rzLkWTNIMwtzuc+qqZKIjwJTqBzab+D0PAbfwyT4r57LCWzTOOedyyls0zjnncsorGueccznlFY1zzrmc8orGOedcTnlF45xzLqe8onHOOZdT/w8bm/MhAn277gAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3db5832d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## TEST CASE 2:\n",
    "fs = 44100 \n",
    "f1 = 23\n",
    "f2 = 36\n",
    "f3 = 230\n",
    "f4 = 900\n",
    "f5 = 2300\n",
    "N = 2**13   # With N=2**13 frequency resolution = fs/N = 5,38Hz, to have a better resolution increase nº of samples\n",
    "            # So to have a good visualization of f1 and f2 we should increase the frequency resolution \n",
    "\n",
    "x = np.sin(2*np.pi*f1*np.arange(N)/fs) + np.sin(2*np.pi*f2*np.arange(N)/fs) +\\\n",
    "    np.sin(2*np.pi*f3*np.arange(N)/fs) + np.sin(2*np.pi*f4*np.arange(N)/fs) +\\\n",
    "    np.sin(2*np.pi*f5*np.arange(N)/fs)\n",
    "\n",
    "x_filt = suppress_freq_dft_model(x,fs,N)\n",
    "\n",
    "w = get_window('hamming', N)\n",
    "\n",
    "mX, pX = dftAnal(x,w,N)\n",
    "mX_filt, pX_filt = dftAnal(x_filt,w,N)\n",
    "\n",
    "# Plotting original magnitude vs filter magnitude\n",
    "f = fs*np.arange(N/2+1)/(N)\n",
    "plt.plot(f,mX)\n",
    "plt.plot(f,mX_filt)\n",
    "plt.ylim(-120,3)\n",
    "plt.xlim(0,2500)\n",
    "plt.ylabel('Amplitude [dB]')\n",
    "plt.xlabel('Frequency [Hz]')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 5 - Window-size, FFT-size and zero-padding\n",
    "\n",
    "Complete the function `zp_fft_size_expt()`to take an input signal, compute three different magnitude spectra (with different parameters) and return them. \n",
    "\n",
    "This function should provide some insights into the effects window-size, FFT-size, and zero-padding on the spectrum of a signal. \n",
    "\n",
    "Te input signal should be of size 512 samples, the sampling rate should be 1000Hz, and the analysis window used should be hamming. The three set of analysis parameters should be:\n",
    "\n",
    "1. window-size = 256, FFT-size = 256 (no zero-padding)\n",
    "2. window-size = 512, FFT-size = 512 (no zero-padding)\n",
    "2. window-size = 256, FFT-size = 512 (zero-padding of 256 samples)\n",
    "\n",
    "Use `dftAnal()` to obtain the positive half of the magnitude spectrum (in dB). Return the 3 magnitude spectra in dB. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [],
   "source": [
    "def zp_fft_size_expt(x):\n",
    "    \"\"\"\n",
    "    Inputs:\n",
    "        x (numpy array) = input signal (512 samples long)\n",
    "    Output:\n",
    "        magnitude spectrum with window-size = 256, FFT-size = 256 (numpy array) \n",
    "        magnitude spectrum with window-size = 512, FFT-size = 512 (numpy array) \n",
    "        magnitude spectrum with window-size = 256, FFT-size = 512 (numpy array) \n",
    "    \"\"\"\n",
    "    \n",
    "    N = len(x)    # should be 512 \n",
    "    w1 = get_window('hamming', N//2)\n",
    "    w2 = get_window('hamming', N)\n",
    "    \n",
    "    (mX1, pX1) = dftAnal(x[:N//2],w1,N//2)\n",
    "    \n",
    "    (mX2, pX2) = dftAnal(x,w2,N)\n",
    "    \n",
    "    (mX3, pX3) = dftAnal(x[:N//2],w1,N)\n",
    "   \n",
    "    \n",
    "    return mX1, mX2, mX3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run and test the function `zp_fft_size_expt()`. Plot the outputs to understand and validate the results. Here is one possible example.\n",
    "\n",
    "_Test case 1:_ Use as input `x = .2*np.cos(2*np.pi*300*n)+.2*np.cos(2*np.pi*600*n)` where `n=np.arange(512)/fs` and the sampling rate `fs=1000`. Call the function with `mX1, mX2, mX3 = zp_fft_size_expt(x)`\n",
    "\n",
    "To understand better, plot the output of `dftAnal()` for each case on a common frequency axis with different colors. You will see that `mX3` is the interpolated version of `mX1` (zero-padding leads to interpolation of the DFT). You will also observe that the 'mainlobe' of the magnitude spectrum in `mX2` will be narrower than that in `mX1` and \n",
    "`mX3`. This shows that having a longer window leads to a narrower mainlobe with better frequency resolution and less spreading of the energy of the sinusoid. \n",
    "\n",
    "If we were to estimate the frequency of the sinusoid using its DFT, a first principles approach is to choose the frequency value of the bin corresponding to the maximum in the DFT magnitude spectrum. If you were to take this approach, which of the magnitude spectra will give you a better estimate of the frequency of the sinusoid? Comment and discuss."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3db5345630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#TEST CASE 1\n",
    "fs = 1000\n",
    "N = 512\n",
    "n = np.arange(N)/fs\n",
    "x = .2*np.cos(2*np.pi*300*n)+.2*np.cos(2*np.pi*600*n)\n",
    "\n",
    "(mX1, mX2, mX3) = zp_fft_size_expt(x)\n",
    "\n",
    "#Plotting \n",
    "f = fs*np.arange(N/2+1)/N\n",
    "plt.plot(f[::2],mX1, color = 'blue')\n",
    "plt.plot(f,mX2, color = 'red')\n",
    "plt.plot(f,mX3, color = 'green')\n",
    "plt.xlabel('Frequency [Hz]')\n",
    "plt.ylabel('Amplitude [dB]')\n",
    "plt.legend(('w-size = 256, FFT-size = 256','w-size = 512, FFT-size = 512','w-size = 256, FFT-size = 512'))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Your explanation for Part 5 here\n",
    "The higher samples we take the higher resolution we will have in the magnitude spectra. So that the frequency intervals we can measure are smaller and therefore we can obtain a better estimation of the frequencies of the track. In this case mX2 and mX3 have higher FFT-size, however the samples taken first in mX3 are 256 so we have to zero-pad to reach the 512 samples of the FFT-size. Zero-padding does not allow us to increase our resolution in frequency, only improves the plotting by increasing the number of points. Therefore the better magnitude spectra will be mX2 with a window-size of 512 samples and a FFT-size of 512 samples.\n"
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