fourier1.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# Beräkning av Fourier serier\n",
    "\n",
    "\n",
    "## Kommandot fourier_series\n",
    "\n",
    "Det grundläggande kommandot har formen `fourier_series(funktion, (variabel, nedre, övre))`. Kommandona finns i sympy. Hela vägen till modulen i sympy är:\n",
    "`sympy.series.fourier.fourier_series(f, limits=None, finite=True)`. Detta innebär att vi skriver `from sympy import fourier_series` och sedan anropar denna med just `fourier_series`. Vi behöver inte de andra sakerna i `sympy.series.fourier`.\n",
    "\n",
    "sympy är en modul för python som gör det möjligt att göra symboliska beräkningar. Symboliska beräkningar är tex:\n",
    "\n",
    "* algebraiska förenklingar: $x²/x=x$\n",
    "\n",
    "* Deriveringar: $Dx²=2x$\n",
    "\n",
    "För att kontrollera vissa parametrar används ipywidgets (som ger tex glidare).\n",
    "\n",
    "**ipywidgets fungerar inte på replit.com.** Så kör på din dators egen python eller använd en kod-cell här i jupyter och skriv program. För att skapa en ny cell så klicka på plustecknet ovan, sidan om spara och klippa. Det som du skriver är borta nästa gång du loggar in, du kan inte förstöra något.\n",
    "Om du vill spara en utskrift så gör t.ex. följande(koden förklaras senare):\n",
    "\n",
    "```\n",
    "from sympy import fourier_series, pi\n",
    "from sympy.abc import x\n",
    "from sympy.plotting import plot  # Vi plottar med sympy.\n",
    "from ipywidgets import interact\n",
    "\n",
    "\n",
    "def g(N):\n",
    "    f = x**2  # Funktionen\n",
    "    s = fourier_series(f, (x, -pi, pi))  # Beräkna fourier serien\n",
    "    s1 = s.truncate(N)\n",
    "    print(s1)\n",
    "    \n",
    "    pl=plot(f,s1, (x,-4,4), show=False)  # Plottar både f och s1\n",
    "    pl.save(\"min.png\")  # Sparar filen med namnet min.png\n",
    "    pl.show()\n",
    "    return N\n",
    "\n",
    "\n",
    "interact(g, N=(1,8))\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vi ber om fourierserien för $f(x)=x^2$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-4*cos(x) + cos(2*x) - 4*cos(3*x)/9 + cos(4*x)/4 + pi**2/3\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 4 \\cos{\\left(x \\right)} + \\cos{\\left(2 x \\right)} - \\frac{4 \\cos{\\left(3 x \\right)}}{9} + \\frac{\\cos{\\left(4 x \\right)}}{4} + \\frac{\\pi^{2}}{3}$"
      ],
      "text/plain": [
       "-4*cos(x) + cos(2*x) - 4*cos(3*x)/9 + cos(4*x)/4 + pi**2/3"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import fourier_series, pi\n",
    "from sympy.abc import x  # Importerar x som en symbol\n",
    "\n",
    "f = x**2  # Funktionen uttryckt symboliskt\n",
    "\n",
    "s = fourier_series(f, (x, -pi, pi))  # Beräkna fourier serien symboliskt\n",
    "\n",
    "s1 = s.truncate(n=5)  # Antal termer\n",
    "\n",
    "print(s1)\n",
    "\n",
    "# Kan få snygga utskrifter i Jupyter\n",
    "s1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Widgets kan användas för att styra antalet termer. Vi använder en widget som kallas glidare. Dra med musen eller klicka och använd piltangenter. Koden i föregående exempel stoppas in i en def som sedan anropas av interact som finns i ipywidgets.\n",
    "\n",
    "'widget' står för (eventuellt) window gadget och enligt svenska datatermsgruppen är den svenska benämningen '(grafisk) gränssnittskomponent'. En widget är ett element i ett grafiskt gränsnitt som tex, kan vara knappar, popup-fönster, ikoner, dialogrutor. Flera widgets bildar ett grafiskt gränssnitt till ett program. Uttrycket 'fönsterpryl' förekommer också. 'gadget' betyder pryl, manick."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "bb333c9e5f1f4baaa7a566644b638570",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=3, description='N', max=6, min=1), Output()), _dom_classes=('widget-inte…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.g(N)>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import fourier_series, pi\n",
    "from sympy.abc import x\n",
    "from ipywidgets import interact\n",
    "\n",
    "def g(N):\n",
    "    f = x**2  # Funktionen\n",
    "    s = fourier_series(f, (x, -pi, pi))  # Beräkna fourier serien\n",
    "    s1 = s.truncate(N)\n",
    "    print(s1)\n",
    "    return N\n",
    "\n",
    "\n",
    "interact(g, N=(1,6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Och vi vill naturligtvis rita fourierserien för att se anpassningen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "16fb1e482e0a4f829e1996d8f85901db",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=4, description='N', max=8, min=1), Output()), _dom_classes=('widget-inte…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.g(N)>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import fourier_series, pi\n",
    "from sympy.abc import x\n",
    "from sympy.plotting import plot  # Vi plottar med sympy.\n",
    "from ipywidgets import interact\n",
    "\n",
    "\n",
    "def g(N):\n",
    "    f = x**2  # Funktionen\n",
    "    s = fourier_series(f, (x, -pi, pi))  # Beräkna fourier serien\n",
    "    s1 = s.truncate(N)\n",
    "    print(s1)\n",
    "    \n",
    "    plot(f,s1, (x,-4,4))  # Plottar både f och s1\n",
    "   \n",
    "    return N\n",
    "\n",
    "\n",
    "interact(g, N=(1,8))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Serier allmänt\n",
    "\n",
    "Om man beräknat ett uttryck för en fourier-serie så kan sympy hantera formeln symboliskt, även indexet, och beräkna summan upp till ett visst antal termer; det görs med list-förståelse (list comprehension). Detta kan användas för att kontrollera att en viss föreslagen fourier-serie är rimlig och i alla fall för ögat syns konvergera mot önskad funktion. \n",
    "\n",
    "Om vi har en serie som ges av uttrycket \n",
    "\n",
    "$$\n",
    "\\frac{1}{2}-\\sum_{n=1}^{\\infty}\\frac{1}{\\pi n}\\sin\\left(2\\pi nx\\right)\n",
    "$$\n",
    "\n",
    "som approximerar en sågtandad våg så kan vi 'se' hur bra det ser ut.\n",
    "\n",
    "Vi använder kommandot `summation(f, [i,a,b])`. Vi anger funktionen f, indexet är i, gränserna är a respektive b. Summationen är symbolisk men man kan skriva plot och sympy tar hand om omvandlingen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-(sin(2*pi*x) + sin(4*pi*x)/2 + sin(6*pi*x)/3 + sin(8*pi*x)/4 + sin(10*pi*x)/5 + sin(12*pi*x)/6)/pi + 0.5\n"
     ]
    },
    {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7ff68b794850>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import summation, symbols, cos, sin, pi\n",
    "from sympy.plotting import plot\n",
    "\n",
    "\n",
    "# I sympy måste vi noga ange det som ska vara symboler\n",
    "i = symbols('i', integer=True)  # Symboler för heltal\n",
    "x = symbols('x')  # Behöver lite fler symboler\n",
    "\n",
    "\n",
    "# Här följer den summa av fouriertermer jag vill beräkna, se ovan. Satt summan från 1 till 6.\n",
    "SF=1/2-summation(1 / (pi*i) * sin(2*pi*i*x), [i, 1, 6])\n",
    "\n",
    "print(SF)\n",
    "\n",
    "plot(SF,(x,-1,1))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fourier och Taylor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Didaktiskt kan det vara värt att jämföra Taylorutveckling med Fourier-serie. Vi behöver importera `series` från sympy för att göra Taylorutveckling.\n",
    "\n",
    "\n",
    "`series(cos(x),x)` ger utvecklingen av cosinus kring 0, variabeln är x.\n",
    "\n",
    "\n",
    "`series(cos(x),x,x0=2,n=9)` ger utvecklingen kring 2 och med 9 termer. Förinställt är utveckling kring 0 och n=6.\n",
    "\n",
    "\n",
    "Vi väljer i exemplet nedan att utgå från $f(x)=e^x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "e87fb63309844efd990b9b00166d03b4",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=2, description='N', max=8, min=2), IntSlider(value=2, description='M', m…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.g(N=2, M=2)>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import fourier_series, pi, series, simplify\n",
    "from sympy.abc import x\n",
    "from sympy.functions import exp\n",
    "from sympy.plotting import plot\n",
    "\n",
    "from ipywidgets import interact\n",
    "\n",
    "\n",
    "def g(N=2,M=2):\n",
    "    f = exp(x)  # Funktionen\n",
    "    \n",
    "    # Fourier\n",
    "    s = fourier_series(f, (x, -2, 2)).truncate(N)  # Beräkna fourier serien\n",
    "    ss = simplify(s)  # Hjälper ibland\n",
    "    print(\"s= \", s,\"\\n\")  # Kontroll\n",
    "    print(\"ss= \", ss,\"\\n\")  # Kontroll\n",
    "    \n",
    "    # Taylor\n",
    "    r = series(f,x,x0=0,n=M).removeO()  # Beräkna Taylor serien; ta bort resttermen\n",
    "    print(\"r= \", r)  # Kontroll\n",
    "    \n",
    "    # Plot\n",
    "    plt = plot(f,s,r,(x,-2,2),show = False)\n",
    "    plt[0].label = \"$exp(x)$\"  # De olika graferna nås med index\n",
    "    plt[1].label = \"$N\\, Fourier$\"  # Observera LaTeX\n",
    "    plt[2].label = \"$M\\, Taylor$\"\n",
    "    \n",
    "    plt[0].line_color = \"r\"\n",
    "    plt[1].line_color = \"b\"\n",
    "    plt[2].line_color = \"g\"\n",
    "    plt.legend = True\n",
    "    plt.show()\n",
    "    return N,M\n",
    "\n",
    "\n",
    "interact(g, N=(2,8), M=(2,8))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Uppgifter\n",
    "\n",
    "**Uppgift 1**\n",
    "Plotta delsummor för olika n. Vilken funktion är det som återskapas? Använd kommandot summation såsom illustrerats.\n",
    "\n",
    "\n",
    "$$\n",
    "\\sum_{n=0}^{N}\\:\\frac{4}{\\pi\\left(2n+1\\right)}\\sin\\left(\\left(2n+1\\right)x\\right)\n",
    "$$\n",
    "\n",
    "[Lösningsexempel](./uppg/fourier1Uppgift1.ipynb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Uppgift 2**\n",
    "\n",
    "Stegfunktionen heter Heaviside(). Se exempel i nästa cell.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n",
      "0\n",
      "0\n",
      "0\n",
      "0\n",
      "1/2\n",
      "1\n",
      "1\n",
      "1\n",
      "1\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7faceddd0f10>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import Heaviside\n",
    "from sympy.abc import x\n",
    "\n",
    "for i in range(-5,5):\n",
    "    print(Heaviside(i))\n",
    "\n",
    "plot(Heaviside(x))  # plotkommando i sympy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plotta Heaviside, fyrkantfunktionen och en sinus gånger fyrkantsfunktionen.\n",
    "Du måste ha:\n",
    "    \n",
    "```\n",
    "from sympy import Heaviside,symbols, sin\n",
    "from sympy.abc import x\n",
    "from sympy.plotting import plot\n",
    "```\n",
    "\n",
    "Ange fönstret med a respektive b. Fyrkant ska konstrueras från Heaviside. Gör en def för varje funktion, 3 stycken.\n",
    "\n",
    "[Lösningsförslag](./uppg/fourier1Uppgift2.ipynb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Uppgift 3**\n",
    "\n",
    "Att plotta ett spektrum innebär att man plottar fourierkoefficienterna i en graf. Man har då vinkelfrekvensen på x-axeln eller multipel av grund-vinkelfrekvensen och amplituden/koefficienten på y-axeln.\n",
    "\n",
    "För att få ut termerna (inte koefficienterna) kan man skriva\n",
    "\n",
    "```\n",
    "cosout = []\n",
    "    cosout.append(s.a0)\n",
    "    for i in range(1,N):\n",
    "        cosout.append(s.an.coeff(i))\n",
    "    print(cosout)\n",
    "```\n",
    "\n",
    "\n",
    "Där s i koden är från typiskt `s = fourier_series(f, (x, -pi, pi))`. \n",
    "\n",
    "\n",
    "Förhållandet mellan termer och koefficienter kan kännas lite förvirrande. Termerna, tex. $\\frac{5}{3}\\sin(2\\pi x)$ måste skiljas från koefficienten som är $\\frac{5}{3}$ .\n",
    "\n",
    "\n",
    "För att få ut termerna används en metod: s.a0 ger $a_0$ termen. s.an.coeff(i) ger de andra termerna. Lägger allt i en lista cosout. Termerna för sinus ges av s.bn.coeff(i). N anger antalet termer (tex. via glidare). Metoden .coeff(i) ger *termen* inte koefficient som man skulle kunna tro.\n",
    "\n",
    "Sedan måste man se till att  sinus respektive cosinus är 1 för att få ut koefficienterna. Det kan man göra genom att se till att cosinus respektive sinusfunktionerna ger värdet 1: `cosout.append(s.an.coeff(i).subs(x,0))` för cosinus. \n",
    "\n",
    "\n",
    "subs(x,0) innebär att symbolen x ersätts med talet 0. Hur gör man för sinus?\n",
    "\n",
    "\n",
    "\n",
    "Plotta spektrum för $y=x^2+x$\n",
    "\n",
    "I lösningsförslaget är spektrum plottat med matplotlib.pyplot men fourierseriens delsummor med sympy.\n",
    "\n",
    "\n",
    "[Lösningsförslag](./uppg/fourier1Uppgift3.ipynb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
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