diff --git a/graphics/background/bach/instability/frame_1.png b/graphics/background/bach/instability/frame_1.png
new file mode 100644
index 0000000..f4f17e9
Binary files /dev/null and b/graphics/background/bach/instability/frame_1.png differ
diff --git a/graphics/background/bach/instability/frame_18.png b/graphics/background/bach/instability/frame_18.png
new file mode 100644
index 0000000..367d979
Binary files /dev/null and b/graphics/background/bach/instability/frame_18.png differ
diff --git a/graphics/background/bach/instability/frame_25.png b/graphics/background/bach/instability/frame_25.png
new file mode 100644
index 0000000..2e64dee
Binary files /dev/null and b/graphics/background/bach/instability/frame_25.png differ
diff --git a/graphics/background/bach/instability/frame_7.png b/graphics/background/bach/instability/frame_7.png
new file mode 100644
index 0000000..8128b19
Binary files /dev/null and b/graphics/background/bach/instability/frame_7.png differ
diff --git a/graphics/background/sign-based-overlay.typ b/graphics/background/sign-based-overlay.typ
new file mode 100644
index 0000000..1de5ff3
--- /dev/null
+++ b/graphics/background/sign-based-overlay.typ
@@ -0,0 +1,34 @@
+#import "@preview/cetz:0.3.4": canvas
+#import "@preview/cetz-plot:0.1.1": plot, chart
+
+#let ymax = 1/calc.sqrt(2*calc.pi)
+
+#let line_style = (stroke: (paint: black, thickness: 3pt))
+#let dashed = (stroke: (dash: "dashed"))
+#canvas({
+
+ plot.plot(size: (11,6),
+ legend: "north",
+ legend-style: (orientation: ltr, item: (spacing: 0.5)),
+ x-tick-step: none,
+ x-ticks: ((0, [0]), (100, [0])),
+ y-label: $cal(Q)(1, x)$,
+ x-label: $x$,
+ y-tick-step: none,
+ y-ticks: ((0, [0]), (ymax, [1])),
+ axis-style: "left",
+ x-min: -3,
+ x-max: 3,
+ y-min: 0,
+ y-max: ymax,{
+ plot.add(
+ plot.sample-fn(
+ (x) => 1/calc.sqrt(2*calc.pi)*calc.exp(-(calc.pow(x,2)/2)),
+ (-3, 3),
+ 300),
+ style: (stroke: (paint: red, thickness: 3pt)),
+ // label: [PDF einer Normalverteilung]
+ )
+ plot.add(((-3,0), (0,0), (0,ymax), (3,ymax)), style: line_style, /*label: [$cal(Q)(1,x)$]*/)
+ })
+})
diff --git a/graphics/background/two-bit-enroll.typ b/graphics/background/two-bit-enroll.typ
new file mode 100644
index 0000000..c8e8191
--- /dev/null
+++ b/graphics/background/two-bit-enroll.typ
@@ -0,0 +1,26 @@
+#import "@preview/cetz:0.3.4": canvas, draw, palette
+#import "@preview/cetz-plot:0.1.1": plot, chart
+
+#let line_style = (stroke: (paint: black, thickness: 3pt))
+#let dashed = (stroke: (dash: "dashed"))
+#canvas({
+ import draw: *
+ set-style(axes: (shared-zero: false))
+ plot.plot(size: (8,6),
+ x-tick-step: none,
+ x-ticks: ((0.25, [$g_1$]), (0.5, [0]), (0.75, [$g_2$])),
+ y-label: $cal(Q)(2, 1, tilde(x))$,
+ x-label: $tilde(x)$,
+ y-tick-step: none,
+ y-ticks: ((0.25, [00]), (0.5, [01]), (0.75, [10]), (1, [11])),
+ axis-style: "left",
+ //x-min: 0,
+ x-max: 1,
+ y-min: 0,
+ y-max: 1,{
+ plot.add(((0,0.25), (0.25,0.25), (0.5,0.5), (0.75,0.75), (1, 1)), line: "vh", style: line_style)
+ //plot.add(((0,0), (0,0)), style: (stroke: none))
+ plot.add-hline(0.25, 0.5, 0.75, 1, style: dashed)
+ plot.add-vline(0.25, 0.5, 0.75, 1, style: dashed)
+ })
+})
diff --git a/graphics/background/z_distribution.csv b/graphics/background/z_distribution.csv
new file mode 100644
index 0000000..8f17016
--- /dev/null
+++ b/graphics/background/z_distribution.csv
@@ -0,0 +1,1000 @@
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diff --git a/graphics/background/z_distribution.typ b/graphics/background/z_distribution.typ
new file mode 100644
index 0000000..f0ead54
--- /dev/null
+++ b/graphics/background/z_distribution.typ
@@ -0,0 +1,27 @@
+#import "@preview/cetz:0.3.4": canvas
+#import "@preview/cetz-plot:0.1.1": plot, chart
+
+#let data = csv("./z_distribution.csv")
+#let data = data.map(value => value.map(v => float(v)))
+
+#let line_style = (stroke: (paint: black, thickness: 3pt))
+#let dashed = (stroke: (dash: "dashed"))
+#canvas({
+ plot.plot(size: (11,5),
+ legend : "south",
+ legend-style: (orientation: ltr, item: (spacing: 0.5)),
+ x-tick-step: none,
+ x-ticks: ((0, [0]), (100, [0])),
+ y-label: $cal(Q)(1, z), abs(f_"Z" (z))$,
+ x-label: $z$,
+ y-tick-step: none,
+ y-ticks: ((0, [0]), (0.6, [1])),
+ axis-style: "left",
+ x-min: -5,
+ x-max: 5,
+ y-min: 0,
+ y-max: 0.6,{
+ plot.add((data), style: (stroke: (paint: red, thickness: 3pt)), /*label: [Optimierte PDF]*/)
+ plot.add(((-5, 0), (0, 0), (0, 0.6), (5, 0.6)), style: line_style, /*label: [Quantisierer]*/)
+ })
+})
diff --git a/graphics/bruteforce_dist.svg b/graphics/bruteforce_dist.svg
new file mode 100644
index 0000000..194d14a
--- /dev/null
+++ b/graphics/bruteforce_dist.svg
@@ -0,0 +1,3145 @@
+
+
diff --git a/graphics/bruteforce_dist2.svg b/graphics/bruteforce_dist2.svg
new file mode 100644
index 0000000..b427359
--- /dev/null
+++ b/graphics/bruteforce_dist2.svg
@@ -0,0 +1,3145 @@
+
+
diff --git a/graphics/bruteforce_helperdata1.svg b/graphics/bruteforce_helperdata1.svg
new file mode 100644
index 0000000..71cada8
--- /dev/null
+++ b/graphics/bruteforce_helperdata1.svg
@@ -0,0 +1,242 @@
+
+
diff --git a/graphics/bruteforce_helperdata2.svg b/graphics/bruteforce_helperdata2.svg
new file mode 100644
index 0000000..32a366d
--- /dev/null
+++ b/graphics/bruteforce_helperdata2.svg
@@ -0,0 +1,298 @@
+
+
diff --git a/graphics/execution/brute-force-occurs.png b/graphics/execution/brute-force-occurs.png
new file mode 100644
index 0000000..ec84674
Binary files /dev/null and b/graphics/execution/brute-force-occurs.png differ
diff --git a/graphics/execution/brute-force.png b/graphics/execution/brute-force.png
new file mode 100644
index 0000000..7a2219a
Binary files /dev/null and b/graphics/execution/brute-force.png differ
diff --git a/graphics/execution/given_codeword.png b/graphics/execution/given_codeword.png
new file mode 100644
index 0000000..e12d965
Binary files /dev/null and b/graphics/execution/given_codeword.png differ
diff --git a/graphics/execution/helperdata_occurs.png b/graphics/execution/helperdata_occurs.png
new file mode 100644
index 0000000..efbe1f7
Binary files /dev/null and b/graphics/execution/helperdata_occurs.png differ
diff --git a/graphics/execution/recursive.typ b/graphics/execution/recursive.typ
new file mode 100644
index 0000000..c86cec4
--- /dev/null
+++ b/graphics/execution/recursive.typ
@@ -0,0 +1,65 @@
+#import "@preview/fletcher:0.5.8" as fletcher: diagram, node, edge
+#import fletcher.shapes: pill
+#import "@preview/touying:0.6.1": *
+
+#let fletcher-diagram = touying-reducer.with(reduce: fletcher.diagram, cover: fletcher.hide)
+
+#fletcher-diagram(
+ spacing: (-3mm, 8mm), // wide columns, narrow rows
+ node-stroke: 1pt, // outline node shapes
+ edge-stroke: 1pt, // make lines thicker
+ mark-scale: 60%, // make arrowheads smaller
+ node((0,0), [Start], name: ),
+ pause,
+
+ node((-4, 1), [u1], name: ),
+ node((4, 1), [u2], name: ),
+ node(enclose: (, ), stroke: aqua, fill: aqua.lighten(90%), name: ),
+ edge(, , "->"),
+ edge(, , "->"),
+ node((-20mm, 39mm), [n1], shape: pill, name: ),
+ edge(, , "->"),
+ pause,
+
+ node((-6, 2), [u11], name: ),
+ node((-2, 2), [u12], name: ),
+ node((2, 2), [u21], name: ),
+ node((6, 2), [u22], name: ),
+ node(enclose: (, ), stroke: teal, fill: teal.lighten(90%), name: ),
+ node((-20mm, 21.9mm), [n2], shape: pill, name: ),
+ edge(, , "->"),
+ edge(, , "->"),
+ edge(, , "->"),
+ edge(, , "->"),
+ edge(, , "->"),
+ pause,
+
+ node((-7, 3), [u111], name: ),
+ node((-5, 3), [u112], name: ),
+ node((-3, 3), [u121], name: ),
+ node((-1, 3), [u122], name: ),
+ node((1, 3), [u211], name: ),
+ node((3, 3), [u212], name: ),
+ node((5, 3), [u221], name: ),
+ node((7, 3), [u222], name: ),
+
+ node((-20mm, 4.6mm), [n3], shape: pill, name: ),
+
+
+ edge(, , "->"),
+
+
+
+ edge(, , "->"),
+ edge(, , "->"),
+ edge(, , "->"),
+ edge(, , "->"),
+ edge(, , "->"),
+ edge(, , "->"),
+ edge(, , "->"),
+ edge(, , "->"),
+
+
+ node(enclose: (, ), stroke: eastern, fill: eastern.lighten(90%), name: )
+
+)
diff --git a/graphics/execution/recursive_distribution .png b/graphics/execution/recursive_distribution .png
new file mode 100644
index 0000000..b9ad89b
Binary files /dev/null and b/graphics/execution/recursive_distribution .png differ
diff --git a/graphics/given_codeword_dist.svg b/graphics/given_codeword_dist.svg
new file mode 100644
index 0000000..2a39f3d
--- /dev/null
+++ b/graphics/given_codeword_dist.svg
@@ -0,0 +1,3153 @@
+
+
diff --git a/graphics/recursive_distribution.svg b/graphics/recursive_distribution.svg
new file mode 100644
index 0000000..432a1ef
--- /dev/null
+++ b/graphics/recursive_distribution.svg
@@ -0,0 +1,3145 @@
+
+
diff --git a/graphics/recursive_helperdata_dist.svg b/graphics/recursive_helperdata_dist.svg
new file mode 100644
index 0000000..a19a882
--- /dev/null
+++ b/graphics/recursive_helperdata_dist.svg
@@ -0,0 +1,274 @@
+
+
diff --git a/main.typ b/main.typ
index 789a589..849d214 100644
--- a/main.typ
+++ b/main.typ
@@ -14,12 +14,164 @@
),
)
-= Einführung
+= Ausgangslage
-== Ausgangslage
+== 1-Bit Quantisierung
-== Planung
+#figure(
+ include("./graphics/background/sign-based-overlay.typ"),
+ // caption: []
+)
-= Ergebnisse
+- Stufe des Quantisiererfunktion nahe des Erwartungswertes der Wahrscheinlichkeitsverteilung
-== Hier dann
+== Optimierung der 1-Bit Quantisierung
+
+- Eingangswerte als Gewichtete Summen
+
+#v(1em)
+
+$ f(bold(x), bold(h)) = h_1 x_1 + h_2 x_2 + h_3 x_3 $
+
+- $h_i$ als Helperdaten
+
+#v(1em)
+
+- Betragsmäßige Maximierung von $f(bold(x), bold(h))$ für möglichst wenige Werte nahe der $0$
+
+#v(1em)
+
+$ max_(h_1, h_2, h_3) |f(bold(x), bold(h))| $
+
+== Optimierung der 1-Bit Quantisierung (2)
+
+#align(horizon)[
+#figure(
+ include("./graphics/background/z_distribution.typ"),
+ caption: [Optimierte Sign-Based Quantisierung]
+)]
+
+== Verallgemeinerung auf n-Bit
+
+- Definition der Quantisierung höherer Ordnung als mehrstufige Funktion
+
+#v(1em)
+
+#figure(
+ include("./graphics/background/two-bit-enroll.typ")
+)
+
+- Problem: Optimale Position der Quantisierergrenzen $g_1$ und $g_2$
+
+== Verallgemeinerung auf n-Bit --- Optimierungsbedingung
+
+- Zwei Möglichkeiten zur optimalen Positionierung der Linearkombinationen:
+ 1. Beste Approximation zur Mitte einer Quantisiererstufe
+ 2. Maximierung des Abstandes zur nächstgelegenen Grenze der Linearkombination
+
+== Verallgemeinerung auf n-Bit --- Algorithmus
+
+1. Naives Raten der Grenzen $g_(1,2)$ und erste Optimierung
+2. Über eCDF der resultierenden Verteilung neue Grenzen definieren, sodass jedes Symbol gleich wahrscheinlich ist
+
+
+
+== Verallgemeinerung auf n-Bit --- Probleme
+
+- Approximation zur Mitte konvergiert, allerdings keine Verbesserung der Fehlerrate
+- Maximierung des Abstandes konvergiert nicht
+
+#figure(
+grid(
+ columns: (1fr, 1fr),
+ rows: (2),
+ [//#figure(
+ #image("./graphics/background/bach/instability/frame_1.png", width: 80%)
+ #v(-0.5em)
+ //)
+ Iteration 1],
+ [//#figure(
+ #image("./graphics/background/bach/instability/frame_18.png", width: 80%)
+ #v(-0.5em)
+ //)
+ Iteration 18],
+),
+)
+
+= Implementierung
+
+== Mögliche Lösungen
+
+- Rekursiver Ansatz
+
+- Vorgabe des Codeworts
+
+- Brute-Force Ansatz
+
+== Rekursiver Ansatz
+
+1. Teile Verteilung am Erwartungswert auf
+2. Addiere weitere Weights ($0.25, 0.125, ...$)
+3. Maximiere Abstand zur nächstgelegenen Grenze
+4. Teile neue Verteilungen wieder am Erwartungswert auf
+
+== Rekursiver Ansatz
+#align(horizon)[
+#include("./graphics/execution/recursive.typ")
+]
+
+== Rekursiver Ansatz --- Resultierende Verteilung
+
+#align(horizon)[
+#figure(
+ image("./graphics/recursive_distribution.svg", width: 60%)
+)]
+
+== Rekursiver Ansatz --- Helperdaten
+
+#align(horizon)[
+ #figure(
+ image("./graphics/recursive_helperdata_dist.svg", width: 60%)
+ )
+]
+
+== Vorgabe des Codeworts
+
+- Generierung eines zufälligen Schlüssels
+- Nutze Quantile der Normalverteilung als vorgegebene Grenzen
+- Wähle jeweilige Linearkombination, die am besten das Quantil des Vorgegebenen Bits annähert
+
+== Vorgabe des Codeworts --- Verteilung
+
+#figure(
+ image("./graphics/given_codeword_dist.svg")
+)
+
+== Brute-Force Ansatz
+
+- Großer Datensatz mit Grenzen
+- Führe Optimierung und Quantisierung mit jedem Satz Grenzen durch
+- Pearson's Chi-Square Test der Quantisierten Symbole auf Gleichverteilung
+- Maximales Resultat des Chi-Square Tests gibt optimale Grenzen zurück
+
+== Brute-Force Ansatz --- Verteilung
+
+#figure(
+ image("./graphics/bruteforce_dist.svg")
+)
+
+== Brute-Force Ansatz --- Helperdaten
+
+#figure(
+ image("./graphics/bruteforce_helperdata1.svg")
+)
+
+= Fazit
+
+- Rekursiver Ansatz generiert gleichverteilte Ausgansverteilungen
+ - Helperdaten geben Informationen über das Codewort bekannt
+- Vorgabe des Codeworts als Bedingung nicht stark genug
+- Brute-Force Ansatz findet Gleichverteilungen
+ - Helperdaten geben keine Informationen bekannt
+ - Verbesserung der Bitfehlerrate um einen Faktor $10$
+ - Vorberechnete Grenzen können auf beliebige Normalverteilungen angewendet werden