From adc00a39baff866a3b68b52ffc55a4aae71e61c4 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Sun, 31 Oct 2021 00:20:51 +0200
Subject: Fix math error in QAM, improve QAM modulator diagram, rename (wrong)
 QPSK to PSK

---
 doc/thesis/chapters/theory.tex | 18 +++++++++---------
 1 file changed, 9 insertions(+), 9 deletions(-)

(limited to 'doc/thesis/chapters/theory.tex')

diff --git a/doc/thesis/chapters/theory.tex b/doc/thesis/chapters/theory.tex
index e3809fd..9477340 100644
--- a/doc/thesis/chapters/theory.tex
+++ b/doc/thesis/chapters/theory.tex
@@ -52,10 +52,10 @@ The two component \(m_i(t)\) and \(m_q(t)\) are mixed with two different periodi
 \begin{subequations} \label{eqn:orthonormal-conditions}
 	\begin{align}
 		\langle \phi_i | \phi_q \rangle
-			&= \int_T \phi_i^* \phi_q \, dt = \int_T \phi_i \phi_q^* \, dt
+			&= \int_T \phi_i \phi_q^* \, dt
 			= 0, \text{ and } \\
 		\langle \phi_k | \phi_k \rangle
-			&= \int_T \phi_k^* \phi_k \,dt = 1,
+			&= \int_T \phi_k \phi_k^*  \,dt = 1,
 			\text{ where } k \text{ is either } i \text{ or } q.
 	\end{align}
 \end{subequations}
@@ -66,10 +66,10 @@ In practice typically \(\phi_i(t) = \cos(\omega_c t)\) and \(\phi_q(t) = j\sin(\
 %% TODO: is this assumption correct?
 Notice that assuming \(m_i\) and \(m_q\) are constant\footnote{This is an approximation assuming that the signal changes much slower relative to the carrier.} over the carrier's period \(T\),
 \begin{align*}
-	\langle s | \phi_i \rangle = \int_T s^* \phi_i \,dt
-		&= \int m_i \phi_i^* \phi_i + m_q \phi_q^* \phi_i \,dt \\
-		&= m_i \underbrace{\int_T \phi_i^* \phi_i \,dt}_{1}
-			+ m_q \underbrace{\int_T \phi_q^* \phi_i \,dt}_{0} = m_i,
+	\langle s | \phi_i \rangle = \int_T s \phi_i^* \,dt
+		&= \int m_i \phi_i \phi_i^* + m_q \phi_q \phi_i^* \,dt \\
+		&= m_i \underbrace{\int_T \phi_i \phi_i^* \,dt}_{1}
+			+ m_q \underbrace{\int_T \phi_q \phi_i^* \,dt}_{0} = m_i,
 \end{align*}
 which effectively means that it is possible to isolate a single component \(m_i(t)\) out of \(s(t)\). The same of course works with \(\phi_q\) as well resulting in \(\langle s | \phi_q \rangle = m_q\). Thus (remarkably) it is possible to send two signals on the same frequency, without them interfering with each other. Since each signal can represent one of \(\sqrt{M}\) values, by having two we obtain \(\sqrt{M} \cdot \sqrt{M} = M\) possible combinations.
 
@@ -83,8 +83,8 @@ A better way to see what QAM does, is to observe a so called \emph{constellation
 	\end{subfigure}
 	\hfill
 	\begin{subfigure}{.4\linewidth}
-		\input{figures/tikz/qpsk-constellation}
-		\caption{8-PSK\label{fig:qpsk-constellation}}
+		\input{figures/tikz/psk-constellation}
+		\caption{8-PSK\label{fig:psk-constellation}}
 	\end{subfigure}
 	\hfill
 	\caption{
@@ -92,7 +92,7 @@ A better way to see what QAM does, is to observe a so called \emph{constellation
 	}
 \end{figure}
 
-\section{Quadrature phase shift keying (\(M\)-QPSK)}
+\section{Quadrature phase shift keying (\(M\)-PSK)}
 
 PSK is a popular modulation type for data transmission\cite{Meyer2011}. With a bipolar binary signal, the amplitude remains constant and only the phase will be changed with phase jumps of 180 degrees, which can be seen as a multiplication of the carrier signal with $\pm$ 1. That is alow known as binary phase shift keying.
 
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