{
"cells": [
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"source": [
"# Spectral Estimation for 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": [
"## The Welch Method\n",
"\n",
"In the previous section it has been shown that the [periodogram](periodogram.ipynb) as a non-parametric estimator of the power spectral density (PSD) $\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ of a random signal $x[k]$ is not consistent. This is due to the fact that its variance does not converge towards zero even when the length of the random signal is increased towards infinity. In order to overcome this problem, the [Bartlett method](https://en.wikipedia.org/wiki/Bartlett's_method) and [Welch method](https://en.wikipedia.org/wiki/Welch's_method) \n",
"\n",
"1. split the random signal into segments, \n",
"2. estimate the PSD for each segment, and \n",
"3. average over these local estimates. \n",
"\n",
"The averaging reduces the variance of the estimated PSD. While Barlett's method uses non-overlapping segments, Welch's is a generalization using windowed overlapping segments. For the discussion of Welch's method we assume a wide-sense ergodic real-valued random process."
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"### Derivation\n",
"\n",
"The random signal $x[k]$ is split into into $L$ overlapping segments of length $N$, starting at multiples of the step size $M \\in {1,2, \\dots, N}$. These segments are windowed by the window $w[k]$ of length $N$, resulting in a windowed $l$-th segment $x_l[k]$ with $0\\leq l\\leq L-1$. The discrete-time Fourier transformation (DTFT) $X_l(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ of the windowed $l$-th segment is then given as \n",
"\n",
"\\begin{equation}\n",
"X_l(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\sum_{k = 0}^{N-1} x[k + l \\cdot M] \\, w[k] \\; \\mathrm{e}^{\\,-\\mathrm{j}\\,\\Omega\\,k}\n",
"\\end{equation}\n",
"\n",
"where the window $w[k]$ defined within $0\\leq k\\leq N-1$ should be normalized as $\\frac{1}{N} \\sum\\limits_{k=0}^{N-1} | w[k] |^2 = 1$. The latter condition ensures that the power of the signal is maintained in the estimate. The stepsize $M$ determines the overlap between the segments. In general, $N-M$ number of samples overlap between adjacent segments. For $M = N$ no overlap occurs. The overlap is sometimes given as ratio $\\frac{N-M}{N}\\cdot 100\\%$. \n",
"\n",
"Introducing $X_l(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ into the definition of the periodogram yields the periodogram of the $l$-th segment\n",
"\n",
"\\begin{equation}\n",
"\\hat{\\Phi}_{xx,l}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\frac{1}{N} \\,| X_l(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) |^2\n",
"\\end{equation}\n",
"\n",
"The estimated PSD is then given by averaging over the segment's periodograms $\\hat{\\Phi}_{xx,l}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ \n",
"\n",
"\\begin{equation}\n",
"\\hat{\\Phi}_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\frac{1}{L} \\sum_{l = 0}^{L-1} \\hat{\\Phi}_{xx,l}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})\n",
"\\end{equation}\n",
"\n",
"Note, that the total number $L$ of segments has to be chosen such that the last required sample $(L-1)\\cdot M + N - 1$ does not exceed the total length of the random signal. Otherwise the last segment $x_{L-1}[k]$ may be zeropadded towards length $N$.\n",
"\n",
"The Bartlett method uses a rectangular window $w[k] = \\text{rect}_N[k]$ and non-overlapping segments $M=N$. The Welch method uses overlapping segments and a window that must be chosen according to the intended spectral analysis task."
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"### Example\n",
"\n",
"The following example is equivalent to the previous [periodogram example](periodogram.ipynb#Example---Periodogram). We aim at estimating the PSD of a random process which draws samples from normally distributed white noise with zero-mean and unit variance. The true PSD is consequently given as $\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = 1$."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Bias of the Welch estimate: \t\t -0.0114\n",
"Variance of the Welch estimate: \t 0.0255\n"
]
},
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\n",
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