{ "cells": [ { "cell_type": "markdown", "metadata": { "nbsphinx": "hidden" }, "source": [ "# Random Signals\n", "\n", "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## White Noise" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Definition\n", "\n", "[White noise](https://en.wikipedia.org/wiki/White_noise) is a wide-sense stationary (WSS) random signal with constant power spectral density (PSD)\n", "\n", "\\begin{equation}\n", "\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega}) = N_0\n", "\\end{equation}\n", "\n", "where $N_0$ denotes the power per frequency. White noise draws its name from the analogy to white light. It refers typically to an idealized model of a random signal, e.g. emerging from measurement noise. The auto-correlation function (ACF) of white noise can be derived by inverse discrete-time Fourier transformation (DTFT) of the PSD\n", "\n", "\\begin{equation}\n", "\\varphi_{xx}[\\kappa] = \\mathcal{F}_*^{-1} \\{ N_0 \\} = N_0 \\cdot \\delta[\\kappa]\n", "\\end{equation}\n", "\n", "This result implies that white noise has to be a zero-mean random process. It can be concluded from the ACF that two neighboring samples $k$ and $k+1$ are uncorrelated. Hence they show no dependencies in the statistical sense. Although this is often assumed, the probability density function (PDF) of white noise is not necessarily given by the normal distribution. In general, it is required to additionally state the amplitude distribution when denoting a signal as white noise." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example - Amplifier Noise\n", "\n", "Additive white Gaussian noise (AWGN) is often used as a model for amplifier noise. In order to evaluate if this holds for a typical audio amplifier, the noise captured from a microphone preamplifier at full amplification with open connectors is analyzed statistically. For the remainder, a function is defined to estimate and plot the PDF and ACF of a given random signal." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import scipy.stats as stats\n", "%matplotlib inline\n", "\n", "\n", "def estimate_plot_pdf_acf(x, nbins=50, acf_range=30):\n", " \n", " # compute and truncate ACF\n", " acf = 1/len(x) * np.correlate(x, x, mode='full')\n", " acf = acf[len(x)-acf_range-1:len(x)+acf_range-1]\n", " kappa = np.arange(-acf_range, acf_range)\n", " \n", " # plot PSD\n", " plt.figure(figsize = (10, 6))\n", " plt.subplot(121)\n", " plt.hist(x, nbins, density=True)\n", " plt.title('Estimated PDF')\n", " plt.xlabel(r'$\\theta$')\n", " plt.ylabel(r'$\\hat{p}_x(\\theta)$')\n", " plt.grid()\n", "\n", " # plot ACF\n", " plt.subplot(122)\n", " plt.stem(kappa, acf)\n", " plt.title('Estimated ACF')\n", " plt.ylabel(r'$\\hat{\\varphi}_{xx}[\\kappa]$')\n", " plt.xlabel(r'$\\kappa$')\n", " plt.axis([-acf_range, acf_range, 1.1*min(acf), 1.1*max(acf)]);\n", " plt.grid()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now the pre-captured noise is loaded and analyzed" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "noise = np.load('../data/amplifier_noise.npz')['noise']\n", "estimate_plot_pdf_acf(noise, nbins=100, acf_range=150)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Inspecting the PDF reveals that it fits quite well to a [normal distribution](important_distributions.ipynb#Normal-Distribution). The ACF consists of a pronounced peak. from which can be concluded that the samples are approximately uncorrelated. Hence, the amplifier noise can be modeled reasonably well as additive white Gaussian noise. In oder to estimate the parameters of the normal distribution, the captured samples are fitted to a normal distribution" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean: -1.537e-05 \n", "Variance: 1.140e-07\n" ] } ], "source": [ "mean, sigma = stats.norm.fit(noise)\n", "print('Mean: {0:1.3e} \\nVariance: {1:1.3e}'.format(mean, sigma**2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Excercise**\n", "\n", "* What relative level does the amplifier noise have when the maximum amplitude of the amplifier is assumed to be $\\pm 1$?\n", "\n", "Solution: The average power of a mean-free random signal is given by is variance, here $\\sigma_\\text{noise}^2$. Due to the very low mean in comparison to the maximum amplitude, the noise can be assumed to be mean-free. Hence, the relative level of the noise is then given as $10*\\log_{10}\\left( \\frac{\\sigma_\\text{noise}^2}{1} \\right)$. Numerical evaluation yields" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Level of amplifier noise: -69.43 dB\n" ] } ], "source": [ "print('Level of amplifier noise: {:2.2f} dB'.format(10*np.log10(sigma**2/1)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example - Generation of White Noise with Different Amplitude Distributions\n", "\n", "Toolboxes for numerical mathematics like `Numpy` or `scipy.stats` provide functions to draw uncorrelated random samples with a given amplitude distribution." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Uniformly distributed white noise**\n", "\n", "For samples drawn from a zero-mean random process with uniform amplitude distribution, the PDF and ACF are estimated as" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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D02z7HgGAiBjZI3BbTZulFF8wAS4FPhkRAbwauCUzfwSQmQ92qmhJO7duhLYNwEE11w8sp9U6GTgGIDOvKXct7A/cX9soM1cAKwAWL16cS5YsabqIk5Zf0WrdTTt10SAfWdOreXhs1t1Z1j3aXScsmZblTkKjPQJHjtUmMwcjYiPwFOBZQEbEamAucHFmfnj6S5a0s+vGFuMGYEFEHEIR1o4D3lbX5pfAK4GLIuJ3gCcBv+5olZI0OTOBlwAvBB4HvhMRN2Xmd+ob1u4tmDt3bi/1NI5p06ZN1tlGAwObGRoaqkStVVmnValzMjoe2spvpKcAq4E+4MLMvDUizgZuzMyVwKnAP0fEX1EclHBSZmana5W0y2pmj8BIm/XlOLZ9gAcpeuWuzswHACJiFfB8YFRoq91bsHDhwpb2FnRLf38/1tk+n157DQMDA5WotSrrtCp1TkZX9s1k5iqKAwxqp51Zc/k24MWdrkuSSs3sEVgJnAhcA7wZuCozR3aLvi8i9gC2Ai+jOFBBkqakegNqJGmaNblH4AKKg6TWAQ9RBDsy8+GI+ChF8EtgVWZO3yBaSbsMQ5skNdDEHoEngLeMcdt/pTjthyS1jb89KkmSVAGGNkmSpAowtEmSJFWAoU2SJKkCDG2SJEkVYGiTJEmqAEObJElSBRjaJEmSKsDQJkmSVAGGNkmSpAowtEmSJFWAoU2SJKkCDG2SJEkVYGiTJEmqAEObJElSBRjaJEmSKsDQJkmSVAGGNkmSpAowtEmSJFWAoU2SJKkCDG2SJEkVYGiTJEmqAEObJElSBRjaJEmSKsDQJkmSVAGGNkmSpAowtEmSJFWAoU2SJKkCDG2SJEkVYGiTJEmqAEObJElSBRjaJEmSKsDQJkmSVAGGNkmSpAqY2e0CJKldImK/JpoNZ+bAtBcjSW1maJO0M7mn/Itx2vQBT+9MOZLUPoY2STuT2zPz8PEaRMTNnSpGktppUmPaImLPiOhrdzGSNEW/36Y2ktRzmgptETEjIt4WEVdExP3AT4B7I+K2iDgvIg6d3jIlaWKZ+QRARCytnxcRM2rbSFLVNNvT9l3gt4HTgd/KzIMy86nAS4BrgXMj4u3TVKMktWpZRBwJEBF9EfEOii+bklRZzY5pOyozt9VPzMyHgMuAyyJiVlsrk6TJexvwjYi4AngXsAb4k+6WJElT02xomx8R7wYOBR4Cfgh8MzN/MdKgUaiTpC55CXAG8EXgpMzs7245kjR1ze4e/QawFjgfeBXwe8DVEXF+ROw+XcVJ0iS9CfhHYH/gnyLikxHxzi7XJElT0mxPW19mXgAQEQ9l5p9HxEzgr4AVwInTVaAktSoz3wEQEQEsABaVf5JUWc32tF0ZEaeUlxMgMwcz8zwmcfh8RBwTEWsjYl1ELB+jzR+XR6feGhFfavU+JCkLP83MyzLzrG7XI0lT0WxP23uB0yPiRuCAiFgGPE4R2B5s5Q7L87uN7GZdD9wQESsz87aaNgsojlR9cWY+HBFPbeU+JO2aIuIHmfn8qbaRpF7UVGjLzGHgQxHxMeAo4HnAvsCPKQb7tuIIYF1m3gEQERcDS4Hbatr8OXB+Zj5c3v/9Ld6HpF3T70TELePMD2CfThUjSe3UVGiLiCh3MzwOrCz/GrZpYnHzgbtrrq8Hjqxr86xymd+n+J3AszLz3xrc5zJgGcC8efPo7+9v4u4Lpy4abLptq+bNnt7lTxfr7izrHq2V9/AYnt1Em6Gp3okkdUOzu0e/GxGXAd/IzF+OTIyI3SgOrT+R4gS8F7WxrgXAEuBAiiNVF2XmQG2jzFxBcSAEixcvziVLljR9Byctv6JNpY526qJBPrKmej/rat2dZd2j3XXCkindvvY0RLUioi8zDWuSKq3ZAxGOofh2+uWIuKc8QOBO4GfA8cA/ZuZFTS5rA3BQzfUDy2m11gMrM3NbZt4J/JQixEnSZKyIiD0AIuKl3S5Gkiaj2TFtTwCfAj5V/vLB/sBmYFNmtrqf5AZgQUQcQhHWjqM4e3mtyynC4OciYn+K3aV3tHg/kjTiTOCCiBikODn41V2uR5Ja1mxP23Zl79e95a7Kz0TEbGj+22sZ8k4BVgO3A5dk5q0RcXZEHFs2Ww08GBG3Uex2PS0zWzpKVZJqfIDiBOEJXNLlWiRpUqY6MOX9wIWtfnvNzFXAqrppZ9ZcTorTjLx3ivVJEsD7MvOBiNgT+DjwZ90uSJJa1XJPWx2/vUqqgnMiYnZmPgZ8udvFSNJkTLWnzW+vkqqgfq/Ad7pcjyS1rKWetoh4TURcV/4E1SXAoQDlt9e/mI4CJakN3CsgqfLGDW0R8ZyI+GLNpE9RjDN7EcX50f4hIo4H8BxIknpJRLwwIl4cEftQ7BU4C3gXRa+bJFXORLtHr2THH4S/PzO/PzIvIq4BrsMxIpJ6SERcTnEOyLuBw4DvR8QpmflYRLhXQFIlTbR79NXAh2qu3xkRHyx/CQFgG1C93+GRtFNpsFfguRSnFnprZi4Evgd8GtwrIKm6xg1tmbkmM0+omTQM/CFwd0R8D1gH9EeEv1YgqZuuBM6ouX4O8D7g9ohYAxwFHBURr46Ip3ajQEmaqpYORMjMt2Xmc4CnA38JnAUE8M8R8cvxbitJ06h+r8AbKE7c/dsUv4/8GaAP+GPgW50vT5KmblKn/MjMLcBN5Z8kdVVmrgFq9wr8GcXP4J0O/BhYCPxbZjZ9WqKIOIbiVEZ9wGcz85y6+bsDnwdeADxIsSv2rpr5TwduA87KzH+YzOOSpFpTPU+bJPWczPw18PqImE8xvm1jZl7b7O0jog84H3gVsB64ISJWZuZtNc1OBh7OzEMj4jjgXOCtNfM/ir16ktrI0CZpp5WZG4ANk7jpEcC6zLwDICIuBpZS9JyNWEoxRATgUuCTERGZmRHxRuBO4LHJ1i5J9ab6M1aStDOaT3G6kBHry2kN22TmILAReEpE7AX8NfC3HahT0i7EnjZJaq+zgI9l5qaIGLdhRCwDlgHMnTuX/v7+aS9uqjZt2mSdbTQwsJmhoaFK1FqVdVqVOifD0CZJo20ADqq5fiCjd7OOtFkfETOBfSgOSDgSeHNEfBiYAwxHxBOZ+cn6O8nMFRS/LsPChQtzyZIl7X4cbdff3491ts+n117DwMBAJWqtyjqtSp2TYWiTpNFuABZExCEU4ew44G11bVYCJwLXAG8GrsrMBP5gpEFEnAVsahTYJKlVhjZJqpOZgxFxCrCa4pQfF2bmrRFxNnBjZq4ELgC+EBHrgIcogp0kTRtDmyQ1kJmrgFV1086sufwE8JYJlnHWtBQnaZfk0aOSJEkVYGiTJEmqAEObJElSBRjaJEmSKsDQJkmSVAGGNkmSpAowtEmSJFWAoU2SJKkCDG2SJEkVYGiTJEmqAEObJElSBRjaJEmSKsDQJkmSVAGGNkmSpAowtEmSJFWAoU2SJKkCDG2SJEkVYGiTJEmqAEObJElSBRjaJEmSKsDQJkmSVAGGNkmSpAowtEmSJFWAoU2SJKkCDG2SJEkVYGiTJEmqAEObJElSBRjaJEmSKqAroS0ijomItRGxLiKWj9PuTRGREbG4k/VJkiT1mo6HtojoA84HXgMcBhwfEYc1aPdk4C+B6zpboSRJUu/pRk/bEcC6zLwjM7cCFwNLG7T7AHAu8EQni5MkSepF3Qht84G7a66vL6dtFxHPBw7KzCs6WZgkSVKvmtntAupFxAzgo8BJTbRdBiwDmDdvHv39/U3fz6mLBidXYBPmzZ7e5U8X6+4s6x6tlfewJO1quhHaNgAH1Vw/sJw24snAc4H+iAD4LWBlRBybmTfWLigzVwArABYvXpxLlixpuoiTlk9fJ96piwb5yJqey8MTsu7Osu7R7jphybQsV5J2Bt3YPXoDsCAiDomI3YDjgJUjMzNzY2bun5kHZ+bBwLXAqMAmSZK0K+l4aMvMQeAUYDVwO3BJZt4aEWdHxLGdrkeSJKkKurJvJjNXAavqpp05RtslnahJkiSpl/mLCJIkSRVgaJMkSaoAQ5skSVIFGNokSZIqwNAmSZJUAYY2SZKkCjC0SZIkVYChTZIkqQIMbZIkSRVgaJMkSaoAQ5skSVIFGNokSZIqwNAmSZJUAYY2SZKkCjC0SZIkVYChTZIkqQIMbZIkSRVgaJMkSaoAQ5skSVIFGNokSZIqwNAmSZJUAYY2SZKkCjC0SZIkVYChTZIkqQIMbZIkSRVgaJOkBiLimIhYGxHrImJ5g/m7R8RXyvnXRcTB5fRXRcRNEbGm/P8Vna5d0s7J0CZJdSKiDzgfeA1wGHB8RBxW1+xk4OHMPBT4GHBuOf0B4A2ZuQg4EfhCZ6qWtLMztEnSaEcA6zLzjszcClwMLK1rsxT4l/LypcArIyIy8+bMvKecfiswOyJ270jVknZqM7tdgCT1oPnA3TXX1wNHjtUmMwcjYiPwFIqethFvAn6QmVsa3UlELAOWAcydO5f+/v62FD+dNm3aZJ1tNDCwmaGhoUrUWpV1WpU6J8PQJknTICKeQ7HL9NVjtcnMFcAKgIULF+aSJUs6U9wU9Pf3Y53t8+m11zAwMFCJWquyTqtS52S4e1SSRtsAHFRz/cByWsM2ETET2Ad4sLx+IPB14E8y8+fTXq2kXYKhTZJGuwFYEBGHRMRuwHHAyro2KykONAB4M3BVZmZEzAGuAJZn5vc7VrGknZ6hTZLqZOYgcAqwGrgduCQzb42IsyPi2LLZBcBTImId8F5g5LQgpwCHAmdGxA/Lv6d2+CFI2gk5pk2SGsjMVcCqumln1lx+AnhLg9t9EPjgtBcoaZdjT5skSVIFGNokSZIqwNAmSZJUAYY2SZKkCjC0SZIkVYChTZIkqQIMbZIkSRVgaJMkSaoAQ5skSVIFGNokSZIqwNAmSZJUAYY2SZKkCuhKaIuIYyJibUSsi4jlDea/NyJui4hbIuI7EfGMbtQpSZLUKzoe2iKiDzgfeA1wGHB8RBxW1+xmYHFm/i5wKfDhzlYpSZLUW7rR03YEsC4z78jMrcDFwNLaBpn53cx8vLx6LXBgh2uUJEnqKTO7cJ/zgbtrrq8Hjhyn/cnAtxrNiIhlwDKAefPm0d/f33QRpy4abLptq+bNnt7lTxfr7izrHq2V97Ak7Wq6EdqaFhFvBxYDL2s0PzNXACsAFi9enEuWLGl62Sctv6INFTZ26qJBPrKmp1dtQ9bdWdY92l0nLJmW5UrSzqAbW4wNwEE11w8sp+0gIo4CzgBelplbOlSbJElST+rGmLYbgAURcUhE7AYcB6ysbRARhwP/BBybmfd3oUZJkqSe0vHQlpmDwCnAauB24JLMvDUizo6IY8tm5wF7AV+NiB9GxMoxFidJkrRL6MqAmsxcBayqm3ZmzeWjOl6UJElSD/MXESRJkirA0CZJklQBhjZJkqQKMLRJkiRVgKFNkiSpAgxtkiRJFWBokyRJqgBDmyRJUgUY2iRJkirA0CZJklQBhjZJkqQKMLRJkiRVgKFNkiSpAgxtkiRJFWBokyRJqgBDmyRJUgUY2iRJkirA0CZJklQBhjZJkqQKMLRJkiRVgKFNkiSpAgxtkiRJFWBokyRJqgBDmyRJUgUY2iRJkirA0CZJklQBhjZJkqQKMLRJkiRVgKFNkiSpAgxtkiRJFWBokyRJqgBDmyRJUgUY2iRJkipgZrcLkCR13+U3b+C81Wu5Z2AzB8yZzWlHL+SNh8/vdllSW1X9dW5ok6Rd3OU3b+D0r61h87YhADYMbOb0r60BqNQGTRrPzvA6N7RJ0i7uvNVrt2/IRmzeNsR5q9dO68ZsOno9Rpa5YWAz86+9atp6Unqhx6ZTNUzHOu3G+uvW67ydDG2StIu7Z2BzS9PbYbK9HuNt7DvVk9ILPTa99FhbDWDdWn/deJ23mwciSFKFXH7zBl58zlUcsvwKXnzOVVx+84YpL/OAObNbmt4O4/V6jGVkY79hYDPJbzb2I+tgMsvsVO3t1iuPdaLnpJu115uu1/l0vCfHYk+bJPWw2l6MfWbP4rGtg2wbSqB9PRSnHb1wh54PgNmz+jjt6IVTK34ck+n1mGj3Vqd6Unqhx6ZXHutkdjl2a/1Nx+u8Ua/haV/9EX/7zVsZeHxb23f9ajjyAAAQfElEQVT9GtokqQfc9cgwLz7nKl7+7Ll89ye/bhjSBjZvG3W7dozJGbnt+y69ha1Dw8yfpjFGtQF0RgRDmaPazIjgkOVXbN/YAdtvM7p1YcPAZg5ZfsWYy2y1J2Wi3X0HzJnNhgYBYzp7JhvdVzM1THXs2Fj3M/I8TfScNHoem33u2/36m47XeaPQum04efjx4r3a7hBnaJOkBiLiGODjQB/w2cw8p27+7sDngRcADwJvzcy7ynmnAycDQ8B7MnN1M/e5YWAz/3rtL7dfbxTSGmlHD8UbD5/Pl68v7vsrf/H723f5tDJOqT4cAGP2EjbaaNdOH9nYEWy/zXhyjGW22pPSzHirbvRM1mumhnaMHWt0PzD281drZHdp/fPY7HNfH3SAlsfONWpf+zqfqmbee/UhbuQ5mAxDmyTViYg+4HzgVcB64IaIWJmZt9U0Oxl4ODMPjYjjgHOBt0bEYcBxwHOAA4ArI+JZmbnjVq+N2t3DM97Gfk5Nm/F229ZvqCcKoH0Nel+2DU8cDMYzmZ6UZnb3Neqxefmz53Le6rX81Vd+2DBQNNPjNVGb+vlvesF8Lrlh/Zi9Ru04WrL+sTZ6niYy0fM41nNf31tV+3qaqAdrvNdwO43VEzmeqYzfM7RJ0mhHAOsy8w6AiLgYWArUhralwFnl5UuBT0ZElNMvzswtwJ0Rsa5c3jXTUejsWX28/NlzW+oVm8h4G/sPvWjGqA1io0DWSuA68pD9uP7OhyZdbyML953B6r9+Rcs9hs2Ot6rtsTn+iKePO65pvLGIIyYaG9VoGZfdtIGD9pvN/nvt3rDXaLJjxxqFx8OfPoeBgQF++vDwuLdtVbPPfaPX01g9WHMY/zV84L6T/5JTv25e/uy5XHbThlH3NZHJ9o4b2iRptPnA3TXX1wNHjtUmMwcjYiPwlHL6tXW3nTBBHfjo/fzdf35qwsKKXAiZye4z+5izxyx+3b+FRTUbta3/L/gqMFy2OWi/2WzaMgjAwU/Zkwc2beHOBx5jaPg38/ffa3dOuvcRAB4Zp1fsvm8WNZzVYm/LeI/nyTfO5Phtw2wZbG7DFxE8+Ukz2TLGbSKC2TPhpmsvYtsDj/GemnWz7Yrgpv333GF9ANz14GMAfOyxbWPW8ZUXf6rh+trypWHOarL2HVzdx0mzZjA4OMjQqhmTWsbIuvjF9/YG2OG5PTeCbPA89c0I/uO1F4567COvjUbr7ISZM4gcZihnjLnOx3tOxqy9xed+IluugvuA94zTZu/ZswD4xff2bvheGOu9MrNvBkPDyXtq1umMK4Kj9tqd+x/dQuZv2jRa73fsM59/+t2lQNFDd9ckHp+hTZK6JCKWAcsAnrnn3o3blP9kwqwZwf57BE8MJhDM2wN+PrCF4bpeiMzcPkB8y+AQP//1JmbNCGbOgF8MbeVXjw0zsk0Zmf/Y44/TV06cNSPG7SlrtEGajAjYYyb05SD7PSn41WNQu+jax970bcr5s2Ykv3jgsVHrZni4mD6rr7g+0LeNjVuT+8p10hfFMsZ6iI3W15bBya2PLYND7DFzmBkzkoEtkwstmUlfDjIwMMDGrbnDc9voeQqKsWO/2vgEDz66hf33CB7ZUu7C7tvGLwaGG66zLduG2Hs3mD0rx1zn7Xwep8usGUFfFqHsF/c/3PC9MNZ7ZXBodC/j8HDy0KYtzNkdivdksHEr3PdYMly+nobZ8bHtNgNe9/Qh/msS9XcltE1lgK8kdcAG4KCa6weW0xq1WR8RM4F9KD6vmrktAJm5AlgBsPvTFuQn3nDqDkePNrM7743jHMHXyFhjkubPmc33l78CKHYBndVg8HkrZs2IUQcRzJoR7PWkmTuMQXr9BOO+YPTg84lu8/rD59Pf3887/+2xMdfNbn0z2Do0zJy63Y71dY51pGPt+nrxOVe1PK6pdhn9/f184trhKS1jvDr6IhjOHLWLFYrHmsDgcDJ/nPFZAXzumD156ZIlY67zEVN9Hseqs9mDUsYye1Yff/9Hi3h1eb8TPW/Njt8L4M5zXjfm/NrHVjv28IwTWn4InQ9tUxng2+laJe2ybgAWRMQhFIHrOOBtdW1WAidSjFV7M3BVZmZErAS+FBEfpTgQYQFw/UR3ePDeM7ZvfFvR6kDosTZCtWNsRkLiRKfaqNUokNUuo6kAevj8hvMncxsYe90EsLXsNRlrPN4eu83k5jNfzSHLr2i47Nr1NdYRluOpP9KzHcsYa5zUcCZ3nvM6XnzOVaMeb22P6oaBzQQ0fL5rD3YZb52PN7+V27R6NPJ4oqy//vU30biyZg+4mOhAoInWVyu60dM26QG+2a4+eUkaRzlG7RRgNcUegQsz89aIOBu4MTNXAhcAXygPNHiIIthRtruE4jNtEPjv03nk6GQ29o3Ub3hqNzQT9eCMF8i6+ZuOjdbNWKGk3sgGvZnzodWH3LF6iupDbe26accyJqq1mcHvxY73HdfR9nC48WcT3r5dmgl+zZz3r7Ynst5kjvys1+lTvXQjtE1lgO8DHalQ0i4vM1cBq+qmnVlz+QngLWPc9kPAh6a1wFIzG/uJTLThGeucYH//R4t6+oe269dNKxvpkaDT7DnZmukpmmhdTXUZE9Xa7ONPirBTf7/9/Z0Lbc2oXV/1R9/C5F7XE5koOE+36HTnVUS8GTgmM/+svP7fgCMz85SaNj8u26wvr/+8bPNA3bK2D+IFFgKd++G38e1PNQOmdXeWdU/dMzJzbreLaIeIeJQ2fYbNmL33fn177Tc/+mbutr3rpF45PYcGtw5temjD8OZHxj3vwvZlzpi5Ww43d5sua/g6nTX34EXRN3O3cW+ZOTz4yK9/MfL4atdns+urHbVOxni1zpi9934z9577DCLG/d3xHBrcuu3XdzU6qVkvvfdHmcxrdDreKy1YmJlPbuUG3ehpm8oA3x3UDuLtJRFxY2Yu7nYdrbLuzrJu1VlbhfValee/KnVCdWq1zvaKiBtbvc24aXuabB/gGxG7UYwDWVnXZmSAL9QM8O1gjZIkST2l4z1tUxngK0mStKvqynnapjLAtyJ6bpdtk6y7s6xbtaqyXq2z/apSq3W2V8t1dvxABEmSJLWuG2PaJEmS1CJDWxtExH4R8e8R8bPy/33Habt3RKyPiE92ssYxapmw7oh4XkRcExG3RsQtEdG1X6aIiGMiYm1ErIuI5Q3m7x4RXynnXxcRB3e+ytGaqPu9EXFbuX6/ExHP6Ead9Saqu6bdmyIiI6Lnj9bqNRHxgfJ5/2FEfDsiDiinR0R8olz3t0TE83ug1vMi4idlPV+PiDk1804va10bEUd3uc63lJ9Xw/WvyV6qs6ynqfdYN0TEhRFxf3kKrpFpTW/rOlTjQRHx3fLz89aI+MterLOs6UkRcX1E/Kis9W/L6YeU26t15fZrotPRpH9T/AM+DCwvLy8Hzh2n7ceBLwGfrELdwLOABeXlA4B7gTldqLUP+DnwTGA34EfAYXVt3g18prx8HPCVHljHzdT9cmCP8vK7qlJ32e7JwNXAtcDibtddtT9g75rL76l5/b4W+BbFWaNeBFzXA7W+GphZXj535PMCOKx8fewOHFK+bvq6WOfvUJy3s7/2NdmDdTb1HutifS8Fng/8uGZa09u6DtX4NOD55eUnAz8tn+eeqrOsI4C9ysuzgOvK9/YlwHHl9M8A7xpvOfa0tcdS4F/Ky/8CvLFRo4h4ATAP+HaH6prIhHVn5k8z82fl5XuA+4FunMx0+8+fZeZWYOTnz2rVPp5LgVdGRKNTJXbShHVn5ncz8/Hy6rUU5y7stmbWN8AHKDbgT3SyuJ1FZj5Sc3VPfvPrQUuBz2fhWmBORDyt4wXWyMxvZ+ZgebX2dboUuDgzt2TmncA6itdPV2Tm7ZnZ6CTFPVUnzb/HuiIzr6Y4e0OtprZ1nZKZ92bmD8rLjwK3U/yiUk/VCVC+lzeVV2eVfwm8gmJ7BU3Uamhrj3mZeW95+VcUwWwHUZyB+iPA/+pkYROYsO5aEXEExTfCn093YQ00+vmz+t8O2eHnz4CRnz/rpmbqrnUyRQ9Lt01Yd7nL7qDMbPyL2mpKRHwoIu4GTgBGjqJv9XXTae/gN6/TXq91RK/V2Wv1NKOlbUYnlcNhDqfowerJOiOiLyJ+SNH58e8U29KBmi9DE74GunLKjyqKiCuB32ow64zaK5mZEdHokNx3A6syc30nO3/aUPfIcp4GfAE4MTOH21ulACLi7cBi4GXdrmUi5ZeQjwIndbmUnjfeezAzv5GZZwBnRMTpwCnA+ztaYI2Jai3bnAEMAl/sZG21mqlT02uibUYnRcRewGXA/8zMR2q3sb1UZ2YOAc8rx4N+HXh2q8swtDUpM48aa15E3BcRT8vMe8twc3+DZr8P/EFEvBvYC9gtIjZl5rQOPm1D3UTE3sAVFB+I105TqRNp28+fdVgzdRMRR1EE6Zdl5pYO1Taeiep+MvBcoL/8gPwtYGVEHJuZLf80y85svPdgnS9SnL/y/TT5umm3iWqNiJOA1wOvzHIQDl2otYV1Wqsr63QcvVZPM5raZnRSRMyiCGxfzMyvlZN7rs5amTkQEd+lyAVzImJm2ds24WvA3aPtUfuzWycCo77pZeYJmfn0zDyYYhfp56c7sDVhwrrLI1m+TlHvpfXzO6iqP382Yd0RcTjwT8CxmdkrHy7j1p2ZGzNz/8w8uHxNX0tRv4GtBRGxoObqUuAn5eWVwJ+UR5G+CNhYs7unKyLiGOB9FM/z4zWzVgLHRXH09iHAAuD6btQ4gV6rs5nPtF4z4Tajk8oxyxcAt2fmR2tm9VSdABExd+SI64iYDbyKYgzedym2V9BMrdN5tMSu8kcxbuo7wM+AK4H9yumLgc82aH8SvXH06IR1A28HtgE/rPl7XpfqfS3F0UE/p+j1AzibYiMC8CTgqxQDjK8Hntntddxk3VcC99Ws35XdrrmZuuva9uPRo5NZx5cBPwZuAb4JzC+nB3B+ue7X9MK6Ld9Xd9e8Tj9TM++Msta1wGu6XOcfUowN2lK+r1b3Yp1lPaPeY73yB3yZ4mwB28r1efJY24wu1vgSisH8t9S8Ll/ba3WWtf4ucHNZ64+BM8vpzyy3V+vK7dfu4y3HX0SQJEmqAHePSpIkVYChTZIkqQIMbZIkSRVgaJMkSaoAQ5skSVIFGNokSZIqwNAmSZJUAYY2VVb547sfj4hbI2JNRDyz2zVJUrMi4nMR8YaImBMRqyLiD7tdk3qboU1VdjpwR2Y+B/gE8O4u1yNJrVgEPEzx00UfzMyvd7ke9ThDmyopIvYE/jAzP15OuhM4tIslSVLTImIG8DvAl4BVmflfXS5JFWBoU1UdBRwUET+MiB8CFwIPdbkmSWrWAuAeit+ifmdEzOpuOaoCQ5uq6nkUP7j7vMx8HvBtih8LlqQqWAT8e2ZeRfED4n/S5XpUAYY2VdW+wOMAETETeDXwza5WJEnNW0QR1gD+Dji9/CyTxhSZ2e0apJZFxLuB52bmuyPiNOCZmfmubtclSdJ0MbSpkiJiX+BbwP7ANcCyzNzc3aokSZo+hjZJkqQKcEybJElSBRjaJEmSKsDQJkmSVAGGNkmSpAowtEmSJFWAoU2SJKkCDG2SJEkVYGiTJEmqgP8Pd5F2uLTbRRIAAAAASUVORK5CYII=\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "np.random.seed(3)\n", "estimate_plot_pdf_acf(np.random.uniform(size=10000)-1/2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Lets listen to uniformly distributed white noise" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "from scipy.io import wavfile\n", "fs = 44100\n", "\n", "x = np.random.uniform(size=5*fs)-1/2\n", "wavfile.write('uniform_white_noise.wav', fs, np.int16(x*32768))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "[./uniform_white_noise.wav](./uniform_white_noise.wav)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Laplace distributed white noise**\n", "\n", "For samples drawn from a zero-mean random process with with Laplace amplitude distribution, the PDF and ACF are estimated as" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "estimate_plot_pdf_acf(np.random.laplace(size=10000, loc=0, scale=1/np.sqrt(2)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise**\n", "\n", "* Do both random processes represent white noise?\n", "* Estimate the power spectral density $N_0$ of both examples.\n", "* How does the ACF change if you lower the length `size` of the random signal. Why?\n", "\n", "Solution: Both processes represent white noise since the ACF can be approximated reasonably well as Dirac impulse $\\delta[\\kappa]$. The weight of the Dirac impulse is equal to $N_0$. In case of the uniformly distributed white noise $N_0 \\approx \\frac{1}{12}$, in case of the Laplace distributed white noise $N_0 \\approx 1$. Decreasing the length `size` of the signal increases the statistical uncertainties in the estimate of the ACF. " ] }, { "cell_type": "markdown", "metadata": { "nbsphinx": "hidden" }, "source": [ "**Copyright**\n", "\n", "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples, 2016-2018*." ] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.4" } }, "nbformat": 4, "nbformat_minor": 1 }