Work

content/BACH.typ

@@ -16,4 +16,21 @@ Before we take a look at the higher order quantization cases, we will start with
 )<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 $X$ 
+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 $X$ that reside near $0$ are to be considered more unreliable than values that are further away from the x-value 0.
+This means that the quantizer used here is very unreliable without generated helper-data.
+
+Now, to increase the reliability of this quantizer, we can try to move our input values further away from the value $x = 0$. 
+To do so, we can define a new input value $x^"lin"$ as a linear combination of two realizations of $X$, $x_1$ and $x_2$ with a set of weights $h_1$ and $h_2$: 
+$
+x^"lin" = h_1 dot x_1 + h_2 dot x_2 .
+$<eq:lin_combs>
+We can define the vector of all possible linear combinations $bold(x^"lin")$ as the vector-matrix multiplication of the two input values $x_i$ and the matrix of all weight combinations: 
+$
+bold(x^"lin") &= vec(x_1, x_2) dot mat(delim: "[", h_1, -h_1, h_1, -h_1; h_2, h_2, -h_2, -h_2)\
+&= vec(x_1, x_2) dot mat(delim: "[", +1, -1, +1, -1; +1, +1, -1, -1)
+$
+We will choose the optimal weights based on the highest absolute value of $bold(x^"lin")$, as that value will be the furthest away from $0$. 
+We may encounter two entries in $bold(x^"lin")$ that both have the same highest absolute value.
+
+Lets take a look at the resulting random distribution of this process: 
+

glossary.typ

@@ -7,5 +7,6 @@
   (key: "hda", short: "HDA", plural: "HDAs", long: "helper data algorithm", longplural: "helper data algorithms"),
   (key: "ecdf", short: "eCDF", plural: "eCDFs", long: "empirical Cumulative Distribution Function", longplural: "empirical Cumulative Distribution Functions"),
   (key: "ber", short: "BER", plural: "BERs", long: "bit error rate", longplural: "bit error rates"),
-  (key: "smhdt", short: "SMHD", plural: "SMHDs", long: "S-Metric Helper Data Method")
+  (key: "smhdt", short: "SMHD", plural: "SMHDs", long: "S-Metric Helper Data Method"),
+  (key: "puf", short: "PUF", plural: "PUFs", long: "physical unclonale function", longplural: "physical unclonale functions")
 ))

pseudocode/find_quantizers.typ

@@ -1,10 +0,0 @@
-#import "@preview/lovelace:0.3.0": *
-
-
-#pseudocode-list(booktabs: true, numbered-title: [Find reconstruction quantizers])[
-  + *start* at the center of the first metric 
-    +  
-  + *while* $(i = 1) <= a$:
-    + *return* $i dot p$ 
-  + *end*
-  ]