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"# Woche 10 #\n",
"\n",
"Diese Wochen beschäftigen wir uns mit Matrizen und Vektoren.\n",
"In diesem Notebook rechnen wir ein paar (Teil)aufgaben aus dem ÜB."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"'''IMPORTS'''\n",
"import os;\n",
"import sys; \n",
"\n",
"# NOTE: need this to force jupyter to reload imports:\n",
"for key in list(sys.modules.keys()):\n",
" if key.startswith('src.'):\n",
" del sys.modules[key];\n",
"\n",
"os.chdir(os.path.dirname(_dh[0]));\n",
"sys.path.insert(0, os.getcwd());\n",
"\n",
"from src.thirdparty.maths import *;\n",
"from src.thirdparty.misc import *;\n",
"from src.thirdparty.render import *;\n",
"from src.maths.diagrams import *;\n",
"\n",
"np.random.seed(8007253); # zur Wiederholbarkeit"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"'''BEISPIEL-SIGNAL'''\n",
"n = 100;\n",
"u = np.zeros((n,), dtype=complex);\n",
"u[8] = -0.7\n",
"u[58] = 1j * sqrt(2);\n",
"\n",
"display_signal(u, 'Beispiel eines Vektors als Signal dargestellt');"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"'''SEMINARAUFGABE 10.4 a)'''\n",
"n = 3;\n",
"t = linspace(0, n-1, n);\n",
"i, j = np.meshgrid(t, t);\n",
"theta = 2*pi * i * j / n;\n",
"\n",
"# definiere die Matrix F_n + die Vektoren u_nk\n",
"F_n = (1/sqrt(n)) * exp(-1j * theta);\n",
"u_n = [ (1/sqrt(n)) * exp(1j * theta[:, k]) for k in range(n) ];\n",
"\n",
"print(dedent(\n",
" f'''\n",
" Matrix:\n",
"\n",
" F_{n} = \\n{np.round(F_n, 3)}\n",
" '''\n",
"));\n",
"print('');\n",
"\n",
"# NOTE: Programmiersprachen können nicht exakt berechnen.\n",
"# Darum entstehen winzige Fehler, die sich durch Runden entfernen lassen.\n",
"print('Matrix-Vektor-Produkte:');\n",
"for k in range(n):\n",
" result = F_n @ u_n[k];\n",
" print(dedent(\n",
" f'''\n",
"\n",
" u_{{n {k}}} = {np.round(u_n[k], 3)}\n",
" F_n · u_{{n {k}}} = {np.round(result, 3)}\n",
" '''\n",
" ));"
]
},
{
"attachments": {},
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"source": [
"## Seminaraufgabe 10.4 b) ##\n",
"\n",
"Für $i\\in\\{0,1,\\ldots,n-1\\}$ ist die $i$-te von $\\mathcal{F}_{n}\\,u_{n,k}$\n",
"\n",
"$$\n",
" (\\mathcal{F}_{n}\\,u_{n,k})_{i}\n",
" = \\sum_{j=0}^{n-1}\n",
" (\\mathcal{F}_{n})_{ij}\\,(u_{n,k})_{j}\n",
" = \\sum_{j=0}^{n-1}\n",
" \\tfrac{1}{\\sqrt{n}}\n",
" \\exp(-\\imath\\tfrac{2\\pi\\,ij}{n})\n",
" \\cdot\n",
" \\tfrac{1}{\\sqrt{n}}\n",
" \\exp(\\imath\\tfrac{2\\pi\\,kj}{n})\n",
" = \\frac{1}{n}\\sum_{j=0}^{n-1}\n",
" \\underbrace{\n",
" \\exp(\\imath\\tfrac{2\\pi\\,(k-i)j}{n})\n",
" }_{= r^{j}}\n",
"$$\n",
"\n",
"wobei $r \\colonequals \\exp(\\imath\\tfrac{2\\pi\\,(k-i)}{n})$.\n",
"Wir haben, dass\n",
" $r = 1$\n",
" $\\Leftrightarrow$\n",
" $\\tfrac{k-i}{n} \\in \\Z$\n",
" $\\Leftrightarrow$\n",
" $i = k$,\n",
"da $0\\leq i,k \\leq n-1$.\n",
"
\n",
"Es gilt außerdem $r^{n} = \\exp(\\imath 2\\pi\\,(k-i)) = 1$.\n",
"
\n",
"Aus der o.s. Berechnung erhält man also\n",
"\n",
"$$\n",
" (\\mathcal{F}_{n}\\,u_{n,k})_{i}\n",
" = \\tfrac{1}{n}\\displaystyle\\sum_{j=0}^{n-1}r^{j}\n",
" = \\tfrac{1}{n}\n",
" \\begin{cases}\n",
" \\displaystyle\\sum_{j=0}^{n-1}1 &: &i = k\\\\\n",
" \\frac{1 - r^{n}}{1 - r} &: &\\text{sonst}\\\\\n",
" \\end{cases}\n",
" = \\tfrac{1}{n}\n",
" \\begin{cases}\n",
" n &: &i = k\\\\\n",
" \\frac{1 - 1}{1 - r} &: &\\text{sonst}\\\\\n",
" \\end{cases}\n",
" = \\delta_{ik}.\n",
"$$\n",
"\n",
"Darum $\n",
" \\mathcal{F}_{n}\\,u_{n,k}\n",
" = \\left(\\begin{matrix}\n",
" 0\\\\\n",
" 0\\\\\n",
" \\vdots\\\\\n",
" 1\\\\\n",
" \\vdots\\\\\n",
" 0\\\\\n",
" \\end{matrix}\\right)\n",
" \\begin{matrix}\n",
" \\\\\n",
" \\\\\n",
" \\\\\n",
" \\leftarrow k\\\\\n",
" \\\\\n",
" \\\\\n",
" \\end{matrix}\n",
" = \\mathbf{e}_{k}\n",
"$.\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"'''SEMINARAUFGABE 10.4 d)'''\n",
"n = 4;\n",
"t = linspace(0, n-1, n);\n",
"i, j = np.meshgrid(t, t);\n",
"theta = 2*pi * i * j / n;\n",
"\n",
"# definiere die Matrix F_n + die Vektoren u_nk\n",
"F_n = (1/sqrt(n)) * exp(-1j * theta);\n",
"u_n = [ (1/sqrt(n)) * exp(1j * theta[:, k]) for k in range(n) ];\n",
"\n",
"# Eingabe des Vektors:\n",
"F_mal_u = np.asarray([100, -3, 0, -3]); # = F_n @ u\n",
"\n",
"'''\n",
"NOTE: Folgende Koeffizienten wurden falsch erraten.\n",
"TODO: Wie lauten die richtigen Werte? -> Aufgabe!\n",
"'''\n",
"c = [10,10,40,50];\n",
"\n",
"# Aufschreibung des Signals »bzgl. der harmonischen Basis«:\n",
"# u = c[0]u_n[0] + c[1]u_n[1] + c[2]u_n[2] + c[3]u_n[3]\n",
"u_guess = sum([ c[k] * u_n[k] for k in range(n) ]);\n",
"\n",
"# Anzeige der Lösung:\n",
"display_signal(u_guess, 'Geschätztes Signal');\n",
"\n",
"# Verifikation von Lösung:\n",
"print(dedent(\n",
" f'''\n",
" %error := ‖u_guess – u‖ / ‖u‖\n",
" = ‖F_n · (u_guess – u)‖ / ‖F_n · u‖\n",
" ... wieso?? [Stichwort: unitär]\n",
" = ‖F_n · u_guess – F_n · u‖ / ‖F_n · u‖\n",
" = {linalg.norm(F_n @ u_guess - F_mal_u)/linalg.norm(F_mal_u):.2%}\n",
" '''\n",
"));\n"
]
}
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