added sections 2.1 and 2.2

applications/apertif/doc/AAF_APERTIF/ASTRON_RP_1549_Apertif_Anti_Aliasing_Filter.tex

@@ -55,6 +55,15 @@ S.J.~Wijnholds
 Creation
 }
 
+% to avoid figures floating too far off
+\renewcommand\floatpagefraction{1.0}
+\renewcommand\topfraction{1.0}
+\renewcommand\bottomfraction{1.0}
+\renewcommand\textfraction{0.0}
+
+% for math
+\newcommand\jcmplx{\mathrm{j}}
+
 \begin{document}
 
 \maketitle
@@ -76,7 +85,7 @@ In the APERTIF signal chains, Nyquist-sampled polyphase filter banks are used to
 In this document, we
 \begin{enumerate}
 \item assess the impact of the aliasing effect at the subband edges on spectral line observations with the goal of determining the need for corrective processing steps to mitigate this effect;
-\item discuss possible solutions and their impact the APERTIF signal chain and observing strategy;
+\item discuss possible solutions and their impact on the APERTIF signal chain and observing strategy;
 \item select the most suitable solution and describe it in sufficient detail to provide a good starting point for implementation in the APERTIF system.
 \end{enumerate}
 
@@ -85,8 +94,39 @@ In this document, we
 In the next section, we present a theoretical framework to describe the FIR filter response and use it to demonstrate the effect it has on the observed spectra. This section is concluded with an assessment on the potential impact on APERTIF observations. In Sec.~\ref{sec:corrective_measures}, we give an overview of possible corrective measures and assess their impact on the APERTIF signal chain and observing strategy. Based on that assessment, a preferred solution is proposed, which is described in more detail in Sec.~\ref{sec:AAF}. This detailed description includes an assessment of the level to which aliasing effects are reduced as well as an updated overview of the APERTIF signal chain that is sufficiently detailed to start work on the implementation of the solution. The code of the key routines used for the simulations presented in this report is provided in the Appendix.
 
 \section{FIR filter response}
+
 \subsection{Theoretical description}
+
+The polyphase filter bank used in APERTIF to form the 781-kHz wide subbands exploits the fast Fourier transform to provide a series of identical finite impulse response (FIR) filters centred at equidistant center frequencies and decimate the time series for each subband to the Nyquist rate. Each subband is formed by applying the same low-pas FIR filter with impulse response $h_0 \left ( n T \right )$ to the baseband signal $x \left ( n T \right )$, where $n$ is the sample index and $T$ is the sampling interval. As the FIR filter is a low-pass filter with a response centred around 0 Hz, the signal at frequency $f_0$ needs to be mixed to 0 Hz before applying the FIR filter. The output signal of the FIR filter can thus be described as
+\begin{equation}
+y \left ( f_0, n T \right ) = \sum_{m=0}^{N_f-1} h_0 \left ( m T \right ) x \left ( (n - m) T \right ) e^{-2\pi \jcmplx f_0 (n - m) T},
+\label{eq:FIR_filter}
+\end{equation}
+where $N_f$ is the length of the FIR filter. Note that this equation describes the output of the FIR filter before decimation.
+
+In Eq.~\eqref{eq:FIR_filter}, we recognise a convolution. As the Fourier transform of a convolution of two function is equal to the product of the Fourier transform of the individual functions, the frequency domain equivalent of Eq.~\eqref{eq:FIR_filter} is
+\begin{eqnarray}
+\mathcal{F} \left \{ y \left ( f_0, n T \right ) \right \} & = &\mathcal{F} \left \{ h_0 \left ( m T \right ) \right \} \mathcal{F} \left \{ x \left (n T \right ) e^{-2\pi \jcmplx f_0 n T} \right \}\\
+Y \left ( f_0, f_s \right ) & = & H \left ( f \right ) X \left ( f_s - f_0 \right ), \label{eq:FIR_freq_response}
+\end{eqnarray}
+where $\mathcal{F} \left \{ . \right \}$ denotes the Fourier transform and $f_s$ denotes the signal frequency. If no further spectral analysis is done on the output of the FIR filter, all power is usually assigned to the center frequency $f_0$. In the case of APERTIF, the subband filter is followed by another polyphase filter bank to form the 12-kHz wide channels. As a result, the dependence on $f = f_s - f_0$ becomes apparent as will be demonstrated in the next section.
+
 \subsection{Impact on spectra}
+
+Eq.~\eqref{eq:FIR_freq_response} describes the voltage output of the FIR filter as function of center frequency $f_0$ and signal frequency $f$. As we usually do not have a reference signal to provide a phase reference, we normally measure the output response in the power domain, i.e., we measure
+\begin{equation}
+\left | Y \left ( f_0, f_s \right ) \right |^2 = \left | H \left ( f \right ) \right |^2 \left | X \left ( f_s - f_0 \right ) \right |^2.
+\end{equation}
+
+\begin{figure}
+\centering
+\includegraphics[width=0.48\textwidth]{sample_spectrum.eps}
+\includegraphics[width=0.48\textwidth]{sample_spectrum_filtered.eps}
+\caption{Sample spectrum at 12-kHz spectral resolution without distorting effects (left) and the same spectrum as would be obtained using the output of the subband filter (right). \label{fig:impact_on_spectrum}}
+\end{figure}
+
+To illustrate the impact of the imperfect FIR filter response $\left | H \left ( f \right ) \right |^2$ on power spectra, we generated the artificial spectrum shown in the left panel of Fig.~\ref{fig:impact_on_spectrum}. The right panel shows the spectrum that would be obtained from the output of the subband filter, clearly showing the subband edges. If the measurement noise would be even lower, also the passband ripple of the FIR filter would be clearly visible.
+
 \subsection{Impact on observations}
 % source smearing? missed lindes? velocity field reconstruction issues?