{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# IA Paper 4 - Mathematics - Examples paper 8" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Question 1\n", "\n", "A train of digital pulses is periodic with period $2\\pi$ and has the form\n", "\n", "$$\n", "f(t) = \n", "\\begin{cases}\n", " 1 & \\text{if} \\ 0 < t \\le T \\\\\n", " 0 & \\text{if} \\ T < t \\le 2 \\pi\n", "\\end{cases}\n", "$$\n", "\n", "Express $f(t)$ as a Fourier series, and evaluate the coefficients. During transmission by a long cable, high-frequency components of the signal are attenuated. Explain briefly how the Fourier series allows this low-pass filtering effect to be studied.\n", "\n", "Now consider a specific example of a cable that transmits perfectly all frequency components below 1 kHz but attenuates completely all frequency components above 1 kHz. The digital pulse train has period $2 \\pi$ ms and $T = \\pi$ ms. Use Python to plot the filtered signal.\n", "\n", "**Python Hint**\n", "\n", "When $T = \\pi$ ms, the digital pulse train is just a square wave and we can use the supplied code to study its Fourier series. Its fundamental frequency is $1000/(2\\pi) = 159$ Hz and its sixth harmonic is $159 \\times 6 = 955$ Hz. So the cable will pass the first six terms of the Fourier series and attenuate the rest. To view the filtered signal, we need only change the program to `num_terms = 6`. Try modifying the program to plot the Fourier series for other functions in this examples paper. For functions of period $2\\pi$, you need only change the expressions for `d`, `an` and `bn`. For other functions, you could just force the period to $2\\pi$, i.e. set $\\omega = 1$ in question 4, $L = \\pi$ in question 6 and $T = 2\\pi$ in Question 8." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We first import some basic packages, including `numpy` and `matplotlib`. " ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "# Import numpy and matplotlib\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Fourier series for the square wave is:\n", "\n", "$$\n", "f(t) = \\frac{1}{2}a_0 + \\sum_{n=1}^{\\infty} a_n \\cos (n t) + \\sum_{n=1}^{\\infty} b_n \\sin (n t)\n", "$$\n", "\n", "with:\n", "\n", "\\begin{align}\n", "a_0 &= \\frac{T}{\\pi} \n", "\\\\[1ex]\n", "a_n &= \\frac{\\sin(n T)}{n \\pi} \n", "\\\\[1ex]\n", "b_n &= \\frac{1 - \\cos(n T)}{n \\pi}\n", "\\end{align}\n", "\n", "We now create some 'time' points in the interval $[-2 \\pi, 2 \\pi)$ at which to evaluate the Fourier series:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Generate array for t\n", "num_points = 200\n", "dt = 4*np.pi/(num_points)\n", "t = np.arange(-2*np.pi, 2*np.pi + dt, dt)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next, we approxinate the Fourier series using a finite number of terms in the series:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# Initialise parameters T and a_0\n", "T = np.pi\n", "a_0 = T/np.pi\n", "\n", "# Set number of terms for Fourier series\n", "num_terms = 20\n", "f = 0.5*a_0*np.ones(len(t))\n", "\n", "# Calcualte the Fourier series of f(t)\n", "for n in range(1, num_terms):\n", " a_n = (np.sin(n*T))/(n*np.pi)\n", " b_n = (1 - np.cos(n*T))/(n*np.pi)\n", " f += a_n * np.cos(n*t) + b_n*np.sin(n*t)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's plot our results using 20 terms from the Fourier series for $f(t)$." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Plot the Fourier function\n", "plt.plot(t, f)\n", "\n", "# Label axis\n", "plt.xlabel(\"t\")\n", "plt.ylabel(\"f(t)\")\n", "\n", "# Set axis limit\n", "plt.axis([t[0], t[-1], -0.5, 1.5]);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can create an interactive plot to explore the effect of $n$ on the approximation of $f(t)$. " ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "78e1178e00c24683a0005d14948b459d", "version_major": 2, "version_minor": 0 }, "text/plain": [ "interactive(children=(IntSlider(value=10, description='num_terms', max=200), Output()), _dom_classes=('widget-…" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Let's make an interactive plot\n", "from ipywidgets import interact, IntSlider\n", "\n", "def plot_taylor_series(num_terms):\n", " # Initialise f array\n", " f = 0.5*a_0*np.ones(len(t))\n", "\n", " # Calculate the Fourier series\n", " for n in range(1, num_terms):\n", " a_n = (np.sin(n*T))/(n*np.pi)\n", " b_n = (1 - np.cos(n*T))/(n*np.pi)\n", " f += a_n*np.cos(n*t) + b_n*np.sin(n*t)\n", "\n", " # Plot the function\n", " plt.plot(t, f)\n", "\n", " # Label axis\n", " plt.xlabel(\"t\")\n", " plt.ylabel(\"f(t)\")\n", "\n", " # Set axis limit\n", " plt.axis([t[0], t[-1], -0.5, 1.5])\n", "\n", "interact(plot_taylor_series, num_terms=IntSlider(min=0, max=200, step=1, value=10));" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.7.6" }, "widgets": { "application/vnd.jupyter.widget-state+json": { "state": { "002d4ec6578a4233a5d2cb388936c2fe": { "model_module": "@jupyter-widgets/base", "model_module_version": "1.2.0", "model_name": "LayoutModel", "state": {} }, "1a7b394971534d7884e2f8a9d3e1c8eb": { "model_module": "@jupyter-widgets/base", "model_module_version": "1.2.0", "model_name": "LayoutModel", "state": {} }, "32e678156aa2423eaaa19ab7143b00b1": { "model_module": "@jupyter-widgets/controls", "model_module_version": "1.5.0", "model_name": "IntSliderModel", "state": { "description": "num_terms", "layout": "IPY_MODEL_a4b87a55c67843c3befd2952deb38fe6", "max": 200, "style": "IPY_MODEL_632e60e68d2a479ebf8b1107198519f8", "value": 10 } }, "410948bcd3454439ad8132669ed59fe4": { "model_module": "@jupyter-widgets/output", "model_module_version": "1.0.0", "model_name": "OutputModel", "state": { "layout": "IPY_MODEL_633215326d264447b6b515fe1fffbeec", "outputs": [ { "data": { "image/png": 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\n", 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\n", 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