{ "cells": [ { "cell_type": "markdown", "metadata": { "nbsphinx": "hidden" }, "source": [ "# Discrete Signals\n", "\n", "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Comunications 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": [ "## Standard Signals\n", "\n", "A discrete signal $x[k] \\in \\mathbb{C}$ is a sequence of values that depends on the discrete index $k \\in \\mathbb{Z}$. It may have been derived by [sampling](../sampling/ideal.ipynb) $x[k] := x(k T)$ from a continuous signal $x(t)$. As for the case of continuous signals, certain [standard signals](../continuous_signals/standard_signals.ipynb) play an important role in the theory and practical application of discrete signals and systems. These standard signals are introduced and illustrated in the following. Some of the signals are derived by sampling their continuous counterparts, some of them by postulating equivalent properties. Discrete signals are commonly denoted by using square brackets for their arguments." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Complex Exponential Signal\n", "\n", "The discrete complex exponential signal is defined by the [complex exponential function](https://en.wikipedia.org/wiki/Exponential_function#Complex_plane)\n", "\n", "\\begin{equation}\n", "x[k] = e^{(\\Sigma + j \\Omega) \\, k} = z^k\n", "\\end{equation}\n", "\n", "where $z = e^{\\Sigma + j \\Omega}$ denotes the complex frequency $z \\in \\mathbb{C}$ with $\\Sigma, \\Omega \\in \\mathbb{R}$. The discrete complex exponential signal can be related to the [continuous complex exponential signal](../continuous_signals/standard_signals.ipynb#Complex-Exponential-Signal) $x(t) = e^{(\\sigma + j \\omega) t}$ by sampling\n", "\n", "\\begin{equation}\n", "x[k] = x(k T) = e^{(\\sigma + j \\omega) k T} = e^{( \\sigma T + j \\omega T ) k}\n", "\\end{equation}\n", "\n", "where $T$ denotes the sampling interval. Comparison to above definition of the discrete signal reveals that $\\Sigma = \\sigma T$ and $\\Omega = \\omega T$. Due to this relation, the latter is termed as *normalized frequency* $\\Omega$. Using [Euler's formula](https://en.wikipedia.org/wiki/Euler's_formula), the definition of the complex exponential signal can be reformulated as\n", "\n", "\\begin{equation}\n", "x[k] = e^{\\Sigma k} \\cos[\\Omega k] + j e^{\\Sigma k} \\sin[\\Omega k]\n", "\\end{equation}\n", "\n", "The real/imaginary part of the exponential signal is given by a weighted discrete cosine/sine with normalized frequency $\\Omega$. The normalized frequency $\\Omega$ is ambiguous due to the periodicity of the cosine/sine function for discrete $k$. For instance\n", "\n", "\\begin{equation}\n", "\\cos[\\Omega k] = \\cos[(\\Omega + n \\cdot 2 \\pi) \\cdot k]\n", "\\end{equation}\n", "\n", "with $n \\in \\mathbb{Z}$. It can be concluded that the normalized frequency $\\Omega$ is unique for $-\\pi < \\Omega < \\pi$. This also becomes evident when considering the sampling of a continuous exponential signal, as shown above. For [critical sampling](../sampling/ideal.ipynb#Sampling-Theorem-for-Low-Pass-Signals) the sampling frequency $\\omega_\\text{s} = 2 \\cdot \\omega$. From $T = \\frac{2 \\pi}{\\omega_\\text{s}}$ follows $\\Omega = \\omega T = \\pi$. It can be concluded that the normalized frequency $\\Omega = \\pm \\pi$ represents the highest/lowest normalized frequency a sampled signal can represent.\n", "\n", "The complex exponential function is only periodic with respect to the discrete index $k$, if\n", "\n", "\\begin{equation}\n", "x[k] = x[k + n \\cdot N_\\text{p}]\n", "\\end{equation}\n", "\n", "holds for $n \\in \\mathbb{Z}$ and $N_\\text{p} \\in \\mathbb{N}$. The periodicity of the complex exponential function is given as\n", "\n", "\\begin{equation}\n", "N_\\text{p} = \\frac{2 \\pi}{\\Omega}\n", "\\end{equation}\n", "\n", "It follows from the requirement $N_\\text{p} \\in \\mathbb{N}$ for a periodic sequence, that not all normalized frequencies $\\Omega$ result in a periodic discrete complex exponential signal. Only $\\Omega = \\frac{2 \\pi}{N_\\text{p}}$ with $N_\\text{p} \\in \\mathbb{N}$ is periodic with period $N_\\text{p}$. Sampling of a continuous complex exponential signal may result in an aperiodic discrete complex exponential signal.\n", "\n", "The complex exponential signal can be expressed in terms of its magnitude and phase\n", "\n", "\\begin{equation}\n", "x[k] = z^k = |z|^k \\cdot e^{j \\varphi(z) k}\n", "\\end{equation}\n", "\n", "where by comparison with its definition $|z| = |e^\\Sigma|$ and $\\varphi(z) = \\Omega$. This finding allows an interpretation of the complex frequency $z$. Its phase $\\varphi(z)$ is equal to the normalized frequency $\\Omega$ of its harmonic part $e^{j \\Omega k}$. This is weighted by the magnitude $|z|^k$. With increasing index $k >0$, the magnitude of the complex exponential signal is\n", "\n", "* exponentially decaying for $|z| < 1$ ($\\Sigma < 0$)\n", "* constantly one for $|z| = 1$ ($\\Sigma = 0$)\n", "* exponentially growing for $|z| > 1$ ($\\Sigma > 0$)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example**\n", "\n", "This example illustrates the discrete complex exponential signal and its parameters. The Python module [NumPy](http://www.numpy.org/) provides functionality for numerical mathematics. In discrete signal processing it is common to define discrete signals as vectors holding the values $x[k]$ for a given set of indexes $k$. Two functions are defined as prerequisite. One for the generation of the exponential signal and one to plot its real and imaginary part." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "\n", "\n", "def exponential_signal(k, Sigma, Omega):\n", " return np.exp((Sigma + 1j * Omega) * k)\n", "\n", "\n", "def plot_signal(k, x):\n", " plt.figure(figsize=(10, 4))\n", " plt.subplot(121)\n", " plt.stem(k, np.real(x))\n", " plt.xlabel('$k$')\n", " plt.ylabel(r'$\\Re \\{ x[k] \\}$')\n", "\n", " plt.subplot(122)\n", " plt.stem(k, np.imag(x))\n", " plt.xlabel('$k$')\n", " plt.ylabel(r'$\\Im \\{ x[k] \\}$')\n", " plt.tight_layout()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The vector `x` is populated by the samples of the exponential signal for a given set of indexes $k$, $\\Sigma = 0.025$ and $\\Omega = 0.5$. Its first ten values are printed for illustration." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([ 1. +0.j , 0.89979867+0.49156225j,\n", " 0.5680042 +0.88461412j, 0.07624651+1.07518404j,\n", " -0.45991338+1.00492907j, -0.90781465+0.67815778j,\n", " -1.15020718+0.16395806j, -1.11555049-0.41786919j,\n", " -0.79836212-0.92436066j, -0.26398437-1.22418317j])" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k = np.arange(31)\n", "x = exponential_signal(k, 0.025, 0.5)\n", "x[:10]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Discrete signals are commonly visualized by '*stem*'-plots. The Python module [`matplotlib`](http://matplotlib.org/) is used for this purpose. The real and imaginary part of the complex exponential signal is shown by the left and right plot, respectively." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "application/pdf": "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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plot_signal(k, x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now the case of an aperiodic harmonic exponential signal with $N_\\text{p} \\notin \\mathbb{N}$ is illustrated for $\\Sigma = 0$. Again the left and right plot show the real and imaginary part." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "application/pdf": "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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Np = 20.55\n", "x = exponential_signal(k, 0, 2 * np.pi / Np)\n", "plot_signal(k, x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise**\n", "\n", "* Change the values of $\\Sigma$ in the first example in order to create a signal with increasing/constant/decaying amplitude.\n", "* Check if the stated non-uniqueness of the normalized frequency $\\Omega$ holds by adding multiples of $2 \\pi$ and plotting the resulting signal." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Dirac Impulse\n", "\n", "The discrete Dirac impulse $\\delta[k]$ cannot be derived by sampling from its [continuous counterpart](../continuous_signals/standard_signals.ipynb#Dirac-Impulse), since the [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) is a distribution and not a function in the conventional sense. The discrete Dirac impulse is defined as\n", "\n", "\\begin{equation}\n", "\\delta[k] = \\begin{cases}\n", "1 & \\text{for } k = 0 \\\\\n", "0 & \\text{otherwise}\n", "\\end{cases}\n", "\\end{equation}\n", "\n", "This function is also known as [*Kronecker delta*](https://en.wikipedia.org/wiki/Kronecker_delta), the signal as *impulse sequence* or *unit impulse*. The discrete Dirac impulse $\\delta[k]$ maintains the essential properties of its continuous counterpart $\\delta(t)$. In particular\n", "\n", "1. **Sifting property**\n", " \\begin{equation}\n", " \\sum_{\\kappa = -\\infty}^{\\infty} x[\\kappa] \\cdot \\delta[k - \\kappa] = x[k]\n", " \\end{equation}\n", " The sifting property implies $\\sum_{\\kappa = -\\infty}^{\\infty} x[\\kappa] = 1$.\n", " \n", "2. **Multiplication**\n", " \\begin{equation}\n", " x[k] \\cdot \\delta[k - \\kappa] = x[\\kappa] \\cdot \\delta[k - \\kappa]\n", " \\end{equation}\n", " \n", "3. **Linearity**\n", " \\begin{equation}\n", " a \\cdot \\delta[k] + b \\cdot \\delta[k] = (a+b) \\cdot \\delta[k]\n", " \\end{equation}\n", "\n", "The discrete Dirac impulse is an important signal in the theory of signals and systems. It is used for instance for the characterization of discrete LTI systems by their impulse response." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example**\n", "\n", "The properties of the discrete Dirac impulse are illustrated. First a function is defined for the discrete Dirac impulse and $\\delta[k]$ is plotted" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(-0.1, 1.1)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" }, { "data": { "application/pdf": "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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def dirac(k):\n", " return np.where(k == 0, 1.0, 0.0)\n", "\n", "\n", "k = np.arange(-10, 11)\n", "x = dirac(k)\n", "plt.stem(k, x)\n", "plt.xlabel('$k$')\n", "plt.ylabel('$\\delta[k]$')\n", "plt.ylim([-0.1, 1.1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let's check the multiplication property by defining a cosine signal $y[k] = \\cos[k]$ and computing the signal $w[k] = y[k] \\cdot \\delta[k-6]$" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "application/pdf": "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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = dirac(k - 6)\n", "y = np.cos(k)\n", "w = x*y\n", "\n", "plt.figure(figsize=(10, 4))\n", "plt.subplot(121)\n", "plt.stem(k, y)\n", "plt.xlabel('$k$')\n", "plt.ylabel(r'$\\cos[\\Omega k]$')\n", "plt.ylim([-1.1, 1.1])\n", "\n", "plt.subplot(122)\n", "plt.stem(k, w)\n", "plt.xlabel('$k$')\n", "plt.ylabel('$\\cos[\\Omega k] \\cdot \\delta[k - \\kappa]$')\n", "plt.ylim([-1.1, 1.1])\n", "plt.tight_layout()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Heaviside Signal\n", "\n", "The discrete Heaviside signal is defined similar to its [continuous Heaviside signal](../continuous_signals/standard_signals.ipynb#Heaviside-Signal) as\n", "\n", "\\begin{equation}\n", "\\epsilon[k] = \\begin{cases} 1 & k \\geq 0 \\\\ 0 & k < 0 \\end{cases}\n", "\\end{equation}\n", "\n", "The Heaviside signal is used to represent a signal that switches on at a specified time and stays on indefinitely. The Heaviside signal can be related to the Dirac impulse by\n", "\n", "\\begin{equation}\n", "\\epsilon[k] = \\sum_{\\kappa = -\\infty}^{k} \\delta[\\kappa]\n", "\\end{equation}\n", "\n", "The Dirac impulse can be expressed in terms of the Heaviside signal\n", "\n", "\\begin{equation}\n", "\\delta[k] = \\epsilon[k] - \\epsilon[k-1]\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example**\n", "\n", "In the following, a function is defined for the Heaviside signal $\\epsilon[k]$ and the signal is plotted for illustration." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(-0.1, 1.1)" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" }, { "data": { "application/pdf": "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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def heaviside(k):\n", " return np.where(k >= 0, 1.0, 0.0)\n", "\n", "\n", "k = np.arange(-5, 20)\n", "x = heaviside(k)\n", "\n", "plt.stem(k, x)\n", "plt.xlabel('$k$')\n", "plt.ylabel('$\\epsilon[k]$')\n", "plt.ylim([-0.1, 1.1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Rectangular Signal\n", "\n", "The discrete rectangular signal is defined as\n", "\n", "\\begin{equation}\n", "\\text{rect}_N[k] = \\begin{cases} 1 & \\text{for } 0 \\leq k < N \\\\ 0 & \\text{otherwise} \\end{cases}\n", "\\end{equation}\n", "\n", "where $N \\in \\mathbb{N}$ denotes the number of its non-zero samples. Note that the discrete rectangular signal is not even symmetric as the [continuous rectangular signal](../continuous_signals/standard_signals.ipynb#Rectangular-Signal). The rectangular signal is used to represent signals which are non-zero for a limited period of time. The rectangular signal can be related to the Heaviside signal by\n", "\n", "\\begin{equation}\n", "\\text{rect}[k] = \\epsilon[k] - \\epsilon[k - N]\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example**\n", "\n", "A function is defined for the rectangular signal and $\\text{rect}_5[k]$ is plotted for illustration." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(-0.1, 1.1)" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" }, { "data": { "application/pdf": "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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def rect(k, N):\n", " return np.where((0 <= k) & (k < N), 1.0, 0.0)\n", "\n", "\n", "k = np.arange(-5, 20)\n", "x = rect(k, 10)\n", "\n", "plt.stem(k, x)\n", "plt.xlabel('$k$')\n", "plt.ylabel('$\\mathrm{rect}[k]$')\n", "plt.ylim([-0.1, 1.1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise**\n", "\n", "* Use $\\text{rect}_N[t]$ to construct a cosine signal $x[k] = \\cos[\\Omega k] \\cdot \\text{rect}_N[t]$ of one period length with $\\Omega= \\frac{2 \\pi}{10}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Sign Signal\n", "\n", "The discrete sign signal is defined analogously to the [continuous sign signal](../continuous_signals/standard_signals.ipynb#Sign-Signal) as\n", "\n", "\\begin{equation}\n", "\\text{sgn}[k] = \\begin{cases} 1 & k>0 \\\\ 0 & k=0 \\\\ -1 & k < 0 \\end{cases}\n", "\\end{equation}\n", "\n", "The sign signal is used to represent the absolute value of a signal $x[k]$ as\n", "\n", "\\begin{equation}\n", "|x[k]| = x[k] \\cdot \\text{sgn}(x[k])\n", "\\end{equation}\n", "\n", "It is related to the Heaviside signal by\n", "\n", "\\begin{equation}\n", "\\text{sgn}[k] = \\epsilon[k] - \\epsilon[-k]\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example**\n", "\n", "The sign signal is realized by using the [`numpy.sign`](http://docs.scipy.org/doc/numpy/reference/generated/numpy.sign.html#numpy.sign) function. It is plotted for illustration." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(-1.1, 1.1)" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" }, { "data": { "application/pdf": "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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "k = np.arange(-10, 11)\n", "x = np.sign(k)\n", "\n", "plt.stem(k, x)\n", "plt.xlabel('$k$')\n", "plt.ylabel('$\\mathrm{sgn}[k]$')\n", "plt.ylim([-1.1, 1.1])" ] }, { "cell_type": "markdown", "metadata": { "nbsphinx": "hidden" }, "source": [ "**Copyright**\n", "\n", "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own 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: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*." ] } ], "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 }