From ab6ed09c5dd6d56bb8f9751ccbcc62dbd9b38a32 Mon Sep 17 00:00:00 2001
From: tv3141 <tv3141@users.noreply.github.com>
Date: Sat, 23 Sep 2017 10:12:17 +0100
Subject: [PATCH] latexify g and h

---
 01-g-h-filter.ipynb | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/01-g-h-filter.ipynb b/01-g-h-filter.ipynb
index f7e6db8..a4142b3 100644
--- a/01-g-h-filter.ipynb
+++ b/01-g-h-filter.ipynb
@@ -4026,7 +4026,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Choice of g and h"
+    "## Choice of $g$ and $h$"
    ]
   },
   {
@@ -4341,14 +4341,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Exercise: Varying g"
+    "## Exercise: Varying $g$"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Now let's look at the effect of varying g. Before you perform this exercise, recall that g is the scale factor for choosing between the measurement and prediction. What do you think the effect of a large value of g will be? A small value? \n",
+    "Now let's look at the effect of varying $g$. Before you perform this exercise, recall that $g$ is the scale factor for choosing between the measurement and prediction. What do you think the effect of a large value of $g$ will be? A small value?\n",
     "\n",
     "Now, let the `noise_factor=50` and `dx=5`. Plot the results of $g = 0.1\\mbox{, } 0.4,\\mbox{ and } 0.8$."
    ]
@@ -4464,7 +4464,7 @@
    "source": [
     "Here we can see the effects of ignoring the signal. We not only filter out noise, but legitimate changes in the signal as well. \n",
     "\n",
-    "Maybe we need a 'Goldilocks' filter, where g is not too large, not too small, but just right? Well, not exactly. As alluded to earlier, different filters choose g and h in different ways depending on the mathematical properties of the problem. For example, the Benedict-Bordner filter was invented to minimize the transient error in this example, where $\\dot{x}$ makes a step jump. We will not discuss this filter in this book, but here are two plots chosen with different allowable pairs of g and h. This filter design minimizes transient errors for step jumps in $\\dot{x}$ at the cost of not being optimal for other types of changes in $\\dot{x}$."
+    "Maybe we need a 'Goldilocks' filter, where $g$ is not too large, not too small, but just right? Well, not exactly. As alluded to earlier, different filters choose $g$ and $h$ in different ways depending on the mathematical properties of the problem. For example, the Benedict-Bordner filter was invented to minimize the transient error in this example, where $\\dot{x}$ makes a step jump. We will not discuss this filter in this book, but here are two plots chosen with different allowable pairs of $g$ and $h$. This filter design minimizes transient errors for step jumps in $\\dot{x}$ at the cost of not being optimal for other types of changes in $\\dot{x}$."
    ]
   },
   {
@@ -4501,14 +4501,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Varying h"
+    "## Varying $h$"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Now let's leave g unchanged and investigate the effect of modifying h. We know that h affects how much we favor the measurement of $\\dot{x}$ vs our prediction. But what does this *mean*? If our signal is changing a lot (quickly relative to the time step of our filter), then a large $h$ will cause us to react to those transient changes rapidly. A smaller $h$ will cause us to react more slowly.\n",
+    "Now let's leave $g$ unchanged and investigate the effect of modifying $h$. We know that $h$ affects how much we favor the measurement of $\\dot{x}$ vs our prediction. But what does this *mean*? If our signal is changing a lot (quickly relative to the time step of our filter), then a large $h$ will cause us to react to those transient changes rapidly. A smaller $h$ will cause us to react more slowly.\n",
     "\n",
     "We will look at three examples. We have a noiseless measurement that slowly goes from 0 to 1 in 50 steps. Our first filter uses a nearly correct initial value for $\\dot{x}$ and a small $h$. You can see from the output that the filter output is very close to the signal. The second filter uses the very incorrect guess of $\\dot{x}=2$. Here we see the filter 'ringing' until it settles down and finds the signal. The third filter uses the same conditions but it now sets $h=0.5$. If you look at the amplitude of the ringing you can see that it is much smaller than in the second chart, but the frequency is greater. It also settles down a bit quicker than the second filter, though not by much."
    ]