{ "metadata": { "name": "", "signature": "sha256:ff5882201349459a8ca39449dacab805ee2d5f2b22e098fe97f0e97f84d7ca31" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": { "internals": { "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "Working with NumPy FFT Results: Scaling and Folding" ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "Imports\n", "-------" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns\n", "sns.set_context(\"talk\")" ], "language": "python", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "outputs": [], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "Create a Test Signal\n", "--------------------\n", "$f_s$ is the sampling frequency, while $f$ is a base frequency for the signal content. We create a signal that contains components at a couple of multiples of this base frequency. Note the amplitudes here since we will be trying to extract those correctly from the FFT later." ] }, { "cell_type": "code", "collapsed": false, "input": [ "f_s = 50.0 # Hz\n", "f = 1.0 # Hz\n", "time = np.arange(0.0, 3.0, 1/f_s)\n", "x = 5 * np.sin(2 * np.pi * f * time) + 2 * np.sin(10 * 2 * np.pi * f * time)" ], "language": "python", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "outputs": [], "prompt_number": 2 }, { "cell_type": "code", "collapsed": false, "input": [ "plt.plot(time, x)\n", "plt.xlabel(\"Time (sec)\")\n", "plt.ylabel(\"x\")" ], "language": "python", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 3, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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EkgknDR2nnko1BKkQiSC0plBGy4edeYGUAY3gABA+r1PU1W7m06HNVur9xnOM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"text": [ "" ] } ], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "Compute the FFT\n", "---------------\n", "The FFT and a matching vector of frequencies" ] }, { "cell_type": "code", "collapsed": false, "input": [ "fft_x = np.fft.fft(x)\n", "n = len(fft_x)\n", "freq = np.fft.fftfreq(n, 1/f_s)\n", "print n\n", "print freq" ], "language": "python", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "150\n", "[ 0. 0.33333333 0.66666667 1. 1.33333333\n", " 1.66666667 2. 2.33333333 2.66666667 3. 3.33333333\n", " 3.66666667 4. 4.33333333 4.66666667 5. 5.33333333\n", " 5.66666667 6. 6.33333333 6.66666667 7. 7.33333333\n", " 7.66666667 8. 8.33333333 8.66666667 9. 9.33333333\n", " 9.66666667 10. 10.33333333 10.66666667 11. 11.33333333\n", " 11.66666667 12. 12.33333333 12.66666667 13. 13.33333333\n", " 13.66666667 14. 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-5. -4.66666667\n", " -4.33333333 -4. -3.66666667 -3.33333333 -3. -2.66666667\n", " -2.33333333 -2. -1.66666667 -1.33333333 -1. -0.66666667\n", " -0.33333333]\n" ] } ], "prompt_number": 4 }, { "cell_type": "code", "collapsed": false, "input": [ "plt.plot(np.abs(fft_x))" ], "language": "python", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 5, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "Swap Half Spaces\n", "----------------\n", "Note that frequencies in the FFT and the `freq` vector go from zero to some larger positive number then from a large negative number back toward zero. We can swap that so that the DC component is in the center of the vector while maintaining a two-sided spectrum." ] }, { "cell_type": "code", "collapsed": false, "input": [ "fft_x_shifted = np.fft.fftshift(fft_x)\n", "freq_shifted = np.fft.fftshift(freq)" ], "language": "python", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "outputs": [], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "plt.plot(freq_shifted, np.abs(fft_x_shifted))\n", "plt.xlabel(\"Frequency (Hz)\")" ], "language": "python", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 7, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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6hRFxLXAQuAV4cUrpLBDA7aXX3UUWTF4C3FrtcKX14MT8ZhVns4jD6rZC6ZeG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\nqmcvg7tLgTsmtt0OPGoPf6Z0Xhpl7rY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"text": [ "" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "Fold Negative Frequencies and Scale\n", "------------------------------\n", "It's actually more common to look at just the first half of the unshifted FFT and frequency vectors and fold all the amplitude information into the positive frequencies. Furthermore, to get ampltude right, we must normalize by the length of the original FFT. Note the factor of $2/n$ in the following which accomplishes both the folding and scaling." ] }, { "cell_type": "code", "collapsed": false, "input": [ "half_n = np.ceil(n/2.0)\n", "fft_x_half = (2.0 / n) * fft_x[:half_n]\n", "freq_half = freq[:half_n]" ], "language": "python", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "outputs": [], "prompt_number": 8 }, { "cell_type": "code", "collapsed": false, "input": [ "plt.plot(freq_half, np.abs(fft_x_half))\n", "plt.xlabel(\"Frequency (Hz)\")\n", "plt.ylabel(\"Amplitude\")" ], "language": "python", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 9, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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GBj4z+/rAQ4nSPfgGP0/c/dACrm0hxrZ0OUsXAAAEaFyF74WFj79J0k8r3Wj5FqVr/x4u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\nN30O6/gAoEriCXz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"text": [ "" ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": { "internals": { "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "-" } }, "source": [ "Now the spectrum contains spikes at the correct amplitudes at only positive frequencies, which are the only ones with physicality." ] } ], "metadata": {} } ] }