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Algunas resoluciones para los trabajos prácticos propuestos por la cátedra de la materia "Señales y Sistemas"
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| #!/usr/bin/env python3.8 | |
| from sympy import Basic, Piecewise, Symbol, Eq, Mod, Heaviside, sympify, plot as splot | |
| import numpy | |
| from typing import Hashable, Tuple, Iterable, NamedTuple, Callable, Type | |
| from matplotlib import pyplot as mplot | |
| from sympy.physics.units.definitions import Hz | |
| from sympy.physics.units.quantities import Quantity | |
| from sympy.physics.units.prefixes import kilo | |
| kilohertz = khz = kHz = Quantity("kilohertz", abbrev="kHz") | |
| kHz.set_global_relative_scale_factor(kilo, Hz) | |
| # Class used for discrete functions | |
| # .zeroth is the index of the element for which n = 0, | |
| # negative numbers imply that n = 0 is outside the list | |
| class DiscreteSpace(NamedTuple): | |
| values: Tuple[Basic, ...] | |
| zeroth: Basic | |
| @property | |
| def n(self) -> Iterable[Basic]: | |
| return (i - self.zeroth for i in range(len(self.values))) | |
| def padded(self, right: int, filler: Basic = 0) -> "DiscreteSpace": | |
| pad = (filler,) | |
| zeroth = max(0, -right) + self.zeroth | |
| return DiscreteSpace(pad * -right + self.values + pad * right, zeroth) | |
| def centered(self, filler: Basic = 0) -> "DiscreteSpace": | |
| return self.padded((self.zeroth*2 + 1) - len(self.values), filler) | |
| def aligned_pair(self, | |
| other: "DiscreteSpace", | |
| filler: Basic = 0) -> Tuple["DiscreteSpace", "DiscreteSpace"]: | |
| lower_diff = other.zeroth - self.zeroth | |
| upper_diff = (len(other.values) - other.zeroth) - (len(self.values) - self.zeroth) | |
| new_zeroth = max(other.zeroth, self.zeroth) | |
| self_padded = (filler,) * lower_diff + self.values + (filler,) * upper_diff | |
| other_padded = (filler,) * -lower_diff + other.values + (filler,) * -upper_diff | |
| return (DiscreteSpace(self_padded, new_zeroth), | |
| DiscreteSpace(other_padded, new_zeroth)) | |
| def as_piecewise(self, var: Symbol, filler: Basic = 0) -> Piecewise: | |
| arguments = [(value, Eq(var, n)) for value, n in zip(self.values, self.n)] | |
| arguments += [(filler, Eq(Mod(var, 1), 0))] | |
| return Piecewise(*arguments) | |
| @classmethod | |
| def from_sampling(cls, zeroth: Basic, nsamples: int, f: Callable[[Basic], Basic]): | |
| samples = (f(i - zeroth) for i in range(nsamples)) | |
| return cls(tuple(samples), zeroth) | |
| # We'll use this to plot discrete spaces as scatter plots | |
| def plot_discs(discs: DiscreteSpace, show=True, **kwargs): | |
| mplot.scatter(list(discs.n), discs.values, **kwargs) | |
| mplot.show() if show else None | |
| def splot_colors(*exprs: Tuple[Basic, str], plotf: Type[splot], **options): | |
| to_plot, colors = zip(*exprs) | |
| show = options.pop("show", True) | |
| plot = plotf(*to_plot, show=False, **options) | |
| for i, color in enumerate(colors): | |
| plot[i].line_color = color | |
| plot.show() if show else None | |
| return plot | |
| # Converts an expression with units for the variable var into | |
| # a dimentionless lambda | |
| def ulambdify(expr: Basic, var: Symbol, unit_in, unit_out): | |
| return lambda x: (expr.subs(var, x*unit_in) / unit_out) | |
| # Splits a function of ``var`` into even and odd parts | |
| def eo_split(f: Basic, var: Symbol): | |
| print(f, var) | |
| return ((f + f.subs(var, -var)) / 2, | |
| (f - f.subs(var, -var)) / 2) | |
| def boxcar(t_on: Basic, t_off: Basic, t: Symbol, unit = 1): | |
| return Heaviside((t - t_on) / unit) - Heaviside((t - t_off) / unit) |
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| jupyter | |
| sympy | |
| matplotlib | |
| numpy |
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| { | |
| "nbformat": 4, | |
| "nbformat_minor": 2, | |
| "metadata": { | |
| "language_info": { | |
| "name": "python", | |
| "codemirror_mode": { | |
| "name": "ipython", | |
| "version": 3 | |
| }, | |
| "version": "3.8.2-final" | |
| }, | |
| "orig_nbformat": 2, | |
| "file_extension": ".py", | |
| "mimetype": "text/x-python", | |
| "name": "python", | |
| "npconvert_exporter": "python", | |
| "pygments_lexer": "ipython3", | |
| "version": 3, | |
| "kernelspec": { | |
| "name": "python38264bit12d9a4114b3e4b5084767bfb1d8ea7ac", | |
| "display_name": "Python 3.8.2 64-bit" | |
| } | |
| }, | |
| "cells": [ | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "import sympy\n", | |
| "from sympy import (DiracDelta, Piecewise, Symbol, Dummy,\n", | |
| " integrate, oo, Basic, Rational as R,\n", | |
| " plot as splot)\n", | |
| "import numpy\n", | |
| "import matplotlib\n", | |
| "from matplotlib import pyplot as mplot\n", | |
| "from functools import partial\n", | |
| "from common import DiscreteSpace, plot_discs, splot_colors\n", | |
| "from typing import Tuple\n", | |
| "splot_colors = partial(splot_colors, plotf=splot)\n", | |
| "matplotlib.style.use(\"dark_background\")" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "# Performs a convolution on two continuous functions\n", | |
| "def convolve(f1: Basic,\n", | |
| " f2: Basic,\n", | |
| " var: Symbol,\n", | |
| " limits: Tuple[Basic, Basic] = (-oo, oo),\n", | |
| " tau: Symbol = Dummy(\"tau\")):\n", | |
| " return integrate(\n", | |
| " f1.subs(var, tau) * f2.subs(var, var - tau),\n", | |
| " (tau , limits[0], limits[1])\n", | |
| " )" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 3, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "def convolve_discrete(a: DiscreteSpace, \n", | |
| " b: DiscreteSpace) -> \\\n", | |
| " Tuple[DiscreteSpace, DiscreteSpace]:\n", | |
| " aligned_a, aligned_b = a.aligned_pair(b)\n", | |
| " convolved = numpy.convolve(aligned_a.values,\n", | |
| " aligned_b.values)\n", | |
| " return DiscreteSpace(tuple(convolved),\n", | |
| " aligned_a.zeroth + aligned_b.zeroth)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 4, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<Figure size 432x288 with 1 Axes>", | |
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| }, | |
| "metadata": {} | |
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| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "Piecewise((0, t < -1), (t + 1, t < 0), (1 - t, t < 1), (0, True))", | |
| "text/latex": "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: t < -1 \\\\t + 1 & \\text{for}\\: t < 0 \\\\1 - t & \\text{for}\\: t < 1 \\\\0 & \\text{otherwise} \\end{cases}$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<Figure size 432x288 with 1 Axes>", | |
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\n" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "Piecewise((0, t < 0), (1, t < 1), (3 - 2*t, t < 3/2), (0, True))", | |
| "text/latex": "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: t < 0 \\\\1 & \\text{for}\\: t < 1 \\\\3 - 2 t & \\text{for}\\: t < \\frac{3}{2} \\\\0 & \\text{otherwise} \\end{cases}$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<Figure size 432x288 with 1 Axes>", | |
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| |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "Piecewise((1, Eq(n, 0)), (2, Eq(n, 1)), (1/2, Eq(n, 2)), (1, Eq(n, 3)), (0, True))", | |
| "text/latex": "$\\displaystyle \\begin{cases} 1 & \\text{for}\\: n = 0 \\\\2 & \\text{for}\\: n = 1 \\\\\\frac{1}{2} & \\text{for}\\: n = 2 \\\\1 & \\text{for}\\: n = 3 \\\\0 & \\text{otherwise} \\end{cases}$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
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| "text/plain": "<Figure size 432x288 with 1 Axes>", | |
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| |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "Piecewise((1, Eq(n, -1)), (1/2, Eq(n, 0)), (3/2, Eq(n, 1)), (0, Eq(n, 2)), (1, Eq(n, 3)), (0, True))", | |
| "text/latex": "$\\displaystyle \\begin{cases} 1 & \\text{for}\\: n = -1 \\\\\\frac{1}{2} & \\text{for}\\: n = 0 \\\\\\frac{3}{2} & \\text{for}\\: n = 1 \\\\0 & \\text{for}\\: n = 2 \\\\1 & \\text{for}\\: n = 3 \\\\0 & \\text{otherwise} \\end{cases}$" | |
| }, | |
| "metadata": {} | |
| } | |
| ], | |
| "source": [ | |
| "t = Symbol('t')\n", | |
| "n = Symbol('n', integer=True)\n", | |
| "x_t = Piecewise((0, t < -1),\n", | |
| " (1 + t, t < 0),\n", | |
| " (1 - t, t < 1),\n", | |
| " (0, True))\n", | |
| "g_t = Piecewise((0, t < 0),\n", | |
| " (1, t < 1),\n", | |
| " (3 - 2*t, t < R(\"1.5\")),\n", | |
| " (0, True))\n", | |
| "x_n = DiscreteSpace((1, 2, R(\".5\"), 1), 0)\n", | |
| "h_n = DiscreteSpace((1, R(\".5\"), R(\"1.5\"), 0, 1), 1)\n", | |
| "splot(x_t, xlim=(-2,2))\n", | |
| "display(x_t)\n", | |
| "splot(g_t, xlim=(-2,2))\n", | |
| "display(g_t)\n", | |
| "plot_discs(x_n.centered(0), color=\"r\")\n", | |
| "display(x_n.as_piecewise(n))\n", | |
| "plot_discs(h_n.centered(0), color=\"g\")\n", | |
| "display(h_n.as_piecewise(n))\n", | |
| "" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 5, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "sol_1a = convolve(x_t, DiracDelta(3 - t), t)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 6, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<Figure size 432x288 with 1 Axes>", | |
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\n" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "Piecewise((0, t > 4), (4 - t, t > 3), (t - 2, t > 2), (0, True))", | |
| "text/latex": "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: t > 4 \\\\4 - t & \\text{for}\\: t > 3 \\\\t - 2 & \\text{for}\\: t > 2 \\\\0 & \\text{otherwise} \\end{cases}$" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 6 | |
| } | |
| ], | |
| "source": [ | |
| "splot_colors((x_t, 'r'), (sol_1a, 'b'), xlim=(-2,4.5))\n", | |
| "sympy.simplify(sol_1a.rewrite(Piecewise, H0=1))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 7, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "# Takes a long time to compute\n", | |
| "sol_1b = convolve(x_t, g_t, t)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 8, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
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