{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"nbsphinx": "hidden"
},
"source": [
"# Characterization of Discrete Systems\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": [
"## Linear Convolution\n",
"\n",
"It was shown previously, that the convolution is an important operation in the theory of signals and linear time-invariant (LTI) systems. The linear convolution of two discrete signals $s[k]$ and $g[k]$ is defined as\n",
"\n",
"\\begin{equation}\n",
"\\sum_{\\kappa = -\\infty}^{\\infty} s[\\kappa] \\cdot g[k - \\kappa] = s[k] * g[k]\n",
"\\end{equation}\n",
"\n",
"where $*$ is a common short-hand notation of the convolution. The general properties of the linear convolution are discussed, followed by a geometrical interpretation of the operation."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Properties\n",
"\n",
"For the discrete signals $s[k]$, $g[k]$, $h[k] \\in \\mathbb{C}$ the convolution shows the following properties \n",
"\n",
"1. The Dirac impulse is the [identity element](https://en.wikipedia.org/wiki/Identity_element) of the convolution\n",
" \\begin{equation}\n",
" s[k] * \\delta[k] = s[k]\n",
" \\end{equation}\n",
" \n",
"2. The convolution is [commutative](https://en.wikipedia.org/wiki/Commutative_property)\n",
" \\begin{equation}\n",
" s[k] * g[k] = g[k] * s[k]\n",
" \\end{equation}\n",
" \n",
"3. The convolution is [associative](https://en.wikipedia.org/wiki/Associative_property)\n",
" \\begin{equation}\n",
" \\left( s[k] * g[k] \\right) * h[k] = s[k] * \\left( g[k] * h[k] \\right) \n",
" \\end{equation}\n",
"\n",
"5. The convolution is [distributive](https://en.wikipedia.org/wiki/Distributive_property)\n",
" \\begin{equation}\n",
" s[k] * \\left( g[k] + h[k] \\right) = s[k] * g[k] + s[k] * h[k]\n",
" \\end{equation}\n",
"\n",
"5. Multiplication with a scalar $a \\in \\mathbb{C}$\n",
" \\begin{equation}\n",
" a \\cdot \\left( s[k] * g[k] \\right) = \\left( a \\cdot s[k] \\right) * g[k] = s[k] * \\left( a \\cdot g[k] \\right)\n",
" \\end{equation}\n",
"\n",
"The first property is a consequence of the sifting property of the Dirac pulse, the second to fifth property can be proven by considering the definition of the convolution."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Geometrical Interpretation\n",
"\n",
"The convolution can be [interpreted in a graphical manner](https://en.wikipedia.org/wiki/Convolution#Visual_explanation). This provides valuable insights into its calculation and allows to estimate the result. The calculation of the linear convolution \n",
"\n",
"\\begin{equation}\n",
"y[k] = x[k] * h[k] = \\sum_{\\kappa = -\\infty}^{\\infty} x[\\kappa] \\cdot h[k - \\kappa]\n",
"\\end{equation}\n",
"\n",
"\n",
"can be decomposed into four subsequent steps:\n",
"\n",
"1. substitute $k$ by $\\kappa$ in both $x[k]$ and $h[k]$,\n",
"\n",
"2. time-reverse $h[\\kappa]$ (reflection at vertical axis),\n",
"\n",
"3. shift $h[- \\kappa]$ by $k$ to the to yield $h[k - \\kappa]$, i.e. a shift to **right** for $k>0$ or a shift to **left** for $k<0$,\n",
"\n",
"4. check for which $k = -\\infty \\dots \\infty$ the mirrored & shifted $h[k - \\kappa]$ overlap with $x[\\kappa]$, calculate the specific sum for all the relevant $k$ to yield $y[k]$\n",
"\n",
"The graphical interpretation of the convolution is illustrated by means of the following example."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Example**\n",
"\n",
"The procedure is illustrated with the signals\n",
"\n",
"\\begin{align}\n",
"h[k] &= \\epsilon[k] \\cdot e^{- \\frac{k}{2}} \\\\\n",
"x[k] &= \\frac{3}{4} \\text{rect}_N[k] \n",
"\\end{align}\n",
"\n",
"for $N=6$. Before proceeding, some helper functions and the signals are defined."
]
},
{
"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 heaviside(k):\n",
" return np.where(k >= 0, 1.0, 0.0)\n",
"\n",
"\n",
"def rect(k, N):\n",
" return np.where((0 <= k) & (k < N), 1.0, 0.0)\n",
"\n",
"\n",
"def x(k):\n",
" return 3/4 * rect(k, 6)\n",
"\n",
"\n",
"def h(k):\n",
" return heaviside(k) * np.exp(- k/2)\n",
"\n",
"\n",
"def plot_signals(k, x, h, xlabel, hlabel, klabel):\n",
" plt.figure(figsize=(8, 4))\n",
" plt.stem(k, x, linefmt='C0-', markerfmt='C0o', label=xlabel)\n",
" plt.stem(k, h, linefmt='C1-', markerfmt='C1o', label=hlabel)\n",
" plt.xlabel(klabel)\n",
" plt.legend()\n",
" plt.ylim([0, 1.5])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's compute and plot the signals $x[k]$ and $h[k]$"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
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],
"text/plain": [
"<Figure size 768x384 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"k = np.arange(-20, 21)\n",
"\n",
"plot_signals(k, x(k), h(k), r'$x[k]$', r'$h[k]$', r'$k$')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The **first step** is to substitute $k$ by $\\kappa$ in both $x[k]$ and $h[k]$ to yield $x[\\kappa]$ and $h[\\kappa]$. Note, the horizontal axis of the plot represents now $\\kappa$, which is our temporal helper variable for the integration"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
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" </defs>\n",
" <g transform=\"translate(447.403125 27.298437)scale(0.1 -0.1)\">\n",
" <use transform=\"translate(0 0.015625)\" xlink:href=\"#DejaVuSans-Oblique-120\"/>\n",
" <use transform=\"translate(59.179688 0.015625)\" xlink:href=\"#DejaVuSans-91\"/>\n",
" <use transform=\"translate(98.193359 0.015625)\" xlink:href=\"#DejaVuSans-Oblique-954\"/>\n",
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" </g>\n",
" </g>\n",
" <g id=\"line2d_107\">\n",
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" </g>\n",
" </g>\n",
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" </g>\n",
" <g id=\"line2d_109\">\n",
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"\" style=\"fill:none;stroke:#d62728;stroke-linecap:square;stroke-width:1.5;\"/>\n",
" </g>\n",
" <g id=\"text_20\">\n",
" <!-- $h[\\kappa]$ -->\n",
" <defs>\n",
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" </defs>\n",
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" </g>\n",
" </g>\n",
" </g>\n",
" </g>\n",
" </g>\n",
" <defs>\n",
" <clipPath id=\"pc8b29bbdd5\">\n",
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"</svg>\n"
],
"text/plain": [
"<Figure size 768x384 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"kappa = np.arange(-20, 21)\n",
"\n",
"x1 = x(kappa)\n",
"h1 = h(kappa)\n",
"\n",
"plot_signals(kappa, x1, h1, r'$x[\\kappa]$', r'$h[\\kappa]$', r'$\\kappa$')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The **second step** is to time-reverse $h[\\kappa]$ to yield $h[-\\kappa]$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
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],
"text/plain": [
"<Figure size 768x384 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"h2 = h(kappa[::-1])\n",
"\n",
"plot_signals(k, x1, h2, r'$x[\\kappa]$', r'$h[-\\kappa]$', r'$\\kappa$')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the **third step** the impulse response $h[-\\kappa]$ is shifted by $k$ samples to yield $h[k - \\kappa]$. The shift is performed to the **right** for $k>0$ and to the **left** for $k<0$.\n",
"\n",
"For the **fourth step** it is often useful to split the calculation of the result according to the overlap between $h[k-\\kappa]$ and $x[\\kappa]$. For the given signals three different cases may be considered\n",
"\n",
"1. no overlap for $k<0$,\n",
"2. partial overlap for $0 \\leq k < 6$, and\n",
"3. full overlap for $k \\geq 6$ (note that the chosen impulse response decays asymptotically).\n",
"\n",
"The first case, no overlap, is illustrated for $k= - 5$"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
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" </g>\n",
" <g id=\"text_20\">\n",
" <!-- $h[-5 -\\kappa]$ -->\n",
" <defs>\n",
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"</svg>\n"
],
"text/plain": [
"<Figure size 768x384 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"h3 = h(-5 + kappa[::-1])\n",
"\n",
"plot_signals(k, x1, h3, r'$x[\\kappa]$', r'$h[-5 -\\kappa]$', r'$\\kappa$')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"From this it becomes clear that the convolution result for the first case is given as\n",
"\n",
"\\begin{equation}\n",
"y[k] = 0 \\qquad \\text{for } k < 0\n",
"\\end{equation}\n",
"\n",
"The second case, partial overlap, is illustrated for $k = 3$"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
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],
"text/plain": [
"<Figure size 768x384 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"h4 = h(3 + kappa[::-1])\n",
"\n",
"plot_signals(k, x1, h4, r'$x[\\kappa]$', r'$h[3 -\\kappa]$', r'$\\kappa$')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Hence, for the second case the convolution sum degenerates to\n",
"\n",
"\\begin{equation}\n",
"y[k] = \\frac{3}{4} \\sum_{\\kappa=0}^{k} e^{-\\frac{k - \\kappa}{2}} \\qquad \\text{for } 0 \\leq k < 6\n",
"\\end{equation}\n",
"\n",
"The third case, full overlap, is illustrated for $k = 10$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
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],
"text/plain": [
"<Figure size 768x384 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"h5 = h(10 + kappa[::-1])\n",
"\n",
"plot_signals(k, x1, h5, r'$x[\\kappa]$', r'$h[10 -\\kappa]$', r'$\\kappa$')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For the third case the convolution sum degenerates to\n",
"\n",
"\\begin{equation}\n",
"y[k] = \\frac{3}{4} \\sum_{\\kappa=0}^{5} e^{-\\frac{k - \\kappa}{2}} \\qquad \\text{for } k \\geq 6\n",
"\\end{equation}\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The overall result is composed from the three individual results. As alternative and in order to plot the result, the convolution is evaluated numerically"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(0, 2)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
},
{
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"source": [
"def y(k):\n",
" return np.convolve(x(k), h(k), mode='same')\n",
"\n",
"\n",
"plt.stem(k, y(k), linefmt='C2-', markerfmt='C2o')\n",
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"plt.ylabel(r'$y[k]$')\n",
"plt.ylim([0, 2])"
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"The entire process is illustrated in the following animation. The upper plot shows the integrands $h[k-\\kappa]$ and $x[\\kappa]$ of the convolution sum, the lower plot the result $y[k] = x[k] * h[k]$ of the convolution. The red dot in the lower plot indicates the particular time instant $k$ for which the result of the convolution sum is computed. The time instant $k$ is varied in the animation."
]
},
{
"cell_type": "code",
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"metadata": {},
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" <br>\n",
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" name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
" onchange=\"anim6f0add05a4bf4d85ae363537be8dfbe0.set_frame(parseInt(this.value));\"></input>\n",
" <br>\n",
" <button onclick=\"anim6f0add05a4bf4d85ae363537be8dfbe0.slower()\"><i class=\"fa fa-minus\"></i></button>\n",
" <button onclick=\"anim6f0add05a4bf4d85ae363537be8dfbe0.first_frame()\"><i class=\"fa fa-fast-backward\">\n",
" </i></button>\n",
" <button onclick=\"anim6f0add05a4bf4d85ae363537be8dfbe0.previous_frame()\">\n",
" <i class=\"fa fa-step-backward\"></i></button>\n",
" <button onclick=\"anim6f0add05a4bf4d85ae363537be8dfbe0.reverse_animation()\">\n",
" <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
" <button onclick=\"anim6f0add05a4bf4d85ae363537be8dfbe0.pause_animation()\"><i class=\"fa fa-pause\">\n",
" </i></button>\n",
" <button onclick=\"anim6f0add05a4bf4d85ae363537be8dfbe0.play_animation()\"><i class=\"fa fa-play\"></i>\n",
" </button>\n",
" <button onclick=\"anim6f0add05a4bf4d85ae363537be8dfbe0.next_frame()\"><i class=\"fa fa-step-forward\">\n",
" </i></button>\n",
" <button onclick=\"anim6f0add05a4bf4d85ae363537be8dfbe0.last_frame()\"><i class=\"fa fa-fast-forward\">\n",
" </i></button>\n",
" <button onclick=\"anim6f0add05a4bf4d85ae363537be8dfbe0.faster()\"><i class=\"fa fa-plus\"></i></button>\n",
" <form action=\"#n\" name=\"_anim_loop_select6f0add05a4bf4d85ae363537be8dfbe0\" class=\"anim_control\">\n",
" <input type=\"radio\" name=\"state\"\n",
" value=\"once\" > Once </input>\n",
" <input type=\"radio\" name=\"state\"\n",
" value=\"loop\" checked> Loop </input>\n",
" <input type=\"radio\" name=\"state\"\n",
" value=\"reflect\" > Reflect </input>\n",
" </form>\n",
"</div>\n",
"\n",
"\n",
"<script language=\"javascript\">\n",
" /* Instantiate the Animation class. */\n",
" /* The IDs given should match those used in the template above. */\n",
" (function() {\n",
" var img_id = \"_anim_img6f0add05a4bf4d85ae363537be8dfbe0\";\n",
" var slider_id = \"_anim_slider6f0add05a4bf4d85ae363537be8dfbe0\";\n",
" var loop_select_id = \"_anim_loop_select6f0add05a4bf4d85ae363537be8dfbe0\";\n",
" var frames = new Array(20);\n",
" \n",
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"\"\n",
"\n",
"\n",
" /* set a timeout to make sure all the above elements are created before\n",
" the object is initialized. */\n",
" setTimeout(function() {\n",
" anim6f0add05a4bf4d85ae363537be8dfbe0 = new Animation(frames, img_id, slider_id, 100.0,\n",
" loop_select_id);\n",
" }, 0);\n",
" })()\n",
"</script>\n"
],
"text/plain": [
"<matplotlib.animation.FuncAnimation at 0x1178ab8d0>"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from animation import animate_discrete_convolution\n",
"plt.rcParams['animation.html'] = 'jshtml'\n",
"\n",
"kappa = np.arange(-5, 15)\n",
"anim = animate_discrete_convolution(x, h, y, k, kappa)\n",
"anim"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Finite-Length Signals\n",
"\n",
"The length of a discrete signal $x[k]$ is defined as the total number of samples in between the first sample which is not zero and the last sample which is zero, plus one. A signal of finite-length or finite-length signal is a signal whose length is finite. The Dirac impulse $\\delta[k]$ is for instance a finite-length signal of length one.\n",
"\n",
"The convolution of two finite-length signals is of practical importance since the convolution can only be evaluated numerically for finite-length signals. Any infinite-length signal can be truncated to a finite length by multiplying it with the rectangular signal $\\text{rect}_N[k]$. It is hence sufficient to consider the convolution of two rectangular signals of length $N \\in \\mathbb{N}$ and $M \\in \\mathbb{N}$\n",
"\n",
"\\begin{equation}\n",
"x[k] = \\text{rect}_N[k] * \\text{rect}_M[k]\n",
"\\end{equation}\n",
"\n",
"in order to derive insights on the convolution of two arbitrary finite-length signals. By following above geometrical interpretation of the linear convolution, the result for $N \\leq M$ can be found as\n",
"\n",
"\\begin{equation}\n",
"x[k] = \\begin{cases}\n",
"0 & \\text{for } k < 0 \\\\\n",
"k+1 & \\text{for } 0 \\leq k < N \\\\\n",
"N & \\text{for } N \\leq k < M \\\\\n",
"N+M-1-k & \\text{for } M \\leq k < N+M-1\\\\\n",
"0 & \\text{for } k \\geq N+M\n",
"\\end{cases}\n",
"\\end{equation}\n",
"\n",
"The convolution of two rectangular signals results in a finite-length signal. The length of the signal is $N+M-1$. This result can be generalized to the convolution of two arbitrary finite-length signals by following above reasoning. The convolution of two finite-length signals with length $N$ and $M$, respectively results in a finite-length signal of length $N+M-1$.\n",
"\n",
"For two causal signals $s[k]$ and $g[k]$ of finite length $N$ and $M$ the convolution reads\n",
"\n",
"\\begin{equation}\n",
"s[k] * g[k] = \\sum_{\\kappa = 0}^{N-1} s[\\kappa] \\cdot g[k - \\kappa] = \\sum_{\\kappa = 0}^{M-1} s[k - \\kappa] \\cdot g[\\kappa]\n",
"\\end{equation}\n",
"\n",
"for $0 \\leq k < N+M-1$. The computation of each output sample requires at least $N$ multiplications and $N-1$ additions. The numerical complexity of the convolution for the computation of $N$ output samples is therefore [in the order of](https://en.wikipedia.org/wiki/Big_O_notation) $\\mathcal{O} ( N^2 )$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Example**\n",
"\n",
"The convolution of two rectangular signals $x[k] = \\text{rect}_N[k] * \\text{rect}_M[k]$ of length $N$ and $M$ is computed in the following. The resulting signal is plotted for illustration."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-5, 25)"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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"source": [
"N = 7\n",
"M = 15\n",
"\n",
"x = np.convolve(np.ones(N), np.ones(M), mode='full')\n",
"\n",
"plt.stem(x)\n",
"plt.xlabel('$k$')\n",
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},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Exercise**\n",
"\n",
"* Compute the convolution of two rectangular signals of equal length analytically. Check your results by modifying the numerical example."
]
},
{
"cell_type": "markdown",
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"nbsphinx": "hidden"
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"source": [
"**Copyright**\n",
"\n",
"The notebooks are provided as [Open Educational Resource](https://de.wikipedia.org/wiki/Open_Educational_Resources). Feel free to use the notebooks 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: *Lecture Notes on Signals and Systems* by Sascha Spors."
]
}
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