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fakuivan/common.py

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Algunas resoluciones para los trabajos prácticos propuestos por la cátedra de la materia "Señales y Sistemas"
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common.py
#!/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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parcial2.ipynb
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requirements.txt
jupyter
sympy
matplotlib
numpy
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tp1.ipynb
{
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"metadata": {
"language_info": {
"name": "python",
"codemirror_mode": {
"name": "ipython",
"version": 3
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"version": "3.8.1-final"
},
"orig_nbformat": 2,
"file_extension": ".py",
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{
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"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import sympy\n",
"import matplotlib\n",
"from functools import partial\n",
"from sympy import (Piecewise, Symbol, Rational as R, \n",
" plot as splot, Heaviside, lambdify)\n",
"from common import DiscreteSpace, plot_discs, splot_colors\n",
"matplotlib.style.use(\"dark_background\")\n",
"splot_colors = partial(splot_colors, plotf=splot)"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
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"text/plain": "Piecewise((0, t < -1), (t + 1, t < 0), (1 - t, t < 1), (0, True))",
"text/latex": "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: t < -1 \\\\t + 1 & \\text{for}\\: t < 0 \\\\1 - t & \\text{for}\\: t < 1 \\\\0 & \\text{otherwise} \\end{cases}$"
},
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},
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"data": {
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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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"text/plain": "Piecewise((1, Eq(n, 0)), (2, Eq(n, 1)), (1/2, Eq(n, 2)), (1, Eq(n, 3)), (0, True))",
"text/latex": "$\\displaystyle \\begin{cases} 1 & \\text{for}\\: n = 0 \\\\2 & \\text{for}\\: n = 1 \\\\\\frac{1}{2} & \\text{for}\\: n = 2 \\\\1 & \\text{for}\\: n = 3 \\\\0 & \\text{otherwise} \\end{cases}$"
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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))"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"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((0, t < 1), (t - 1, t < 2), (3 - t, t < 3), (0, True))",
"text/latex": "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: t < 1 \\\\t - 1 & \\text{for}\\: t < 2 \\\\3 - t & \\text{for}\\: t < 3 \\\\0 & \\text{otherwise} \\end{cases}$"
},
"metadata": {}
}
],
"source": [
"sol_3a = x_t.subs(t, t - 2)\n",
"splot_colors((x_t, 'r'), (sol_3a, 'b'), xlim = (-1.5, 3))\n",
"display(sol_3a.simplify())"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"data": {
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},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": "Piecewise((0, t > 1), (1, t > 0), (2*t + 1, t > -1/2), (0, True))",
"text/latex": "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: t > 1 \\\\1 & \\text{for}\\: t > 0 \\\\2 t + 1 & \\text{for}\\: t > - \\frac{1}{2} \\\\0 & \\text{otherwise} \\end{cases}$"
},
"metadata": {}
}
],
"source": [
"sol_3b = g_t.subs(t, 1 - t)\n",
"splot_colors((g_t, 'r'), (sol_3b, 'b'), xlim=(-1, 2))\n",
"display(sol_3b.simplify())"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"data": {
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},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": "Piecewise((0, t > 6), (3 - t/2, t > 4), (t/2 - 1, t > 2), (0, True))",
"text/latex": "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: t > 6 \\\\3 - \\frac{t}{2} & \\text{for}\\: t > 4 \\\\\\frac{t}{2} - 1 & \\text{for}\\: t > 2 \\\\0 & \\text{otherwise} \\end{cases}$"
},
"metadata": {}
}
],
"source": [
"sol_3c = x_t.subs(t, 2 - t / 2)\n",
"splot_colors((x_t, 'r'), (sol_3c, 'b'), xlim=(-1.5, 6.5))\n",
"display(sol_3c.simplify())"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"data": {
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},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": "Piecewise((0, t > 1), (t, (t > 1/2) & (t < 1)), (2*t - 1, t > 1/2), (0, t < -1), (t + 1, t < 0), (1 - t, t < 1), (0, True))",
"text/latex": "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: t > 1 \\\\t & \\text{for}\\: t > \\frac{1}{2} \\wedge t < 1 \\\\2 t - 1 & \\text{for}\\: t > \\frac{1}{2} \\\\0 & \\text{for}\\: t < -1 \\\\t + 1 & \\text{for}\\: t < 0 \\\\1 - t & \\text{for}\\: t < 1 \\\\0 & \\text{otherwise} \\end{cases}$"
},
"metadata": {}
}
],
"source": [
"sol_3d = (x_t + g_t.subs(t, 2 - t)) * Heaviside(1 - t)\n",
"splot_colors((x_t, 'r'), \n",
" (g_t.subs(t, 2 - t), 'g'),\n",
" (Heaviside(1 - t), 'm'),\n",
" (sol_3d, 'b'),\n",
" xlim=(-1.5, 2.5))\n",
"display(sol_3d.rewrite(Piecewise, H0=1).simplify())"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"data": {
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},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": "Piecewise((1, Eq(n, 3)), (2, Eq(n, 4)), (1/2, Eq(n, 5)), (1, Eq(n, 6)), (0, True))",
"text/latex": "$\\displaystyle \\begin{cases} 1 & \\text{for}\\: n = 3 \\\\2 & \\text{for}\\: n = 4 \\\\\\frac{1}{2} & \\text{for}\\: n = 5 \\\\1 & \\text{for}\\: n = 6 \\\\0 & \\text{otherwise} \\end{cases}$"
},
"metadata": {}
}
],
"source": [
"sol_3e = x_n.as_piecewise(n, 0).subs(n, n - 3)\n",
"plot_discs(DiscreteSpace.from_sampling(0, 9, \n",
" lambdify(n, sol_3e)))\n",
"display(sol_3e.simplify())"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"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, 4)), (1/2, Eq(n, 3)), (3/2, Eq(n, 2)), (0, Eq(n, 1)), (1, Eq(n, 0)), (0, True))",
"text/latex": "$\\displaystyle \\begin{cases} 1 & \\text{for}\\: n = 4 \\\\\\frac{1}{2} & \\text{for}\\: n = 3 \\\\\\frac{3}{2} & \\text{for}\\: n = 2 \\\\0 & \\text{for}\\: n = 1 \\\\1 & \\text{for}\\: n = 0 \\\\0 & \\text{otherwise} \\end{cases}$"
},
"metadata": {}
}
],
"source": [
"sol_3f = h_n.as_piecewise(n, 0).subs(n, 3 - n)\n",
"plot_discs(DiscreteSpace.from_sampling(2, 9, \n",
" lambdify(n, sol_3f)))\n",
"display(sol_3f.simplify())"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"data": {
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},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": "Piecewise((0, n > 3), (1, Eq(n, 0) | Eq(n, 1)), (0, True))",
"text/latex": "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: n > 3 \\\\1 & \\text{for}\\: n = 0 \\vee n = 1 \\\\0 & \\text{otherwise} \\end{cases}$"
},
"metadata": {}
}
],
"source": [
"sol_3g = (x_n.as_piecewise(n, 0).subs(n, 3*n)*\n",
" Heaviside(3 - n).rewrite(Piecewise, H0=1))\n",
"plot_discs(DiscreteSpace.from_sampling(2, 6, \n",
" lambdify(n, sol_3g)))\n",
"display(sol_3g.simplify())"
]
}
]
}
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