master > master: notebook - wk10

notebooks/week-10.ipynb

@@ -0,0 +1,223 @@
+{
+ "cells": [
+  {
+   "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": {},
+   "cell_type": "markdown",
+   "metadata": {},
+   "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",
+    "<br>\n",
+    "Es gilt außerdem $r^{n} = \\exp(\\imath 2\\pi\\,(k-i)) = 1$.\n",
+    "<br>\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",
+    "c = [10,10,40,50];\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"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.8"
+  },
+  "vscode": {
+   "interpreter": {
+    "hash": "98590ff4fe04c8543246b2a01debd3de3c5ca9b666f43f1fa87d5110c692004c"
+   }
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

requirements.txt

@@ -1,4 +1,4 @@
-pip>=22.3
+pip>=22.3.1
 wheel>=0.37.1
 
 # jupyter
@@ -16,7 +16,7 @@ codetiming>=1.3.0
 
 # testing + dev
 coverage[toml]>=6.4
-pytest>=7.1.1
+pytest>=7.2.0
 pytest-asyncio>=0.18.3
 pytest-cov>=3.0.0
 pytest-lazy-fixture>=0.6.3
@@ -33,7 +33,6 @@ lorem>=0.1.1
 safetywrap>=1.5.0
 typing>=3.7.4.3
 nptyping>=2.1.1
-typing-extensions>=3.10.0.2
 
 # config
 python-dotenv>=0.20.0

src/maths/diagrams/__init__.py

@@ -6,6 +6,7 @@
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 from src.maths.diagrams.sets import *;
+from src.maths.diagrams.signal import *;
 
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # EXPORTS
@@ -14,4 +15,5 @@ from src.maths.diagrams.sets import *;
 __all__ = [
     'Function',
     'Functions',
+    'display_signal',
 ];

src/maths/diagrams/signal.py

@@ -0,0 +1,48 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# IMPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+from __future__ import annotations;
+
+from src.thirdparty.code import *;
+from src.thirdparty.types import *;
+from src.thirdparty.maths import *;
+from src.thirdparty.plots import *;
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# EXPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+__all__ = [
+    'display_signal',
+];
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# CONSTANTS / VARIABLES
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+def display_signal(
+    u: np.ndarray,
+    title: str,
+    max_labels = 10,
+):
+    n = len(u);
+    if n <= max_labels:
+        x_axis = list(range(n));
+    else:
+        skip = int(n/max_labels);
+        x_axis = [ k*skip for k in range(max_labels) ];
+
+    fig, axs = mplot.subplots(1, 1, constrained_layout=True);
+    mplot.title(title);
+    mplot.xlabel('»Zeit« [=Indizes]');
+    mplot.ylabel('Werte [=Einträge]');
+    axs.stem(linspace(0,n-1,n), np.real(u), label='reeller Teil', linefmt='red');
+    axs.stem(linspace(0,n-1,n) + 0.05, np.imag(u), label='imaginärer Teil', linefmt='lime');
+    mplot.legend();
+    axs.set_xticks(x_axis, labels=[ str(index) for index in x_axis]);
+    mplot.show();
+    return;

src/thirdparty/maths.py

@@ -6,10 +6,17 @@
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 from fractions import Fraction;
+from numpy import cos;
+from numpy import exp;
+from numpy import linalg;
+from numpy import linspace;
+from numpy import pi;
+from numpy import sin;
+from numpy import sqrt;
+from typing import TypeVar;
 import math;
 import numpy as np;
 import random;
-from typing import TypeVar;
 
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # MODIFICATIONS
@@ -39,8 +46,15 @@ def sample(
 
 __all__ = [
     'Fraction',
+    'cos',
+    'exp',
+    'linalg',
+    'linspace',
     'math',
     'np',
+    'pi',
     'random',
     'sample',
+    'sin',
+    'sqrt',
 ];

src/thirdparty/render.py

@@ -19,11 +19,11 @@ from qiskit.visualization import array_to_latex;
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 __all__ = [
+    'Latex',
     'array_to_latex',
+    'display',
     'display_latex',
     'display_markdown',
     'display_png',
-    'display',
-    'Latex',
     'widgets',
 ];

src/thirdparty/types.py

@@ -38,8 +38,8 @@ from typing import Generic;
 from typing import Optional;
 from typing import Type;
 from typing import TypeVar;
-from typing_extensions import Concatenate;
+from typing import Concatenate;
-from typing_extensions import ParamSpec;
+from typing import ParamSpec;
 
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # EXPORTS