{ "metadata": { "name": "More_Fourier_Transform" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Introduction\n", "--------------\n", "\n", "* Aliasing\n", "* Decimation\n", "* Interpolation " ] }, { "cell_type": "code", "collapsed": false, "input": [ "%qtconsole" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 273 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following plot shows that two different signal frequencies (i.e.,\n", "$f=0$, $f=f_s$) can generate the *exact* same samples." ] }, { "cell_type": "code", "collapsed": false, "input": [ "n = arange(5)\n", "fs = 0.25\n", "t = linspace(0,n.max()/fs,200)\n", "\n", "fig,ax = subplots()\n", "\n", "ax.stem(n/fs,cos(2*pi*n),basefmt='',label='samples')\n", "ax.plot(t,cos(2*pi*fs*t),'--g',label='f=fs')\n", "ax.plot(t,t*0+1,'r--',label='f=0')\n", "ax.set_ylim(top=1.5,bottom=-2.5)\n", "ax.set_xlim(left=-.5,right=t.max()*1.1)\n", "ax.grid()\n", "ax.set_xlabel('time',fontsize=18)\n", "ax.set_ylabel('Amplitude',fontsize=18)\n", "ax.set_title('Two signals have identical samples',fontsize=18)\n", "ax.legend(loc=0)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 274, "text": [ "" ] }, { "output_type": "display_data", "png": 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hceho0VHqGPXpf5RXl8NhgwOuzrwKV3NXqWPUp/9RXl0O23W2uD33NuxM7KSO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\nrl0LhUKh1ZaRkQG5XI6amhosWbIEGRkZJopOmPTlnGEYvP/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"text": [ "" ] } ], "prompt_number": 274 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is the *aliasing* problem and it is a fact of sampling. In fact, its implications are even more problematic." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from __future__ import division\n", "from matplotlib.patches import FancyArrow\n", "import mpl_toolkits.mplot3d.art3d as art3d\n", "from mpl_toolkits.mplot3d.art3d import Poly3DCollection\n", "import matplotlib.gridspec as gridspec\n", "\n", "def dftmatrix(Nfft=32,N=None):\n", " 'construct DFT matrix'\n", " k= np.arange(Nfft)\n", " if N is None: N = Nfft\n", " n = arange(N)\n", " U = matrix(exp(1j* 2*pi/Nfft *k*n[:,None])) # use numpy broadcasting to create matrix\n", " return U/sqrt(Nfft)\n", "\n", "def facet_filled(x,alpha=0.5,color='b'):\n", " 'construct 3D facet from adjacent points filled to zero'\n", " a,b=x\n", " a0= a*array([1,1,0])\n", " b0= b*array([1,1,0])\n", " ve = vstack([a,a0,b0,b]) # create closed polygon facet\n", " poly = Poly3DCollection([ve]) # create facet\n", " poly.set_alpha(alpha)\n", " poly.set_color(color)\n", " return poly\n", "\n", "def drawDFTView(X,ax=None,fig=None):\n", " 'above code as a function. Draws 3D diagram given DFT matrix'\n", " a=2*pi/len(X)*arange(len(X))\n", " d=vstack([cos(a),sin(a),array(abs(X)).flatten()]).T\n", " if ax is None and fig is None:\n", " fig = plt.figure()\n", " fig.set_size_inches(6,6)\n", " \n", " if ax is None: # add ax to existing figure\n", " ax = fig.add_subplot(1, 1, 1, projection='3d')\n", " \n", " ax.axis([-1,1,-1,1])\n", " ax.set_zlim([0,d[:,2].max()])\n", " ax.set_aspect(1)\n", " ax.view_init(azim=-30)\n", " a=FancyArrow(0,0,1,0,width=0.02,length_includes_head=True)\n", " ax.add_patch(a)\n", " b=FancyArrow(0,0,0,1,width=0.02,length_includes_head=True)\n", " ax.add_patch(b)\n", " art3d.patch_2d_to_3d(a)\n", " art3d.patch_2d_to_3d(b)\n", " ax.axis('off')\n", "\n", " sl=[slice(i,i+2) for i in range(d.shape[0]-2)] # collect neighboring points\n", " for s in sl:\n", " poly=facet_filled(d[s,:])\n", " ax.add_collection3d(poly)\n", " \n", " # edge polygons \n", " ax.add_collection3d(facet_filled(d[[-1,0],:]))\n", " ax.add_collection3d(facet_filled(d[[-2,-1],:]))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 275 }, { "cell_type": "code", "collapsed": false, "input": [ "from scipy.signal import chirp\n", "\n", "f0 = 0 # start frequency\n", "t1 = 2 # end-time\n", "f1 = 10 # frequency at end-time\n", "fs = 40 # sample rate\n", "\n", "t = arange(0,t1,1/fs)\n", "x = chirp(t,f0,t1,f1)\n", "\n", "#drawDFTView(fft.fft(x,512),ax=None,fig=None)\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 276 }, { "cell_type": "code", "collapsed": false, "input": [ "def dftmatrix(Nfft=32,N=None):\n", " 'construct DFT matrix'\n", " k= np.arange(Nfft)\n", " if N is None: N = Nfft\n", " n = arange(N)\n", " U = matrix(exp(1j* 2*pi/Nfft *k*n[:,None])) # use numpy broadcasting to create matrix\n", " return U/sqrt(Nfft)\n", "\n", "plot(U[:,1].real)\n", "\n", "U = dftmatrix(128*2,32)\n", "drawDFTView(U.H*(U[:,0]+U[:,100]).real)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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PzDU2en5d/XfouvJ6OAz85CdZO0S9HoquRfbfXywGQH78lZXy+LLL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\n1j/4IH1HtPJyEX/ztli39Pr0EfGeO9d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"text": [ "" ] } ], "prompt_number": 277 }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\\Omega_k = \\frac{2\\pi}{N} k$$\n", "\n", "and for sampled frequency,\n", "\n", "$$f_k = \\frac{f_s}{N} k$$\n", "\n", "\n", "$$\\delta f = f_s/N$$\n", "\n", "$$\\Omega_{k+N} = \\Omega_k$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "Nf = 64\n", "fs = 64 # delta_f = 1 Hz\n", "f = 10\n", "t = arange(0,1,1/fs)\n", "deltaf = 1/2.\n", "\n", "fig,ax = subplots(2,1,sharex=True,sharey=True)\n", "\n", "x=cos(2*pi*f*t) + cos(2*pi*(f+2)*t)\n", "X = fft.fft(x,Nf)\n", "ax[0].plot(linspace(0,fs,len(X)),abs(X),'-o')\n", "ax[0].set_title(r'$\\delta f = 2$',fontsize=18)\n", "ax[0].set_ylabel(r'$|X(k)|$',fontsize=18)\n", "ax[0].grid()\n", "\n", "x=cos(2*pi*f*t) + cos(2*pi*(f+deltaf)*t)\n", "X = fft.fft(x,Nf)\n", "ax[1].plot(linspace(0,fs,len(X)),abs(X),'-o')\n", "ax[1].set_title(r'$\\delta f = 1/2$',fontsize=18)\n", "ax[1].set_ylabel(r'$|X(k)|$',fontsize=18)\n", "ax[1].set_xlabel('Frequency (Hz)',fontsize=18)\n", "ax[1].set_xlim(xmax = fs/2)\n", "ax[1].grid()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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WN2ypxvv372PcuHGYNm2aPoeSl5cXJBIJJBIJZsyYgePHjzeobiFR80lO6CgUwN27coN9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\np2natGl6TUJvJwBo27YtRo8ejVOnTjV6GwnWOYh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"text": [ "" ] } ], "prompt_number": 282 }, { "cell_type": "code", "collapsed": false, "input": [ "Nf = 64*2\n", "fig,ax = subplots(2,1,sharex=True,sharey=True)\n", "fig.set_size_inches((6,6))\n", "\n", "X = fft.fft(x,Nf)\n", "ax[0].plot(linspace(0,fs,len(X)),abs(X),'-o',ms=3.)\n", "ax[0].set_title(r'$N=%d$'%Nf,fontsize=18)\n", "ax[0].set_ylabel(r'$|X(k)|$',fontsize=18)\n", "ax[0].grid()\n", "\n", "\n", "Nf = 64*4\n", "X = fft.fft(x,Nf)\n", "ax[1].plot(linspace(0,fs,len(X)),abs(X),'-o',ms=3.)\n", "ax[1].set_title(r'$N=%d$'%Nf,fontsize=18)\n", "ax[1].set_ylabel(r'$|X(k)|$',fontsize=18)\n", "ax[1].set_xlabel('Frequency (Hz)',fontsize=18)\n", "ax[1].set_xlim(xmax = fs/2)\n", "ax[1].grid()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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WDD2OlXqq5gDarDvo8RyVx2j5gPFy8umaw+jRo9m0aZNTW1JSEtHR0Rw4cIC+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\nOHHC0aYqeN3Za6+9RkxMDH5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5NYfLly8zZswY+vXrx6hRo6hTp06lDmqz2Vi9ejV///vf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\ngg==\n", "text": [ "" ] } ], "prompt_number": 283 }, { "cell_type": "code", "collapsed": false, "input": [ "t = arange(0,2,1/fs)\n", "x=cos(2*pi*f*t) + cos(2*pi*(f+deltaf)*t)\n", "\n", "Nf = 64*2\n", "fig,ax = subplots(2,1,sharex=True,sharey=True)\n", "fig.set_size_inches((6,6))\n", "\n", "X = fft.fft(x,Nf)\n", "ax[0].plot(linspace(0,fs,len(X)),abs(X),'-o',ms=3.)\n", "ax[0].set_title(r'$N=%d$'%Nf,fontsize=18)\n", "ax[0].set_ylabel(r'$|X(k)|$',fontsize=18)\n", "ax[0].grid()\n", "\n", "Nf = 64*8\n", "X = fft.fft(x,Nf)\n", "ax[1].plot(linspace(0,fs,len(X)),abs(X),'-o',ms=3.)\n", "ax[1].set_title(r'$N=%d$'%Nf,fontsize=18)\n", "ax[1].set_ylabel(r'$|X(k)|$',fontsize=18)\n", "ax[1].set_xlabel('Frequency (Hz)',fontsize=18)\n", "ax[1].set_xlim(xmax = fs/2)\n", "ax[1].grid()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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0jEYjgwcPtq7LyckhJCQEf39/QkNDycvLs2OEbUMPtdLqZaXG9Bwcoaykh2NU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\nuXNMW7FCnt/PT755S8QijrrRxP79bGlH5axUWFuT+hhVpKe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"text": [ "" ] } ], "prompt_number": 298 }, { "cell_type": "code", "collapsed": false, "input": [ "#t = arange(0,10,1/fs).reshape(10,-1)\n", "#x=cos(2*pi*f*t) + cos(2*pi*(f+deltaf)*t)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 285 }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\\frac{\\sin \\left( N_s \\frac{2\\pi}{N} k\\right)}{\\sin \\left( \\frac{2\\pi}{N} k \\right)}$$" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Summary" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this section, we considered the Discrete Fourier Transform (DFT) using a matrix/vector approach. We used this approach to develop an intuitive visual vocabulary for the DFT with respect to high/low frequency and real-valued signals. We recognized that zero-padding an input signal is the same as analyzing more discrete frequencies in the transform domain.\n", "\n", "As usual, the corresponding IPython notebook for this post is available for download [here](https://github.com/unpingco/Python-for-Signal-Processing/blob/master/Fourier_Transform.ipynb). \n", "\n", "Comments and corrections welcome!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "References\n", "---------------\n", "\n", "* Oppenheim, A. V., and A. S. Willsky. \"Signals and Systems.\" Prentice-Hall, (1997)." ] } ], "metadata": {} } ] }