From 6f854dfb6674e61617ee6928ccd7d11750e206b8 Mon Sep 17 00:00:00 2001 From: "Kasper D. Fischer" Date: Sun, 24 Apr 2022 18:42:45 +0200 Subject: [PATCH] add notebooks for exercise 2 * filter basics * Fourier transformation --- .../1-filter_basics.ipynb | 581 ++++++++++++++++++ .../2-fourier_transform.ipynb | 554 +++++++++++++++++ README.md | 4 + 3 files changed, 1139 insertions(+) create mode 100644 02-FFT_and_Basic_Filtering/1-filter_basics.ipynb create mode 100644 02-FFT_and_Basic_Filtering/2-fourier_transform.ipynb diff --git a/02-FFT_and_Basic_Filtering/1-filter_basics.ipynb b/02-FFT_and_Basic_Filtering/1-filter_basics.ipynb new file mode 100644 index 0000000..063d231 --- /dev/null +++ b/02-FFT_and_Basic_Filtering/1-filter_basics.ipynb @@ -0,0 +1,581 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
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Signal Processing
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Filtering Basics - Solution
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\n", + "\n", + "Seismo-Live: http://seismo-live.org\n", + "\n", + "##### Authors:\n", + "* Stefanie Donner ([@stefdonner](https://github.com/stefdonner))\n", + "* Celine Hadziioannou ([@hadzii](https://github.com/hadzii))\n", + "* Ceri Nunn ([@cerinunn](https://github.com/cerinunn))\n", + "\n", + "\n", + "---" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "outputs": [], + "source": [ + "# Cell 0 - Preparation: load packages, set some basic options \n", + "#%matplotlib inline\n", + "from obspy import *\n", + "from obspy.clients.fdsn import Client\n", + "import numpy as np\n", + "import matplotlib.pylab as plt\n", + "plt.style.use('ggplot')\n", + "plt.rcParams['figure.figsize'] = 15, 4\n", + "plt.rcParams['lines.linewidth'] = 0.5" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Basics in filtering" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## The Filter\n", + "\n", + "In seismology, filters are used to correct for the instrument response, avoid aliasing effects, separate 'wanted' from 'unwanted' frequencies, identify harmonic signals, model a specific recording instrument, and much more ...\n", + "\n", + "There is no clear classification of filters. Roughly speaking, we can distinguish linear vs. non-linear, analog (circuits, resistors, conductors) vs. digital (logical components), and continuous vs. discrete filters. In seismology, we generally avoid non-linear filters, because their output contains frequencies which are not in the input signal. Analog filters can be continuous or discrete, while digital filters are always discrete. Discrete filters can be subdivided into infinite impulse response (IIR) filters, which are recursive and causal, and finite impulse response (FIR) filters, which are non-recursive and causal or acausal. We will explain more about these types of filters below.\n", + "Some filters have a special name, such as Butterworth, Chebyshev or Bessel filters, but they can also be integrated in the classification described above." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "### Response Funtion\n", + "\n", + "A filter is characterised by its frequency response function $T(j\\omega)$ which relate the Fourier transformation of the output signal $Y(j\\omega)$ and the Fourier transformation of the input signal $X(j\\omega)$:\n", + "$$ Y(j\\omega) = T(j\\omega)X(j\\omega)$$\n", + "\n", + "For a simple lowpass filter it is given as:\n", + "$$ |T(j\\omega)| = \\sqrt{ \\frac{1}{1+(\\frac{\\omega}{\\omega_c})^{2n}} } $$\n", + "\n", + "with $\\omega$ indicating the frequency samples, $\\omega_c$ the corner frequency of the filter, and $n$ the order of the filter (also called the number of corners of the filter). For a lowpass filter, all frequencies lower than the corner frequency are allowed to pass the filter. This is the pass band of the filter. On the other hand, the range of frequencies above the corner frequency is called stop band. In between lies the transition band, a small band of frequencies in which the passed amplitudes are gradually decreased to zero. The steepness of the slope of this transition band is defined by the order of the filter: the higher the order, the steeper the slope, the more effectively 'unwanted' frequencies get removed." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "### Time Domain vs. Frequency Domain\n", + "\n", + "In the time domain, filtering means to convolve the data with the impulse response function of the filter. Doing this operation in the time domain is mathematically complex, computationally expensive and slow. Therefore, the digital application of filters is almost always done in the frequency domain, where it simplifies to a much faster multiplication between data and filter response. The procedure is as follows: transfer the signal into the frequency domain via FFT, multiply it with the filter's frequency response function (i.e. the FFT of the impulse response function), and transfer the result back to the time-domain. As a consequence, when filtering, we have to be aware of the characteristics and pit-falls of the Fourier transformation." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Filter types\n", + "\n", + "There are 4 main types of filters: a lowpass, a highpass, a bandpass, and a bandstop filter. Low- and highpass filters only have one corner frequency, allowing frequencies below and above this corner frequency to pass the filter, respectively. In contrast, bandpass and bandstop filters have two corner frequencies, defining a frequency band to pass and to stop, respectively.\n", + "Here, we want to see how exactly these filters act on the input signal. In Cell 1, the vertical component of the M$_w\\,$9.1 Tohoku earthquake, recorded at Bochum - Germany, is downloaded and pre-processed. In Cell 2, the four basic filters are applied to these data and plotted together with the filter functions and the resulting amplitude spectrum.\n", + "\n", + "1) Look at the figure and explain what the different filters do.\n", + "\n", + "2) Change the order of the filter (i.e the number of corners). What happens and why?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "outputs": [], + "source": [ + "# Cell 1: prepare data from Tohoku-Oki earthquake. \n", + "client = Client(\"BGR\")\n", + "t1 = UTCDateTime(\"2011-03-11T05:00:00.000\")\n", + "st = client.get_waveforms(\"GR\", \"BUG\", \"\", \"LHZ\", t1, t1 + 6 * 60 * 60, \n", + " attach_response = True)\n", + "st.remove_response(output=\"VEL\")\n", + "st.detrend('linear')\n", + "st.detrend('demean')\n", + "st[0].plot();" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "skip" + } + }, + "outputs": [], + "source": [ + "# Cell 1b: print stats\n", + "print(st[0].stats)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 2 - filter types\n", + "npts = st[0].stats.npts # number of samples in the trace\n", + "dt = st[0].stats.delta # sample interval\n", + "fNy = 1. / (2. * dt) # Nyquist frequency \n", + "time = np.arange(0, npts) * dt # time axis for plotting\n", + "freq = np.linspace(0, fNy, npts // 2 + 1) # frequency axis for plotting\n", + "corners = 4 # order of filter\n", + "\n", + "# several filter frequencies for the different filter types\n", + "f0 = 0.04\n", + "fmin1 = 0.04\n", + "fmax1 = 0.07\n", + "fmin2 = 0.03\n", + "fmax2 = 0.07\n", + "\n", + "# filter functions\n", + "LP = 1 / ( 1 + (freq / f0) ** (2 * corners))\n", + "HP = 1 - 1 / (1 + (freq / f0) ** (2 * corners))\n", + "wc = fmax1 - fmin1\n", + "wb = 0.5 * wc + fmin1\n", + "BP = 1/(1 + ((freq - wb) / wc) ** (2 * corners))\n", + "wc = fmax2 - fmin2\n", + "wb = 0.5 * wc + fmin2\n", + "BS = 1 - ( 1 / (1 + ((freq - wb) / wc) ** (2 * corners)))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "outputs": [], + "source": [ + "# Cell 2b: filtered traces\n", + "stHP = st.copy()\n", + "stHP.filter('highpass', freq=f0, corners=corners, zerophase=True)\n", + "stLP = st.copy()\n", + "stLP.filter('lowpass', freq=f0, corners=corners, zerophase=True)\n", + "stBP = st.copy()\n", + "stBP.filter('bandpass', freqmin=fmin1, freqmax=fmax1, corners=corners, zerophase=True)\n", + "stBS = st.copy()\n", + "stBS.filter('bandstop', freqmin=fmin2, freqmax=fmax2, corners=corners, zerophase=True)\n", + "\n", + "# amplitude spectras\n", + "Ospec = np.fft.rfft(st[0].data)\n", + "LPspec = np.fft.rfft(stLP[0].data)\n", + "HPspec = np.fft.rfft(stHP[0].data)\n", + "BPspec = np.fft.rfft(stBP[0].data)\n", + "BSspec = np.fft.rfft(stBS[0].data)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "rise": { + "scroll": true + }, + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 2c - plot\n", + "plt.rcParams['figure.figsize'] = 17, 17\n", + "tx1 = 3000\n", + "tx2 = 8000\n", + "fx2 = 0.12\n", + "\n", + "fig = plt.figure()\n", + "\n", + "ax1 = fig.add_subplot(5,3,1)\n", + "ax1.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(time, st[0].data, 'k')\n", + "plt.xlim(tx1, tx2)\n", + "plt.title('time-domain data')\n", + "plt.ylabel('original data \\n amplitude [ms$^-1$]')\n", + "\n", + "ax3 = fig.add_subplot(5,3,3)\n", + "plt.plot(freq, abs(Ospec), 'k')\n", + "plt.title('frequency-domain data \\n amplitude spectrum')\n", + "plt.ylabel('amplitude')\n", + "plt.xlim(0,fx2)\n", + "\n", + "ax4 = fig.add_subplot(5,3,4)\n", + "ax4.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(time, stLP[0].data, 'k')\n", + "plt.xlim(tx1, tx2)\n", + "plt.ylabel('LOWPASS \\n amplitude [ms$^-1$]')\n", + "\n", + "ax5 = fig.add_subplot(5,3,5)\n", + "plt.plot(freq, LP, 'k', linewidth=1.5)\n", + "plt.xlim(0,fx2)\n", + "plt.ylim(-0.1,1.1)\n", + "plt.title('filter function')\n", + "plt.ylabel('amplitude [%]')\n", + "\n", + "ax6 = fig.add_subplot(5,3,6)\n", + "plt.plot(freq, abs(LPspec), 'k')\n", + "plt.ylabel('amplitude ')\n", + "plt.xlim(0,fx2)\n", + "\n", + "ax7 = fig.add_subplot(5,3,7)\n", + "ax7.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(time, stHP[0].data, 'k')\n", + "plt.xlim(tx1, tx2)\n", + "plt.ylabel('HIGHPASS \\n amplitude [ms$^-1$]')\n", + "\n", + "ax8 = fig.add_subplot(5,3,8)\n", + "plt.plot(freq, HP, 'k', linewidth=1.5)\n", + "plt.xlim(0,fx2)\n", + "plt.ylim(-0.1,1.1)\n", + "plt.ylabel('amplitude [%]')\n", + "\n", + "ax9 = fig.add_subplot(5,3,9)\n", + "plt.plot(freq, abs(HPspec), 'k')\n", + "plt.ylabel('amplitude ')\n", + "plt.xlim(0,fx2)\n", + "\n", + "ax10 = fig.add_subplot(5,3,10)\n", + "ax10.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(time, stBP[0].data, 'k')\n", + "plt.xlim(tx1, tx2)\n", + "plt.ylabel('BANDPASS \\n amplitude [ms$^-1$]')\n", + "\n", + "ax11 = fig.add_subplot(5,3,11)\n", + "plt.plot(freq, BP, 'k', linewidth=1.5)\n", + "plt.xlim(0,fx2)\n", + "plt.ylim(-0.1,1.1)\n", + "plt.ylabel('amplitude [%]')\n", + "\n", + "ax12 = fig.add_subplot(5,3,12)\n", + "plt.plot(freq, abs(BPspec), 'k')\n", + "plt.ylabel('amplitude ')\n", + "plt.xlim(0,fx2)\n", + "\n", + "ax13 = fig.add_subplot(5,3,13)\n", + "ax13.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(time, stBS[0].data, 'k')\n", + "plt.xlim(tx1, tx2)\n", + "plt.xlabel('time [sec]')\n", + "plt.ylabel('BANDSTOPP \\n amplitude [ms$^-1$]');\n", + "\n", + "ax14 = fig.add_subplot(5,3,14)\n", + "plt.plot(freq, BS, 'k', linewidth=1.5)\n", + "plt.xlim(0,fx2)\n", + "plt.ylim(-0.1,1.1)\n", + "plt.ylabel('amplitude [%]')\n", + "plt.xlabel('frequency [Hz]')\n", + "\n", + "ax15 = fig.add_subplot(5,3,15)\n", + "plt.plot(freq, abs(BSspec), 'k')\n", + "plt.xlabel('frequency [Hz]')\n", + "plt.ylabel('amplitude ')\n", + "plt.xlim(0,fx2)\n", + "\n", + "plt.subplots_adjust(wspace=0.3, hspace=0.4)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Causal vs. acausal\n", + "\n", + "Filters can be causal or acausal. The output of a causal filter depends only on past and present input, while the output also depends on future input. Thus, an acausal filter is always symmetric and a causal one not. In this exercise, we want to see the effects of such filters on the signal. In Cell 3, the example seismogram of ObsPy is loaded and lowpass filtered several times with different filter order $n$ and causality.\n", + "\n", + "3) Explain the effects of the different filters. You can also play with the order of the filter (variable $ncorners$).\n", + "\n", + "4) Zoom into a small window around the first onset (change the variables $start$, $end$ and $amp$). Which filter would you use for which purpose?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 3 - filter effects\n", + "stf = read() # load example seismogram\n", + "tr = stf[0] # select the first trace in the Stream object\n", + "tr.detrend('demean') # preprocess data\n", + "tr.detrend('linear')\n", + "tr.filter(\"highpass\", freq=2) # removing long-period noise\n", + "print(tr)\n", + "t = tr.times() # time vector for x axis\n", + "\n", + "f = 15.0 # frequency for filters (intial: 10 Hz)\n", + "start = 4 # start time to plot in sec (initial: 4)\n", + "end = 8 # end time to plot in sec (initial: 8)\n", + "amp = 1500 # amplitude range for plotting (initial: 1500) \n", + "ncorners = 4 # number of corners/order of the filter (initial: 4)\n", + "\n", + "tr_filt = tr.copy() # causal filter / not zero phase. Order = 2\n", + "tr_filt.filter('lowpass', freq=f, zerophase=False, corners=2)\n", + "tr_filt2 = tr.copy() # causal filter / not zero phase. Order = set by ncorners\n", + "tr_filt2.filter('lowpass', freq=f, zerophase=False, corners=ncorners)\n", + "tr_filt3 = tr.copy() # acausal filter / zero phase. Order = set by ncorners\n", + "tr_filt3.filter('lowpass', freq=f, zerophase=True, corners=ncorners)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "skip" + } + }, + "outputs": [], + "source": [ + "# Cell 3b: plot - comment single lines to better see the remaining ones\n", + "plt.rcParams['figure.figsize'] = 15, 4\n", + "plt.plot(t, tr.data, 'k', label='original', linewidth=1.)\n", + "plt.plot(t, tr_filt.data, 'b', label='causal, n=2', linewidth=1.2)\n", + "plt.plot(t, tr_filt2.data, 'r', label='causal, n=%s' % ncorners, linewidth=1.2)\n", + "plt.plot(t, tr_filt3.data, 'g', label='acausal, n=%s' % ncorners, linewidth=1.2)\n", + "\n", + "plt.xlabel('time [s]')\n", + "plt.xlim(start, end) \n", + "plt.ylim(-amp, amp)\n", + "plt.ylabel('amplitude [arbitrary]')\n", + "plt.legend(loc='lower right')\n", + "\n", + "plt.show();" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Frequency ranges - bandpass filter\n", + "\n", + "We will now look at an event which took place in Kazakhstan on July 8, 1989. We want to see how filtering helps to derive information from a signal. The signal has been recorded by a Chinese station and is bandpassed in several different frequency bands.\n", + "\n", + "5) What do you see in the different frequency bands?\n", + "\n", + "6) Play with the channel used for filtering in Cell 5. What do you not see? Can you guess what kind of event it is?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 4 - get + preprocess data\n", + "c = Client(\"IRIS\")\n", + "tmp1 = UTCDateTime(\"1989-10-19T09:50:00.0\")\n", + "tmp2 = UTCDateTime(\"1989-10-19T10:20:00.0\")\n", + "dat = c.get_waveforms(\"CD\", \"WMQ\", \"\", \"BH*\", tmp1, tmp2, attach_response = True)\n", + "dat.detrend('linear')\n", + "dat.detrend('demean')\n", + "dat.remove_response(output=\"VEL\")\n", + "dat.detrend('linear')\n", + "dat.detrend('demean')\n", + "print(dat)\n", + "dat.plot();" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 5 - filter data in different frequency ranges\n", + "\n", + "chanel = 2\n", + "tm = dat[chanel].times()\n", + "xmin = 0\n", + "xmax = 700\n", + "\n", + "dat1 = dat[chanel].copy()\n", + "dat2 = dat[chanel].copy()\n", + "dat2.filter(type=\"bandpass\", freqmin=0.01, freqmax=0.05)\n", + "dat3 = dat[chanel].copy()\n", + "dat3.filter(type=\"bandpass\", freqmin=0.05, freqmax=0.1)\n", + "dat4 = dat[chanel].copy()\n", + "dat4.filter(type=\"bandpass\", freqmin=0.1, freqmax=0.5)\n", + "dat5 = dat[chanel].copy()\n", + "dat5.filter(type=\"bandpass\", freqmin=0.5, freqmax=1)\n", + "dat6 = dat[chanel].copy()\n", + "dat6.filter(type=\"bandpass\", freqmin=1., freqmax=5.)\n", + "dat7 = dat[chanel].copy()\n", + "dat7.filter(type=\"bandpass\", freqmin=5., freqmax=9.99)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 5b: plot\n", + "plt.rcParams['figure.figsize'] = 17, 21\n", + "fig = plt.figure()\n", + "ax1 = fig.add_subplot(7,1,1)\n", + "ax1.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(tm, dat1.data, 'k')\n", + "plt.xlim(xmin, xmax)\n", + "plt.title('unfiltered')\n", + "plt.ylabel('amplitude \\n [m/s]')\n", + "ax2 = fig.add_subplot(7,1,2)\n", + "ax2.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(tm, dat2.data, 'k')\n", + "plt.xlim(xmin, xmax)\n", + "plt.title('0.01 - 0.05 Hz')\n", + "plt.ylabel('amplitude \\n [m/s]')\n", + "ax3 = fig.add_subplot(7,1,3)\n", + "ax3.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(tm, dat3.data, 'k')\n", + "plt.xlim(xmin, xmax)\n", + "plt.title('0.05 - 0.1 Hz')\n", + "plt.ylabel('amplitude \\n [m/s]')\n", + "ax4 = fig.add_subplot(7,1,4)\n", + "ax4.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(tm, dat4.data, 'k')\n", + "plt.xlim(xmin, xmax)\n", + "plt.title('0.1 - 0.5 Hz')\n", + "plt.ylabel('amplitude \\n [m/s]')\n", + "ax5 = fig.add_subplot(7,1,5)\n", + "ax5.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(tm, dat5.data, 'k')\n", + "plt.xlim(xmin, xmax)\n", + "plt.title('0.5 - 1.0 Hz')\n", + "plt.ylabel('amplitude \\n [m/s]')\n", + "ax6 = fig.add_subplot(7,1,6)\n", + "ax6.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(tm, dat6.data, 'k')\n", + "plt.xlim(xmin, xmax)\n", + "plt.title('1.0 - 5.0 Hz')\n", + "plt.ylabel('amplitude \\n [m/s]')\n", + "ax7 = fig.add_subplot(7,1,7)\n", + "ax7.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n", + "plt.plot(tm, dat7.data, 'k')\n", + "plt.xlim(xmin, xmax)\n", + "plt.title('5.0 - 10.0 Hz')\n", + "plt.xlabel('time [sec]')\n", + "plt.ylabel('amplitude \\n [m/s]')\n", + "\n", + "plt.subplots_adjust(hspace=0.3)\n", + "plt.show();" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "celltoolbar": "Slideshow", + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "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": "3.9.7" + }, + "rise": { + "scroll": true + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/02-FFT_and_Basic_Filtering/2-fourier_transform.ipynb b/02-FFT_and_Basic_Filtering/2-fourier_transform.ipynb new file mode 100644 index 0000000..8d91c63 --- /dev/null +++ b/02-FFT_and_Basic_Filtering/2-fourier_transform.ipynb @@ -0,0 +1,554 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
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Signal Processing
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Fourier Transformation
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\n", + "\n", + "Seismo-Live: http://seismo-live.org\n", + "\n", + "##### Authors:\n", + "* Stefanie Donner ([@stefdonner](https://github.com/stefdonner))\n", + "* Celine Hadziioannou ([@hadzii](https://github.com/hadzii))\n", + "* Ceri Nunn ([@cerinunn](https://github.com/cerinunn))\n", + "\n", + "Some code used in this tutorial is taken from [stackoverflow.com](http://stackoverflow.com/questions/4258106/how-to-calculate-a-fourier-series-in-numpy/27720302#27720302). We thank [Giulio Ghirardo](https://www.researchgate.net/profile/Giulio_Ghirardo) for his kind permission to use his code here.\n", + "\n", + "---" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "code_folding": [ + 0 + ], + "slideshow": { + "slide_type": "fragment" + } + }, + "outputs": [], + "source": [ + "# Cell 0 - Preparation: load packages, set some basic options \n", + "%matplotlib inline\n", + "from scipy import signal\n", + "from obspy.signal.invsim import cosine_taper \n", + "from matplotlib import rcParams\n", + "import numpy as np\n", + "import matplotlib.pylab as plt\n", + "plt.style.use('ggplot')\n", + "plt.rcParams['figure.figsize'] = 15, 3" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "

Tutorial on Fourier transformation in 1D

\n", + "
" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## The Fourier transformation\n", + "\n", + "In the world of seismology, we use the *Fourier transformation* to transform a signal from the time domain into the frequency domain. That means, we split up the signal and separate the content of each frequency from each other. Doing so, we can analyse our signal according to energy content per frequency. We can extract information on how much amplitude each frequency contributes to the final signal. In other words: we get a receipt of the ingredients we need to blend our measured signal. " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "The *Fourier transformation* is based on the *Fourier series*. With the *Fourier series* we can approximate an (unknown) function $f(x)$ by another function $g_n(x)$ which consists of a sum over $N$ basis functions weighted by some coefficients. The basis functions need to be orthogonal. $sin$ and $cos$ functions seem to be a pretty good choice because any signal can be filtered into several sinusoidal paths. In the period range of $[-T/2 ; T/2]$ the *Fourier series* is defined as:\n", + "\n", + "$$\n", + "f(t) \\approx g_n(t) = \\frac{1}{2} a_0 + \\sum_{k=1}^N \\left[ a_k \\cos \\left(\\frac{2\\pi k t}{T} \\right) + b_k \\sin\\left(\\frac{2\\pi k t}{T}\\right)\\right]\n", + "$$\n", + "\n", + "$$ \n", + "a_k = \\frac{2}{T} \\int_{-T/2}^{T/2} f(t) \\cos\\left(\\frac{2\\pi k t}{T}\\right)dt\n", + "$$\n", + "\n", + "$$\n", + "b_k = \\frac{2}{T} \\int_{-T/2}^{T/2} f(t) \\sin\\left(\\frac{2\\pi k t}{T}\\right)dt\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "At this stage, we consider continuous, periodic and infinite functions. The more basis functions are used to approximate the unknown function, the better is the approximation, i.e. the more similar the unknown function is to its approximation. \n", + "\n", + "For a non-periodic function the interval of periodicity tends to infinity. That means, the steps between neighbouring frequencies become smaller and smaller and thus the infinite sum of the *Fourier series* turns into an integral and we end up with the integral form of the *Fourier transformation*:\n", + "\n", + "$$\n", + "F(\\omega) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} f(t) e^{-i\\omega t} dt \\leftrightarrow f(t) = \\int_{-\\infty}^{\\infty} F(\\omega)e^{i\\omega t}dt\n", + "$$\n", + "\n", + "Attention: sign and factor conventions can be different in the literature!" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "In seismology, we do not have continuous but discrete time signals. Therefore, we work with the discrete form of the *Fourier transformation*:\n", + "\n", + "$$\n", + "F_k = \\frac{1}{N} \\sum_{j=0}^{N-1} f_j e^{-2\\pi i k j /N} \\leftrightarrow f_k = \\sum_{j=0}^{N-1} F_j e^{2\\pi i k j /N}\n", + "$$\n", + "\n", + "Some intuitive gif animations on what the *Fourier transform* is doing, can be found [here](https://en.wikipedia.org/wiki/File:Fourier_series_and_transform.gif), [here](https://en.wikipedia.org/wiki/File:Fourier_series_square_wave_circles_animation.gif), and [here](https://en.wikipedia.org/wiki/File:Fourier_series_sawtooth_wave_circles_animation.gif).\n", + "Further and more detailed explanations on *Fourier series* and *Fourier transformations* can be found [here](https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/) and [here](www.fourier-series.com).\n", + "\n", + "---" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### The Fourier series and its coefficients \n", + "\n", + "In the following two code cells, we first define a function which calculates the coefficients of the Fourier series for a given function. The function in the next cell does it the other way round: it is creating a function based on given coefficients and weighting factors." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "code_folding": [ + 1 + ], + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 1: code by Giulio Ghirardo \n", + "def fourier_series_coeff(f, T, N):\n", + " \"\"\"Calculates the first 2*N+1 Fourier series coeff. of a periodic function.\n", + "\n", + " Given a periodic, function f(t) with period T, this function returns the\n", + " coefficients a0, {a1,a2,...},{b1,b2,...} such that:\n", + "\n", + " f(t) ~= a0/2+ sum_{k=1}^{N} ( a_k*cos(2*pi*k*t/T) + b_k*sin(2*pi*k*t/T) )\n", + " \n", + " Parameters\n", + " ----------\n", + " f : the periodic function, a callable like f(t)\n", + " T : the period of the function f, so that f(0)==f(T)\n", + " N_max : the function will return the first N_max + 1 Fourier coeff.\n", + "\n", + " Returns\n", + " -------\n", + " a0 : float\n", + " a,b : numpy float arrays describing respectively the cosine and sine coeff.\n", + " \"\"\"\n", + " # From Nyquist theorem we must use a sampling \n", + " # freq. larger than the maximum frequency you want to catch in the signal. \n", + " f_sample = 2 * N\n", + " \n", + " # We also need to use an integer sampling frequency, or the\n", + " # points will not be equispaced between 0 and 1. We then add +2 to f_sample.\n", + " t, dt = np.linspace(0, T, f_sample + 2, endpoint=False, retstep=True)\n", + " y = np.fft.rfft(f) / t.size\n", + " y *= 2\n", + " return y[0].real, y[1:-1].real[0:N], -y[1:-1].imag[0:N]" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "code_folding": [ + 1 + ], + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 2: code by Giulio Ghirardo \n", + "def series_real_coeff(a0, a, b, t, T):\n", + " \"\"\"calculates the Fourier series with period T at times t,\n", + " from the real coeff. a0,a,b\"\"\"\n", + " tmp = np.ones_like(t) * a0 / 2.\n", + " for k, (ak, bk) in enumerate(zip(a, b)):\n", + " tmp += ak * np.cos(2 * np.pi * (k + 1) * t / T) + bk * np.sin(\n", + " 2 * np.pi * (k + 1) * t / T)\n", + " return tmp" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "Now, we can create an arbitrary function, which we use to experiment with in the following example. \n", + "1) When you re-run cell 3 several times, do you always see the same function? Why? What does it tell you about the Fourier series? " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "outputs": [], + "source": [ + "# Cell 3: create periodic, discrete, finite signal\n", + "\n", + "# number of samples (intial value: 3000)\n", + "samp = 3000\n", + "# sample rate (initial value: 1)\n", + "dt = 1\n", + "# period\n", + "T = 1.0 / dt\n", + "length = samp * dt\n", + "# number of coefficients (initial value: 100)\n", + "N = 100\n", + "# weighting factors for coefficients (selected randomly)\n", + "a0 = np.random.rand(1)\n", + "a = np.random.randint(1, high=11, size=N)\n", + "b = np.random.randint(1, high=11, size=N)\n", + "\n", + "t = np.linspace(0, length, samp) # time axis\n", + "sig = series_real_coeff(a0, a, b, t, T)\n", + "\n", + "# plotting\n", + "plt.plot(t, sig, 'r', label='arbitrary, periodic, discrete, finite signal')\n", + "plt.ticklabel_format(axis='y', style='sci', scilimits=(-1,1))\n", + "plt.xlabel('time [sec]')\n", + "plt.ylabel('amplitude')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "Now, we can play with the signal and see what happens when we try to reconstruct it with a limited number of coefficients. \n", + "2) Run the cells 4 and 5. What do you observe? \n", + "3) Increase the number of coefficients $n$ step by step and re-run cells 4 and 5. What do you observe now? Can you explain? \n", + "4) In cell 5 uncomment the lines to make a plot which is not normalized (and comment the other two) and re-run the cell. What do you see now and can you explain it?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "outputs": [], + "source": [ + "# Cell 4: determine the first 'n' coefficients of the function using the code function of cell 1\n", + "T = 1 # period\n", + "n = 100 # number of coeffs to reconstruct\n", + "a0, a, b = fourier_series_coeff(sig, T, n)\n", + "a_ = a.astype(int)\n", + "b_ = b.astype(int)\n", + "print('coefficient a0 = ', int(a0))\n", + "print('array coefficients ak =', a_)\n", + "print('array coefficients bk =', b_)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 5: reconstruct the function using the code in cell 2\n", + "g = series_real_coeff(a0, a, b, t, dt)\n", + "\n", + "# plotting\n", + "#plt.plot(t, sig, 'r', label='original signal') # NOT normalized \n", + "#plt.plot(t, g, 'g', label='reconstructed signal')\n", + "plt.plot(t, sig/max(sig), 'r', label='original signal') # normalized \n", + "plt.plot(t, g/max(g), 'g', label='reconstructed signal')\n", + "\n", + "plt.ticklabel_format(axis='y', style='sci', scilimits=(-1,1))\n", + "plt.xlabel('time [sec]')\n", + "plt.ylabel('amplitude')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Fourier series, convergence and Gibb's phenomenon\n", + "\n", + "As seen above the convergence of the *Fourier series* can be tricky due to the fact that we work with signals of finite length. To analyse this effect in a bit more detail, we define a square wave in cell 6 and try to reconstruct it in cell 7. \n", + "5) First, we use only 5 coefficients to reconstruct the wave. Describe what you see. \n", + "6) Increase the number of coefficients $n$ in cell 7 step by step and re-run the cell. What do you see now? Can you explain it?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 6: define a square wave of 5 Hz\n", + "freq = 5.\n", + "npts = 3000\n", + "dt_ = 0.002\n", + "length = npts * dt_\n", + "t_ = np.linspace(0, length, npts, endpoint=False)\n", + "square = signal.square(2 * np.pi * freq * t_)\n", + "\n", + "plt.plot(t_, square)\n", + "plt.xlabel('time [sec]')\n", + "plt.ylabel('amplitude')\n", + "plt.xlim(0, 1.05)\n", + "plt.ylim(-1.2, 1.2)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "##### You may replace the square function by something else. What about a sawtooth function?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "outputs": [], + "source": [ + "signal.sawtooth?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 7: reconstruct signal using convergence criterion\n", + "n = 5 # number of coefficients (initial: 5)\n", + "T_ = 1/freq # period of signal\n", + "\n", + "# determine coefficients\n", + "a0 = 0\n", + "a = []\n", + "b = []\n", + "for i in range(1,n):\n", + " if (i%2 != 0):\n", + " a_ = 4/(np.pi*i)\n", + " else:\n", + " a_ = 0\n", + " a.append(a_)\n", + " b_ = (2*np.pi*i)/T_\n", + " b.append(b_)\n", + "\n", + "# reconstruct signal\n", + "g = np.ones_like(t_) * a0\n", + "for k, (ak, bk) in enumerate(zip(a, b)):\n", + " g += ak * np.sin(bk*t_)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 7b: plotting\n", + "plt.plot(t_, square, 'r', label='Analytisches Signal') \n", + "plt.plot(t_, g, 'g', label='Reihenentwicklung')\n", + "plt.ticklabel_format(axis='y', style='sci', scilimits=(-1,1))\n", + "plt.xlabel('time [sec]')\n", + "plt.ylabel('amplitude')\n", + "#plt.ylim(0.9,1.1)\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Fourier transformation\n", + "\n", + "Let us now do the Fourier transformation of the signal created in cell 3 and have a look on the amplitude spectra. In computer science the transformation is performed as fast Fourier transformation (FFT). \n", + "\n", + "7) Why do we need to taper the signal before we perform the FFT? \n", + "8) How do you interpret the plot of the amplitude spectra? \n", + "9) Which frequency contributes most to the final signal? " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "# Cell 8: FFT of signal\n", + "# number of sample points need to be the same as in cell 3\n", + "print('samp =',samp,' Need to be the same as in cell 3.')\n", + "# number of sample points need to be the same as in cell 3\n", + "print('T =',T,' Need to be the same as in cell 3.')\n", + "# percentage of taper applied to signal (initial: 0.1)\n", + "taper_percentage = 0.1\n", + "taper = cosine_taper(samp,taper_percentage)\n", + "\n", + "sig_ = square * taper\n", + "Fsig = np.fft.rfft(sig_, n=samp)\n", + "\n", + "# prepare plotting\n", + "xf = np.linspace(0.0, 1.0/(2.0*T), (samp//2)+1)\n", + "rcParams[\"figure.subplot.hspace\"] = (0.8)\n", + "rcParams[\"figure.figsize\"] = (15, 9)\n", + "rcParams[\"axes.labelsize\"] = (15)\n", + "rcParams[\"axes.titlesize\"] = (20)\n", + "rcParams[\"font.size\"] = (12)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false, + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [], + "source": [ + "#Cell 8b: plotting\n", + "plt.subplot(311)\n", + "plt.title('Time Domain')\n", + "plt.plot(t, square, linewidth=1)\n", + "plt.xlabel('Time [s]')\n", + "plt.ylabel('Amplitude')\n", + "\n", + "plt.subplot(312)\n", + "plt.title('Frequency Domain')\n", + "plt.plot(xf, 2.0/npts * np.abs(Fsig))\n", + "#plt.xlim(0, 0.04) \n", + "plt.xlabel('Frequency [Hz]')\n", + "plt.ylabel('Amplitude')\n", + "plt.show()" + ] + } + ], + "metadata": { + "anaconda-cloud": {}, + "celltoolbar": "Slideshow", + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "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": "3.9.7" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/README.md b/README.md index dcd4c19..d8621e7 100644 --- a/README.md +++ b/README.md @@ -9,3 +9,7 @@ The content of this repository is licensed under Creative Commons Attribution 4. ### 01 - Python Introduction This has been forked from [Seismo Live](http://seismo-live.org). The source code is available at https://github.com/krischer/seismo_live (licensed under a ***CC BY-NC-SA 4.0 License***. © 2015-2019 Lion Krischer). + +### 02 - Fourier-Transform and Basic Filtering + +This has been forked from [Seismo Live](http://seismo-live.org). The source code is available at https://github.com/krischer/seismo_live (licensed under a ***CC BY-NC-SA 4.0 License***. © 2015-2019 Lion Krischer).