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\begin{document}

\begin{center}
{\Large
Effect of Phase Center Offsets in LCP/RCP Correlation Maps
}
~\\
~\\
\end{center}

\section{Defining H/V/LCP/RCP Waveforms}
Waveform in the vertical polarization.  The $n$ index represents antenna number and the $i$ index is for time bin.
\begin{equation}
\label{eq:vwaveform}
v_n(t_i) = \sum_k e^{-j \omega_k t_i} V(\omega_k) \Delta \omega
\end{equation}
Now horizontal polarization, giving all of the H-pol phase centers a time offset relative to the V-pol phase centers.
\begin{equation}
\label{eq:hwaveform}
h_n(t_i) = \sum_k e^{-j \left( \omega_k t_i + \omega_k t_0 \right) } H(\omega_k) \Delta \omega
\end{equation}

\begin{equation}
\label{eq:rwaveform}
r_n(t_i) = \dfrac{1}{\sqrt{2}} \left[ v(t_i) + j h(t_i) \right]
\end{equation}

\begin{equation}
\label{eq:lwaveform}
\ell_n(t_i) = \dfrac{1}{\sqrt{2}} \left[ v(t_i) - j h(t_i) \right]
\end{equation}

Substituting Eqs.~\ref{eq:vwaveform} and ~\ref{eq:hwaveform} into Eqs. ~\ref{eq:rwaveform} and ~\ref{eq:lwaveform}:
\begin{equation}
r_n(t_i) = \dfrac{1}{\sqrt{2}}  \sum_k  \Delta \omega e^{-j \omega_kt_i}  \left[ V(\omega_k) + j e^{-j\omega_k t_0}  H(\omega_k) \right]
\end{equation}

\section{Cross-Correlations with LCP/RCP Waveforms}
Now consider two antennas, and antenna 1 has a delay $T$ with respect to 2.  Then,
\begin{equation}
r_1(t_i) = \dfrac{1}{\sqrt{2}}  \sum_k  \Delta \omega e^{-j \omega_k t_i}  \left[ V(\omega_k) + j e^{-j\omega_k t_0}  H(\omega_k) \right]
\end{equation}

And since
\begin{equation}
\label{eq:vwaveform}
v_2(t_i) = \sum_k e^{-j \omega_k (t_i+T)} V(\omega_k) \Delta \omega
\end{equation}
\begin{equation}
\label{eq:hwaveform}
h_2(t_i) = \sum_k e^{-j \left[ \omega_k  (t_i+t_0+T) \right] } H(\omega_k) \Delta \omega
\end{equation}
then
\begin{equation}
r_2(t_i) = \dfrac{1}{\sqrt{2}}  \sum_k  \Delta \omega e^{-j \omega_k (t_i+T)}  \left[ V(\omega_k) + j e^{-j\omega_k t_0}  H(\omega_k) \right]
\end{equation}


Cross-correlating the RCP waveforms from antennas 1 and 2 ($r_1$ and $r_2$), and ignoring the normalization factor in the denominator for now, the get the following as a function of delay $\tau$ between the two RCP waveforms:
\begin{equation}
\label{eq:C12rr}
C^{rr}_{12}(\tau) = \sum_{k^{\prime}} \Delta t ~ r_1(t_i) r^*_2 (t_i+\tau)
\end{equation}
where the sum is over the region where the waveforms overlap for a given $\tau$.
Then substituting $r_1(t_i)$ and $r_2(t_i+\tau)$ into Eq.~\ref{eq:C12rr},
\begin{multline}
C^{rr}_{12}(\tau) = \dfrac{1}{2}  \left[  \sum_{k_1}  \Delta \omega e^{-j \omega_{k_1} t_i}  \left[ V(\omega_{k_1}) + j e^{-j\omega_{k_1} t_0}  H(\omega_{k_1}) \right] \right] \times \\
\left[  \sum_{k_2}  \Delta \omega e^{+j \omega_{k_2} (t_i+T+\tau)}  \left[ V(\omega_{k_2}) + j e^{-j\omega_{k_2} t_0}  H(\omega_{k_2}) \right]   \right]
\end{multline}
Collecting terms, we get:
\begin{multline}
C^{rr}_{12}(\tau) = \sum_{k_1}  \sum_{k_2}  (\Delta \omega)^2 e^{-j\left[ \omega_{k_1} t_i-\omega_{k_2} (t_i+T+\tau) \right]} \times \\
\left[ V(\omega_{k_1} )V^* (\omega_{k_2})  -j e^{j \omega_{k_2} t_0 } V(\omega_{k_1} )H^*(\omega_{k_2}) +j e^{-j \omega_{k_1} t_0} H(\omega_{k_1}) V^* (\omega_{k_2}) +H(\omega_{k_1}) H^* (\omega_{k_2}) \right]
\end{multline}

Likewise the LCP waveforms for antennas 1 and 2, where again antenna 1 has a delay $T$ with respect to 2:
\begin{equation}
\ell_1(t_i) = \dfrac{1}{\sqrt{2}}  \sum_k  \Delta \omega e^{-j \omega_k t_i}  \left[ V(\omega_k) - j e^{-j\omega_k t_0}  H(\omega_k) \right]
\end{equation}
\begin{equation}
\ell_2(t_i) = \dfrac{1}{\sqrt{2}}  \sum_k  \Delta \omega e^{-j \omega_k (t_i+T)}  \left[ V(\omega_k) - j e^{-j\omega_k t_0}  H(\omega_k) \right]
\end{equation}
Then, 
\begin{equation}
\label{eq:C12rr}
C^{\ell\ell}_{12}(\tau) = \sum_{k^{\prime}} \Delta t ~ \ell_1(t_i) \ell^*_2 (t_i+\tau)
\end{equation}
\begin{multline}
C^{\ell \ell}_{12}(\tau) = \sum_{k_1}  \sum_{k_2}  (\Delta \omega)^2 e^{-j\left[ \omega_{k_1} t_i-\omega_{k_2} (t_i+T+\tau) \right]} \times \\
\left[ V(\omega_{k_1} )V^* (\omega_{k_2})  +j e^{j \omega_{k_2} t_0 } V(\omega_{k_1} )H^*(\omega_{k_2}) -j e^{-j \omega_{k_1} t_0} H(\omega_{k_1}) V^* (\omega_{k_2}) +H(\omega_{k_1}) H^* (\omega_{k_2}) \right]
\end{multline}



\end{document}