{ "cells": [ { "cell_type": "markdown", "metadata": { "nbsphinx": "hidden" }, "source": [ "# Realization of Recursive Filters\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": [ "## Introduction\n", "\n", "Computing the output $y[k] = \\mathcal{H} \\{ x[k] \\}$ of a [linear time-invariant](https://en.wikipedia.org/wiki/LTI_system_theory) (LTI) system is of central importance in digital signal processing. This is often referred to as [*filtering*](https://en.wikipedia.org/wiki/Digital_filter) of the input signal $x[k]$. We already have discussed the realization of [non-recursive filters](../nonrecursive_filters/introduction.ipynb). This section focuses on the realization of recursive filters." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Recursive Filters\n", "\n", "Linear difference equations with constant coefficients represent linear time-invariant (LTI) systems\n", "\n", "\\begin{equation}\n", "\\sum_{n=0}^{N} a_n \\; y[k-n] = \\sum_{m=0}^{M} b_m \\; x[k-m]\n", "\\end{equation}\n", "\n", "where $y[k] = \\mathcal{H} \\{ x[k] \\}$ denotes the response of the system to the input signal $x[k]$, $N$ the order, $a_n$ and $b_m$ constant coefficients, respectively. Above equation can be rearranged with respect to the output signal $y[k]$ by extracting the first element ($n=0$) of the left-hand sum\n", "\n", "\\begin{equation}\n", "y[k] = \\frac{1}{a_0} \\left( \\sum_{m=0}^{M} b_m \\; x[k-m] - \\sum_{n=1}^{N} a_n \\; y[k-n] \\right)\n", "\\end{equation}\n", "\n", "It is evident that the output signal $y[k]$ at time instant $k$ is given as a linear combination of past output samples $y[k-n]$ superimposed by a linear combination of the actual $x[k]$ and past $x[k-m]$ input samples. Hence, the actual output $y[k]$ is composed from the two contributions\n", "\n", "1. a [non-recursive part](../nonrecursive_filters/introduction.ipynb#Non-Recursive-Filters), and\n", "2. a recursive part where a linear combination of past output samples is fed back.\n", "\n", "The impulse response of the system is given as the response of the system to a Dirac impulse at the input $h[k] = \\mathcal{H} \\{ \\delta[k] \\}$. Using above result and the properties of the discrete Dirac impulse we get\n", "\n", "\\begin{equation}\n", "h[k] = \\frac{1}{a_0} \\left( b_k - \\sum_{n=1}^{N} a_n \\; h[k-n] \\right)\n", "\\end{equation}\n", "\n", "Due to the feedback, the impulse response will in general be of infinite length. The impulse response is termed as [infinite impulse response](https://en.wikipedia.org/wiki/Infinite_impulse_response) (IIR) and the system as recursive system/filter." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Transfer Function\n", "\n", "Applying a $z$-transform to the left- and right-hand side of the difference equation and rearranging terms yields the transfer function $H(z)$ of the system\n", "\n", "\\begin{equation}\n", "H(z) = \\frac{Y(z)}{X(z)} = \\frac{\\sum_{m=0}^{M} b_m \\; z^{-m}}{\\sum_{n=0}^{N} a_n \\; z^{-n}}\n", "\\end{equation}\n", "\n", "The transfer function is given as a [rational function](https://en.wikipedia.org/wiki/Rational_function) in $z$. The polynominals of the numerator and denominator can be expressed alternatively by their roots as\n", "\n", "\\begin{equation}\n", "H(z) = \\frac{b_M}{a_N} \\cdot \\frac{\\prod_{\\mu=1}^{P} (z - z_{0\\mu})^{m_\\mu}}{\\prod_{\\nu=1}^{Q} (z - z_{\\infty\\nu})^{n_\\nu}}\n", "\\end{equation}\n", "\n", "where $z_{0\\mu}$ and $z_{\\infty\\nu}$ denote the $\\mu$-th zero and $\\nu$-th pole of degree $m_\\mu$ and $n_\\nu$ of $H(z)$, respectively. The total number of zeros and poles is denoted by $P$ and $Q$. Due to the symmetries of the $z$-transform, the transfer function of a real-valued system $h[k] \\in \\mathbb{R}$ exhibits complex conjugate symmetry\n", "\n", "\\begin{equation}\n", "H(z) = H^*(z^*)\n", "\\end{equation}\n", "\n", "Poles and zeros are either real valued or complex conjugate pairs for real-valued systems ($b_m\\in\\mathbb{R}$, $a_n\\in\\mathbb{R}$). For the poles of a causal and stable system $H(z)$ the following condition has to hold\n", "\n", "\\begin{equation}\n", "\\max_{\\nu} | z_{\\infty\\nu} | < 1\n", "\\end{equation}\n", "\n", "Hence, all poles have to be located inside the unit circle $|z| = 1$. Amongst others, this implies that $M \\leq N$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example\n", "\n", "The following example shows the pole/zero diagram, the magnitude and phase response, and impulse response of a recursive filter with so-called [Butterworth](https://en.wikipedia.org/wiki/Butterworth_filter) lowpass characteristic." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from matplotlib.markers import MarkerStyle\n", "from matplotlib.patches import Circle\n", "import scipy.signal as sig\n", "\n", "N = 5 # order of recursive filter\n", "L = 128 # number of computed samples\n", "\n", "\n", "def zplane(z, p, title='Poles and Zeros'):\n", " \"Plots zero and pole locations in the complex z-plane\"\n", " ax = plt.gca()\n", " \n", " ax.plot(np.real(z), np.imag(z), 'bo', fillstyle='none', ms = 10)\n", " ax.plot(np.real(p), np.imag(p), 'rx', fillstyle='none', ms = 10)\n", " unit_circle = Circle((0,0), radius=1, fill=False,\n", " color='black', ls='solid', alpha=0.9)\n", " ax.add_patch(unit_circle)\n", " ax.axvline(0, color='0.7')\n", " ax.axhline(0, color='0.7')\n", " \n", " plt.title(title)\n", " plt.xlabel(r'Re{$z$}')\n", " plt.ylabel(r'Im{$z$}')\n", " plt.axis('equal')\n", " plt.xlim((-2, 2))\n", " plt.ylim((-2, 2))\n", " plt.grid()\n", "\n", "\n", "# compute coefficients of recursive filter\n", "b, a = sig.butter(N, 0.2, 'low')\n", "# compute transfer function\n", "Om, H = sig.freqz(b, a)\n", "# compute impulse response\n", "k = np.arange(L)\n", "x = np.where(k==0, 1.0, 0)\n", "h = sig.lfilter(b, a, x)\n", "\n", "# plot pole/zero-diagram\n", "plt.figure(figsize=(5, 5))\n", "zplane(np.roots(b), np.roots(a))\n", "# plot magnitude response\n", "plt.figure(figsize=(10, 3))\n", "plt.plot(Om, 20 * np.log10(abs(H)))\n", "plt.xlabel(r'$\\Omega$')\n", "plt.ylabel(r'$|H(e^{j \\Omega})|$ in dB')\n", "plt.grid()\n", "plt.title('Magnitude response')\n", "# plot phase response\n", "plt.figure(figsize=(10, 3))\n", "plt.plot(Om, np.unwrap(np.angle(H)))\n", "plt.xlabel(r'$\\Omega$')\n", "plt.ylabel(r'$\\varphi (\\Omega)$ in rad')\n", "plt.grid()\n", "plt.title('Phase response')\n", "# plot impulse response (magnitude)\n", "plt.figure(figsize=(10, 3))\n", "plt.stem(20*np.log10(np.abs(np.squeeze(h))))\n", "plt.xlabel(r'$k$')\n", "plt.ylabel(r'$|h[k]|$ in dB')\n", "plt.grid()\n", "plt.title('Impulse response (magnitude)');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise**\n", "\n", "* Does the system have an IIR?\n", "* What happens if you increase the order `N` of the filter?\n", "\n", "Solution: It can be concluded from the last illustration, showing the magnitude of the impulse response $|h[k]|$ on a logarithmic scale, that the magnitude of the impulse response decays continuously for increasing $k$ but does not become zero at some point. This behavior continues with increasing $k$ as can be observed when increasing the number `L` of computed samples in above example. The magnitude response $|H(e^{j \\Omega})|$ of the filter decays faster with increasing order `N` of the filter." ] }, { "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": "3.7.1" } }, "nbformat": 4, "nbformat_minor": 1 }