DegrauImpulsoDiscreto.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "2d4af1f9-d488-4b7e-8f07-7270db48d81e",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "# Sinais degrau e impulso unitário em tempo discreto\n",
    "\n",
    "Renato Naville Watanabe"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5852e163-e642-4b0f-8f6e-e352d6ca6158",
   "metadata": {},
   "source": [
    "## Preparação do ambiente"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "29d45909-6338-4cb5-9398-0723f2e0200f",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23dfb6ee-14a5-49c0-9829-7db3c0529c66",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## Sinal impulso unitário"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2e95b19e-00ec-4e34-88bd-b599c03a9e25",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "É o sinal mais simples que existe em tempio discreto. Para representá-lo, usamos a letra $\\delta$, e é definido da seguinte maneira:\n",
    "\n",
    "$$\\delta[n] = \\left\\{\\begin{array}{cc}1& ,n = 0 \\\\ 0& ,c.c \\end{array}\\right. $$\n",
    "\n",
    "O seu gráfico é o seguinte:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "4509d9dc-7741-4398-b2d2-ceb872ed88cd",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Vwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCrtItwsWLBA6enpiouLk8vl0oYNG1o07vXXX1dUVJRGjRoV3AIBAEDECHu4Wbp0qQoKClRcXKxNmzZp0KBBysnJ0ZEjR5odt2fPHv3Xf/2XbrjhhhBVCgAAIkHYw83cuXM1fvx45eXlqX///lq4cKE6d+6sxYsXNzmmoaFB9913n5588kldcsklIawWAAC0d2ENN/X19aqoqFB2dravLTo6WtnZ2SovL29y3PTp05WYmKhx48ad8z7q6urk8Xj8LgAAwF5hDTdHjx5VQ0ODkpKS/NqTkpLkdrsbHbN27Vq99NJLWrRoUYvuo6SkRE6n03dJS0s777oBAED7Ffa3pQJRU1Oj+++/X4sWLVJCQkKLxhQWFqq6utp32b9/f5CrBAAA4dQhnHeekJCgmJgYVVVV+bVXVVUpOTn5rP6fffaZ9uzZo5EjR/ravF6vJKlDhw7auXOnLr30Ur8xDodDDocjCNUDAID2KKxHbmJjYzVkyBCVlZX52rxer8rKypSVlXVW/yuvvFKffPKJKisrfZcf/ehHuvnmm1VZWclbTgAAILxHbiSpoKBAY8eO1dChQ5WZmal58+aptrZWeXl5kqTc3Fz16tVLJSUliouLU0ZGht/4bt26SdJZ7QAA4MIU9nAzevRoffHFFyoqKpLb7dbgwYNVWlrqO8l43759io6OqFODAABAGEUZY0y4iwglj8cjp9Op6upqxcfHh7scAG3kRP1p9S9aJUnaNj1HnWPD/n83AG0okNdvDokAAACrEG4AAIBVCDcAAMAqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKoQbAABgFcINAACwCuEGAABYhXADAACsQrgBAABWIdwAAACrEG4AAIBVCDcAAMAqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKoQbAABgFcINAACwCuEGAABYhXADAACsQrgBAABWIdwAAACrEG4AAIBVCDcAAMAqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFZpF+FmwYIFSk9PV1xcnFwulzZs2NBk30WLFumGG25Q9+7d1b17d2VnZzfbHwAAXFjCHm6WLl2qgoICFRcXa9OmTRo0aJBycnJ05MiRRvuvWbNGY8aM0fvvv6/y8nKlpaXptttu08GDB0NcOQAAaI+ijDEmnAW4XC5de+21ev755yVJXq9XaWlpevjhhzV16tRzjm9oaFD37t31/PPPKzc395z9PR6PnE6nqqurFR8ff971A2gfTtSfVv+iVZKkbdNz1Dm2Q5grAtCWAnn9DuuRm/r6elVUVCg7O9vXFh0drezsbJWXl7dojhMnTujUqVPq0aNHo7fX1dXJ4/H4XQAAgL3CGm6OHj2qhoYGJSUl+bUnJSXJ7Xa3aI4pU6YoNTXVLyB9W0lJiZxOp++SlpZ23nUDAID2K+zn3JyPGTNm6PXXX9dbb72luLi4RvsUFhaqurrad9m/f3+IqwQAAKEU1jelExISFBMTo6qqKr/2qqoqJScnNzt2zpw5mjFjht59910NHDiwyX4Oh0MOh6NN6gUAAO1fWI/cxMbGasiQISorK/O1eb1elZWVKSsrq8lxs2bN0lNPPaXS0lINHTo0FKUCAIAIEfaPExQUFGjs2LEaOnSoMjMzNW/ePNXW1iovL0+SlJubq169eqmkpESSNHPmTBUVFem1115Tenq679ycLl26qEuXLmFbBwAAaB/CHm5Gjx6tL774QkVFRXK73Ro8eLBKS0t9Jxnv27dP0dH/PMD0wgsvqL6+Xv/6r//qN09xcbGeeOKJUJYOAADaobB/z02o8T03gJ34nhvAbhHzPTcAAABtjXADAACsQrgBAABWIdwAAACrEG4AAIBVCDcAAMAqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKoQbAABgFcINAACwCuEGAABYhXADAACsQrgBAABWIdwAAACrEG4AAIBVCDcAAMAqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKoQbAABgFcINAACwCuEGAABYhXADAACsQrgBAABWIdwAAACrEG4AAIBVCDcAAMAqhBsAAGCVDuEuwBYNXqMNu4/pSM1JJXaNU2bfHoqJjmq38wZz7kisOZhzU3Po5g4WHufgzxupc1Nz6OYORLsINwsWLNDs2bPldrs1aNAgzZ8/X5mZmU32X7ZsmaZNm6Y9e/bo8ssv18yZM3XHHXeEsGJ/pVsO68m/bNPh6pO+thRnnIpH9teIjJR2Ny81h25uag7d3MHC4xzZNQdzbmoO3dyBijLGmJDe43csXbpUubm5WrhwoVwul+bNm6dly5Zp586dSkxMPKv/unXrdOONN6qkpER33XWXXnvtNc2cOVObNm1SRkbGOe/P4/HI6XSqurpa8fHx511/6ZbDmvjqJn33QTyTU1/42TWt2tRgzRvMuSOx5mDOTc2hm1uSTtSfVv+iVZKkbdNz1Dn2/P/vxuMc/HkjdW5qDt3cZwTy+h32cONyuXTttdfq+eeflyR5vV6lpaXp4Ycf1tSpU8/qP3r0aNXW1uqdd97xtf3gBz/Q4MGDtXDhwnPeX1uGmwav0fUz3/smpRojR0O93+1RkpLi4/RuwU0BHZZr8Bplz/1Abs/JRm9v7bzBnDsSaw7m3NQcurnPOFF/WkN+/a4kqeLx7PMONzzOkV1zMOem5pbNXRcTK0VFKUpSsjNOa6fccl5vUUVMuKmvr1fnzp315ptvatSoUb72sWPH6vjx43r77bfPGtO7d28VFBTokUce8bUVFxdr+fLl+tvf/nZW/7q6OtXV1fmuezwepaWltUm4Kf/sS41Z9JEkyXG6Tsvf+dV5zQcAgC1G3fW06jo4fNf/OP4Hyrr0olbPF0i4CeunpY4ePaqGhgYlJSX5tSclJcntdjc6xu12B9S/pKRETqfTd0lLS2ub4iUdqWk8AQMAAH+hfM1sFycUB1NhYaEKCgp8188cuWkLiV3jfH+ui4nVqLuebrTfkrxMufr2aPG863cf089f3nDOfoHOG8y5I7HmYM5NzaGbO1h4nIM/b6TOTc0tm7suJtbv+rdfM4MtrOEmISFBMTExqqqq8muvqqpScnJyo2OSk5MD6u9wOORwOBq97Xxl9u2hFGec3NUnZaKi/A6/SfK9z5h5VS9FB/A+Y+ZVndTjIuc38zZye2vnDebckVhzMOem5tDNHSw8zpFdczDnpuZWzh3C/7iE9W2p2NhYDRkyRGVlZb42r9ersrIyZWVlNTomKyvLr78krV69usn+wRQTHaXikf0l/fOM8DPOXC8e2T/gE6iCNW8w547EmoM5NzWHbu5g4XEO/ryROjc1h27u1gr7NxQXFBRo0aJFeuWVV7R9+3ZNnDhRtbW1ysvLkyTl5uaqsLDQ13/SpEkqLS3Vb37zG+3YsUNPPPGENm7cqIceeigs9Y/ISNELP7tGyU7/w23Jzrjz+uhbsOal5tDNTc2hmztYeJwju+Zgzk3NoZu7NcL+UXBJev75531f4jd48GD99re/lcvlkiQNHz5c6enpWrJkia//smXL9Pjjj/u+xG/WrFkt/hK/tv6emzP4Jsngzxupc1Nz6OYOFh7n4M8bqXNTc+jmjpiPgodDsMINAAAInoj5KDgAAEBbI9wAAACrEG4AAIBVCDcAAMAqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYJ66+Ch8OZL2T2eDxhrgQAALTUmdftlvywwgUXbmpqaiRJaWlpYa4EAAAEqqamRk6ns9k+F9xvS3m9Xh06dEhdu3ZVVFTb/lifx+NRWlqa9u/fb+XvVtm+Psn+NbK+yGf7Gllf5AvWGo0xqqmpUWpqqqKjmz+r5oI7chMdHa2LL744qPcRHx9v7V9ayf71SfavkfVFPtvXyPoiXzDWeK4jNmdwQjEAALAK4QYAAFiFcNOGHA6HiouL5XA4wl1KUNi+Psn+NbK+yGf7Gllf5GsPa7zgTigGAAB248gNAACwCuEGAABYhXADAACsQrgBAABWIdwE4Omnn9awYcPUuXNndevWrdE++/bt05133qnOnTsrMTFRjz32mE6fPt3svMeOHdN9992n+Ph4devWTePGjdNXX30VhBUEZs2aNYqKimr08vHHHzc5bvjw4Wf1nzBhQggrb7n09PSzap0xY0azY06ePKn8/HxddNFF6tKli/7lX/5FVVVVIao4MHv27NG4cePUt29fderUSZdeeqmKi4tVX1/f7Lj2vIcLFixQenq64uLi5HK5tGHDhmb7L1u2TFdeeaXi4uI0YMAArVy5MkSVBq6kpETXXnutunbtqsTERI0aNUo7d+5sdsySJUvO2qu4uLgQVRyYJ5544qxar7zyymbHRNL+SY3/mxIVFaX8/PxG+7f3/fvwww81cuRIpaamKioqSsuXL/e73RijoqIipaSkqFOnTsrOztY//vGPc84b6PM4UISbANTX1+vuu+/WxIkTG729oaFBd955p+rr67Vu3Tq98sorWrJkiYqKipqd97777tPWrVu1evVqvfPOO/rwww/14IMPBmMJARk2bJgOHz7sd3nggQfUt29fDR06tNmx48eP9xs3a9asEFUduOnTp/vV+vDDDzfbf/LkyfrLX/6iZcuW6YMPPtChQ4f005/+NETVBmbHjh3yer168cUXtXXrVj377LNauHChfvnLX55zbHvcw6VLl6qgoEDFxcXatGmTBg0apJycHB05cqTR/uvWrdOYMWM0btw4bd68WaNGjdKoUaO0ZcuWEFfeMh988IHy8/P10UcfafXq1Tp16pRuu+021dbWNjsuPj7eb6/27t0boooD9/3vf9+v1rVr1zbZN9L2T5I+/vhjv/WtXr1aknT33Xc3OaY9719tba0GDRqkBQsWNHr7rFmz9Nvf/lYLFy7U+vXr9b3vfU85OTk6efJkk3MG+jxuFYOAvfzyy8bpdJ7VvnLlShMdHW3cbrev7YUXXjDx8fGmrq6u0bm2bdtmJJmPP/7Y1/a///u/Jioqyhw8eLDNaz8f9fX1pmfPnmb69OnN9rvpppvMpEmTQlPUeerTp4959tlnW9z/+PHjpmPHjmbZsmW+tu3btxtJpry8PAgVtr1Zs2aZvn37Ntunve5hZmamyc/P911vaGgwqamppqSkpNH+//Zv/2buvPNOvzaXy2V+8YtfBLXOtnLkyBEjyXzwwQdN9mnq36P2qLi42AwaNKjF/SN9/4wxZtKkSebSSy81Xq+30dsjaf8kmbfeest33ev1muTkZDN79mxf2/Hjx43D4TB//OMfm5wn0Odxa3Dkpg2Vl5drwIABSkpK8rXl5OTI4/Fo69atTY7p1q2b35GQ7OxsRUdHa/369UGvORB//vOf9eWXXyovL++cff/nf/5HCQkJysjIUGFhoU6cOBGCCltnxowZuuiii3T11Vdr9uzZzb6NWFFRoVOnTik7O9vXduWVV6p3794qLy8PRbnnrbq6Wj169Dhnv/a2h/X19aqoqPB77KOjo5Wdnd3kY19eXu7XX/rmORlJeyXpnPv11VdfqU+fPkpLS9OPf/zjJv+9aQ/+8Y9/KDU1VZdcconuu+8+7du3r8m+kb5/9fX1evXVV/Xv//7vzf5QcyTt37ft3r1bbrfbb4+cTqdcLleTe9Sa53FrXHA/nBlMbrfbL9hI8l13u91NjklMTPRr69Chg3r06NHkmHB56aWXlJOTc84fHr333nvVp08fpaam6u9//7umTJminTt36k9/+lOIKm25//zP/9Q111yjHj16aN26dSosLNThw4c1d+7cRvu73W7Fxsaedc5VUlJSu9uvxuzatUvz58/XnDlzmu3XHvfw6NGjamhoaPQ5tmPHjkbHNPWcjIS98nq9euSRR3TdddcpIyOjyX79+vXT4sWLNXDgQFVXV2vOnDkaNmyYtm7dGvQfCQ6Uy+XSkiVL1K9fPx0+fFhPPvmkbrjhBm3ZskVdu3Y9q38k758kLV++XMePH9fPf/7zJvtE0v5915l9CGSPWvM8bo0LPtxMnTpVM2fObLbP9u3bz3nSWyRpzZoPHDigVatW6Y033jjn/N8+X2jAgAFKSUnRrbfeqs8++0yXXnpp6wtvoUDWV1BQ4GsbOHCgYmNj9Ytf/EIlJSXt+uvRW7OHBw8e1IgRI3T33Xdr/PjxzY4N9x5Cys/P15YtW5o9J0WSsrKylJWV5bs+bNgwXXXVVXrxxRf11FNPBbvMgNx+++2+Pw8cOFAul0t9+vTRG2+8oXHjxoWxsuB46aWXdPvttys1NbXJPpG0f5Hkgg83jz76aLOpWpIuueSSFs2VnJx81hnfZz5Fk5yc3OSY755Edfr0aR07dqzJMeerNWt++eWXddFFF+lHP/pRwPfncrkkfXPUIBQvjOezpy6XS6dPn9aePXvUr1+/s25PTk5WfX29jh8/7nf0pqqqKmj71ZhA13jo0CHdfPPNGjZsmH7/+98HfH+h3sPGJCQkKCYm5qxPpjX32CcnJwfUv7146KGHfB8uCPR/7x07dtTVV1+tXbt2Bam6ttOtWzddccUVTdYaqfsnSXv37tW7774b8NHOSNq/M/tQVVWllJQUX3tVVZUGDx7c6JjWPI9bpc3O3rmAnOuE4qqqKl/biy++aOLj483JkycbnevMCcUbN270ta1atapdnVDs9XpN3759zaOPPtqq8WvXrjWSzN/+9rc2rqztvfrqqyY6OtocO3as0dvPnFD85ptv+tp27NjRrk8oPnDggLn88svNPffcY06fPt2qOdrLHmZmZpqHHnrId72hocH06tWr2ROK77rrLr+2rKysdntCqtfrNfn5+SY1NdV8+umnrZrj9OnTpl+/fmby5MltXF3bq6mpMd27dzfPPfdco7dH2v59W3FxsUlOTjanTp0KaFx73j81cULxnDlzfG3V1dUtOqE4kOdxq2pts5kuAHv37jWbN282Tz75pOnSpYvZvHmz2bx5s6mpqTHGfPOXMiMjw9x2222msrLSlJaWmp49e5rCwkLfHOvXrzf9+vUzBw4c8LWNGDHCXH311Wb9+vVm7dq15vLLLzdjxowJ+fqa8u677xpJZvv27WfdduDAAdOvXz+zfv16Y4wxu3btMtOnTzcbN240u3fvNm+//ba55JJLzI033hjqss9p3bp15tlnnzWVlZXms88+M6+++qrp2bOnyc3N9fX57vqMMWbChAmmd+/e5r333jMbN240WVlZJisrKxxLOKcDBw6Yyy67zNx6663mwIED5vDhw77Lt/tEyh6+/vrrxuFwmCVLlpht27aZBx980HTr1s33CcX777/fTJ061df/r3/9q+nQoYOZM2eO2b59uykuLjYdO3Y0n3zySbiW0KyJEycap9Np1qxZ47dXJ06c8PX57hqffPJJs2rVKvPZZ5+ZiooKc88995i4uDizdevWcCyhWY8++qhZs2aN2b17t/nrX/9qsrOzTUJCgjly5IgxJvL374yGhgbTu3dvM2XKlLNui7T9q6mp8b3WSTJz5841mzdvNnv37jXGGDNjxgzTrVs38/bbb5u///3v5sc//rHp27ev+frrr31z3HLLLWb+/Pm+6+d6HrcFwk0Axo4daySddXn//fd9ffbs2WNuv/1206lTJ5OQkGAeffRRv+T+/vvvG0lm9+7dvrYvv/zSjBkzxnTp0sXEx8ebvLw8X2BqD8aMGWOGDRvW6G27d+/2ewz27dtnbrzxRtOjRw/jcDjMZZddZh577DFTXV0dwopbpqKiwrhcLuN0Ok1cXJy56qqrzDPPPON3lO276zPGmK+//tr8x3/8h+nevbvp3Lmz+clPfuIXFtqTl19+udG/s98+aBtpezh//nzTu3dvExsbazIzM81HH33ku+2mm24yY8eO9ev/xhtvmCuuuMLExsaa73//+2bFihUhrrjlmtqrl19+2dfnu2t85JFHfI9HUlKSueOOO8ymTZtCX3wLjB492qSkpJjY2FjTq1cvM3r0aLNr1y7f7ZG+f2esWrXKSDI7d+4867ZI278zr1nfvZxZg9frNdOmTTNJSUnG4XCYW2+99ax19+nTxxQXF/u1Nfc8bgtRxhjTdm9yAQAAhBffcwMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAq3QIdwEAcL6GDx+ugQMHKi4uTn/4wx8UGxurCRMm6Iknngh3aQDCgCM3AKzwyiuv6Hvf+57Wr1+vWbNmafr06Vq9enW4ywIQBvwqOICIN3z4cDU0NOj//u//fG2ZmZm65ZZbNGPGjDBWBiAcOHIDwAoDBw70u56SkqIjR46EqRoA4US4AWCFjh07+l2PioqS1+sNUzUAwolwAwAArEK4AQAAViHcAAAAq/BpKQAAYBWO3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKv8f02br9q/uUf0AAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = np.arange(-10,11)\n",
    "x = np.zeros_like(n)\n",
    "x[n==0] = 1\n",
    "\n",
    "plt.stem(n,x)\n",
    "plt.xlabel('n')\n",
    "plt.ylabel('x');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d113ffd4-95ad-4c10-be64-7cefe2b5b32e",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Para obter um impulso unitário em outra posição que não $n=0$, basta atrasarmos (ou adiantarmos) o sinal $\\delta[n]$. Se queremos que o sinal seja 1 em $n=k$, basta fazermos $\\delta[n-k]$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7fcc67b5-cfac-4f14-8c40-2cfbd10c07ee",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "---\n",
    "\n",
    "### Exemplo 1\n",
    "\n",
    "Por exemplo, se queremos um impulso unitário em $n=2$, fazemos $\\delta[n-2]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "02ba1adc-5603-45a7-96f6-2e3fce814ead",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = np.arange(-10,11)\n",
    "x = np.zeros_like(n)\n",
    "x[n==2] = 1\n",
    "\n",
    "plt.stem(n,x)\n",
    "plt.xlabel('n')\n",
    "plt.ylabel('x');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0c02a26d-e817-4fa6-9d1d-2593d8f53591",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e925e7d-55de-4408-b486-67d3cc2bd667",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## Sinal degrau\n",
    "\n",
    "Outro sinal que usaremos bastante é o sinal degrau de tempo discreto. Esse sinal é sempre igual a 1 para $n\\geq 0$. Para esse sinal usaremos a letra $u$:\n",
    "\n",
    "$$u[n] =   \\left\\{\\begin{array}{cc}1& ,n \\geq 0 \\\\ 0& ,c.c \\end{array}\\right. $$\n",
    "\n",
    "O seu gráfico é mostrado abaixo: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "87c08f03-e0ac-4bfb-9905-b2e70cac8d12",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = np.arange(-10,11)\n",
    "x = np.zeros_like(n)\n",
    "x[n>=0] = 1\n",
    "\n",
    "plt.stem(n,x)\n",
    "plt.xlabel('n')\n",
    "plt.ylabel('x');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4a0b34a2-4d71-4816-acff-7023fd995027",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Os sinais degrau e impulso unitário em tempo discreto estão bastante relacionados. O sinal impulso unitário é o sinal degrau subtraido do mesmo sinal degrau atrasado de 1 passo.\n",
    "\n",
    "$$\\delta[n] = u[n] - u[n-1]$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f4d6d7d-7bbe-49ca-a0d7-3b9becffd22a",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    },
    "tags": []
   },
   "source": [
    "## Exercícios\n",
    "\n",
    "**1)** Faça o gráfico dos seguintes  sinais:\n",
    "\n",
    "a) $x_1[n]=u[n] - u[n-3]$\n",
    "\n",
    "b) $x_2[n] = u[n] - 2u[n-3]+u[n-4]$\n",
    "\n",
    "c) $x_3[n] = nu[n]-2(n-1)u[n-1]+(n-3)u[n-3]$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f965ad92-2c15-46fe-a9af-85e3dee97b22",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    },
    "tags": []
   },
   "source": [
    "**2)** Faça o gráfico dos seguintes sinais:\n",
    "\n",
    "a) $x_1[n] = u[n] - \\delta[n-3]-u[n-4]$\n",
    "\n",
    "b) $x_2[n] = nu[n] - nu[n-5] + 3\\delta[n-2]$\n",
    "\n",
    "c) $x_3[n] = 2\\delta[n-1] - 4\\delta[n-3]$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0129c8a2-de0e-478b-b267-9c8c0dd8dcb6",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    },
    "tags": []
   },
   "source": [
    "**3)** Considere o seguinte sinal de tempo discreto:\n",
    "\n",
    "$$x[n] = 1-\\displaystyle\\sum_{k=2^\\infty}\\delta[n-2-k] $$\n",
    "\n",
    "Determine os valores dos números inteiros $M$ e $n_0$ para escrever $x[n]$ na forma:\n",
    "\n",
    "$$x[n]=u[Mn-n_0] $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4ad17ae3-b797-4a0a-afe4-c641ab91106f",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    },
    "tags": []
   },
   "source": [
    "**4)** Considere o seguinte sinal:\n",
    "\n",
    "$$x[n] = 0.5^{n-1}(u[n+2]- u[n-8])$$\n",
    "\n",
    "Encontre $a$ e $b$ na expressão abaixo:\n",
    "\n",
    "$$x[n-k] = \\left\\{\\begin{array}{cc}0.5^{n-k-1}&\\text{ , se } a< k < b \\\\ 0& \\text{ , c.c.} \\end{array}\\right. $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "67555f96-2adf-4788-bd8b-9e7dbfb6b9f9",
   "metadata": {},
   "source": [
    "**5)** Escreva os seguintes sinais com apenas uma expressão (sem o uso de chaves com condições):\n",
    "\n",
    "a) $x_1[n] = \\left\\{\\begin{array}{cc} n & , 0 \\leq n \\leq 3\\\\n^2&, 3< n \\leq 7\\\\ 0& \\text{c.c.} \\end{array}\\right.$\n",
    "\n",
    "\n",
    "b) $x_2[n] = \\left\\{\\begin{array}{cc} \\cos\\left(\\frac{\\pi}{6}n\\right) & , -3 \\leq n \\leq 3\\\\ 0&\\text{c.c.} \\end{array}\\right.$\n",
    "\n",
    "$x[n] = \\left\\{\\begin{array}{cc} 1 & , 0 \\leq n \\leq 2\\\\2&, 2< n \\leq 7\\\\ 1&7 < n \\leq 10 \\\\0&\\text{c.c.} \\end{array}\\right.$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3756de4b-629f-47a9-be93-050e708a3b64",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## Referências\n",
    "\n",
    "- Oppenheim, M.  e Willsky, S., Sistemas e Sinais, 2010\n",
    "- Ulaby, FT e Yagle, AE, Signals & Systems: Theory and Applications, 2018, [ss2.eecs.umich.edu](ss2.eecs.umich.edu)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8968544a-1148-4dc0-9c85-2b462776831f",
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    "## Solução dos exercícios"
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   "source": []
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