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Diffstat (limited to 'doc/thesis/chapters/theory.tex')
| -rw-r--r-- | doc/thesis/chapters/theory.tex | 18 |
1 files changed, 9 insertions, 9 deletions
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. |