Not very much progress today

content/BACH.typ

@@ -12,6 +12,8 @@ Before we take a look at the higher order quantization cases, we will start with
 
 #figure(
   include("./../graphics/quantizers/bach/sign-based-overlay.typ"),
-  caption: [Nice graph]
+  caption: [1-bit quantizer with the PDF of a normal distribution]
-)
+)<fig:1-bit_normal>
 
+If we overlay the PDF of a zero-mean Gaussian distributed variable $X$ with a sign-based quantizer function as shown in @fig:1-bit_normal, we can see that the expected value of the Gaussian distribution overlaps with the decision threshold of the sign-based quantizer.
+Considering that the margin of error of the value $x$ is comparable with the one shown in @fig:tmhd_example_enroll, we can conclude that values of $X$ 

graphics/quantizers/bach/sign-based-overlay.typ

@@ -1,27 +1,32 @@
 #import "@preview/cetz:0.2.2": *
 
-
+#let ymax = 1/calc.sqrt(2*calc.pi)
 
 #let line_style = (stroke: (paint: black, thickness: 2pt))
 #let dashed = (stroke: (dash: "dashed"))
 #canvas({
   plot.plot(size: (8,4),
+    legend: "legend.north",
+    legend-style: (orientation: ltr, item: (spacing: 0.5)),
     x-tick-step: none,
     x-ticks: ((0, [0]), (100, [0])),
-    y-label: $cal(Q)(1, x)$,
+    y-label: $cal(Q)(1, x), xi(x)$,
     x-label: $x$,
-    y-tick-step: 1,
+    y-tick-step: none,
+    y-ticks: ((0, [0]), (ymax, [1])),
     axis-style: "left",
     x-min: -3,
     x-max: 3,
     y-min: 0,
-    y-max: 1,{
+    y-max: ymax,{
-    plot.add(((-3,0), (0,0), (0,1), (3,1)), style: line_style)
+    plot.add(
-    plot.add(plot.sample-fn(
+      plot.sample-fn(
-  (x) => 1/calc.sqrt(2*calc.pi)*calc.exp(-(calc.pow(x,2)/2)),
+        (x) => 1/calc.sqrt(2*calc.pi)*calc.exp(-(calc.pow(x,2)/2)),
-  (-3, 3),
+        (-3, 3),
-  300 
+        300),
-))
+      style: (stroke: (paint: red, thickness: 2pt)),
-    
+      label: [PDF of a normal distribution]
+    )
+    plot.add(((-3,0), (0,0), (0,ymax), (3,ymax)), style: line_style, label: [$cal(Q)(1,x)$])
   })
 })