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    "# Synchronization"
   ]
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    "Currently, the samples are still represented as complex numbers.\n",
    "To continue working with them, we convert them to their magnitudes:\n",
    "```{math}\n",
    "s = a + bi \\mapsto |s| = \\sqrt{a^2 + b^2}\n",
    "```\n",
    "\n",
    "Here is a visualization of the signal as processed so far:\n",
    "```{figure} img/reference/filtered_full_scaled.webp\n",
    "---\n",
    "name: fig:filtered_full_scaled.en\n",
    "---\n",
    "Filtered signal.\n",
    "We observe that the sync lines are bent.\n",
    "```\n",
    "\n",
    "The sync patterns mark the beginning of each line.\n",
    "To \"straighten out\" the image, we have to find them.\n",
    "\n",
    "We proceed as follows:\n",
    "- Sync A has the pattern `000011001100110011001100110011000000000`.\n",
    "  We have 4 samples per pixel, and thus also per 0/1 in the pattern.\n",
    "  We want to find this pattern.\n",
    "  This is easiest if we scale both the pattern and our samples into the range between $-1$ and $1$.\n",
    "  For every `0` in the pattern, we take four $-1$s, and for every `1` we take four $1$s.\n",
    "  We save this sequence of 39 * 4 = 156 values as $p$.\n",
    "- We now look at blocks $x$ of 8320 samples.\n",
    "  We find the minimum value $x_{min}$ and the maximum value $x_{max}$ within the block, and scale it such that the minimum is at $-1$ and the maximum at $1$:\n",
    "  ```{math}\n",
    "  x[n] \\mapsto -1 + 2 \\cdot \\frac{x[n] - x_{min}}{x_{max} - x_{min}}\n",
    "  ```\n",
    "- We want to find the position that best matches the pattern.\n",
    "  To do this, we iterate over all positions $i$ in the block from 0 to (8320-156).\n",
    "  For each position, we compute the *correlation*\n",
    "  ```{math}\n",
    "  z = \\sum_{n=0}^{156} x[i + n] \\cdot p[n]\n",
    "  ```\n",
    "  The position with the highest $z$ value is the one that matches best.\n",
    "- This way, we can find the starting position of each line.\n",
    "  From there, we can take every fourth sample to get the pixel values for this line."
   ]
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