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"# Random Signals and LTI-Systems\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).*"
]
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"## The Wiener Filter\n",
"\n",
"The [Wiener filter](https://en.wikipedia.org/wiki/Wiener_filter), named after [*Norbert Wiener*](https://en.wikipedia.org/wiki/Norbert_Wiener), aims at estimating an unknown random signal by filtering a noisy distorted observation of the signal. It has a wide range of applications in signal processing. For instance, noise reduction, system identification, deconvolution and signal detection. The Wiener filter is frequently used as basic building block for algorithms that denoise audio signals, like speech, or remove noise from a image."
]
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"source": [
"### Signal Model\n",
"\n",
"The following signal model is underlying the Wiener filter\n",
"\n",
"![Illustration: Signal model for the Wiener filter](model_wiener_filter.png)\n",
"\n",
"The random signal $s[k]$ is subject to distortion by the linear time-invariant (LTI) system $G(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ and additive noise $n[k]$, resulting in the observed signal $x[k] = s[k] * g[k] + n[k]$. The additive noise $n[k]$ is assumed to be uncorrelated from $s[k]$. It is furthermore assumed that all random signals are wide-sense stationary (WSS). This distortion model holds for many practical problems, like e.g. the measurement of a physical quantity by a sensor.\n",
"\n",
"The basic concept of the Wiener filter is to apply the LTI system $H(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ to the observed signal $x[k]$ such that the output signal $y[k] = x[k] * h[k]$ matches $s[k]$ as close as possible. In order to quantify the deviation between $y[k]$ and $s[k]$, the error signal\n",
"\n",
"\\begin{equation}\n",
"e[k] = y[k] - s[k]\n",
"\\end{equation}\n",
"\n",
"is introduced. The error $e[k]$ equals zero, if the output signal $y[k]$ perfectly matches $s[k]$. In general, this goal cannot be achieved in a strict sense. As alternative one could aim at minimizing the linear average of the error $e[k]$. However, this quantity is not very well suited for optimization since it is signed. Instead, the quadratic average of the error $e[k]$ is used in the Wiener filter and other techniques. This measure is known as [*mean squared error*](https://en.wikipedia.org/wiki/Mean_squared_error) (MSE). It is defined as\n",
"\n",
"\\begin{equation}\n",
"E \\{ |e[k]|^2 \\} = \\frac{1}{2 \\pi} \\int\\limits_{-\\pi}^{\\pi} \\Phi_{ee}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) \\,\\mathrm{d}\\Omega = \\varphi_{ee}[\\kappa] \\Big\\vert_{\\kappa = 0}\n",
"\\end{equation}\n",
"\n",
"the equalities are deduced from the properties of the power spectral density (PSD) and the auto-correlation function (ACF). Above equation is referred to as [*cost function*](https://en.wikipedia.org/wiki/Loss_function). We aim at minimizing the cost function, hence minimizing the MSE between the original signal $s[k]$ and its estimate $y[k]$. The solution is referred to as [minimum mean squared error](https://en.wikipedia.org/wiki/Minimum_mean_square_error) (MMSE) solution."
]
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"### Transfer Function of the Wiener Filter\n",
"\n",
"At first, the Wiener filter shall only have access to the observed signal $x[k]$ and selected statistical measures. It is assumed that the cross-power spectral density $\\Phi_{xs}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ between the observed signal $x[k]$ and the original signal $s[k]$, and the PSD of the observed signal $\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ are known. This knowledge can either be gained by estimating both from measurements taken at an actual system or by using suitable statistical models.\n",
"\n",
"The optimal filter is found by minimizing the MSE $E \\{ |e[k]|^2 \\}$ with respect to the transfer function $H(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$. The solution of this optimization problem goes beyond the scope of this notebook and can be found in the literature, e.g. [[Girod et. al](../index.ipynb#Literature)]. The transfer function of the Wiener filter is given as\n",
"\n",
"\\begin{equation}\n",
"H(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\frac{\\Phi_{sx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})}{\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})} = \\frac{\\Phi_{xs}(\\mathrm{e}^{\\,-\\mathrm{j}\\,\\Omega})}{\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})}\n",
"\\end{equation}\n",
"\n",
"for $\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) \\neq 0$.\n",
"No knowledge on the actual distortion process is required besides the assumption of an LTI system and additive noise. Only the PSDs $\\Phi_{sx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ and $\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ have to be known in order to estimate $s[k]$ from $x[k]$ in the MMSE sense. Care has to be taken that the filter $H(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ is causal and stable in practical applications."
]
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"**Example - Denoising of an Deterministic Signal**\n",
"\n",
"The following example considers the estimation of the original signal from a distorted observation. It is assumed that the original signal is a deterministic signal $s[k] = \\sin[\\Omega_0\\,k]$ which is distorted by an LTI system and additive wide-sense ergodic normal distributed zero-mean white noise with PSD $N_0 = 0.1$. The PSDs $\\Phi_{sx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ and $\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ are estimated from $s[k]$ and $x[k]$ using the [Welch technique](../spectral_estimation_random_signals/welch_method.ipynb). The Wiener filter is applied to the observation $x[k]$ in order to compute the estimate $y[k]$ of $s[k]$. Note that the discrete signals have been illustrated by continuous lines for ease of illustration."
]
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\n",
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"text/plain": [
"