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  \begin{document}
  \title{Filter optimization for real time digital processing of radiofrequency signals: application
  to oscillator metrology}
  
  \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
  G. Goavec-M\'erou\IEEEauthorrefmark{1},
  P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}
  \IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }
  \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
  Email: \{pyb2,jmfriedt\}@femto-st.fr}
  }
  \maketitle
  \thispagestyle{plain}
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  ewtheorem{definition}{Definition}
  
  \begin{abstract}
  Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
  radiofrequency signal processing. Applied to oscillator characterization in the context
  of ultrastable clocks, stringent filtering requirements are defined by spurious signal or
  noise rejection needs. Since real time radiofrequency processing must be performed in a
  Field Programmable Array to meet timing constraints, we investigate optimization strategies
  to design filters meeting rejection characteristics while limiting the hardware resources
  required and keeping timing constraints within the targeted measurement bandwidths.
  \end{abstract}
  
  \begin{IEEEkeywords}
  Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
  \end{IEEEkeywords}
  
  \section{Digital signal processing of ultrastable clock signals}
  
  Analog oscillator phase noise characteristics are classically performed by downconverting
  the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband,
  followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In
  a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by
  multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
  
  \begin{figure}[h!tb]
  \begin{center}
  \includegraphics[width=.8\linewidth]{images/schema}
  \end{center}
  \caption{Fully digital oscillator phase noise characterization: the Device Under Test
  (DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and
  downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals
  and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite
  Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
  the spectral characteristics of the phase fluctuations.}
  \label{schema}
  \end{figure}
  
  As with the analog mixer,
  the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as
  well as the generation of the frequency sum signal in addition to the frequency difference.
  These unwanted spectral characteristics must be rejected before decimating the data stream
  for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the
  downconverter
  and the decimation processing blocks are core characteristics of an oscillator characterization
  system, and must reject out-of-band signals below the targeted phase noise -- typically in the
  sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will
  use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency
  datastream: optimizing the performance of the filter while reducing the needed resources is
  hence tackled in a systematic approach using optimization techniques. Most significantly, we
  tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with
  tunable number of coefficients and tunable number of bits representing the coefficients and the
  data being processed.
  
  \section{Finite impulse response filter}
  
  We select FIR filter for their unconditional stability and ease of design. A FIR filter is defined
  by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
  outputs $y_k$

  \begin{align}
      y_n=\sum_{k=0}^N b_k x_{n-k}
      \label{eq:fir_equation}
  \end{align}

  
  As opposed to an implementation on a general purpose processor in which word size is defined by the
  processor architecture, implementing such a filter on an FPGA offer more degrees of freedom since
  not only the coefficient values and number of taps must be defined, but also the number of bits
  defining the coefficients and the sample size. For this reason, and because we consider pipeline
  processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency
  signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but
  the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language (VHDL) level.
  Since latency is not an issue in a openloop phase noise characterization instrument, the large
  numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
  is not considered as an issue as would be in a closed loop system.
  
  The coefficients are classically expressed as floating point values. However, this binary
  number representation is not efficient for fast arithmetic computation by an FPGA. Instead,
  we select to quantify these floating point values into integer values. This quantization
  will result in some precision loss.

  \begin{figure}[h!tb]
  \includegraphics[width=\linewidth]{images/demo_filtre}
  \caption{Impact of the quantization resolution of the coefficients: the quantization is
  set to 6~bits -- with the horizontal black lines indicating $\pm$1 least significant bit -- setting
  the 30~first and 30~last coefficients out of the initial 128~band-pass
  filter coefficients to 0 (red dots).}
  \label{float_vs_int}
  \end{figure}
  
  The tradeoff between quantization resolution and number of coefficients when considering
  integer operations is not trivial. As an illustration of the issue related to the
  relation between number of fiter taps and quantization, Fig. \ref{float_vs_int} exhibits
  a 128-coefficient FIR bandpass filter designed using floating point numbers (blue). Upon
  quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the
  taps become null, making the large number of coefficients irrelevant and allowing to save
  processing resource by shrinking the filter length. This tradeoff aimed at minimizing resources
  to reach a given rejection level, or maximizing out of band rejection for a given computational
  resource, will drive the investigation on cascading filters designed with varying tap resolution
  and tap length, as will be shown in the next section. Indeed, our development strategy closely
  follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
  in which basic blocks are defined and characterized before being assembled \cite{hide}
  in a complete processing chain. In our case, assembling the filter blocks is a simpler block
  combination process since we assume a single value to be processed and a single value to be
  generated at each clock cycle. The FIR filters will not be considered to decimate in the
  current implementation: the decimation is assumed to be located after the FIR cascade at the
  moment.

  \section{Methodology description}
  We want create a new methodology to develop any Digital Signal Processing (DSP) chain
  and for any hardware platform (Altera, Xilinx...). To do this we have defined an
  abstract model to represent some basic operations of DSP.
  
  For the moment, we are focused on only two operations: the filtering and the shift of data.
  We have chosen this basic operation because the shifting and the filtering have already be studied in
  lot of works {\color{red} mettre les nouvelles référence ici} hence it will be easier
  to check and validate our results.
  
  However having only two operations is insufficient to work with complex DSP but
  in this paper we only want demonstrate the relevance and the efficiency of our approach.
  In future work it will be possible to add more operations and we are able to
  model any DSP chain.
  
  We will apply our methodology on very simple DSP chain. We generate a digital signal
  thanks at generator of Pseudo-Random Number (PRN) or thanks at an Analog to Digital
  Converter (ADC). Once we have a digital signal, we filter it to decrease the noise level.
  Finally we stored some burst of filtered samples before post-processing it.
  % TODO: faire un schéma
  In this particular case, we want optimize the filtering step to have the best noise
  rejection for constrain number of resource or to have the minimal resources
  consumption for a given rejection objective.
  
  The first step of our approach is to model the DSP chain and since we just optimize
  the filtering, we have not modeling the PRN generator or the ADC. The filtering can be
  done by two ways. The first one we use only one FIR filter with lot of coefficients
  to rejection the noise, we called this approach a monolithic approach. And the second one
  we select different FIR filters with less coefficients the monolithic filter and we cascaded
  it to filtering the signal.
  
  After each filter we leave the possibility of shifting the filtered data to consume
  less resources. Hence in the case of cascaded filter, we define a stage as a filter
  and a shifter (the shift could be omitted if we do not need to divide the filtered data).
  
  \subsection{Model of a FIR filter}
  A cascade of filter are composed of $n$ stage. In stage $i$ ($1 \leq i \leq n$)
  the FIR has $C_i$ coefficients and each coefficients are integer values with $\pi^C_i$
  bits and the filtered data are shifted of $\pi^S_i$ bits. We define also $\pi^-_i$ as
  the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage}
  shows a filtering stage.
  
  \begin{figure}
    \centering
    \begin{tikzpicture}[node distance=2cm]
      
  ode[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ;
      
  ode[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ;
      
  ode (Start) [left of=FIR] { } ;
      
  ode (End) [right of=Shift] { } ;
  
      
  ode[draw,fit=(FIR) (Shift)] (Filter) { } ;
  
      \draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ;
      \draw[->] (FIR) -- (Shift) ;
      \draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ;
    \end{tikzpicture}
    \caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)}
    \label{fig:fir_stage}
  \end{figure}

  FIR $i$ can reject $F(C_i, \pi_i^C)$ dB. $F$ is determined numerically.
  To measure this rejection, we use GNU Octave software to design FIR filter coefficients thanks to two
  algorithms (\texttt{firls} and \texttt{fir1}).
  For each configuration $(C_i, \pi_i^C)$, we first create a FIR with floating point coefficients and a given $C_i$ number of coefficients.
  Then, the floating point coefficients are discretized into integers. In order to ensure that the coefficients are coded on $\pi_i^C$~bits effectively,
  the coefficients are normalized by their absolute maximum before being scaled to integer coefficients.
  At least one coefficient is coded on $\pi_i^C$~bits, and in practice only $b_{C_i/2}$ is coded on $\pi_i^C$~bits while the other are coded on very fewer bits.
  
  With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter.
  Comparing the performance between FIRs requires however a unique criterion. As shown in figure~\ref{fig:fir_mag},
  the FIR magnitude exhibits two parts.
  
  \begin{figure}
    \centering
    \begin{tikzpicture}[scale=0.3]
      \draw[<->] (0,15) -- (0,0) -- (21,0) ;
      \draw[thick] (0,12) -- (8,12) -- (20,0) ;
  
      \draw (0,14) node [left] { $P$ } ;
      \draw (20,0) node [below] { $f$ } ;
  
      \draw[>=latex,<->] (0,14) -- (8,14) ;
      \draw (4,14) node [above] { passband } node [below] { $40\%$ } ;
  
      \draw[>=latex,<->] (8,14) -- (12,14) ;
      \draw (10,14) node [above] { transition } node [below] { $20\%$ } ;
  
      \draw[>=latex,<->] (12,14) -- (20,14) ;
      \draw (16,14) node [above] { stopband } node [below] { $40\%$ } ;
  
      \draw[>=latex,<->] (16,12) -- (16,8) ;
      \draw (16,10) node [right] { rejection } ;
  
      \draw[dashed] (8,-1) -- (8,14) ;
      \draw[dashed] (12,-1) -- (12,14) ;
  
      \draw[dashed] (8,12) -- (16,12) ;
      \draw[dashed] (12,8) -- (16,8) ;
  
    \end{tikzpicture}
  
  %  \includegraphics[width=.5\linewidth]{images/fir_magnitude}
  \caption{Shape of the filter transmitted power $P$ as a function of frequency $f$:
  the passband is considered to occupy the initial 40\% of the Nyquist frequency range,
  the stopband the last 40\%, allowing 20\% transition width.}
  \label{fig:fir_mag}

  \end{figure}

  In the transition band, the behavior of the filter is left free, we only care about the passband and the stopband.
  Our first criterion considers the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion does not work because we do not consider the shape of the passband.
  A second criterion considers the maximum rejection within the stopband minus the mean of the absolute value of passband rejection. With this criterion, the results are significantly improved as shown in figure~\ref{fig:custom_criterion}.

  \begin{figure}
  \centering
  \includegraphics[width=\linewidth]{images/mean_criterion}
  \caption{Mean criterion comparison between monolithic filter and cascade filters}
  \label{fig:mean_criterion}

  \end{figure}

  \begin{figure}
  \centering
  \includegraphics[width=\linewidth]{images/custom_criterion}
  \caption{Custom criterion comparison between monolithic filter and cascade filters}
  \label{fig:custom_criterion}
  \end{figure}

  Although we have a efficient criterion to estimate the rejection of one set of coefficient
  we have a problem when we sum two or more criterion. If the FIR filter coefficients are the same
  between the stage, we have:
  $$F_{total} = F_1 + F_2$$
  But when we choose two different set of coefficient, the previous equality are not
  true. The figure~\ref{fig:sum_rejection} illustrates the problem. The red and blue curves
  are two different filter coefficient and we can see that their maximum on the stopband
  are not at the same frequency. So when we sum the rejection criteria (the dotted yellow line)
  we do not meet the dashed yellow line. Define the rejection of cascaded filters
  is more difficult than just take the summation between all the rejection criteria of each filter.
  However this summation gives us an upper bound for rejection although in fact we obtain
  better rejection than expected.
  
  \begin{figure}
  \centering
  \includegraphics[width=\linewidth]{images/sum_rejection}
  \caption{Rejection of two cascaded filters}
  \label{fig:sum_rejection}
  \end{figure}

  \section{Experiments with fixed area space}

  \begin{figure}
  \centering
  \includegraphics[width=\linewidth]{images/max_rejection/prn_500}
  \caption{Experimental results for design with PRN as data input and 500 a.u. as max arbitrary space}
  \label{fig:prn_500}
  \end{figure}

  \begin{figure}
  \centering
  \includegraphics[width=\linewidth]{images/max_rejection/prn_1000}
  \caption{Experimental results for design with PRN as data input and 1000 a.u. as max arbitrary space}
  \label{fig:prn_1000}

  \end{figure}

  \begin{figure}
  \centering
  \includegraphics[width=\linewidth]{images/max_rejection/prn_2000}
  \caption{Experimental results for design with PRN as data input and 2000 a.u. as max arbitrary space}
  \label{fig:prn_2000}
  \end{figure}
  
  \begin{table}
  \centering
  \begin{tabular}{|c|c|ccc|c|c|}
  \hline
  \multicolumn{2}{|c|}{\multirow{2}{*}{Stage}}  & \multicolumn{3}{c|}{Stage}  & \multirow{2}{*}{Rejection} & \multirow{2}{*}{Area} \\ \cline{3-5}
  \multicolumn{2}{|c|}{}                        & i = 1 & i = 2 & i = 3       &                            &                       \\ \hline
        & C                                     & 19    & -     & -           &                            &                       \\
  n = 1 & $pi^C$                                & 7     & -     & -           & 33 dB                      & 437 a.u.              \\
        & $pi^S$                                & 0     & -     & -           &                            &                       \\ \hline
        & C                                     & 11    & 19    & -           &                            &                       \\
  n = 2 & $pi^C$                                & 5     & 7     & -           & 53 dB                      & 478 a.u.              \\
        & $pi^S$                                & 16    & 0     & -           &                            &                       \\ \hline
        & C                                     & 9     & 15    & 11          &                            &                       \\
  n = 3 & $pi^C$                                & 4     & 6     & 5           & 57 dB                      & 499 a.u.              \\
        & $pi^S$                                & 16    & 3     & 0           &                            &                       \\ \hline

  \end{tabular}

  \caption{Solver results for design with PRN as data input and 500 a.u. as max arbitrary space}
  \label{tbl:prn_500}

  \end{table}

  \begin{table}
  \centering
  {\scalefont{0.85}
  \begin{tabular}{|c|c|ccccc|c|c|}
  \hline
  \multicolumn{2}{|c|}{\multirow{2}{*}{Stage}}  & \multicolumn{5}{c|}{Stage}            & \multirow{2}{*}{Rejection} & \multirow{2}{*}{Area} \\ \cline{3-7}
  \multicolumn{2}{|c|}{}                        & i = 1 & i = 2 & i = 3 & i = 4 & i = 5 &                            &                       \\ \hline
        & C                                     & 37    & -     & -     & -     & -     &                            &                       \\
  n = 1 & $pi^C$                                & 11    & -     & -     & -     & -     & 56 dB                      & 999 a.u.              \\
        & $pi^S$                                & 0     & -     & -     & -     & -     &                            &                       \\ \hline
        & C                                     & 11    & 39    & -     & -     & -     &                            &                       \\
  n = 2 & $pi^C$                                & 5     & 13    & -     & -     & -     & 82 dB                      & 972 a.u.              \\
        & $pi^S$                                & 16    & 0     & -     & -     & -     &                            &                       \\ \hline
        & C                                     & 9     & 31    & 19    & -     & -     &                            &                       \\
  n = 3 & $pi^C$                                & 7     & 8     & 7     & -     & -     & 93 dB                      & 990 a.u.              \\
        & $pi^S$                                & 19    & 2     & 0     & -     & -     &                            &                       \\ \hline
        & C                                     & 9     & 19    & 17    & 11    & -     &                            &                       \\
  n = 4 & $pi^C$                                & 4     & 7     & 7     & 5     & -     & 99 dB                      & 992 a.u.              \\
        & $pi^S$                                & 16    & 3     & 3     & 0     & -     &                            &                       \\ \hline
        & C                                     & 9     & 15    & 11    & 11    & 11    &                            &                       \\
  n = 5 & $pi^C$                                & 4     & 7     & 5     & 5     & 5     & 99 dB                      & 998 a.u.              \\
        & $pi^S$                                & 16    & 3     & 2     & 1     & 1     &                            &                       \\ \hline
  \end{tabular}
  }
  \caption{Solver results for design with PRN as data input and 1000 a.u. as max arbitrary space}
  \label{tbl:prn_1000}
  \end{table}

  \begin{table}
  \centering
  {\scalefont{0.85}
  \begin{tabular}{|c|c|ccccc|c|c|}
  \hline
  \multicolumn{2}{|c|}{\multirow{2}{*}{Stage}}  & \multicolumn{5}{c|}{Stage}            & \multirow{2}{*}{Rejection} & \multirow{2}{*}{Area} \\ \cline{3-7}
  \multicolumn{2}{|c|}{}                        & i = 1 & i = 2 & i = 3 & i = 4 & i = 5 &                            &                       \\ \hline
        & C                                     & 39    & -     & -     & -     & -     &                            &                       \\
  n = 1 & $pi^C$                                & 13    & -     & -     & -     & -     & 61 dB                      & 1131 a.u.             \\
        & $pi^S$                                & 0     & -     & -     & -     & -     &                            &                       \\ \hline
        & C                                     & 37    & 39    & -     & -     & -     &                            &                       \\
  n = 2 & $pi^C$                                & 11    & 13    & -     & -     & -     & 117 dB                     & 1974 a.u.             \\
        & $pi^S$                                & 17    & 0     & -     & -     & -     &                            &                       \\ \hline
        & C                                     & 15    & 35    & 35    & -     & -     &                            &                       \\
  n = 3 & $pi^C$                                & 9     & 11    & 11    & -     & -     & 138 dB                     & 1985 a.u.             \\
        & $pi^S$                                & 19    & 3     & 0     & -     & -     &                            &                       \\ \hline
        & C                                     & 11    & 27    & 27    & 23    & -     &                            &                       \\
  n = 4 & $pi^C$                                & 5     & 9     & 9     & 9     & -     & 148 dB                     & 1993 a.u.             \\
        & $pi^S$                                & 16    & 3     & 2     & 0     & -     &                            &                       \\ \hline
        & C                                     & 11    & 27    & 31    & 11    & 11    &                            &                       \\
  n = 5 & $pi^C$                                & 5     & 9     & 8     & 5     & 5     & 153 dB                     & 2000 a.u.             \\
        & $pi^S$                                & 16    & 3     & 1     & 0     & 1     &                            &                       \\ \hline
  \end{tabular}
  }
  \caption{Solver results for design with PRN as data input and 2000 a.u. as max arbitrary space}
  \label{tbl:prn_2000}
  \end{table}

  \begin{table}
  \centering
  \begin{tabular}{|c|c|c|c|c|}\hline
  Input  & Stages & Computation time        & Vivado time      &  Redpitaya time  \\\hline\hline
         & 1      & 0.02~s                  & $\approx$ 20 min & $\approx$ 1 min  \\
  PRN    & 2      & 1.70~s                  & $\approx$ 20 min & $\approx$ 1 min  \\
         & 3      & 19~s                    & $\approx$ 20 min & $\approx$ 1 min  \\\hline
  \end{tabular}
  \caption{Time to compute and deploy the designs for PRN 500}
  \label{tbl:time_prn_500}
  \end{table}

  \begin{table}
  \centering
  \begin{tabular}{|c|c|c|c|c|}\hline
  Input  & Stages & Computation time        & Vivado time      &  Redpitaya time  \\\hline\hline
         & 1      & 0.07~s                  & $\approx$ 20 min & $\approx$ 1 min  \\
         & 2      & 1.31~s                  & $\approx$ 20 min & $\approx$ 1 min  \\
  PRN    & 3      & 119~s ($\approx$ 2~min) & $\approx$ 20 min & $\approx$ 1 min  \\
         & 4      & 270~s ($\approx$ 5~min) & $\approx$ 20 min & $\approx$ 1 min  \\
         & 5      & 5998~s ($\approx$ 2~h)  & $\approx$ 20 min & $\approx$ 1 min  \\\hline
  \end{tabular}
  \caption{Time to compute and deploy the designs for PRN 1000}
  \label{tbl:time_prn_1000}
  \end{table}

  \begin{table}
  \centering
  \begin{tabular}{|c|c|c|c|c|}\hline
  Input  & Stages & Computation time          & Vivado time      &  Redpitaya time  \\\hline\hline
         & 1      & 0.07~s                    & $\approx$ 20 min & $\approx$ 1 min  \\
         & 2      & 0.75~s                    & $\approx$ 20 min & $\approx$ 1 min  \\
  PRN    & 3      & 36~s                      & -                & -                \\
         & 4      & 14500~s ($\approx$ 4~h)   & $\approx$ 20 min & $\approx$ 1 min  \\
         & 5      & 74237~s ($\approx$ 20~h)  & $\approx$ 20 min & $\approx$ 1 min  \\\hline
  \end{tabular}
  \caption{Time to compute and deploy the designs for PRN 2000}
  \label{tbl:time_prn_2000}
  \end{table}

  \section{Experiments with fixed rejection target}

  \begin{figure}
  \centering
  \includegraphics[width=\linewidth]{images/min_area/prn_50}
  \caption{Results for design with PRN as data input and 50 dB as aimed rejection level}
  \label{fig:prn_500}
  \end{figure}

  \begin{figure}
  \centering
  \includegraphics[width=\linewidth]{images/min_area/prn_100}
  \caption{Results for design with PRN as data input and 50 dB as aimed rejection level}
  \label{fig:prn_100}

  \end{figure}

  \begin{figure}
  \centering
  \includegraphics[width=\linewidth]{images/min_area/prn_150}
  \caption{Results for design with PRN as data input and 2000 a.u. as max arbitrary space}
  \label{fig:prn_150}
  \end{figure}

  
  \section{Conclusion}

  \section*{Acknowledgement}
  
  This work is supported by the ANR Programme d'Investissement d'Avenir in
  progress at the Time and Frequency Departments of the FEMTO-ST Institute
  (Oscillator IMP, First-TF and Refimeve+), and by R\'egion de Franche-Comt\'e.

  The authors would like to thank E. Rubiola, F. Vernotte, and G. Cabodevila

  for support and fruitful discussions.
  
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