{ "cells": [ { "cell_type": "markdown", "metadata": { "nbsphinx": "hidden" }, "source": [ "# Continuous Signals\n", "\n", "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Elementary Operations\n", "\n", "Operations like superposition, temporal shifting and scaling are used to construct signals with a more complex structure than the previously introduced [standard signals](standard_signals.ipynb). A set of elementary operations are introduced that are frequently used in signal processing." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Superposition\n", "\n", "The weighted superposition $x(t)$ of two signals $x_\\text{A}(t)$ and $x_\\text{B}(t)$ is given as\n", "\n", "\\begin{equation}\n", "x(t) = A \\cdot x_\\text{A}(t) + B \\cdot x_\\text{B}(t)\n", "\\end{equation}\n", "\n", "with the complex weights $A, B \\in \\mathbb{C}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example**\n", "\n", "The following example illustrates the superposition of two harmonic signals $x_\\text{A}(t) = A \\cdot \\cos(\\omega_\\text{A} t)$ and $x_\\text{B}(t) = B \\cdot \\cos(\\omega_\\text{B} t)$ with weights $A$, $B$ and angular frequencies $\\omega_\\text{A}$ and $\\omega_\\text{B}$." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import sympy as sym\n", "%matplotlib inline\n", "sym.init_printing()\n", "\n", "t = sym.symbols('t', real=True)\n", "\n", "A = .3\n", "omA = 3\n", "B = .5\n", "omB = 5\n", "\n", "x = A*sym.cos(omA*t) + B*sym.cos(omB*t)\n", "\n", "sym.plot(x, (t, -5, 5), ylim=[-1.2, 1.2], ylabel=r'$x(t)$')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Temporal Shift\n", "\n", "The temporal shift of a signal $x(t)$ by the time $\\tau$ is a frequently applied operation in signal processing. For instance, to model the propagation of signals from an actuator to a sensor.\n", "\n", "The temporally shifted signal $x(t)$ is defined as\n", "\n", "\\begin{equation}\n", "y(t) = x(t-\\tau)\n", "\\end{equation}\n", "with $\\tau \\in \\mathbb{R}$. The signal $x(t)$ is\n", "\n", "* shifted to the right (*delayed*) for $\\tau > 0$\n", "* shifted to the left (*leading*) for $\\tau < 0$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example**\n", "\n", "In order to illustrate the temporal shifting of signals, the construction of a staircase signal by a superposition of shifted [rectangular signals](standard_signals.ipynb#Rectangular-Signal) is considered\n", "\n", "\\begin{equation}\n", "x(t) = \\text{rect}\\left(t - \\frac{1}{2} \\right) + \\frac{2}{3} \\cdot \\text{rect}\\left(t-\\frac{3}{2} \\right) + \\frac{1}{3} \\cdot \\text{rect} \\left(t-\\frac{5}{2} \\right)\n", "\\end{equation}" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "rect = sym.Heaviside(t + 1/2) - sym.Heaviside(t - 1/2)\n", "x = rect.subs(t, t-1/2) + 2/3*rect.subs(t, t-3/2) + 1/3*rect.subs(t, t-5/2)\n", "\n", "sym.plot(x, (t, -1, 5), ylim=[-0.2, 1.2], ylabel='$x(t)$');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise**\n", "\n", "* Add another step to the beginning of the staircase signal by modifying above example." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Temporal Scaling\n", "\n", "The temporal scaling of a signal $x(t)$ is defined as\n", "\n", "\\begin{equation}\n", "y(t) = x(a \\cdot t)\n", "\\end{equation}\n", "\n", "with $a \\in \\mathbb{R}$. The signal $x(t)$ is\n", "\n", "* stretched for $0 < a < 1$\n", "* compressed for $a > 1$\n", "* time-reversed and scaled for $a < 0$\n", "\n", "An application of temporal scaling in signal processing is the adaption of the time scale for standard signals and the modeling of the [Doppler effect](https://en.wikipedia.org/wiki/Doppler_effect)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example**\n", "\n", "The following example illustrates the temporal scaling of the staircase signal $y(t) = x(a \\cdot t)$ introduced in the previous example. The original $x(t)$ is plotted in gray, the scaled signal $y(t)$ in blue. Here stretching is realized, such that $y(t)$ is twice as long as $x(t)$." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "a = .5\n", "y = x.subs(t, a*t)\n", "\n", "px = sym.plot(x, (t, -3, 7), ylim=[-0.2, 1.2],\n", " ylabel=r'$y(t)$', show=False, line_color='gray')\n", "py = sym.plot(y.subs(a, 1/2), (t, -3, 7),\n", " ylim=[-0.2, 1.2], ylabel=r'$y(t)$', show=False)\n", "px.extend(py)\n", "px.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise**\n", "\n", "* Modify above example such that the signal is compressed.\n", "* Modify above example such that the signal is scaled and time reversed. What scaling factors `a` lead to stretching/compression in this context?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Temporal Flipping" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The temporal flipping of a signal $x(t)$ is defined as\n", "\n", "\\begin{equation}\n", "y(t) = x(\\tau - t)\n", "\\end{equation}\n", "\n", "for $\\tau \\in \\mathbb{R}$. As $x(\\tau - t) = x(- (t - \\tau))$ the flipping operation can also be represented as a time-reversal of the signal $x(t)$ followed by a shift of $\\tau$ of the reversed signal. For $\\tau = 0$ this results in only a time-reversal of the signal. \n", "\n", "The temporal flipping operation can be interpreted geometrically as a mirroring of the signal $x(t)$ at the vertical axis $t=\\frac{\\tau}{2}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example**\n", "\n", "The following example illustrates the temporal flipping $y(t) = x(\\tau - t)$ of the staircase signal $x(t)$ introduced before." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "application/pdf": 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04YKmLpVCgcUl8Jl8dXvovk8ZeGoZcnYEEUPJYAlquhZNWLQ8n5BOAeL/fsPuLeShkvPKnhv5G5zt8DuzbuEnanYi0XIVMtSzNMcYCBNFHjx5RaZw4rPWd9U0EtRmC06WAa5OP4wOAGAiXlmA7K5EOUvSjqWfb7zH9w9AAFO0CmVuZHN0cmVhbQplbmRvYmoKMjAgMCBvYmoKPDwgL0Jhc2VGb250IC9EZWphVnVTYW5zIC9DaGFyUHJvY3MgMjEgMCBSCi9FbmNvZGluZyA8PAovRGlmZmVyZW5jZXMgWyA0MCAvcGFyZW5sZWZ0IC9wYXJlbnJpZ2h0IDQ2IC9wZXJpb2QgNDggL3plcm8gL29uZSAvdHdvIDUyIC9mb3VyIDU0IC9zaXgKNTYgL2VpZ2h0IDExNiAvdCBdCi9UeXBlIC9FbmNvZGluZyA+PgovRmlyc3RDaGFyIDAgL0ZvbnRCQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRm9udERlc2NyaXB0b3IgMTkgMCBSCi9Gb250TWF0cml4IFsgMC4wMDEgMCAwIDAuMDAxIDAgMCBdIC9MYXN0Q2hhciAyNTUgL05hbWUgL0RlamFWdVNhbnMKL1N1YnR5cGUgL1R5cGUzIC9UeXBlIC9Gb250IC9XaWR0aHMgMTggMCBSID4+CmVuZG9iagoxOSAwIG9iago8PCAvQXNjZW50IDkyOSAvQ2FwSGVpZ2h0IDAgL0Rlc2NlbnQgLTIzNiAvRmxhZ3MgMzIKL0ZvbnRCQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRm9udE5hbWUgL0RlamFWdVNhbnMgL0l0YWxpY0FuZ2xlIDAKL01heFdpZHRoIDEzNDIgL1N0ZW1WIDAgL1R5cGUgL0ZvbnREZXNjcmlwdG9yIC9YSGVpZ2h0IDAgPj4KZW5kb2JqCjE4IDAgb2JqClsgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAKNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCAzMTggNDAxIDQ2MCA4MzggNjM2Cjk1MCA3ODAgMjc1IDM5MCAzOTAgNTAwIDgzOCAzMTggMzYxIDMxOCAzMzcgNjM2IDYzNiA2MzYgNjM2IDYzNiA2MzYgNjM2IDYzNgo2MzYgNjM2IDMzNyAzMzcgODM4IDgzOCA4MzggNTMxIDEwMDAgNjg0IDY4NiA2OTggNzcwIDYzMiA1NzUgNzc1IDc1MiAyOTUKMjk1IDY1NiA1NTcgODYzIDc0OCA3ODcgNjAzIDc4NyA2OTUgNjM1IDYxMSA3MzIgNjg0IDk4OSA2ODUgNjExIDY4NSAzOTAgMzM3CjM5MCA4MzggNTAwIDUwMCA2MTMgNjM1IDU1MCA2MzUgNjE1IDM1MiA2MzUgNjM0IDI3OCAyNzggNTc5IDI3OCA5NzQgNjM0IDYxMgo2MzUgNjM1IDQxMSA1MjEgMzkyIDYzNCA1OTIgODE4IDU5MiA1OTIgNTI1IDYzNiAzMzcgNjM2IDgzOCA2MDAgNjM2IDYwMCAzMTgKMzUyIDUxOCAxMDAwIDUwMCA1MDAgNTAwIDEzNDIgNjM1IDQwMCAxMDcwIDYwMCA2ODUgNjAwIDYwMCAzMTggMzE4IDUxOCA1MTgKNTkwIDUwMCAxMDAwIDUwMCAxMDAwIDUyMSA0MDAgMTAyMyA2MDAgNTI1IDYxMSAzMTggNDAxIDYzNiA2MzYgNjM2IDYzNiAzMzcKNTAwIDUwMCAxMDAwIDQ3MSA2MTIgODM4IDM2MSAxMDAwIDUwMCA1MDAgODM4IDQwMSA0MDEgNTAwIDYzNiA2MzYgMzE4IDUwMAo0MDEgNDcxIDYxMiA5NjkgOTY5IDk2OSA1MzEgNjg0IDY4NCA2ODQgNjg0IDY4NCA2ODQgOTc0IDY5OCA2MzIgNjMyIDYzMiA2MzIKMjk1IDI5NSAyOTUgMjk1IDc3NSA3NDggNzg3IDc4NyA3ODcgNzg3IDc4NyA4MzggNzg3IDczMiA3MzIgNzMyIDczMiA2MTEgNjA1CjYzMCA2MTMgNjEzIDYxMyA2MTMgNjEzIDYxMyA5ODIgNTUwIDYxNSA2MTUgNjE1IDYxNSAyNzggMjc4IDI3OCAyNzggNjEyIDYzNAo2MTIgNjEyIDYxMiA2MTIgNjEyIDgzOCA2MTIgNjM0IDYzNCA2MzQgNjM0IDU5MiA2MzUgNTkyIF0KZW5kb2JqCjIxIDAgb2JqCjw8IC9laWdodCAyMiAwIFIgL2ZvdXIgMjMgMCBSIC9vbmUgMjUgMCBSIC9wYXJlbmxlZnQgMjYgMCBSCi9wYXJlbnJpZ2h0IDI3IDAgUiAvcGVyaW9kIDI4IDAgUiAvc2l4IDI5IDAgUiAvdCAzMCAwIFIgL3R3byAzMSAwIFIKL3plcm8gMzIgMCBSID4+CmVuZG9iagozIDAgb2JqCjw8IC9GMSAyMCAwIFIgL0YyIDE0IDAgUiA+PgplbmRvYmoKNCAwIG9iago8PCAvQTEgPDwgL0NBIDAgL1R5cGUgL0V4dEdTdGF0ZSAvY2EgMSA+PgovQTIgPDwgL0NBIDEgL1R5cGUgL0V4dEdTdGF0ZSAvY2EgMSA+PiA+PgplbmRvYmoKNSAwIG9iago8PCA+PgplbmRvYmoKNiAwIG9iago8PCA+PgplbmRvYmoKNyAwIG9iago8PCAvRGVqYVZ1U2Fucy1taW51cyAyNCAwIFIgPj4KZW5kb2JqCjIgMCBvYmoKPDwgL0NvdW50IDEgL0tpZHMgWyAxMCAwIFIgXSAvVHlwZSAvUGFnZXMgPj4KZW5kb2JqCjMzIDAgb2JqCjw8IC9DcmVhdGlvbkRhdGUgKEQ6MjAxOTA0MDkxMTQwMjMrMDInMDAnKQovQ3JlYXRvciAobWF0cGxvdGxpYiAzLjAuMywgaHR0cDovL21hdHBsb3RsaWIub3JnKQovUHJvZHVjZXIgKG1hdHBsb3RsaWIgcGRmIGJhY2tlbmQgMy4wLjMpID4+CmVuZG9iagp4cmVmCjAgMzQKMDAwMDAwMDAwMCA2NTUzNSBmIAowMDAwMDAwMDE2IDAwMDAwIG4gCjAwMDAwMDk2NzAgMDAwMDAgbiAKMDAwMDAwOTQ0MCAwMDAwMCBuIAowMDAwMDA5NDgzIDAwMDAwIG4gCjAwMDAwMDk1ODIgMDAwMDAgbiAKMDAwMDAwOTYwMyAwMDAwMCBuIAowMDAwMDA5NjI0IDAwMDAwIG4gCjAwMDAwMDAwNjUgMDAwMDAgbiAKMDAwMDAwMDM5MiAwMDAwMCBuIAowMDAwMDAwMjA4IDAwMDAwIG4gCjAwMDAwMDI4MjQgMDAwMDAgbiAKMDAwMDAwMzgzOSAwMDAwMCBuIAowMDAwMDAzNjMxIDAwMDAwIG4gCjAwMDAwMDMzMDggMDAwMDAgbiAKMDAwMDAwNDg5MiAwMDAwMCBuIAowMDAwMDAyODQ1IDAwMDAwIG4gCjAwMDAwMDMwOTYgMDAwMDAgbiAKMDAwMDAwODIyNyAwMDAwMCBuIAowMDAwMDA4MDI3IDAwMDAwIG4gCjAwMDAwMDc2NDQgMDAwMDAgbiAKMDAwMDAwOTI4MCAwMDAwMCBuIAowMDAwMDA0OTM0IDAwMDAwIG4gCjAwMDAwMDUzOTkgMDAwMDAgbiAKMDAwMDAwNTU2MSAwMDAwMCBuIAowMDAwMDA1NzMxIDAwMDAwIG4gCjAwMDAwMDU4ODMgMDAwMDAgbiAKMDAwMDAwNjEwMyAwMDAwMCBuIAowMDAwMDA2MzI1IDAwMDAwIG4gCjAwMDAwMDY0NDYgMDAwMDAgbiAKMDAwMDAwNjgzNiAwMDAwMCBuIAowMDAwMDA3MDQwIDAwMDAwIG4gCjAwMDAwMDczNjEgMDAwMDAgbiAKMDAwMDAwOTczMCAwMDAwMCBuIAp0cmFpbGVyCjw8IC9JbmZvIDMzIDAgUiAvUm9vdCAxIDAgUiAvU2l6ZSAzNCA+PgpzdGFydHhyZWYKOTg4NAolJUVPRgo=\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "tau = -1\n", "y = x.subs(t, tau - t)\n", "\n", "px = sym.plot(x, (t, -5, 5), ylim=[-0.2, 1.2],\n", " ylabel=r'$y(t)$', show=False, line_color='gray')\n", "py = sym.plot(y, (t, -5, 5), ylim=[-0.2, 1.2], ylabel=r'$y(t)$', show=False)\n", "px.extend(py)\n", "px.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Excercise**\n", "\n", "* For what value $\\tau$ does the signal start at $t=0$?\n", "* Realize the temporal flipping by splitting it into two consecutive operations: (i) time-reversal and (ii) temporal shift." ] }, { "cell_type": "markdown", "metadata": { "nbsphinx": "hidden" }, "source": [ "**Copyright**\n", "\n", "The notebook is provided as [Open Educational Resource](https://de.wikipedia.org/wiki/Open_Educational_Resources). Feel free to use the notebook for your own educational purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Signals and Systems* by Sascha Spors." ] } ], "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": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 1 }