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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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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# Señales y Sistemas: Segundo Examen Parcial\n", | |
| "\n", | |
| "## Problemática\n", | |
| "\n", | |
| "### Enunciado\n", | |
| "Partiendo de la señal $x(t)$ de amplitud proporcional a una constante $A$ se desea obtener una señal sinusoidal pura mediante el uso de un filtro de banda de frecuencia ideal, representado por la función de transferencia $H(w)$. La amplitud de $x(t)$ debe variarse mediante $A$ de forma que la onda de salida posea valor RMS de $1 V$.\n", | |
| "\n", | |
| "#### Puntualmente\n", | |
| "\n", | |
| "1. Calcular el valor de $A$ en voltios para obtener la señal de salida deseada.\n", | |
| "2. Graficar los espectros involucrados en el sistema y desarroyar conceptualmente cómo obtuvo el resultado.\n", | |
| "3. En caso de que no pueda obtener una señal senoidal pura de $5 kHz$ explique el por qué y qué parámetros modificaría en $V(t)$ ($x(t)$) para obtener la señal deseada a la salida del sistema." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "# Importando símbolos utilizados en la resolución\n", | |
| "import sympy\n", | |
| "from sympy.physics.units.definitions import Hz, ms, V, s\n", | |
| "from sympy.physics.units import convert_to\n", | |
| "from sympy.physics.units.util import quantity_simplify as qsimplify\n", | |
| "from IPython.display import display, Markdown as Md\n", | |
| "from sympy import Function, Symbol, Eq, Piecewise, Or, And, pi, plot as splot, \\\n", | |
| " Sum, oo, Mod, Integral, exp, I, DiracDelta\n", | |
| "from sympy.abc import t, w, f, n, a\n", | |
| "from common import kHz\n", | |
| "from itertools import takewhile, count" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "### Datos" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<IPython.core.display.Markdown object>", | |
| "text/markdown": "Respuesta en frecuencia del filtro ideal:" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "Eq(H(w), Piecewise((second*volt, (w > 9*pi*kilohertz) & (w < 11*pi*kilohertz)), (0, True)))", | |
| "text/latex": "$\\displaystyle H{\\left(w \\right)} = \\begin{cases} \\text{s} \\text{V} & \\text{for}\\: w > 9 \\pi \\text{kHz} \\wedge w < 11 \\pi \\text{kHz} \\\\0 & \\text{otherwise} \\end{cases}$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<IPython.core.display.Markdown object>", | |
| "text/markdown": "Ancho de banda: ``1*kilohertz``. Centro: ``5*kilohertz``" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<Figure size 432x288 with 1 Axes>", | |
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| "data": { | |
| "text/plain": "<IPython.core.display.Markdown object>", | |
| "text/markdown": "$H(jw)$ en función de $w$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<IPython.core.display.Markdown object>", | |
| "text/markdown": "Señal de entrada:" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "Eq(x(t), Piecewise((A, (t > 0) & (t < 0.2*millisecond)), (0, True)))", | |
| "text/latex": "$\\displaystyle x{\\left(t \\right)} = \\begin{cases} A & \\text{for}\\: t > 0 \\wedge t < 0.2 \\text{ms} \\\\0 & \\text{otherwise} \\end{cases}$" | |
| }, | |
| "metadata": {} | |
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| { | |
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| "data": { | |
| "text/plain": "<Figure size 432x288 with 1 Axes>", | |
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| |
| }, | |
| "metadata": { | |
| "needs_background": "light" | |
| } | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<IPython.core.display.Markdown object>", | |
| "text/markdown": "$x(t)$ en función de $t$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<IPython.core.display.Markdown object>", | |
| "text/markdown": "Período: ``1*millisecond``. Ciclo de trabajo: ``20.0%``" | |
| }, | |
| "metadata": {} | |
| } | |
| ], | |
| "source": [ | |
| "# Definiendo la función de transferencia del filtro H(jw)\n", | |
| "centro = 5 * kHz\n", | |
| "ancho_de_banda = 1 * kHz\n", | |
| "freq_min = centro - ancho_de_banda / 2\n", | |
| "freq_max = centro + ancho_de_banda / 2\n", | |
| "H_w = Piecewise((1*V*s, And(freq_min < f, f < freq_max)),\n", | |
| " (0, True)).subs(f, w/(2*pi))\n", | |
| "display(Md(\"Respuesta en frecuencia del filtro ideal:\"))\n", | |
| "display(Eq(Function(\"H\")(w), H_w))\n", | |
| "display(Md(f\"Ancho de banda: ``1*{ancho_de_banda}``. Centro: ``{centro}``\"))\n", | |
| "splot((H_w/(V*s)).subs(w, w*kHz),\n", | |
| " (w, 0, float((freq_max*2*pi)/kHz) + 4),\n", | |
| " axis_center=(0, 0))\n", | |
| "display(Md(\"$H(jw)$ en función de $w$\"))\n", | |
| "\n", | |
| "# Definiendo la señal de entrada x(t)\n", | |
| "A = Symbol(\"A\")\n", | |
| "cliclo_de_trabajo = 0.2\n", | |
| "período = 1 * ms\n", | |
| "x_t = Piecewise((A, And(t > 0, t < cliclo_de_trabajo * período)),\n", | |
| " (0, True))\n", | |
| "display(Md(\"Señal de entrada:\"))\n", | |
| "display(Eq(Function(\"x\")(t), x_t))\n", | |
| "xp_t = x_t.subs(t, Mod(t, período))\n", | |
| "splot((xp_t.subs(t, t*ms)/A).subs(ms, 1).simplify(),\n", | |
| " (t, 0, período/ms * 2.5))\n", | |
| "display(Md(\"$x(t)$ en función de $t$\"))\n", | |
| "display(Md(f\"Período: ``1*{período}``. Ciclo de trabajo: ``{float(cliclo_de_trabajo)*100}%``\"))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Metodología\n", | |
| "\n", | |
| "Para lograr determinar la salida $y(t)$ generada por la excitación $x(t)$ se propone trabajar en el espacio de la frecuencia ya que las funciones completamente periódicas, mediante la transformada de Fourier, se pueden expresar cómo trenes de impulsos escalados por los coeficientes $c_n$ que forman la serie de Fourier exponencial.\n", | |
| "\n", | |
| "La porpiedad de muestreo del tren de impulsos y la de desactivación de los pulsos de la TF del filtro luego pueden utilizarse para reducir la expresión en series a una suma de terminos finitos, los cuales con un poco de suerte podrán ser antitransformados a una función senoidal pura." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Resolución\n", | |
| "\n", | |
| "En funciones periodicas los impulsos se encuentran distribuidos en el plano de la frecuencia en múliplos de la frecuencia fundamental de $x(t)$, ésto implica que para que exista respuesta, al menos uno de éstos múltiplos debe econtrarse entre los límites de la banda de frecuencia del filtro. Se puede encontrar $n$ para el impulso más cercano al centro del intervalo de la siguiente manera" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 3, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<IPython.core.display.Markdown object>", | |
| "text/markdown": "Período fundamental: ``2*pi*kilohertz``" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<IPython.core.display.Markdown object>", | |
| "text/markdown": "$n$ para el impulso más cercano: ``5``" | |
| }, | |
| "metadata": {} | |
| } | |
| ], | |
| "source": [ | |
| "w_0 = convert_to(2*pi/período, kHz)\n", | |
| "display(Md(f\"Período fundamental: ``{w_0}``\"))\n", | |
| "n_más_cercano = round((centro*2*pi/w_0))\n", | |
| "display(Md(f\"$n$ para el impulso más cercano: ``{n_más_cercano}``\"))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 4, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<IPython.core.display.Markdown object>", | |
| "text/markdown": "Valores de $n$ para impulsos dentro del rango del filtro: ``[5]``" | |
| }, | |
| "metadata": {} | |
| } | |
| ], | |
| "source": [ | |
| "n_no_filtrado = lambda n: And(freq_min*2*pi < w_0*n, w_0*n < freq_max*2*pi)\n", | |
| "ns_finales = list(takewhile(n_no_filtrado, count(n_más_cercano, 1))) + \\\n", | |
| " list(takewhile(n_no_filtrado, count(n_más_cercano-1, -1)))\n", | |
| "\n", | |
| "display(Md(f\"Valores de $n$ para impulsos dentro del rango del filtro: ``{ns_finales}``\"))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "La siguiente figura demuestra de forma gráfica que dentro del intervalo de frecuencias (rojo) solamente se encuentra el armónico (azúl) correspondiente a $n=5$. En el eje orizontal la frecuenca angular ($kHz$), y el el vertical la respuesta en frecuencia ($V{\\cdot}s$)." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 5, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<Figure size 432x288 with 1 Axes>", | |
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\n" | |
| }, | |
| "metadata": { | |
| "needs_background": "light" | |
| } | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<IPython.core.display.Markdown object>", | |
| "text/markdown": "$H(wj)$ en función de $w$ (rojo)" | |
| }, | |
| "metadata": {} | |
| } | |
| ], | |
| "source": [ | |
| "armónicos_plot = splot((H_w/(V*s)).subs(w, w*kHz),\n", | |
| " (w, 0, float((freq_max*2*pi)/kHz) + 4),\n", | |
| " axis_center=(0, 0), color=\"r\", show=False)\n", | |
| "armónicos_plot[0].line_color = \"r\"\n", | |
| "linea_vertical = lambda x_0, x: (x_0-x)*999999999\n", | |
| "lineas = [splot(linea_vertical(w_0*n/kHz,t), \n", | |
| " (t, n*w_0/kHz-1, n*w_0/kHz+1),\n", | |
| " show=False) for n in range(0,n_más_cercano+2)]\n", | |
| "[armónicos_plot.extend(plot) for plot in lineas]\n", | |
| "armónicos_plot.show()\n", | |
| "display(Md(\"$H(wj)$ en función de $w$ (rojo)\"))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "### Una posible solución\n", | |
| "\n", | |
| "Una vez acotado el tren de impulsos a los que pasan el filtro, el gurpo pasa a representar la transformada de Fourier para la respuesta del sistema, no es necesario aplicar la propiedad de muestreo del impulso debido a que éste filtro en específico no altera las señales dentro de la banda (los escalones son unitarios), por lo que los coeficientes $c_n$ pueden utilizarse directamente para resolver la respuesta del sistema mediante una expansión en series de Fourier." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 6, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<IPython.core.display.Markdown object>", | |
| "text/markdown": "Coeficientes $c_n$ por definición:" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "Eq(c_n, Integral(Piecewise((A*exp(-2*I*pi*kilohertz*n*t)/millisecond, (t > 0) & (t < 0.2*millisecond)), (0, True)), (t, 0, millisecond)))", | |
| "text/latex": "$\\displaystyle c_{n} = \\int\\limits_{0}^{\\text{ms}} \\begin{cases} \\frac{A e^{- 2 i \\pi n t \\text{kHz}}}{\\text{ms}} & \\text{for}\\: t > 0 \\wedge t < 0.2 \\text{ms} \\\\0 & \\text{otherwise} \\end{cases}\\, dt$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "Eq(c_n, Piecewise((-I*A/(2*pi*n) + I*A*exp(-0.4*I*pi*n)/(2*pi*n), Ne(n, 0)), (0.2*A, True)))", | |
| "text/latex": "$\\displaystyle c_{n} = \\begin{cases} - \\frac{i A}{2 \\pi n} + \\frac{i A e^{- 0.4 i \\pi n}}{2 \\pi n} & \\text{for}\\: n \\neq 0 \\\\0.2 A & \\text{otherwise} \\end{cases}$" | |
| }, | |
| "metadata": {} | |
| } | |
| ], | |
| "source": [ | |
| "display(Md(\"Coeficientes $c_n$ por definición:\"))\n", | |
| "x_cn_integral = Integral(1/período*x_t*exp(-I*n*w_0*t),(t,0,período)).simplify()\n", | |
| "x_cn = x_cn_integral.doit().subs(kHz*ms, 1)\n", | |
| "display(Eq(Symbol(\"c_n\"),x_cn_integral))\n", | |
| "display(Eq(Symbol(\"c_n\"),x_cn))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Como se puede notar en la expresión de $c_n$, el argumento de la componente exponencial compleja se puede expresar de la forma ${2\\pi}{\\cdot}ct{\\cdot}n$ donde $ct$ es el ciclo de trabajo, lo que da a entender que para valores de $n$ múltiplos del recíproco de $ct$ el término exponencial complejo se vuelve $1$, lo que vuelve cero al término $c_n$ para dicho $n$." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 7, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "Eq(Sum(Piecewise((pi*(-I*A/(2*pi*n) + I*A*exp(-0.4*I*pi*n)/(2*pi*n))*exp(2*I*pi*kilohertz*n*t), Ne(n, 0)), (0.2*pi*A*exp(2*I*pi*kilohertz*n*t), True)), (n, 5, 5)), pi*(-I*A/(10*pi) + I*A*exp(-2.0*I*pi)/(10*pi))*exp(10*I*pi*kilohertz*t))", | |
| "text/latex": "$\\displaystyle \\sum_{n=5}^{5} \\begin{cases} \\pi \\left(- \\frac{i A}{2 \\pi n} + \\frac{i A e^{- 0.4 i \\pi n}}{2 \\pi n}\\right) e^{2 i \\pi n t \\text{kHz}} & \\text{for}\\: n \\neq 0 \\\\0.2 \\pi A e^{2 i \\pi n t \\text{kHz}} & \\text{otherwise} \\end{cases} = \\pi \\left(- \\frac{i A}{10 \\pi} + \\frac{i A e^{- 2.0 i \\pi}}{10 \\pi}\\right) e^{10 i \\pi t \\text{kHz}}$" | |
| }, | |
| "metadata": {} | |
| } | |
| ], | |
| "source": [ | |
| "x_hat_t_sum = Sum(x_cn*1*pi*exp(I*n*w_0*t).subs(kHz*ms, 1), (n, min(ns_finales), max(ns_finales)))\n", | |
| "x_hat_t = x_hat_t_sum.doit()\n", | |
| "display(Eq(x_hat_t_sum, x_hat_t))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Dado que el único armónico que pasa por el filtro se encuentra en $n=5$, y éste es el recíproco del ciclo de trabajo, el resultado de la expansión en series es cero." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 8, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "Eq(pi*(-I*A/(10*pi) + I*A*exp(-2.0*I*pi)/(10*pi))*exp(10*I*pi*kilohertz*t), 0)", | |
| "text/latex": "$\\displaystyle \\pi \\left(- \\frac{i A}{10 \\pi} + \\frac{i A e^{- 2.0 i \\pi}}{10 \\pi}\\right) e^{10 i \\pi t \\text{kHz}} = 0$" | |
| }, | |
| "metadata": {} | |
| } | |
| ], | |
| "source": [ | |
| "display(Eq(x_hat_t, x_hat_t.simplify()))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Conclusiones\n", | |
| "\n", | |
| "1. En base al análisis realizado, resulta imposible con la función de entrada presentada en el enunciado y variando la constante $A$, conseguir una función sinusoidal pura, o cualquier señal distinta de cero.\n", | |
| "3. Para lograr obtener alguna señal distinta de cero con el mismo tipo de señal en la entrada, se debería alteral el ciclo de trabajo, y así que los coeficientes no sean nulos en la banda del filtro. El período puede quedar fijo, ya que $5{\\cdot}w_0$ resulta ser exactamente el centro de la banda del filtro" | |
| ] | |
| } | |
| ], | |
| "metadata": { | |
| "language_info": { | |
| "codemirror_mode": { | |
| "name": "ipython", | |
| "version": 3 | |
| }, | |
| "file_extension": ".py", | |
| "mimetype": "text/x-python", | |
| "name": "python", | |
| "nbconvert_exporter": "python", | |
| "pygments_lexer": "ipython3", | |
| "version": "3.8.2-final" | |
| }, | |
| "orig_nbformat": 2, | |
| "kernelspec": { | |
| "name": "python38264bit12d9a4114b3e4b5084767bfb1d8ea7ac", | |
| "display_name": "Python 3.8.2 64-bit" | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 2 | |
| } |
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| jupyter | |
| sympy | |
| matplotlib | |
| numpy |
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