{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Scipy Signal Processing: IIR Filter Design" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
" ], "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import addutils.toc ; addutils.toc.js(ipy_notebook=True)" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from addutils import css_notebook\n", "import bokeh.plotting as bk\n", "css_notebook()" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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\\n\"+\n \"

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\\n\"+\n \"
    \\n\"+\n \"
  • re-rerun `output_notebook()` to attempt to load from CDN again, or
    • \\n\"+\n \"
  • use INLINE resources instead, as so:
    • \\n\"+\n \"
\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
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This notebook is not a thorough introduction to IIR filter design." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Documentation and reference:\n", "* [Signal Processing (scipy.signal)](http://docs.scipy.org/doc/scipy/reference/signal.html#module-scipy.signal)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A filter is usually defined in the frequency domain. The following figure depicts the frequency domain specification:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The specification figure illustrates a low-pass filter but the terminology applies to all filter types, lowpass, highpass, bandpass, and stopband.\n", "\n", "* **Passband (Wp):** This is the frequency range which we desire to let the signal through with minimal attenuation. In the scipy functions this is in normalized frequency, 1 > Wp > 0.\n", "* **Stopband (Ws):** This is the frequency range which the signal should be attenuated. Again, in scipy this is in normalized frequency, 1 > Ws > 0.\n", "* **Passband ripple (Rp, gpass):* The max variation in the passband, in decibals.\n", "* **Stopband attenuation (As, gstop):** The max level in the stopband or in other words the min attenuation in the stopband, in decibals.\n", "\n", "NOTE: The cutoff frequency is the -3dB point. If the cutoff frequency is required the algorithm will work to meet the -3dB point at the Wc frequency. The pass frequency, Wp, is the last, first, or first and last frequencies in the passband which the Rp value occurs (the rest are less).\n", "\n", "The above specifications are the information that will be used and passed to the IIR filter design functions. In addition to our filter specification information we need to select our \"filter\" type. The following table summarizes the different IIR filter design approaches supported by the `scipy.signal` design functions:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2 Filter Design" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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\\n\"+\n", " \"
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\\n\"+\n", " \"\\n\"+\n", " \"from bokeh.resources import INLINE\\n\"+\n", " \"output_notebook(resources=INLINE)\\n\"+\n", " \"\\n\"+\n", " \"
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\\n\"+\n \"

\\n\"+\n \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n \"

\\n\"+\n \"
    \\n\"+\n \"
  • re-rerun `output_notebook()` to attempt to load from CDN again, or
    • \\n\"+\n \"
  • use INLINE resources instead, as so:
    • \\n\"+\n \"
\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
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i < inline_js.length; i++) {\n inline_js[i].call(root, root.Bokeh);\n }if (force === true) {\n display_loaded();\n }} else if (Date.now() < root._bokeh_timeout) {\n setTimeout(run_inline_js, 100);\n } else if (!root._bokeh_failed_load) {\n console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n root._bokeh_failed_load = true;\n } else if (force !== true) {\n var cell = $(document.getElementById(\"c3a6ee3f-caa8-4759-bb21-a18564ee5e37\")).parents('.cell').data().cell;\n cell.output_area.append_execute_result(NB_LOAD_WARNING)\n }\n\n }\n\n if (root._bokeh_is_loading === 0) {\n console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n run_inline_js();\n } else {\n load_libs(js_urls, function() {\n console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n run_inline_js();\n });\n }\n}(window));" }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "from scipy import signal\n", "from scipy.signal import filter_design as fd\n", "\n", "import bokeh.plotting as bk\n", "bk.output_notebook()" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Specification for our filter\n", "Wp = 0.270 # Cutoff frequency \n", "Ws = 0.412 # Stop frequency \n", "Rp = 0.1 # passband maximum loss (gpass)\n", "As = 60 # stoppand min attenuation (gstop)\n", "\n", "Filters = {'ellip' : (), 'cheby2' : (), 'butter' : (), 'cheby1' : (), 'bessel' : ()}\n", "\n", "# The ellip and cheby2 filter design\n", "Filters['ellip'] = fd.iirdesign(Wp, Ws, Rp, As, ftype='ellip')\n", "Filters['cheby2'] = fd.iirdesign(Wp, Ws, Rp, As, ftype='cheby2')\n", "\n", "# The butter and cheby1 need less constraint spec\n", "Rpl = Rp*10; Asl = As/4.\n", "Filters['butter'] = fd.iirdesign(Wp, Ws, Rp, As, ftype='butter')\n", "Filters['cheby1'] = fd.iirdesign(Wp, Ws, Rp, As, ftype='cheby1')\n", "\n", "# The bessel max order of 8 for this cutoff, can't use\n", "# iirdesign have to use iirfilter.\n", "Filters['bessel'] = fd.iirfilter(8, Wp, btype='lowpass', ftype='bessel')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3 Frequency and Phase Response" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To define plot colors, any [legal HTML name for colors](http://en.wikipedia.org/wiki/Web_colors) can be used:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": true }, "outputs": [], "source": [ "plotcolors = ['DarkRed', 'Crimson', 'Red',\n", " 'Olive', 'Green', 'SeaGreen',\n", " 'SteelBlue', 'Blue', 'Navy',\n", " 'Fuchsia', 'DarkViolet', 'Indigo']" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Plotting:\n", "\n", "* `a, b` are respectively numerator and denominator of the filter\n", "* `w` are the frequencies at which the frequency response `'h'` is computed" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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"xmn1, xmx1, ymn1, ymx1 = 0.1, 0.5, -75., 5.\n", "xmn2, xmx2, ymn2, ymx2 = 0.1, 0.5, -22., -0.\n", "\n", "fig1 = bk.figure(plot_width=350, plot_height=400,\n", " title='Frequency response',\n", " y_axis_label='Magnitude (db)',\n", " x_axis_label='Normalized Frequency',\n", " x_range=(xmn1, xmx1), y_range=(ymn1, ymx1))\n", "fig2 = bk.figure(plot_width=350, plot_height=400,\n", " title='Phase response',\n", " x_axis_label='Normalized Frequency',\n", " x_range=(xmn2, xmx2), y_range=(ymn2, ymx2))\n", "\n", "for i, filtername in enumerate(Filters):\n", " b, a = Filters[filtername]\n", " w,h = signal.freqz(b,a)\n", " h_dB = 20*np.log10(np.abs(h))\n", " h_Phase = np.unwrap(np.arctan2(np.imag(h),np.real(h)))\n", " fig1.line(w/np.max(w),h_dB, color=plotcolors[i].lower(), legend=filtername)\n", " fig2.line(w/np.max(w),h_Phase,color=plotcolors[i].lower(), legend=filtername)\n", "\n", "fig1.line([Wp, Wp], [ymx1, ymn1], line_dash=[5])\n", "fig1.line([Ws, Ws], [ymx1, ymn1], line_dash=[5])\n", "fig1.line([xmn1, xmx1], [-As, -As], line_dash=[5])\n", "fig1.legend.location = \"bottom_left\"\n", "fig2.legend.location = \"bottom_left\"\n", "\n", "bk.show(bk.gridplot([[fig1, fig2]]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can have a better look at the Passband Ripple:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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= bk.figure(title='Frequency Response', \n", " x_axis_label='Normalized Frequency', \n", " y_axis_label='Magnitude (db)',\n", " plot_width=700, plot_height=400,\n", " x_range=(xmn1, xmx1), y_range=(ymn1, ymx1))\n", "\n", "for i, filtername in enumerate(Filters):\n", " b, a = Filters[filtername]\n", " w,h = signal.freqz(b,a)\n", " h_dB = 20*np.log10(np.abs(h))\n", " fig.line(w/np.max(w),h_dB, color=plotcolors[i].lower(), legend=filtername)\n", "\n", "xmn1, xmx1, ymn1, ymx1 = 0.1, 0.3, -0.11, 0.01\n", "fig.line([Wp, Wp], [ymx1, ymn1], line_dash=[5])\n", "fig.line([Ws, Ws], [ymx1, ymn1], line_dash=[5])\n", "fig.line([xmn1, xmx1], [-As, -As], line_dash=[5])\n", "fig.legend.location = \"bottom_right\"\n", "bk.show(fig)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4 Step and Impulse Response" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can have a look to the inpulse and step response for any filter (bessel in this example):" ] }, { "cell_type": "code", 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Application\",\"version\":\"0.12.13\"}};\n", " var render_items = [{\"docid\":\"77698ccc-397f-404f-a827-602092819eb7\",\"elementid\":\"f8cb3792-5fa6-4a52-8115-b46e38d4f7c5\",\"modelid\":\"e26805e1-1f77-4af9-830c-2cb296739551\"}];\n", " root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n", "\n", " }\n", " if (root.Bokeh !== undefined) {\n", " embed_document(root);\n", " } else {\n", " var attempts = 0;\n", " var timer = setInterval(function(root) {\n", " if (root.Bokeh !== undefined) {\n", " embed_document(root);\n", " clearInterval(timer);\n", " }\n", " attempts++;\n", " if (attempts > 100) {\n", " console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n", " clearInterval(timer);\n", " }\n", " }, 10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "e26805e1-1f77-4af9-830c-2cb296739551" } }, "output_type": "display_data" } ], "source": [ "from bokeh.models.tickers import AdaptiveTicker\n", "b, a = Filters['bessel']\n", "impulse = np.zeros(50); impulse[0] = 1\n", "x = np.arange(0, 50)\n", "response = signal.lfilter(b, a, impulse)\n", "step = np.cumsum(response)\n", "d1 = bk.figure(title=\"Impulse response\", plot_width=350, plot_height=400)\n", "d1.xaxis.axis_label = 'n (samples)'\n", "d1.yaxis.axis_label = 'Amplitude'\n", "d1.segment(range(len(response)), 0, range(len(response)), response,\n", " line_width=2, line_color=\"green\")\n", "d1.circle(range(len(response)), response,\n", " size=5, fill_color=\"darkred\", line_color=\"green\", line_width=1)\n", "d2 = bk.figure(title=\"Step response\", plot_width=350, plot_height=400)\n", "d2.xaxis.axis_label = 'n (samples)'\n", "d2.yaxis.axis_label = 'Amplitude'\n", "d2.segment(range(len(step)), 0, range(len(step)), step, \n", " line_width=2, line_color=\"SteelBlue\")\n", "d2.circle(range(len(step)), step, \n", " size=5, fill_color=\"orange\", line_color=\"SteelBlue\", line_width=1)\n", "\n", "bk.show(bk.gridplot([[d1, d2]]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "---\n", "\n", "Visit [www.add-for.com]() for more tutorials and updates.\n", "\n", "This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License." ] } ], "metadata": { "kernelspec": { "display_name": "Python [conda env:addfor_tutorials]", "language": "python", "name": "conda-env-addfor_tutorials-py" }, "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 }