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week1/13-01-2022/13-01-2022.tex

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+%        File: 13-01-2022.tex
+%     Created: 07:32:59 Thu, 13 Jan 2022 EST
+% Last Change: 07:32:59 Thu, 13 Jan 2022 EST
+%
+\documentclass[letterpaper]{article}
+\usepackage{amsmath}
+\usepackage{graphicx}
+\usepackage{cancel}
+\usepackage{amssymb}
+\usepackage{listings}
+\usepackage[shortlabels]{enumitem}
+\usepackage{lipsum}
+\usepackage{soul}
+\usepackage{geometry}
+
+\geometry{portrait, margin=1in}
+
+\date{ 2022-01-13 }
+\title{%
+        Statistics\\
+    \large EEET--425--01: Digital System Processing}
+ \author{Blizzard MacDougall}
+ \begin{document}
+ \maketitle
+ \pagenumbering{arabic}
+
+ SNR (signal to noise ratio) is the mean over the standard deviation.
+ Alternatively, power of signal over power over noise.
+
+ Variance:
+ \begin{equation}
+     \hat{\sigma}^2=\frac{1}{N-1}\sum^{N-1}_{i=0}(x_i-\mu)^2
+ \end{equation}
+
+ Standard deviation is the sqrt of the variance.
+
+ Power is functionally equivalent to variance.
+
+ When sampling a signal, the following is used for calculations:
+
+\begin{equation}
+    \begin{split}
+        \hat{\mu}=\frac1N\sum^{N-1}_{i=0}x_i\\
+        \hat{\sigma}^2=\frac{1}{N-1}\sum^{N-1}_{i=0}(x_i-\hat{\mu})^2
+    \end{split}
+\end{equation}
+
+When you get new samples, you need to remember all the old samples as well.
+
+Variance can also be calculated as:
+\begin{equation}
+    \sigma^2=\frac1{N-1}\left( \sum x^2 - \frac{\left( \sum x\right)^2}{N}\right)
+\end{equation}
+
+ \end{document}