{ "cells": [ { "cell_type": "markdown", "metadata": { "internals": { "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Fourier series\n", "\n", "> Marcos Duarte \n", "> Laboratory of Biomechanics and Motor Control ([http://demotu.org/](http://demotu.org/)) \n", "> Federal University of ABC, Brazil" ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "## Sinusoids\n", "\n", "The sine and cosine functions (with the same amplitude and frequency) are different only by a constant phase $(^\\pi/_2)$: \n", "\n", "$$\\cos(t)=\\sin(t+^\\pi/_2) \\quad , \\quad \\sin(t)=\\cos(t-^\\pi/_2)$$\n", "\n", "This means for instance that we can use only sines to represent both sines and cosines. Because of that we can refer to both sine and cosine waves as sinusoids. The relation between the sine and cosine functions can be verified using the trigonometric identities, for example:\n", "\n", "$$ \\sin(\\alpha + ^\\pi/_2) = \\sin{\\alpha} \\cos{^\\pi/_2} + \\cos{\\alpha} \\sin{^\\pi/_2} = \\cos{\\alpha} $$\n", "\n", "$$ \\cos(\\alpha - ^\\pi/_2) = \\cos{\\alpha} \\cos{^\\pi/_2} + \\sin{\\alpha} \\sin{^\\pi/_2} = \\sin{\\alpha} $$\n", "\n", "Sinusoids are periodic functions, that is, if their period is $T$, then $ x(t+T) = x(t)$.\n", "\n", "A cosine is an even function and the sine an odd function, as illustrated next." ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "In the XIX century, the French mathematician [Jean-Baptiste Joseph Fourier](http://en.wikipedia.org/wiki/Joseph_Fourier) made the discovery that any (in fact, almost any) complicated but periodic function could be represented as a sum of sines and cosines with varying amplitudes and frequencies, which are known as the [Fourier series](http://en.wikipedia.org/wiki/Fourier_series), in his honor. \n", "Based on the Fourier series, he also proposed a mathematical transformation to transform functions between time (or spatial) domain and frequency domain, known as the [Fourier transform](http://en.wikipedia.org/wiki/Fourier_transform). \n", "Nowadays, the Fourier transform is typically implemented employing an algorithm known as the [Fast Fourier Transform (FFT)](http://en.wikipedia.org/wiki/Fast_Fourier_transform). \n", "\n", "Let's look at some basic concepts about sinusoids and then we will look at the Fourier series. \n", "You should be familiar with [basic trigonometry](http://nbviewer.ipython.org/github/demotu/BMC/blob/master/notebooks/TrigonometryBasics.ipynb) and the [basic properties of signals](http://nbviewer.ipython.org/github/demotu/BMC/blob/master/notebooks/SignalBasicProperties.ipynb) before proceeding." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "ExecuteTime": { "end_time": "2017-12-30T09:05:21.889326Z", "start_time": "2017-12-30T09:05:21.635703Z" }, "internals": {}, "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [ "import numpy as np\n", "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "import sys\n", "sys.path.insert(1, r'./../functions') # directory of BMC Python functions" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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/+UZ1oyKEEL5sz/k9TN0y1VGe1nwaFQtWNDEia9I0jfmPzyd/zvwAHI8+zuiN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\nE9XxrhDZ2auvQrNm6rGuq2YE9s6ohbXNf3w+lQq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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from even_odd_plot import even_odd_plot\n", "ax = even_odd_plot()" ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "### Integral of sinusoids\n", "\n", "The integral of a sine or cosine function over one period (as matter of fact, over any integer multiple of the period) is zero:\n", "\n", "$$ \\int_{t_0}^{t_0+T} \\sin(k \\frac{2 \\pi}{T} t) \\:\\mathrm{d}t = 0 $$\n", "\n", "$$ \\int_{t_0}^{t_0+T} \\cos(k \\frac{2 \\pi}{T} t) \\:\\mathrm{d}t = 0 $$\n", "\n", "Where $k$ is an integer and $ \\sin(k2\\pi t/T) $ or $ \\cos(k2\\pi t/T) $ are the harmonics with periodicities $T/k$. \n", "\n", "This means that the total area over one period under the sine or cosine curve is zero; in other words, the average value of these functions is zero (the average value is the integral (area) over one period divided by the period), as illustrated in the next figure." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "internals": {}, "run_control": { "breakpoint": false }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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zjSR9tcQbid9CoKNSqq1SKgS4Gqiwy1ZsbKwXDikC2R1XDubZS7oBMHXzKtbs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\nmttZRgjhBMLCVPFf79xe3WfPwiOPwKVLxsblyk6eVN+DvKQvMFBt+dp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AAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from sinusoid_int_plot import sinusoid_int_plot\n", "ax = sinusoid_int_plot()" ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "### Orthogonality of sinusoids\n", "\n", "Another important property of sinusoids is that they are orthogonal at different frequencies and cosines and sines are always orthogonal to each other. That is, if we multiply two sinusoids and integrate over one period the value is zero if the frequencies are different:\n", "\n", "$$ \\int_{t_0}^{t_0+T} \\sin(m \\frac{2 \\pi}{T} t)\\sin(n \\frac{2 \\pi}{T} t) \\:\\mathrm{d}t = \\left\\{ \n", "\\begin{array}{l l}\n", " 0 & \\quad \\text{if } m \\neq n \\\\\n", " T/2 & \\quad \\text{if } m = n \\neq 0\n", "\\end{array} \\right. $$\n", "\n", "$$ \\int_{t_0}^{t_0+T} \\cos(m \\frac{2 \\pi}{T} t)\\cos(n \\frac{2 \\pi}{T} t) \\:\\mathrm{d}t = \\left\\{ \n", "\\begin{array}{l l}\n", " 0 & \\quad \\text{if } m \\neq n \\\\\n", " T/2 & \\quad \\text{if } m = n \\neq 0\n", "\\end{array} \\right. $$\n", "\n", "And:\n", "\n", "$$ \\int_{t_0}^{t_0+T} \\sin(m \\frac{2 \\pi}{T} t)\\cos(n \\frac{2 \\pi}{T} t) \\:\\mathrm{d}t = \n", "\\begin{array}{l l}\n", " 0, & \\quad \\quad \\text{for all } n, m\n", "\\end{array} $$\n", "\n", "The term orthogonal here means independent and has a similar meaning as in linear algebra. It's not that cosines and sines are geometrically perpendicular to each other as are two orthogonal vectors in 3D space, but rather the fact that one vector (or sinusoid) cannot be expressed in terms of the other. For example, the following set forms an orthogonal set:\n", "\n", "$$ 1, \\cos x, \\sin x, cos 2x, \\sin 2x, \\dots $$\n", "\n", "From linear algebra, we know that three orthogonal vectors in 3D space can form a basis, a set of linearly independent vectors that, in a linear combination, can represent every vector. In this sense, cosines and sines can also form a basis and they can be used in a linear combination to represent other mathematical functions. This is the essence of the Fourier series." ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "### Euler's formula\n", "\n", "There is an interesting relationship between the sine, cosine, and exponential functions and complex numbers, known as the [Euler's formula](http://en.wikipedia.org/wiki/Euler%27s_formula):\n", "\n", "$$ e^{i\\theta} = \\cos \\theta + i \\sin \\theta $$\n", "\n", "Where $e$ is the Napier's constant or Euler's number ($e=2.71828...$, the limit of $(1 + 1/n)^n$ as $n$ approaches infinity, and it's also the inverse of the [natural logarithm](http://en.wikipedia.org/wiki/Natural_logarithm) function: $ln(e)=1$) and $i$ is the [imaginary unit](http://en.wikipedia.org/wiki/Imaginary_number) ($\\sqrt{-1}=i$). \n", "Using the Euler's formula, we can express the cosine and sine functions as:\n", "\n", "$$ \\cos \\theta = \\frac{e^{i\\theta} + e^{-i\\theta}}{2} $$\n", "\n", "$$ \\sin \\theta = \\frac{e^{i\\theta} - e^{-i\\theta}}{2i} $$\n", "\n", "Of note, a special case of the Euler's formula, known as the [Euler's identity](http://en.wikipedia.org/wiki/Euler's_identity), has been referred as one of the most beautiful equations [[1](http://en.wikipedia.org/wiki/Mathematical_beauty)]:\n", "\n", "$$ e^{i\\pi} + 1 = 0 $$\n", "\n", "It may sound strange why one would use exponential functions and complex numbers instead of the 'simpler' sine and cosine functions, but it turns out that calculating the Fourier series is simpler with the exponential form." ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "## Fourier series\n", "\n", "Without further ado, a periodic function can be represented as a sum of sines and cosines, known as the [Fourier series](http://en.wikipedia.org/wiki/Fourier_series):\n", "\n", "$$ x(t) \\sim \\sum_{k=0}^\\infty a_k \\cos\\left(k\\frac{2\\pi}{T}t\\right) + b_k \\sin\\left(k\\frac{2\\pi}{T}t\\right) $$\n", "\n", "Where $a_k$ and $b_k$ are the Fourier coefficients. \n", "\n", "When $k=0$, $\\cos(0t)=1$ and $\\sin(0t)=0$, then the Fourier series can be written as:\n", "\n", "$$ x(t) \\sim a_0 + \\sum_{k=1}^\\infty a_k \\cos\\left(k\\frac{2\\pi}{T}t\\right) + b_k \\sin\\left(k\\frac{2\\pi}{T}t\\right) $$" ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "### Determination of the Fourier coefficients\n", "\n", "Let's deduce how to calculate the Fourier coefficients $a_0$, $a_k$, and $b_k$. \n", "The coefficient $a_0$ can be computed if we integrate the Fourier series over one period:\n", "\n", "$$ \\int_{t_0}^{t_0+T} x(t) \\mathrm{d}t = \\int_{t_0}^{t_0+T} \\left[ a_0 dt + \\sum_{k=1}^\\infty a_k \\cos\\left(k\\frac{2\\pi}{T}t\\right) + b_k \\sin\\left(k\\frac{2\\pi}{T}t\\right) \\right] \\:\\mathrm{d}t $$\n", "\n", "On the right side, the first term is the integral of a constant and the integral of the summation is zero because it's the integral over one period of sines and cosines. Then:\n", "\n", "$$ a_0 = \\frac{1}{T} \\int_{t_0}^{t_0+T} x(t) \\:\\mathrm{d}t $$\n", "\n", "Which is the average value of the function $x(t)$ over the period $T$. \n", "\n", "The coefficients $a_n$ and $b_n$ can be computed if we multiply the Fourier series respectively by $cos(\\cdot)$ and $sin(\\cdot)$ and integrate over one period (deduction not shown):\n", "\n", "$$ a_k = \\frac{2}{T} \\int_{t_0}^{t_0+T} x(t) \\cos\\left(k\\frac{2\\pi}{T}t\\right) \\:\\mathrm{d}t $$\n", "\n", "$$ b_k = \\frac{2}{T} \\int_{t_0}^{t_0+T} x(t) \\sin\\left(k\\frac{2\\pi}{T}t\\right) \\:\\mathrm{d}t $$\n", "\n", "Note that $a_n$ and $b_n$ have the integrals multiplied by 2 while $a_0$ does not have this multiplication. Because of that, to make the equation for $a_0$ consistent with the equations for $a_n$ and $b_n$, it's common to also multiply by 2 the equation for $a_0$ and then express the Fourier series as:\n", "\n", "$$ x(t) \\sim \\frac{a_0}{2} + \\sum_{k=1}^\\infty a_k \\cos\\left(k\\frac{2\\pi}{T}t\\right) + b_k \\sin\\left(k\\frac{2\\pi}{T}t\\right) $$" ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "### Convergence of the Fourier series\n", "\n", "The Fourier series is an expansion of a function and the following theorem states that the Fourier series actually converges to the function:\n", "\n", "> The Fourier series of a piecewise continuous function $f(x)$ converges for all values of $x$ where $f(x)$ is continuous. At the values where $f(x)$ is discontinuous, the Fourier series is equal to the average of the right and left limits of $f(x)$:\n", "\n", "$$ \\frac{1}{2}\\left[f(x^+) + f(x^-)\\right] $$" ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "### Example: Fourier series for a square wave function\n", "\n", "Consider the following periodic function (a square wave with period $T=2\\:s$):\n", "\n", "$$ x(t) = \\left\\{ \n", "\\begin{array}{l l}\n", " -1 & \\quad \\text{if } -1 \\leq t < 0\\\\\n", " +1 & \\quad \\text{if } 0 \\leq t < +1\n", "\\end{array} \\right. $$\n", " \n", "$$ x(t + 2) = x(t), \\quad \\text{for all } t$$\n", "\n", "Note that this function is discontinuous at $ t=\\dots,-3,-2,-1,\\;0,\\;1,\\;2,\\;3,\\dots$.\n", "\n", "Which has the following plot:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "internals": {}, "run_control": { "breakpoint": false }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from square_wave_plot import square_wave_plot\n", "ax = square_wave_plot()\n", "%config InlineBackend.close_figures=False # hold plot for next cell" ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "The Fourier coefficients are given by ($a_0=0$ because the average value of $x(t)$ is zero):\n", "\n", "$$ a_k = \\frac{2}{2}\\left[\\int_{-1}^0 -\\cos\\left(k\\frac{2\\pi}{2}t\\right) \\:\\mathrm{d}t + \\int_{0}^1 \\cos\\left(k\\frac{2\\pi}{2}t\\right) \\:\\mathrm{d}t \\right] $$\n", "\n", "$$ b_k = \\frac{2}{2}\\left[\\int_{-1}^0 -\\sin\\left(k\\frac{2\\pi}{2}t\\right) \\:\\mathrm{d}t + \\int_{0}^1 \\sin\\left(k\\frac{2\\pi}{2}t\\right) \\:\\mathrm{d}t \\right] $$\n", "\n", "Which results in:\n", "\n", "$$ a_k = 0 $$\n", "\n", "$$ b_k = \\left\\{ \n", "\\begin{array}{c l}\n", " 0 & \\quad \\text{if k is even} \\\\\n", " \\frac{4}{k\\pi} & \\quad \\text{if k is odd}\n", "\\end{array} \\right. $$\n", "\n", "Then, the Fourier series for $x(t)$ is:\n", "\n", "$$ x(t) = \\frac{4}{\\pi} \\left[ \\sin(\\pi t) + \\frac{\\sin(3\\pi t)}{3} + \\frac{\\sin(5\\pi t)}{5} + \\frac{\\sin(7\\pi t)}{7} + \\frac{\\sin(9\\pi t)}{9} + \\dots \\right] $$\n", "\n", "The fact that the Fourier series of the given square wave has only sines is expected because the given square wave is an odd function, $x(t) = -x(-t)$. Then, its Fourier series should have only odd functions (sines). \n", "\n", "Let's plot it:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "internals": {}, "run_control": { "breakpoint": false }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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301qyhK5p185x+Ny5FD5ihBNRYnZLL2y7ojrY4PRDHObL4cMWo80ZAwZQnFmz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\ndjoTF0f52KGDewmkpEgZEUHXrFhB5zIypGzdms6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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "t = np.linspace(-3, 3, 1000)\n", "x = 4/np.pi*(np.sin(1*np.pi*t)/1 + np.sin(3*np.pi*t)/3 + np.sin(5*np.pi*t)/5 +\n", " np.sin(7*np.pi*t)/7 + np.sin(9*np.pi*t)/9)\n", "\n", "ax.plot(t, x, color='r', linewidth=2, label='Fourier series')\n", "ax.set_title('Square wave function and the first 5 non-zero terms of its Fourier series',\n", " fontsize=14)\n", "%config InlineBackend.close_figures=True # hold off the figure\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And here are plots for each of the first five non-zero components:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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yBSlRUVZLdQORcXGoP3gwggBse/FFnD5wgCmVfJDyTz+NKpMmIeXxx7FhyhRc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\np5UuTfTaa0QbNxKNGUNUrhxvDwjg7f/9R5Sby+ldunAaQFS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i9ufk4JzVDPYVfv0VtaZN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baCDh2MVuMe3mjPGzcAJfpDp06GSnEbb7Rnbi7w55+83qWLsVq8BOkASiQSia+hvGD//ddxek4O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\nr7nGG3aNVN+jj7ph0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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "t = np.linspace(-3, 3, 100)\n", "fig, axs = plt.subplots(5, 2, figsize=(9, 5), sharex=True, sharey=True, squeeze=True)\n", "axs = axs.flat\n", "x2 = t*0\n", "for k, ax in enumerate(axs):\n", " if ~k%2:\n", " xf = 4/np.pi*np.sin((k+1)*np.pi*t)/(k+1)\n", " ax.plot(t, xf, 'r', linewidth=2)\n", " x2 = x2 + xf\n", " else:\n", " ax.plot(t, x2, 'r', linewidth=2)\n", " ax.xaxis.set_ticks_position('bottom')\n", " ax.set_ylim(-2, 2)\n", " ax.set_yticks([-1.5, 0, 1.5])\n", " ax.grid()\n", "ax.set_xlabel('Time (s)', fontsize=16)\n", "axs[-2].set_xlabel('Time (s)', fontsize=16)\n", "axs[4].set_ylabel('Amplitude', fontsize=16)\n", "axs[0].set_title('First five harmonics')\n", "axs[1].set_title('Sum of the harmonics')\n", "fig.tight_layout()\n", "plt.subplots_adjust(hspace=0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This animation resumes the steps we have done to compute the Fourier series of a square wave function: \n", "
\"Fourier
Figure. Fourier series of a square wave function (from Wikipedia).
" ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "### Example: Fourier series for a triangular wave function\n", "\n", "Consider the following periodic function (a triangular wave with period $T=2\\pi$):\n", "\n", "$$ x(t) = \\left\\{ \n", "\\begin{array}{l l}\n", " 1+t/\\pi & \\quad \\text{if } -\\pi \\leq t < 0\\\\\n", " 1-t/\\pi & \\quad \\text{if } 0 \\leq t < +\\pi\n", "\\end{array} \\right. $$\n", " \n", "$$ x(t + 2\\pi) = x(t), \\quad \\text{for all } t$$\n", "\n", "Which has the following plot:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "internals": {}, "run_control": { "breakpoint": false }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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XroSjoyM2btyIjRs34tChQ03WNW7cOLz99tsIDAxEQoI2E+bBd5Rjwt+PIq+k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\nqMbCLdGoqmk67xzghd/c2U+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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from triangular_wave_plot import triangular_wave_plot\n", "ax = triangular_wave_plot()\n", "%config InlineBackend.close_figures=False # hold plot for next cell" ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "The Fourier coefficients are given by:\n", "\n", "$$ a_0 = \\frac{1}{2\\pi} \\left[ \\int_{-\\pi}^0 \\left(1+t/\\pi\\right) \\:\\mathrm{d}t + \\int_{0}^{\\pi} \\left(1-t/\\pi\\right) \\:\\mathrm{d}t \\right] $$\n", "\n", "$$ a_k = \\frac{1}{\\pi} \\left[ \\int_{-\\pi}^0 \\left(1+t/\\pi\\right)\\cos\\left(k\\frac{2\\pi}{T}t\\right) \\:\\mathrm{d}t + \\int_{0}^{\\pi} \\left(1-t/\\pi\\right)\\cos\\left(k\\frac{2\\pi}{T}t\\right) \\:\\mathrm{d}t \\right] $$\n", "\n", "$$ b_k = \\frac{1}{\\pi} \\left[ \\int_{-pi}^0 \\left(1+t/\\pi\\right)\\sin\\left(k\\frac{2\\pi}{T}t\\right) \\mathrm{d}t + \\int_{0}^{\\pi} \\left(1-t/\\pi\\right)\\sin\\left(k\\frac{2\\pi}{T}t\\right) \\:\\mathrm{d}t \\right] $$\n", "\n", "Which results in:\n", "\n", "$$ a_0 = 1/2 $$\n", "\n", "$$ a_k = \\left\\{ \n", "\\begin{array}{c l}\n", " \\frac{4}{k^2\\pi^2} & \\quad \\text{if k is odd} \\\\\n", " 0 & \\quad \\text{if k is even}\n", "\\end{array} \\right. $$\n", "\n", "$$ b_k = 0 $$\n", "\n", "Then, the Fourier series for $x(t)$ is:\n", "\n", "$$ x(t) = \\frac{1}{2} + \\frac{4}{\\pi^2} \\left[ \\cos(t) + \\frac{\\cos(3 t)}{3^2} + \\frac{\\cos(5 t)}{5^2} + \\frac{\\cos(7 t)}{7^2} + \\frac{\\cos(9 t)}{9^2} + \\dots \\right] $$\n", "\n", "The given triangular wave is an even function, $x(t)=x(-t)$, and then its Fourier series has only even functions (cosines).\n", "\n", "Let's plot it:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "internals": {}, "run_control": { "breakpoint": false }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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wrC0NTGlSmJnl2/I2oS0/3LvG2WlLY0ixQmFapJTc6jaAZN7unMhWjFflKtO1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\n6bXrGv12zgNgfcVm9On2UzNiCYJa6euqaO7RhxMFSmMeG0X5VXPZFPL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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "t = np.linspace(-3*np.pi, 3*np.pi, 1000)\n", "x = 1/2 + 4/np.pi**2*(np.cos(t) + np.cos(3*t)/9 + np.cos(5*t)/25 +\n", " np.cos(7*t)/49 + np.cos(9*t)/81)\n", "\n", "ax.plot(t, x, color='r', linewidth=2, label='Fourier series')\n", "ax.set_title('Triangular wave function and the first 6 non-zero terms of its Fourier series',\n", " fontsize=14)\n", "#ax.legend(frameon=False, loc='center right', framealpha=.5, fontsize=14)\n", "%config InlineBackend.close_figures=True # hold off the figure\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And here are plots for each of the first six non-zero components:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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cjjJGmegCTktL0/T09HiLETY7d+6kWrXkCnGUbDInm7zgZI42s2fP3qiq9eItRyBOd0Wf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LmkxIXNm9VquHKlJjksWaJ9Zy3lc/nlOqd16qiyctFFkJx86n6vv65lXTIy4JZbtO5ddS+j4wqC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\n1woL+X1xMfD88/xzL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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "t = np.linspace(-3*np.pi, 3*np.pi, 100)\n", "fig, axs = plt.subplots(6, 2, figsize=(9, 5), sharex=True, squeeze=True)\n", "axs = axs.flat\n", "for k, ax in enumerate(axs):\n", " if k == 0:\n", " x2 = .5*t/t\n", " ax.plot(t, x2, 'r', linewidth=2)\n", " ax.set_ylim(-.65, .65)\n", " ax.set_yticks([-.5, 0, .5])\n", " elif ~k%2:\n", " xf = 4/((k-1)*np.pi)**2*np.cos((k-1)*t)\n", " ax.plot(t, xf, 'r', linewidth=2)\n", " x2 = x2 + xf\n", " ax.set_ylim(-.65, .65)\n", " ax.set_yticks([-.5, 0, .5])\n", " else:\n", " ax.plot(t, x2, 'r', linewidth=2)\n", " ax.set_ylim(-.15, 1.15)\n", " ax.set_yticks([0, 0.5, 1])\n", " ax.xaxis.set_ticks_position('bottom')\n", " ax.set_xlim((-3*np.pi, 3*np.pi))\n", " ax.grid()\n", "ax.set_xticks(np.linspace(-3*np.pi-0.1, 3*np.pi+0.1, 7))\n", "ax.set_xticklabels(['$-3\\pi$', '$-2\\pi$', '$-\\pi$', '$0$', '$\\pi$', '$2\\pi$', '$3\\pi$'])\n", "ax.set_xlabel('Time (s)', fontsize=16)\n", "axs[-2].set_xlabel('Time (s)', fontsize=16)\n", "axs[4].set_ylabel('Amplitude', fontsize=16)\n", "axs[0].set_title('Average value and first five harmonics')\n", "axs[1].set_title('Sum of the harmonics')\n", "fig.tight_layout()\n", "plt.subplots_adjust(hspace=0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Amplitude and phase of the frequency components in the Fourier series\n", "\n", "The $a_k$ and $b_k$ coefficients are the amplitudes of the sine and cosine terms, each with frequency $k/T$, of the Fourier series. \n", "We can express the Fourier series in a more explicit form with a single amplitude and phase using the trigonometric identities:\n", "\n", "$$ x(t) = c_0 + \\sum_{k=1}^\\infty c_k \\sin\\left(k\\frac{2\\pi}{T}t + \\theta_k \\right) $$\n", "\n", "or\n", "\n", "$$ x(t) = c_0 + \\sum_{k=1}^\\infty c_k \\cos\\left(k\\frac{2\\pi}{T}t - \\theta_k \\right) $$\n", "\n", "Where:\n", "\n", "$$ c_0=a_0 \\quad , \\quad c_k = \\sqrt{a_k^2 + b_k^2} \\quad , \\quad \\theta_k = \\tan^{-1}\\left(\\frac{b_k}{a_k}\\right) $$\n", "\n", "The energy or power of each harmonic, which is proportional to the amplitude squared, is given by:\n", "\n", "$$ P_k = c_k^2 = a_k^2 + b_k^2 $$\n", "\n", "The plot of the amplitude (or power) and phase of each harmonic as a function of the corresponding frequency is called the amplitude (or power) spectrum and phase spectrum, respectively. These spectra are very useful to visualize the frequency content of a function." ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "### The complex form of the Fourier series\n", "\n", "Using the Euler's formula, the complex form of the Fourier series is:\n", "\n", "$$ x(t) \\sim \\sum_{k=-\\infty}^\\infty c_k \\exp\\left(ik\\frac{2\\pi}{T}t\\right) $$\n", "\n", "It can be shown that the relation between $c_k$, $a_k$, and $b_k$ coefficients is:\n", "\n", "$$ c_0 = \\frac{a_0}{2} \\quad , \\quad c_k = \\frac{a_k-i b_k}{2} \\quad , \\quad c_{-k} = \\frac{a_k+i b_k}{2}$$\n", "And:\n", "$$ a_k = 2\\text{ Real}\\{c_k\\} \\quad \\text{and} \\quad b_k = -2\\text{ Imag}\\{c_k\\} $$\n", "\n", "The complex Fourier coefficients, $c_k$, can be determined similarly to $a_k$ and $b_k$:\n", "\n", "$$ c_k = \\frac{1}{T} \\int_{t_0}^{t_0+T} x(t)\\exp\\left(-ik\\frac{2\\pi}{T}t\\right) \\:\\mathrm{d}t $$\n", "\n", "And the amplitudes and phases of the Fourier series in complex form are given by:\n", "\n", "$$ |c_k| \\, = \\, |c_{-k}| \\, = \\, \\frac{1}{2}\\sqrt{a_k^2 + b_k^2} \\quad , \\quad \\phi_k \\, = \\, \\tan^{-1}\\left(-\\frac{b_k}{a_k}\\right)\\,=\\, \\tan^{-1}\\left(\\frac{\\text{Imag}\\{c_k\\}}{\\text{Real}\\{c_k\\}}\\right)$$\n", "\n", "The complex form is more often used to express the Fourier series and in the calculation of the amplitude (or power) and phase spectra." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example: complex form of the Fourier series for a square wave function\n", "\n", "Let's calculate again the Fourier series of the square wave function but now using the complex form.\n", "\n", "The $c_k$ coefficients are given by ($c_0=0$ because the average value of $x(t)$ is zero):\n", "\n", "$$ c_k = \\frac{1}{2}\\left[ \\int_{-1}^0 -\\exp\\left(-ik\\frac{2\\pi}{2}t\\right) \\:\\mathrm{d}t + \\int_{0}^1 \\exp\\left(-ik\\frac{2\\pi}{2}t\\right) \\:\\mathrm{d}t\\right] $$\n", "\n", "$$ = \\frac{i\\:exp(-ik\\pi)-i}{k\\pi} $$\n", "\n", "$$ = \\frac{i}{k\\pi}\\left(\\cos(k\\pi) - 1\\right) $$\n", "\n", "Which results in:\n", "\n", "$$ c_k = \\left\\{ \n", "\\begin{array}{c l}\n", " 0 & \\quad \\text{if k is even} \\\\\n", " -\\frac{2i}{k\\pi} & \\quad \\text{if k is odd}\n", "\\end{array} \\right. $$\n", "\n", "If you compare with the previous calculation you will notice some differences. \n", "First, now the expression is multiplied by 2 instead of by 4. But remember that now $k$ goes from $-\\infty$ to $\\infty$, i.e., we will now add terms for $k=\\pm1,\\pm2,\\pm3,\\dots$, meaning that the expression above will be counted twice for each $|k|$. \n", "Second, the $c_k$ coefficients are purely imaginary; the imaginary part is negative when $k$ is positive and positive when $k$ is negative.\n", "\n", "So, the amplitude $\\text{A}_k$ and phase $\\phi_k$ of each Fourier coefficient for $k > 0$ are:\n", "\n", "$$ \\text{A}_k = \\left\\{ \n", "\\begin{array}{c l}\n", " 0 & \\quad \\text{if k is even} \\\\\n", " \\frac{2}{k\\pi} & \\quad \\text{if k is odd}\n", "\\end{array} \\right. $$\n", "\n", "$$ \\phi_k = \\left\\{ \n", "\\begin{array}{c l}\n", " 0 & \\quad \\text{if k is even} \\\\\n", " -\\frac{\\pi}{2} & \\quad \\text{if k is odd}\n", "\\end{array} \\right. $$\n", "\n", "And the negative of these expressions for $k < 0$.\n", "\n", "Let's plot the corresponding amplitude and phase spectra:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "C:\\Miniconda3\\lib\\site-packages\\ipykernel_launcher.py:3: RuntimeWarning: divide by zero encountered in true_divide\n", " This is separate from the ipykernel package so we can avoid doing imports until\n" ] }, { "data": { "image/png": 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OHdLgwYMVHR2tXr16ac6cOYEuCUGoqKhIffr00U9/+tNAl1Jj/gfAgwdrNl6PDh06pHXr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\nk0PAANCoGGPSrbU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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "k = np.arange(-9,10)\n", "fk = k/2\n", "ampk = 2/(k*np.pi)\n", "ampk[1::2] = 0\n", "phik = -np.pi/2*(k+.1)/abs(k+.1)*180/np.pi\n", "phik[1::2] = 0\n", "\n", "fig, ax = plt.subplots(2, 1, figsize=(9, 5), sharex=True)\n", "ax[0].stem(fk, ampk, markerfmt='ro', linefmt='r-')\n", "ax[0].set_ylim(-.7, .7)\n", "ax[0].annotate('Frequency (Hz)', xy=(3, -.3), xycoords = 'data', color='k',\n", " size=14, xytext=(0, 0), textcoords = 'offset points')\n", "ax[0].set_title('Amplitude', fontsize=16)\n", "ax[1].stem(fk, phik, markerfmt='ro', linefmt='r-')\n", "ax[1].set_title('Phase ( $^o$)', fontsize=16)\n", "ax[1].set_xlim(-4.55, 4.55)\n", "ax[1].set_ylim(-120, 120)\n", "ax[1].set_yticks([-90, 0, 90])\n", "ax[1].annotate('Frequency (Hz)', xy=(3, 15), xycoords = 'data', color='k',\n", " size=14, xytext=(0, 0), textcoords = 'offset points')\n", "for axi in ax:\n", " axi.spines['left'].set_position('zero')\n", " axi.spines['right'].set_color('none')\n", " axi.spines['bottom'].set_position('zero')\n", " axi.spines['top'].set_color('none')\n", " axi.spines['right'].set_color('none')\n", " axi.xaxis.set_ticks_position('bottom')\n", " axi.yaxis.set_ticks_position('left')\n", " axi.tick_params(axis='both', direction='inout', which='both', length=5)\n", " axi.locator_params(axis='y', nbins=4)\n", "fig.tight_layout()\n", "plt.subplots_adjust(hspace=0.2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Usually we don't care for the negative frequencies of real signals and we are more interested in the power of the signal:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Pk = ampk[int((ampk.size)/2):]**2\n", "\n", "fig, ax = plt.subplots(1, 1, figsize=(9, 3))\n", "ax.stem(fk[int((fk.size)/2):], Pk, markerfmt='ro', linefmt='r-')\n", "ax.plot(fk[int((fk.size)/2):], Pk, 'ro', clip_on=False)\n", "ax.set_ylabel('Power', fontsize=16)\n", "ax.set_xlabel('Frequency (Hz)', fontsize=16)\n", "ax.set_xlim(0, 4.55)\n", "ax.grid()\n", "fig.tight_layout()" ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "## Problems\n", "\n", " 1. Using a trigonometric identity, verify that $\\cos(x)=\\sin(x+\\pi/2)$. \n", " 2. Using the Euler's formula, deduce the expressions for sine and cosine. \n", " 3. Using the complex exponential forms, show that the integrals of the sine and cosine functions are zero. \n", " 4. Verify that the results of the integrals for the orthogonality of sinusoids are indeed correct. \n", " 5. Deduce the Fourier series for the triangular wave function using the complex form. \n", " 6. Deduce the Fourier series for the function $x(t) = t$ in the interval $[0, 1]$. \n", " 7. Deduce the Fourier series for the saw-tooth function $x(t) = 2t/T$ in the interval $0 \\leq t < T/2$ and $x(t)=2(t-T)/T$ in the interval $T/2 \\leq t < T$. \n", " 8. Calculate the power and phase spectra for the functions $x(t) = 2\\sin(4\\pi t)$ and $x(t) = 2\\cos(4\\pi t)$." ] }, { "cell_type": "markdown", "metadata": { "internals": {}, "slideshow": { "slide_type": "-" } }, "source": [ "## References\n", "\n", "- James JF (2011) [A Student’s Guide to Fourier Transform](http://books.google.com.br/books?id=_T99VW0ARfkC). 3rd edition. Cambridge.\n", "- Lyons RG (2010) [Understanding Digital Signal Processing](http://books.google.com.br/books?id=UBU7Y2tpwWUC&hl). 3rd edition. Prentice Hall. \n", "- Robertson G, Caldwell G, Hamill J, Kamen G (2013) [Research Methods in Biomechanics](http://books.google.com.br/books?id=gRn8AAAAQBAJ). 2nd Edition. Human Kinetics. \n", "- Tang KT (2007) [Mathematical Methods for Engineers and Scientists 3](http://books.google.com.br/books?id=gG-ybR3uIGsC). Springer. \n", "- Tolstov GP (1976) [Fourier Series](http://books.google.com.br/books/about/Fourier_Series.html?id=XqqNDQeLfAkC). Courier Dover Publications. 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