GDET3 Reale Abtastung.ipynb 23.6 KB
 Christian Rohlfing committed Dec 05, 2019 1 2 3 4 5 6 7 8 9 10 11 12 { "cells": [ { "cell_type": "code", "execution_count": null, "metadata": { "jupyter": { "source_hidden": true } }, "outputs": [], "source": [  Hafiz Emin Kosar committed Mar 19, 2020 13  "# Copyright 2020 Institut für Nachrichtentechnik, RWTH Aachen University\n",  Christian Rohlfing committed Dec 05, 2019 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29  "%matplotlib widget\n", "\n", "from ipywidgets import interact, interactive, fixed, HBox, VBox\n", "import ipywidgets as widgets\n", "from IPython.display import clear_output, display, HTML\n", "\n", "from ient_nb.ient_plots import *\n", "from ient_nb.ient_transforms import *\n", "from ient_nb.ient_signals import *\n", "\n", "signals_t = {'cos-Funktion': lambda t: np.cos(2 * np.pi * t),\n", " 'sin-Funktion': lambda t: np.sin(2 * np.pi * t),\n", " 'si-Funktion': lambda t: si(2 * np.pi * t),\n", " 'Rechteckimpuls': lambda t: rect(t / 1.05),\n", " 'Dreieckimpuls': tri}\n", "\n",  Christian Rohlfing committed Feb 20, 2020 30 31  "signals_f = {'cos-Funktion': lambda f, F: np.isin(f/F, f[find_ind_least_diff(f/F, [-1, 1])]/F) * 0.5,\n", " 'sin-Funktion': lambda f, F: np.isin(f/F, f[find_ind_least_diff(f/F, [1])]/F) / 2j - np.isin(f/F, f[find_ind_least_diff(f/F, [-1])]/F) / 2j,\n",  Christian Rohlfing committed Dec 05, 2019 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47  " 'si-Funktion': lambda f, F: 1/(2*np.abs(F))*rect(f / (2*F)),\n", " 'Rechteckimpuls': lambda f, F: 1/np.abs(F)*si(np.pi * f / F),\n", " 'Dreieckimpuls': lambda f, F: 1/np.abs(F)*si(np.pi * f/F) ** 2}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "\n", "# Reale Abtastung\n", "\n",  Iris Heisterklaus committed Dec 16, 2019 48  "Im Gegensatz zur [idealen Abtastung](GDET3%20Ideale%20Abtastung.ipynb) werden hier zwei Verfahren zur realen Abtastung betrachtet. Tatsächlich kann nicht mit einem idealen Dirac an einem definierten Zeitpunkt abgetastet werden. Im Folgenden werden zwei Verfahren der Abtastung, die *Shape-top Abtastung* und die *Flat-top Abtastung* beschrieben.\n",  Christian Rohlfing committed Dec 05, 2019 49 50 51  "\n", "## Shape-top Abtastung\n", "\n",  Iris Heisterklaus committed Dec 16, 2019 52  "Bei der Shape-top Abtastung wird das kontinuierliche Signal $s(t)$ mit Abstand $T=\\frac{1}{r}$ abgetastet. \n",  Iris Heisterklaus committed Jan 06, 2020 53  "Anstatt einer Diracfolge wird eine Folge schmaler Rechteckimpulse mit endlicher Dauer $T_0$ verwendet. Das abgetastete Signal $s_0(t)$ ergibt sich zu\n",  Christian Rohlfing committed Dec 05, 2019 54 55 56 57  "\n", "$$\n", "s_0(t) \n", "= s(t) \\cdot \\sum\\limits_{n=-\\infty}^\\infty \\mathrm{rect}\\left(\\frac{t-nT}{T_0}\\right) \n",  Iris Heisterklaus committed Dec 16, 2019 58  "= s(t) \\cdot \\left[\\mathrm{rect}\\left(\\frac{t}{T_0}\\right) \\ast \\sum\\limits_{n=-\\infty}^\\infty \\delta(t-nT) \\right].\n",  Christian Rohlfing committed Dec 05, 2019 59 60  "$$\n", "\n",  Iris Heisterklaus committed Dec 16, 2019 61  "Im Frequenzbereich folgt durch Transformation\n",  Christian Rohlfing committed Dec 05, 2019 62 63 64 65  "$$\n", "S_0(f) \n", "= S(f) \\ast \\left[T_0 \\mathrm{si}\\left(\\pi T_0 f\\right) \\cdot \\frac{1}{T}\\sum_{k=-\\infty}^\\infty \\delta(f-kr)\\right]\n", "=\n",  Iris Heisterklaus committed Jan 06, 2020 66  "S(f) \\ast \\left[\\frac{T_0}{T} \\sum_{k=-\\infty}^\\infty \\delta\\left(f-\\frac{k}{T}\\right) \\mathrm{si} \\left(\\pi T_0 \\frac{k}{T}\\right) \\right]\\text{.}\n",  Christian Rohlfing committed Dec 05, 2019 67 68  "$$\n", "\n",  Iris Heisterklaus committed Jan 06, 2020 69  "Auch hier entstehen wie bei der idealen Abtastung spektrale Kopien, welche um $\\frac{k}{T}$ zentriert sind. Jede spektrale Kopie wird mit einem von $f$ unabhängigen Faktor $\\mathrm{si} \\left(\\pi T_0 \\frac{k}{T}\\right)$ skaliert.\n",  Christian Rohlfing committed Dec 05, 2019 70  "\n",  Iris Heisterklaus committed Dec 16, 2019 71  "Der Grenzübergang $T_0\\rightarrow 0$ liefert hier die ideale Abtastung: $\\lim\\limits_{T_0\\rightarrow 0}\\left(\\frac{1}{T_0}s_0(t)\\right) = s_\\mathrm{a}(t)$."  Christian Rohlfing committed Dec 05, 2019 72 73 74 75 76 77 78  ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Flat-top Abtastung\n",  Iris Heisterklaus committed Dec 16, 2019 79  "Bei der Flat-top Abtastung wird in Intervallen endlicher Dauer abgetastet und der Signalwert dann um $T=\\frac{1}{r}$ gehalten. Dieses Verfahren wird häufig in Analog-Digital-Wandlern eingesetzt. Das abgetastete Signal $s_0(t)$ ergibt sich somit zu\n",  Christian Rohlfing committed Dec 05, 2019 80 81 82  "\n", "$$\n", "s_0(t) \n",  Christian Rohlfing committed Dec 05, 2019 83  "= \\sum\\limits_{n=-\\infty}^\\infty s(nT) \\mathrm{rect}\\left(\\frac{t}{T}-\\frac{1}{2}-nT\\right)\n",  Christian Rohlfing committed Dec 05, 2019 84  "= s_\\mathrm{a}(t) \\ast \\mathrm{rect}\\left(\\frac{t}{T}-\\frac{1}{2}\\right)\n",  Iris Heisterklaus committed Jan 06, 2020 85  "= \\left[s(t) \\cdot \\sum\\limits_{n=-\\infty}^\\infty \\delta(t-nT)\\right] \\ast \\mathrm{rect}\\left(\\frac{t}{T}\\right)\\ast \\delta\\left(t-\\frac{T}{2}\\right) \\text{.}\n",  Christian Rohlfing committed Dec 05, 2019 86 87  "$$\n", "\n",  Iris Heisterklaus committed Jan 06, 2020 88  "Im Frequenzbereich folgt dann\n",  Christian Rohlfing committed Dec 05, 2019 89 90 91  "$$\n", "S_0(f) \n", "= S_\\mathrm{a}(f) \\cdot T \\cdot \\mathrm{si}(\\pi f T) \\cdot \\mathrm{e}^{-\\mathrm{j}\\pi f T}\n",  Iris Heisterklaus committed Jan 06, 2020 92  "= \\left[S(f) \\ast \\sum\\limits_{k=-\\infty}^\\infty \\delta(f-kr)\\right] \\cdot \\mathrm{si}(\\pi f T) \\cdot \\mathrm{e}^{-\\mathrm{j}\\pi f T}\\text{.}\n",  Christian Rohlfing committed Dec 05, 2019 93 94  "$$\n", "\n",  Iris Heisterklaus committed Jan 06, 2020 95  "In der Demonstration kann der Unterschied zwischen der Shape-top und der Flat-top Abtastung nocheinmal anschaulich betrachtet werden."  Christian Rohlfing committed Dec 05, 2019 96 97 98 99 100 101  ] }, { "cell_type": "markdown", "metadata": {}, "source": [  Iris Heisterklaus committed Dec 16, 2019 102  "## Beispiel\n",  Christian Rohlfing committed Dec 05, 2019 103 104  "### Abtastung\n", "\n",  Iris Heisterklaus committed Jan 06, 2020 105 106 107 108 109 110  "Zunächst wird nocheinmal die [ideale Abtastung](GDET3%20Ideale%20Abtastung.ipynb) wiederholt. In der Abbildung ist als gestrichelte Linie das Signal $s(t)$ dargestellt, das zugehörige Spektrum in blau. Das Signal wird mit Abtastrate $r=2$ abgetastet.\n", "Das abgetastete Signal \n", "$s_\\mathrm{a}(t) = \\sum\\limits_{n=-\\infty}^\\infty s(nT)\\cdot\\delta(t-nT)$ und das Spektrum des abgetasteten Signals \n", "$S_\\mathrm{a}(f) = \\frac{1}{T} \\sum\\limits_{k=-\\infty}^\\infty S(f-kr)$ sind ebenfalls abgebildet. \n", "\n", "Das Spektrum des abgetasteten Signals zeigt die für die Abtastung typischen spektralen Wiederholungen. "  Christian Rohlfing committed Dec 05, 2019 111 112 113 114 115  ] }, { "cell_type": "code", "execution_count": null,  Iris Heisterklaus committed Dec 16, 2019 116 117 118 119 120  "metadata": { "jupyter": { "source_hidden": true } },  Christian Rohlfing committed Dec 05, 2019 121 122 123 124 125 126 127 128 129 130  "outputs": [], "source": [ "# Construction of s(t) and corresponding spectrum S(f)\n", "t,deltat = np.linspace(-10,10,50001, retstep=True) # t-axis\n", "f,deltaf = np.linspace(-50,50,len(t), retstep=True) # f-axis\n", "\n", "F = 0.9 # frequency of the signal\n", "s = lambda t: signals_t['cos-Funktion'](t*F); \n", "S = lambda f: signals_f['cos-Funktion'](f, F);\n", "\n",  Christian Rohlfing committed Dec 05, 2019 131  "# Ideal sampling: Construction of sa(t) and Sa(f)\n",  Christian Rohlfing committed Dec 05, 2019 132 133 134 135 136 137 138 139 140 141 142 143 144 145  "r = 2; T = 1/r; # sampling rate\n", "\n", "## Time domain\n", "nT, snT = ient_sample(t, s(t), T)\n", "\n", "## Frequency domain\n", "Sa = np.zeros_like(S(f))\n", "kMax = 16 # number of k values in sum for Sa(f), should be infinity :)\n", "for k in np.arange(-kMax, kMax+1): # evaluate infinite sum only for 2*kMax+1 elements \n", " Sa += S(f-k/T)\n", "Sa = Sa/T\n", "fSadirac = f[np.where(Sa)]; Sadirac = Sa[np.where(Sa)]\n", "\n", "# Plot\n",  Christian Rohlfing committed Dec 05, 2019 146  "## Time domain\n",  Hafiz Emin Kosar committed Feb 04, 2020 147  "fig, axs = plt.subplots(2,1, **ient_landscape)\n",  Christian Rohlfing committed Dec 05, 2019 148 149 150 151  "ax = axs[0]; ax.set_title('Zeitbereich');\n", "ax.plot(t, s(t), color='rwth', linestyle='--', label=r'$s(t)$');\n", "ient_plot_dirac(ax, nT, snT, 'rot', label=r'$s_\\mathrm{a}(t)$')\n", "ax.set_xlabel(r'$\\rightarrow t$'); ax.set_xlim([-7.5,7.5]); ax.legend(loc=2); ient_grid(ax); ient_axis(ax);\n",  Christian Rohlfing committed Dec 05, 2019 152  "## Frequency domain\n",  Christian Rohlfing committed Dec 05, 2019 153 154 155  "ax = axs[1]; ax.set_title('Frequenzbereich');\n", "ient_plot_dirac(ax, fSadirac, Sadirac, 'rot', label=r'$S_\\mathrm{a}(f)$');\n", "ient_plot_dirac(ax, f[np.where(S(f))], S(f)[np.where(S(f))], 'rwth', label=r'$S(f)$');\n",  Christian Rohlfing committed Dec 05, 2019 156  "ax.set_xlim([-7.5,7.5]); ax.legend(loc=2); ax.set_xlabel(r'$\\rightarrow f$'); ient_grid(ax); ient_axis(ax);\n",  Christian Rohlfing committed Dec 05, 2019 157 158 159 160 161 162 163  "txt,_=ient_annotate_xtick(ax, r'$r=2$', r, -.15, 'black'); txt.get_bbox_patch().set_alpha(1);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [  Iris Heisterklaus committed Jan 06, 2020 164  "Nun wird das abgetastete Signal $s_0(t)$ im Zeitbereich betrachtet, welches mittels Flat-top Abtastung erzeugt wurde:\n",  Christian Rohlfing committed Dec 05, 2019 165 166 167  "$$\n", "s_0(t) \n", "= s_\\mathrm{a}(t) \\ast \\mathrm{rect}\\left(\\frac{t}{T}-\\frac{1}{2}\\right)\n",  Iris Heisterklaus committed Jan 06, 2020 168 169 170 171 172  "= \\sum\\limits_{n=-\\infty}^\\infty s(nT) \\mathrm{rect}\\left(\\frac{t}{T}-\\frac{1}{2}-nT\\right).\n", "$$\n", "\n", "Die Abtastrate ist ebenfalls $r=2$.\n", "Für die Flat-top Abtastung charakteristisch wird der Signalwert bis zum nächsten Abtastwert gehalten. "  Christian Rohlfing committed Dec 05, 2019 173 174 175 176 177  ] }, { "cell_type": "code", "execution_count": null,  Iris Heisterklaus committed Dec 16, 2019 178 179 180 181 182  "metadata": { "jupyter": { "source_hidden": true } },  Christian Rohlfing committed Dec 05, 2019 183 184 185 186 187 188 189 190 191  "outputs": [], "source": [ "# Time-Domain\n", "s0 = np.zeros_like(t) # sum of rects\n", "Nmax = np.ceil(t[-1]/T) # Parts of the infinite rect sum, should be infinity :)\n", "for n in np.arange(-Nmax,Nmax+1):\n", " s0 = s0 + rect((t-n*T)/T-0.5) * s(n*T)\n", "\n", "# Plot\n",  Hafiz Emin Kosar committed Feb 04, 2020 192  "fig, ax = plt.subplots(**ient_landscape); ax.set_title('Zeitbereich')\n",  Christian Rohlfing committed Dec 05, 2019 193 194 195 196 197 198 199 200  "ax.plot(t, s(t), 'rwth', linestyle='--', label=r'$s(t)$'); ax.plot(t, s0, 'rot', label=r'$s_0(t)$')\n", "ax.set_xlabel(r'$\\rightarrow t$'); ax.set_xlim([-2.9, 2.9]); ax.legend(loc=2); ient_grid(ax); ient_axis(ax);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [  Iris Heisterklaus committed Jan 06, 2020 201 202  "Dies hat im Frequenzbereich zur Konsequenz, dass die spektralen Kopien mit $T \\mathrm{si}(\\pi f T) \\cdot \\mathrm{e}^{-j\\pi f T}$ gewichtet werden. \n", "$S_0(f)$ ergibt sich also zu\n",  Christian Rohlfing committed Dec 05, 2019 203 204  "$$\n", "S_0(f) \n",  Iris Heisterklaus committed Jan 06, 2020 205 206 207  "= S_\\mathrm{a}(f) \\cdot T \\mathrm{si}(\\pi f T) \\cdot \\mathrm{e}^{-j\\pi f T}.\n", "$$\n", "Die nachfolgende Abbildung verdeutlicht diesen Effekt."  Christian Rohlfing committed Dec 05, 2019 208 209 210 211 212  ] }, { "cell_type": "code", "execution_count": null,  Iris Heisterklaus committed Dec 16, 2019 213 214 215 216 217  "metadata": { "jupyter": { "source_hidden": true } },  Christian Rohlfing committed Dec 05, 2019 218 219 220 221 222 223 224 225  "outputs": [], "source": [ "# Frequency domain\n", "S_si = T*si(np.pi*T*f)\n", "S_exp = np.exp(-1j*np.pi*f*T)\n", "S0 = Sa * S_si * S_exp\n", "\n", "# Plot\n",  Christian Rohlfing committed Dec 05, 2019 226  "## Magnitude\n",  Hafiz Emin Kosar committed Feb 04, 2020 227  "fig, axs = plt.subplots(2,1, **ient_landscape); \n",  Christian Rohlfing committed Dec 05, 2019 228 229  "ax = axs[0]; ax.set_title('Betrag')\n", "ax.plot(f, np.abs(S_si*S_exp), 'k--', label=r'$|T\\mathrm{si}(\\pi f T)e^{-\\mathrm{j}\\pi f T}|$')\n",  Christian Rohlfing committed Dec 05, 2019 230  "fS0dirac = f[np.where(S0)]; S0dirac = S0[np.where(S0)] # sample S0 and f\n",  Christian Rohlfing committed Dec 05, 2019 231 232  "ient_plot_dirac(ax, fS0dirac, np.abs(S0dirac), 'rot', label=r'$|S_0(f)$|');\n", "ax.set_xlim([-7.5,7.5]); ax.legend(loc=2); ax.set_xlabel(r'$\\rightarrow f$'); ient_grid(ax); ient_axis(ax);\n",  Christian Rohlfing committed Dec 05, 2019 233  "## Phase\n",  Christian Rohlfing committed Dec 05, 2019 234 235 236 237 238 239 240 241 242 243 244 245  "ax = axs[1]; ax.set_title('Phase')\n", "ax.plot(f, np.angle(S_si*S_exp), 'k--', label=r'$\\angle T\\mathrm{si}(\\pi f T)e^{-\\mathrm{j}\\pi f T}$')\n", "ient_stem(ax, fS0dirac, np.angle(S0dirac), 'rot', label=r'$\\angle S_0(f)$')\n", "ax.set_xlim([-7.5,7.5]); ax.legend(loc=2); ax.set_xlabel(r'$\\rightarrow f$'); ient_grid(ax); ient_axis(ax);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Rekonstruktion\n", "\n",  Iris Heisterklaus committed Jan 06, 2020 246  "Durch die Multiplikation von $S_\\mathrm{a}(f)$ mit $T \\cdot \\mathrm{si}(\\pi f T) \\cdot \\mathrm{e}^{-j\\pi f T}$ wird das Spektrum $S_0(f)$ im Basisband verzerrt. So ist es nicht möglich, $S(f)$ mittels eines einfachen idealen Tiefpasses zu rekonstruieren. Zusätzlich ist ein Filter nötig, welches diesen Faktor ausgleicht, dieses ist in der folgenden Abbildung dargestellt.\n",  Christian Rohlfing committed Dec 05, 2019 247 248 249 250 251 252 253 254  "$$\n", "H_\\mathrm{eq}(f) = \\frac{1}{T \\cdot \\mathrm{si}(\\pi f T) \\cdot \\mathrm{e}^{-j\\pi f T}}\n", "$$" ] }, { "cell_type": "code", "execution_count": null,  Iris Heisterklaus committed Dec 16, 2019 255 256 257 258 259  "metadata": { "jupyter": { "source_hidden": true } },  Christian Rohlfing committed Dec 05, 2019 260 261 262 263 264 265 266 267 268 269 270  "outputs": [], "source": [ "# Reconstruction filters #plt.close('all')\n", "## Ideal Low pass to crop the base band\n", "H_lp = rect(f/(r+0.001)) # ideal low pass between -r/2 and r/2\n", "## Equalizing filter to compensate the influence of si(...) and exp(...) terms in S_0(f)\n", "H_eq = 1/(T*si(np.pi*T*f) * np.exp(-1j*np.pi*f*T))\n", "## Overall reconstruction filter\n", "H = H_lp * H_eq\n", "\n", "# Plot\n",  Hafiz Emin Kosar committed Feb 04, 2020 271  "fig,axs = plt.subplots(2,1, **ient_landscape)\n",  Christian Rohlfing committed Dec 05, 2019 272 273 274 275 276 277 278 279 280 281 282 283 284  "ax = axs[0]\n", "ax.plot(f, np.abs(H_eq), 'k--', linewidth=1, label=r'$|H_\\mathrm{eq}(f)|$')\n", "ax.plot(f, np.abs(H), 'k', label=r'$|H(f)|$')\n", "ax.set_xlim([-7.5,7.5]); ax.legend(loc=2); ax.set_ylim([-.1,21]);\n", "ax.set_xlabel(r'$\\rightarrow f$'); ient_grid(ax); ient_axis(ax);\n", "\n", "ax = axs[1]\n", "ax.plot(f, np.angle(H_eq), 'k--', linewidth=1, label=r'$\\angle H_\\mathrm{eq}(f)$')\n", "ax.plot(f, np.angle(H)*H_lp, 'k', label=r'$\\angle H(f)$')\n", "ax.set_xlim([-7.5,7.5]); ax.legend(loc=2); #ax.set_ylim([-.1,11]);\n", "ax.set_xlabel(r'$\\rightarrow f$'); ient_grid(ax); ient_axis(ax);" ] },  Christian Rohlfing committed Dec 05, 2019 285 286 287 288  { "cell_type": "markdown", "metadata": {}, "source": [  Iris Heisterklaus committed Jan 06, 2020 289 290  "Dieses Filter wird nun zur Rekonstruktion angewendet. \n", "Das rekonstruierte Signal unter Verwendung des ausgleichenden Filters ist unten abgebildet. Das Originalsignal konnte erfolgreich rekonstruiert werden. "  Christian Rohlfing committed Dec 05, 2019 291 292 293 294 295  ] }, { "cell_type": "code", "execution_count": null,  Hafiz Emin Kosar committed Feb 04, 2020 296 297 298 299 300  "metadata": { "jupyter": { "source_hidden": true } },  Christian Rohlfing committed Dec 05, 2019 301 302 303 304 305 306 307  "outputs": [], "source": [ "G = S0 * H*T; \n", "g = ient_idft(G); g = np.fft.ifftshift(np.real(g)); # IDFT\n", "G = np.real(G); # discards small imaginary numbers close to zero\n", "\n", "# Plot\n",  Hafiz Emin Kosar committed Feb 04, 2020 308  "fig, axs = plt.subplots(2, 1, **ient_landscape)\n",  Christian Rohlfing committed Dec 05, 2019 309 310 311 312 313 314 315 316 317 318  "ax = axs[0]; fGdirac = f[np.where(G)]; Gdirac = G[np.where(G)]\n", "ient_plot_dirac(ax, fGdirac, Gdirac, 'grun');\n", "ax.set_xlabel(r'$\\rightarrow f$'); ax.set_ylabel(r'$\\uparrow G(f)$', bbox=ient_wbbox);\n", "ax.set_xlim([-7.5,7.5]); ient_grid(ax); ient_axis(ax);\n", "\n", "ax = axs[1]; ax.plot(t, s(t), color='rwth', linestyle='--', label=r'$s(t)$');\n", "ax.plot(t/deltat/(f[-1]*2), g, color='grun', linestyle='-', label=r'$g(t)$');\n", "ax.set_xlabel(r'$\\rightarrow t$'); ax.set_xlim([-7.5,7.5]); ax.legend(loc=2); ient_grid(ax); ient_axis(ax);" ] },  Christian Rohlfing committed Dec 05, 2019 319 320 321 322  { "cell_type": "markdown", "metadata": {}, "source": [  Iris Heisterklaus committed Dec 16, 2019 323  "## Demo\n",  Iris Heisterklaus committed Jan 06, 2020 324 325 326 327 328 329 330 331 332  "In der folgenden interaktiven Demonstration kann nun betrachtet werden, was der Unterschied zwischen der Flat-top und der Shape-top Abtastung ist. \n", "Dargestellt sind untereinander:\n", "* der Zeitbereich mit dem Originalsignal und dem abgestasteten Signal\n", "* der zugehörige Frequenzbereich mit dem Originalspektrum, dem Spektrum des abgetasteten Signal und dem Rekonstruktionsfilter\n", "* das rekonstruierte Spektrum\n", "* das rekonstruierte Signal im Zeitbereich.\n", "\n", "Im Drop-Down-Menü kann die Art der Abtastung (Shape-top oder Flat-top) sowie die abzutastende Funktion (Cosinus-, Sinus-und si-Funktion, sowie Rechteck- oder Dreieckimpuls) ausgewählt werden. \n", "Über den Schieberegler kann $F$ geändert werden, was bei Cosinus-, Sinus- und si-Funktion die Frequenz und bei Rechteck- und Dreieckimpuls die Breite beeinflusst. "  Christian Rohlfing committed Dec 05, 2019 333 334 335 336 337 338 339 340 341 342 343 344 345 346  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "jupyter": { "source_hidden": true } }, "outputs": [], "source": [ "T0 = T/4 # width of rects for shape-top sampling\n", "\n",  Hafiz Emin Kosar committed Mar 05, 2020 347  "plt.close(); fig, axs = plt.subplots(4, 1, **ient_landscape); plt.tight_layout();\n",  Christian Rohlfing committed Dec 05, 2019 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465  "@widgets.interact(sampling_type=widgets.Dropdown(options=['Shape-top', 'Flat-top'], description=r'Art der Abtastung:', style=ient_wdgtl_style),\n", " s_type=widgets.Dropdown(options=list(signals_t.keys()), description=r'Wähle $s(t)$:'),\n", " F=widgets.FloatSlider(min=0.1, max=2, value=0.9, step=.1, description=r'$F$', style=ient_wdgtl_style, continuous_update=False))\n", "def update_plots(sampling_type, s_type, F):\n", " s = lambda t: signals_t[s_type](t*F); \n", " S = lambda f: signals_f[s_type](f, F);\n", " nT, snT = ient_sample(t, s(t), T)\n", " \n", " # Construct sampled signal\n", " if sampling_type == 'Shape-top':\n", " u = np.zeros_like(t) # sum of rects\n", " for n in np.arange(-Nmax,Nmax+1):\n", " u = u + rect((t-n*T)/T0)\n", " s0 = s(t) * u\n", " elif sampling_type == 'Flat-top':\n", " s0 = np.zeros_like(t) # sum of rects\n", " for n in np.arange(-Nmax,Nmax+1):\n", " s0 = s0 + rect((t-n*T)/T-0.5) * s(n*T)\n", " \n", " # Construct sampled spectrum\n", " if sampling_type == 'Shape-top':\n", " S0 = np.zeros_like(S(f))\n", " for k in np.arange(-kMax, kMax+1): # evaluate infinite sum only for 2*kMax+1 elements \n", " S0 += S(f-k/T) * si(np.pi*T0*k/T)\n", " S0 = S0*T0/T\n", " elif sampling_type == 'Flat-top':\n", " Sa = np.zeros_like(S(f))\n", " for k in np.arange(-kMax, kMax+1): # evaluate infinite sum only for 2*kMax+1 elements \n", " Sa += S(f-k/T)\n", " S0 = Sa * si(np.pi*T*f) * np.exp(-1j*np.pi*f*T)\n", " \n", " # Reconstruct g(t)\n", " if sampling_type == 'Shape-top':\n", " H = H_lp\n", " G = S0 * H * T / T0;\n", " elif sampling_type == 'Flat-top':\n", " H = H_lp * H_eq * T\n", " G = S0 * H\n", " g = ient_idft(G); \n", " g = np.fft.ifftshift(np.real(g)); # IDFT\n", " \n", " # Sample for plot\n", " if s_type == 'cos-Funktion' or s_type == 'sin-Funktion':\n", " fS0dirac = f[np.where(S0)]; S0dirac = S0[np.where(S0)]\n", " fSdirac = f[np.where(S(f))]; Sdirac = S(f)[np.where(S(f))]\n", " fGdirac = f[np.where(G)]; Gdirac = G[np.where(G)]\n", " if s_type == 'sin-Funktion':\n", " Sdirac = np.imag(Sdirac); S = lambda f: np.imag(signals_f[s_type](f, F)); \n", " G = np.imag(G); Gdirac = np.imag(Gdirac)\n", " else:\n", " Sdirac = np.real(Sdirac); S0 = np.real(S0); G = np.real(G); Gdirac = np.real(Gdirac)\n", " else:\n", " g /= (len(f)/(2*f[-1])) # Parseval :)\n", " S0dirac = np.zeros_like(f)\n", " G = np.real(G)\n", " \n", " if sampling_type == 'Shape-top': # normalize to T0 for plotting reasons\n", " S0 = S0/T0\n", " S0dirac = S0dirac/T0\n", " \n", " # Plot\n", " if not axs[0].lines: # Call plot() and decorate axes. Usually, these functions take some processing time\n", " ax = axs[0]; ax.set_title('Zeitbereich');\n", " ax.plot(t, s(t), 'rwth', linestyle='--', label=r'$s(t)$'); \n", " ax.plot(t, s0, 'rot', label=r'$s_0(t)$')\n", " ax.set_xlabel(r'$\\rightarrow t$'); \n", " ax.set_xlim([-2.9, 2.9]); ax.legend(loc=2); ient_grid(ax); ient_axis(ax);\n", " \n", " ax = axs[1]; ax.set_title('Frequenzbereich');\n", " ax.plot(f, np.abs(H), '-', color='black', label=r'$|H(f)|$')\n", " if s_type == 'cos-Funktion' or s_type == 'sin-Funktion':\n", " ax.plot(f, np.ones_like(f)*np.nan, '-', color='rot', label=r'$|S_0(f)|$');\n", " ient_plot_dirac(ax, fS0dirac, np.abs(S0dirac), 'rot');\n", " else:\n", " ax.plot(f, np.abs(S0), '-', color='rot', label=r'$|S_0(f)|$'); \n", " ient_plot_dirac(ax, [], [], 'rot');\n", " ax.plot(f, S(f), color='rwth', linestyle='--', linewidth=1, label=r'$S(f)$')\n", " ax.set_xlim([-7.5,7.5]); ax.set_ylim([-1,2]); ax.legend(loc=2);\n", " ax.set_xlabel(r'$\\rightarrow f$'); ient_grid(ax); ient_axis(ax);\n", " txt,_=ient_annotate_xtick(ax, r'$r=2$', r, -.15, 'black'); txt.get_bbox_patch().set_alpha(1);\n", " txt,_=ient_annotate_xtick(ax, r'$f_\\mathrm{g}$', r/2, -.15, 'black'); txt.get_bbox_patch().set_alpha(1);\n", " \n", " ax = axs[2];\n", " if s_type == 'cos-Funktion' or s_type == 'sin-Funktion':\n", " ax.plot(f, np.ones_like(f)*np.nan, '-', color='grun'); \n", " ient_plot_dirac(ax, fGdirac, Gdirac, 'grun');\n", " else:\n", " ax.plot(f, G, '-', color='grun'); \n", " ient_plot_dirac(ax, [], [], 'rot');\n", " ax.set_xlabel(r'$\\rightarrow f$'); ax.set_ylabel(r'$\\uparrow G(f)$', bbox=ient_wbbox);\n", " ax.set_xlim([-7.5,7.5]); ient_grid(ax); ient_axis(ax);\n", " \n", " ax = axs[3]; ax.set_title('Zeitbereich (nach Rekonstruktion)');\n", " ax.plot(t, s(t), color='rwth', linestyle='--', label=r'$s(t)$');\n", " ax.plot(t/deltat/(f[-1]*2), g, 'grun', label=r'$g(t)$');\n", " ax.set_xlabel(r'$\\rightarrow t$'); ax.set_xlim([-7.5,7.5]); ax.legend(loc=2); ient_grid(ax); ient_axis(ax);\n", " plt.tight_layout()\n", " else:\n", " axs[0].lines[0].set_ydata(s(t)); axs[0].lines[1].set_ydata(s0); axs[1].lines[-3].set_ydata(S(f)); \n", " axs[1].lines[0].set_ydata(np.abs(H))\n", " if s_type == 'cos-Funktion' or s_type == 'sin-Funktion': # dirac plot\n", " ient_dirac_set_data(axs[1].containers, fS0dirac, np.abs(S0dirac));\n", " axs[1].lines[1].set_ydata(np.ones_like(f)*np.nan); \n", " ient_dirac_set_data(axs[2].containers, fGdirac, Gdirac); axs[2].lines[0].set_ydata(np.ones_like(f)*np.nan);\n", " else:\n", " axs[1].lines[1].set_ydata(np.abs(S0)); ient_dirac_set_data(axs[1].containers, [], []); \n", " axs[2].lines[0].set_ydata(G); ient_dirac_set_data(axs[2].containers, [], []);\n", " \n", " axs[3].lines[0].set_ydata(s(t)); axs[3].lines[1].set_ydata(g);\n", " \n", " if s_type == 'sin-Funktion': # Adapt labels\n", " axs[1].lines[-3].set_label(r'$\\mathrm{Im}\\{S(f)\\}$');\n", " axs[2].yaxis.label.set_text(r'$\\uparrow \\mathrm{Im}\\{G(f)\\}$');\n", " else:\n", " axs[1].lines[-3].set_label(r'$S(f)$')\n", " axs[2].yaxis.label.set_text(r'$\\uparrow G(f)$');\n", " axs[1].legend(loc=2)\n", " \n",  Christian Rohlfing committed Dec 06, 2019 466  " tmp = np.concatenate((np.abs(S0),np.abs(S0dirac),np.abs(H))); ient_update_ylim(axs[1], tmp, 0.19, np.max(np.abs(tmp))); ient_update_ylim(axs[2], G, 0.19, np.max(G));\n",  Christian Rohlfing committed Dec 05, 2019 467 468 469 470 471  " ient_update_ylim(axs[3], np.concatenate((s(t),g)), 0.19, np.max(np.concatenate((s(t),g))));" ] }, { "cell_type": "markdown",  Iris Heisterklaus committed Jan 06, 2020 472 473 474 475 476 477 478 479 480 481 482 483 484  "metadata": {}, "source": [ "### Aufgaben\n", "Wähle eine beliebige Funktion aus.\n", "* Wie sieht das abgetastete Signal mit Shape-top Abtastung aus und wie mit Flat-top Abtastung?\n", "* Welche Auswirkungen hat die Abtastungsart auf das Spektrum und die Rekonstruktion?\n", "* Variiere $F$. Was passiert?\n", "* Führe diese Beobachtung für unterschiedliche Funktionen aus. \n", "\n" ] }, { "cell_type": "markdown",  Christian Rohlfing committed Dec 05, 2019 485 486  "metadata": {}, "source": [  Hafiz Emin Kosar committed Mar 19, 2020 487  "___\n",  Christian Rohlfing committed Dec 05, 2019 488 489 490  "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources) (OER). Feel free to use the notebook for your own purposes. The code is licensed under the [MIT license](https://opensource.org/licenses/MIT). \n", "\n", "Please attribute the work as follows: \n",  Christian Rohlfing committed Feb 20, 2020 491  "*Christian Rohlfing, Übungsbeispiele zur Vorlesung \"Grundgebiete der Elektrotechnik 3 - Signale und Systeme\"*, gehalten von Jens-Rainer Ohm, 2020, Institut für Nachrichtentechnik, RWTH Aachen University."  Christian Rohlfing committed Dec 05, 2019 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510  ] } ], "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",  Hafiz Emin Kosar committed Feb 04, 2020 511  "version": "3.8.1"  Christian Rohlfing committed Dec 05, 2019 512 513 514 515 516  } }, "nbformat": 4, "nbformat_minor": 4 }