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Formatting for LaTeX/PDF export
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‎random_signals/correlation_functions.ipynb

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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"** Copyright **\n",
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"**Copyright**\n",
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"\n",
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"<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
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" <a rel=\"license\"\n",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.4.3"
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"version": "3.5.0"
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}
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},
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"nbformat": 4,

‎random_signals/distributions.ipynb

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"\n",
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"and\n",
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"\n",
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"$$\\int_{-\\infty}^{\\infty} p_x(\\theta, k) \\, \\mathrm{d}\\theta = P_x(\\infty, k) = 1$$\n",
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"$$\\int\\limits_{-\\infty}^{\\infty} p_x(\\theta, k) \\, \\mathrm{d}\\theta = P_x(\\infty, k) = 1$$\n",
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"\n",
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"The CDF can be derived from the PDF by integration\n",
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"\n",
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"$$P_x(\\theta, k) = \\int_{-\\infty}^{\\theta} p_x(\\theta, k) \\, \\mathrm{d}\\theta$$"
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"$$P_x(\\theta, k) = \\int\\limits_{-\\infty}^{\\theta} p_x(\\theta, k) \\, \\mathrm{d}\\theta$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"** Copyright **\n",
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"**Copyright**\n",
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"\n",
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"<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
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" <a rel=\"license\"\n",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.4.3"
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"version": "3.5.0"
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}
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},
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"nbformat": 4,

‎random_signals/ensemble_averages.ipynb

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"\n",
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"$$ \\mu_x[k] = E\\{ x[k] \\} = \\int\\limits_{-\\infty}^{\\infty} \\theta \\, p_x(\\theta, k) \\, \\mathrm{d}\\theta $$\n",
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"\n",
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"where $\\mu_x[k]$ is a common abbreviation of the linear mean. A process is termed *mean free* if $\\mu_x[k] = 0$."
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"where $\\mu_x[k]$ is a common abbreviation of the linear mean. A process is termed *mean free* if $\\mu_x[k] = 0$.Note that $\\mu_x$ should not be confused with the frequency index variable of the DFT."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"** Copyright **\n",
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"**Copyright**\n",
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"\n",
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"<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
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" <a rel=\"license\"\n",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.4.3"
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"version": "3.5.0"
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}
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},
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"nbformat": 4,

‎random_signals/important_distributions.ipynb

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"where $\\mu_x$ and $\\sigma_x^2$ denote the linear mean and variance, respectively. Normal distributions are often used to represent random variables whose distributions are not known. The central limit theorem states that averages of random variables independently drawn from independent distributions become normally distributed when the number of random variables is sufficiently large. As a result, random signals that are expected to be the sum of many independent processes often have distributions that are nearly normal. The CDF can be derived by integration over $\\theta$\n",
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"\n",
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"\\begin{align}\n",
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"P_x(\\theta) &= \\frac{1}{\\sqrt{2 \\pi} \\sigma_x} \\int_{—\\infty}^{\\theta} \\mathrm{e}^{- \\frac{\\zeta - \\mu_x}{2 \\sigma_x^2}} \\mathrm{d}\\zeta \\\\\n",
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"P_x(\\theta) &= \\frac{1}{\\sqrt{2 \\pi} \\sigma_x} \\int\\limits_{—\\infty}^{\\theta} \\mathrm{e}^{- \\frac{\\zeta - \\mu_x}{2 \\sigma_x^2}} \\mathrm{d}\\zeta \\\\\n",
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"&= \\frac{1}{2} \\left( 1 + \\text{erf}\\left( \\frac{\\theta-\\mu_x}{\\sqrt{2} \\sigma_x} \\right)\\right)\n",
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"\\end{align}\n",
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"\n",
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"** Copyright **\n",
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"**Copyright**\n",
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"\n",
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"<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
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" <a rel=\"license\"\n",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.4.3"
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"version": "3.5.0"
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}
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"nbformat": 4,

‎random_signals/introduction.ipynb

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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"** Copyright **"
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"**Copyright**"
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]
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},
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{
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.5.0"
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}
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"nbformat": 4,

‎random_signals/power_spectral_densities.ipynb

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"\n",
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"2. The quadratic mean of a random signal is given as\n",
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"\n",
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" $$ E\\{ x[k]^2 \\} = \\varphi_{xx}[0] = \\frac{1}{2\\pi} \\int\\limits_{-\\pi}^{\\pi} \\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega}) \\mathrm{d} \\Omega $$\n",
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" $$ E\\{ x[k]^2 \\} = \\varphi_{xx}[0] = \\frac{1}{2\\pi} \\int\\limits_{-\\pi}^{\\pi} \\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega}) \\,\\mathrm{d} \\Omega $$\n",
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"\n",
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" The last relation can be found by introducing the definition of the inverse DTFT."
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"** Copyright **\n",
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"**Copyright**\n",
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"\n",
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"<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
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" <a rel=\"license\"\n",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.5.0"
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}
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},
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"nbformat": 4,

‎random_signals/stationary_ergodic.ipynb

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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"** Copyright **\n",
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"**Copyright**\n",
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"\n",
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"<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
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" <a rel=\"license\"\n",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.4.3"
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"version": "3.5.0"
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}
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"nbformat": 4,

‎random_signals/white_noise.ipynb

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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"** Copyright **\n",
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"**Copyright**\n",
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"\n",
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"<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
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" <a rel=\"license\"\n",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.4.3"
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"version": "3.5.0"
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}
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},
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"nbformat": 4,

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