added equation numbers

Author: mstudinger

Diff for: ATM_centroid_tracker.ipynb

@@ -30,7 +30,9 @@
     "The centroid $t_c$ of a discrete waveform $w\\,(t_i)$ with amplitudes $w$ digitized at times $t_i$ is defined as\n",
     "\n",
     "$$\n",
-    "  t_c=\\frac{\\sum_\\limits{t_{start}}^{t_{end}} w\\,(t_i) \\cdot t_i}{\\sum_\\limits{t_{start}}^{t_{end}} w\\,(t_i)}\n",
+    "\\begin{align}\n",
+    "  t_c=\\frac{\\sum_\\limits{t_{start}}^{t_{end}} w\\,(t_i) \\cdot t_i}{\\sum_\\limits{t_{start}}^{t_{end}} w\\,(t_i)} \\tag{1}\n",
+    "\\end{align}\n",
     "$$\n",
     "\n",
     "where $t_{start}$ and $t_{end}$ are the start and end times of the part of the waveform that is being used for centroid calculation."
@@ -73,23 +75,23 @@
     "\n",
     "$$\n",
     "\\begin{align}\n",
-    " g\\,(t_i)= b + a \\cdot {e ^{-\\frac12 \\frac{(t_i - t_m)^2}{\\sigma^2}}} \\tag{1}\n",
-    " \\end{align}\n",
+    " g\\,(t_i)= b + a \\cdot {e ^{-\\frac12 \\frac{(t_i - t_m)^2}{\\sigma^2}}} \\tag{2}\n",
+    "\\end{align}\n",
     "$$\n",
     "\n",
     "where $b$ is the signal baseline (noise floor), $a$ is the maximum pulse amplitude, $t_m$ is pulse location of the pulse maximum and mean, and $\\sigma$ is the $1-\\sigma$ standard deviation pulse width. For a Gaussian pulse, the mean $t_m$ is identical with the centroid $t_c$, i.e.\n",
     "\n",
     "$$\n",
     "\\begin{align}\n",
-    " t_c = t_m \\tag{2}\n",
+    " t_c = t_m \\tag{3}\n",
     "\\end{align}\n",
     "$$\n",
     "\n",
     "To create a synthetic Gaussian lidar pulse that is similar to an actual ATM lidar pulse, we need to define some parameters first. The pulse specifications of the 1.3 ns ATM lasers are measured in full width at half maximum (FWHM). For a Gaussian pulse, the relationship between FWHM and the standard deviation ($1-\\sigma$) is:\n",
     "\n",
     "$$\n",
     "\\begin{align}\n",
-    "    FWHM=2 \\cdot \\sqrt{2 \\ln 2} \\space \\sigma \\approx 2.35482 \\space \\sigma \\tag{3}\n",
+    "    FWHM=2 \\cdot \\sqrt{2 \\ln 2} \\space \\sigma \\approx 2.35482 \\space \\sigma \\tag{4}\n",
     "\\end{align}\n",
     "$$\n"
    ]