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\begin{document} 12 12 \begin{document}
\title{Filter optimization for real time digital processing of radiofrequency signals: application 13 13 \title{Filter optimization for real time digital processing of radiofrequency signals: application
to oscillator metrology} 14 14 to oscillator metrology}
15 15
\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2}, 16 16 \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
G. Goavec-M\'erou\IEEEauthorrefmark{1}, 17 17 G. Goavec-M\'erou\IEEEauthorrefmark{1},
P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M Friedt\IEEEauthorrefmark{1}} 18 18 P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M Friedt\IEEEauthorrefmark{1}}
\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France } 19 19 \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 \\ 20 20 \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
Email: \{pyb2,jmfriedt\}@femto-st.fr} 21 21 Email: \{pyb2,jmfriedt\}@femto-st.fr}
} 22 22 }
\maketitle 23 23 \maketitle
\thispagestyle{plain} 24 24 \thispagestyle{plain}
\pagestyle{plain} 25 25 \pagestyle{plain}
26 26
\begin{abstract} 27 27 \begin{abstract}
Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to 28 28 Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
radiofrequency signal processing. Applied to oscillator characterization in the context 29 29 radiofrequency signal processing. Applied to oscillator characterization in the context
of ultrastable clocks, stringent filtering requirements are defined by spurious signal or 30 30 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 31 31 noise rejection needs. Since real time radiofrequency processing must be performed in a
Field Programmable Array to meet timing constraints, we investigate optimization strategies 32 32 Field Programmable Array to meet timing constraints, we investigate optimization strategies
to design filters meeting rejection characteristics while limiting the hardware resources 33 33 to design filters meeting rejection characteristics while limiting the hardware resources
required and keeping timing constraints within the targeted measurement bandwidths. 34 34 required and keeping timing constraints within the targeted measurement bandwidths.
\end{abstract} 35 35 \end{abstract}
36 36
\begin{IEEEkeywords} 37 37 \begin{IEEEkeywords}
Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter 38 38 Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
\end{IEEEkeywords} 39 39 \end{IEEEkeywords}
40 40
\section{Digital signal processing of ultrastable clock signals} 41 41 \section{Digital signal processing of ultrastable clock signals}
42 42
Analog oscillator phase noise characteristics are classically performed by downconverting 43 43 Analog oscillator phase noise characteristics are classically performed by downconverting
the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband, 44 44 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 45 45 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 46 46 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}. 47 47 multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
48 48
\begin{figure}[h!tb] 49 49 \begin{figure}[h!tb]
\begin{center} 50 50 \begin{center}
\includegraphics[width=.8\linewidth]{images/schema} 51 51 \includegraphics[width=.8\linewidth]{images/schema}
\end{center} 52 52 \end{center}
\caption{Fully digital oscillator phase noise characterization: the Device Under Test 53 53 \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 54 54 (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 55 55 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 56 56 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 57 57 Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
the spectral characteristics of the phase fluctuations.} 58 58 the spectral characteristics of the phase fluctuations.}
\label{schema} 59 59 \label{schema}
\end{figure} 60 60 \end{figure}
61 61
As with the analog mixer, 62 62 As with the analog mixer,
the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as 63 63 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. 64 64 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 65 65 These unwanted spectral characteristics must be rejected before decimating the data stream
for the phase noise spectral characterization. The characteristics introduced between the downconverter 66 66 for the phase noise spectral characterization. The characteristics introduced between the downconverter
and the decimation processing blocks are core characteristics of an oscillator characterization 67 67 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 68 68 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 69 69 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 70 70 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 71 71 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 72 72 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 73 73 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 74 74 tunable number of coefficients and tunable number of bits representing the coefficients and the
data being processed. 75 75 data being processed.
76 76
\section{Filter optimization} 77 77 \section{Filter optimization}
78 78
A basic approach for implementing the FIR filter is to compute the transfer function of 79 79 A basic approach for implementing the FIR filter is to compute the transfer function of
a monolithic filter: this single filter defines all coefficients with the same resolution 80 80 a monolithic filter: this single filter defines all coefficients with the same resolution
(number of bits) and processes data represented with their own resolution. Meeting the 81 81 (number of bits) and processes data represented with their own resolution. Meeting the
filter shape requires a large number of coefficients, limited by resources of the FPGA since 82 82 filter shape requires a large number of coefficients, limited by resources of the FPGA since
this filter must process data stream at the radiofrequency sampling rate after the mixer. 83 83 this filter must process data stream at the radiofrequency sampling rate after the mixer.
84 84
An optimization problem \cite{leung2004handbook} aims at improving one or many 85 85 An optimization problem \cite{leung2004handbook} aims at improving one or many
performance criteria within a constrained resource environment. Amongst the tools 86 86 performance criteria within a constrained resource environment. Amongst the tools
developed to meet this aim, Mixed-Integer Linear Programming (MILP) provides the framework to 87 87 developed to meet this aim, Mixed-Integer Linear Programming (MILP) provides the framework to
provide a formal definition of the stated problem and search for an optimal use of available 88 88 provide a formal definition of the stated problem and search for an optimal use of available
resources \cite{yu2007design, kodek1980design}. 89 89 resources \cite{yu2007design, kodek1980design}.
90 90
The degrees of freedom when addressing the problem of replacing the single monolithic 91 91 The degrees of freedom when addressing the problem of replacing the single monolithic
FIR with a cascade of optimized filters are the number of coefficients $N_i$ of each filter $i$, 92 92 FIR with a cascade of optimized filters are the number of coefficients $N_i$ of each filter $i$,
the number of bits $c_i$ representing the coefficients and the number of bits $d_i$ representing 93 93 the number of bits $c_i$ representing the coefficients and the number of bits $d_i$ representing
the data fed to the filter. Because each FIR in the chain is fed the output of the previous stage, 94 94 the data fed to the filter. Because each FIR in the chain is fed the output of the previous stage,
the optimization of the complete processing chain within a constrained resource environment is not 95 95 the optimization of the complete processing chain within a constrained resource environment is not
trivial. The resource occupation of a FIR filter is considered as $c_i+d_i+\log_2(N_i)$ which is 96 96 trivial. The resource occupation of a FIR filter is considered as $c_i+d_i+\log_2(N_i)$ which is
the number of bits needed in a worst case condition to represent the output of the FIR. 97 97 the number of bits needed in a worst case condition to represent the output of the FIR.
98 98
99 99
\begin{figure}[h!tb] 100 100 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/noise-rejection.pdf} 101 101 \includegraphics[width=\linewidth]{images/noise-rejection.pdf}
\caption{Rejection as a function of number of coefficients and number of bits} 102 102 \caption{Rejection as a function of number of coefficients and number of bits}
\label{noise-rejection} 103 103 \label{noise-rejection}
\end{figure} 104 104 \end{figure}
105 105
The objective function maximizes the noise rejection while keeping resource occupation below 106 106 The objective function maximizes the noise rejection while keeping resource occupation below
a user-defined threshold. The MILP solver is allowed to choose the number of successive 107 107 a user-defined threshold. The MILP solver is allowed to choose the number of successive
filters, within an upper bound. The last problem is to model the noise rejection. Since filter 108 108 filters, within an upper bound. The last problem is to model the noise rejection. Since filter
noise rejection capability is not modeled with linear equation, a look-up-table is generated 109 109 noise rejection capability is not modeled with linear equation, a look-up-table is generated
for multiple filter configurations in which the $c_i$, $d_i$ and $N_i$ parameters are varied: for each 110 110 for multiple filter configurations in which the $c_i$, $d_i$ and $N_i$ parameters are varied: for each
one of these conditions, the low-pass filter rejection defined as the mean power between 111 111 one of these conditions, the low-pass filter rejection defined as the mean power between
half the Nyquist frequency and the Nyquist frequency is stored as computed by the frequency response 112 112 half the Nyquist frequency and the Nyquist frequency is stored as computed by the frequency response
of the digital filter (Fig. \ref{noise-rejection}). 113 113 of the digital filter (Fig. \ref{noise-rejection}).
114 114
Linear program formalism for solving the problem is well documented: an objective function is 115 115 Linear program formalism for solving the problem is well documented: an objective function is
defined which is linearly dependent on the parameters to be optimized. Constraints are expressed 116 116 defined which is linearly dependent on the parameters to be optimized. Constraints are expressed
as linear equation and solved using one of the available solvers, in our case GLPK\cite{glpk}. 117 117 as linear equation and solved using one of the available solvers, in our case GLPK\cite{glpk}.
118 118
The MILP solver provides a solution to the problem by selecting a series of small FIR with 119 119 The MILP solver provides a solution to the problem by selecting a series of small FIR with
increasing number of bits representing data and coefficients as well as an increasing number 120 120 increasing number of bits representing data and coefficients as well as an increasing number
of coefficients, instead of a single monolithic filter. Fig. \ref{compare-fir} exhibits the 121 121 of coefficients, instead of a single monolithic filter. Fig. \ref{compare-fir} exhibits the
performance comparison between one solution and a monolithic FIR when selecting a cutoff 122 122 performance comparison between one solution and a monolithic FIR when selecting a cutoff
frequency of half the Nyquist frequency: a series of 5 FIR and a series of 10 FIR with the 123 123 frequency of half the Nyquist frequency: a series of 5 FIR and a series of 10 FIR with the
same space usage are provided as selected by the MILP solver. The FIR cascade provides improved 124 124 same space usage are provided as selected by the MILP solver. The FIR cascade provides improved
rejection than the monolithic FIR at the expense of a lower cutoff frequency which remains to 125 125 rejection than the monolithic FIR at the expense of a lower cutoff frequency which remains to
be tuned or compensated for. 126 126 be tuned or compensated for.
127 127
\begin{figure}[h!tb] 128 128 \begin{figure}[h!tb]
% \includegraphics[width=\linewidth]{images/compare-fir.pdf} 129 129 % \includegraphics[width=\linewidth]{images/compare-fir.pdf}
\includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-200dB.pdf} 130 130 \includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-200dB.pdf}
\caption{Comparison of the rejection capability between a series of FIR and a monolithic FIR 131 131 \caption{Comparison of the rejection capability between a series of FIR and a monolithic FIR
with a cutoff frequency set at half the Nyquist frequency.} 132 132 with a cutoff frequency set at half the Nyquist frequency.}
\label{compare-fir} 133 133 \label{compare-fir}
\end{figure} 134 134 \end{figure}
135 135
The resource occupation when synthesizing such FIR on a Xilinx FPGA is summarized as Tab. \ref{t1}. 136 136 The resource occupation when synthesizing such FIR on a Xilinx FPGA is summarized as Tab. \ref{t1}.
137 137
\begin{table}[h!tb] 138 138 \begin{table}[h!tb]
\caption{Resource occupation on a Xilinx Zynq-7000 series FPGA when synthesizing the FIR cascade 139 139 \caption{Resource occupation on a Xilinx Zynq-7000 series FPGA when synthesizing the FIR cascade
identified as optimal by the MILP solver within a finite resource criterion. The last line refers 140 140 identified as optimal by the MILP solver within a finite resource criterion. The last line refers
to available resources on a Zynq-7010 as found on the Redpitaya board. The rejection is the mean 141 141 to available resources on a Zynq-7010 as found on the Redpitaya board. The rejection is the mean
value from 0.6 to 1 Nyquist frequency.} 142 142 value from 0.6 to 1 Nyquist frequency.}
\begin{center} 143 143 \begin{center}
\begin{tabular}{|c|cccc|}\hline 144 144 \begin{tabular}{|c|cccc|}\hline
FIR & BlockRAM & LookUpTables & DSP & rejection (dB)\\\hline\hline 145 145 FIR & BlockRAM & LookUpTables & DSP & rejection (dB)\\\hline\hline
1 (monolithic) & 1 & 4064 & 40 & \\ 146 146 1 (monolithic) & 1 & 4064 & 40 & -71.78 \\
5 & 5 & 12332 & 0 & \\ 147 147 5 & 5 & 12332 & 0 & -216.58 \\
10 & 10 & 12717 & 0 & \\\hline\hline 148 148 10 & 10 & 12717 & 0 & -251.01 \\\hline\hline
Zynq 7010 & 60 & 17600 & 80 & \\\hline 149 149 Zynq 7010 & 60 & 17600 & 80 & \\\hline
\end{tabular} 150 150 \end{tabular}
\end{center} 151 151 \end{center}
%\vspace{-0.7cm} 152 152 %\vspace{-0.7cm}
\label{t1} 153 153 \label{t1}
\end{table} 154 154 \end{table}
155 155
\section{Filter coefficient selection} 156 156 \section{Filter coefficient selection}
157 157
The coefficients of a single monolithic filter are computed as the impulse response 158 158 The coefficients of a single monolithic filter are computed as the impulse response
of the filter transfer function, and practically approximated by a multitude of methods 159 159 of the filter transfer function, and practically approximated by a multitude of methods
including least square optimization (Matlab's {\tt firls} function), Hamming or Kaiser windowing 160 160 including least square optimization (Matlab's {\tt firls} function), Hamming or Kaiser windowing
(Matlab's {\tt fir1} function). Cascading filters opens a new optimization opportunity by 161 161 (Matlab's {\tt fir1} function). Cascading filters opens a new optimization opportunity by
selecting various coefficient sets depending on the number of coefficients. Fig. \ref{2} 162 162 selecting various coefficient sets depending on the number of coefficients. Fig. \ref{2}
illustrates that for a number of coefficients ranging from 8 to 47, {\tt fir1} provides a better 163 163 illustrates that for a number of coefficients ranging from 8 to 47, {\tt fir1} provides a better
rejection than {\tt firls}: since the linear solver increases the number of coefficients along 164 164 rejection than {\tt firls}: since the linear solver increases the number of coefficients along
the processing chain, the type of selected filter also changes depending on the number of coefficients 165 165 the processing chain, the type of selected filter also changes depending on the number of coefficients
and evolves along the processing chain. 166 166 and evolves along the processing chain.
167 167
\begin{figure}[h!tb] 168 168 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/fir1-vs-firls} 169 169 \includegraphics[width=\linewidth]{images/fir1-vs-firls}
\caption{Evolution of the rejection capability of least-square optimized filters and Hamming 170 170 \caption{Evolution of the rejection capability of least-square optimized filters and Hamming
FIR filters as a function of the number of coefficients, for floating point numbers and 8-bit 171 171 FIR filters as a function of the number of coefficients, for floating point numbers and 8-bit
encoded integers.} 172 172 encoded integers.}
\label{2} 173 173 \label{2}
\end{figure} 174 174 \end{figure}
175 175
\section{Conclusion} 176 176 \section{Conclusion}
177 177
We address the optimization problem of designing a low-pass filter chain in a Field Programmable Gate 178 178 We address the optimization problem of designing a low-pass filter chain in a Field Programmable Gate
Array for improved noise rejection within constrained resource occupation, as needed for 179 179 Array for improved noise rejection within constrained resource occupation, as needed for
real time processing of radiofrequency signal when characterizing spectral phase noise 180 180 real time processing of radiofrequency signal when characterizing spectral phase noise
characteristics of stable oscillators. The flexibility of the digital approach makes the result 181 181 characteristics of stable oscillators. The flexibility of the digital approach makes the result
best suited for closing the loop and using the measurement output in a feedback loop for 182 182 best suited for closing the loop and using the measurement output in a feedback loop for
controlling clocks, e.g. in a quartz-stabilized high performance clock whose long term behavior 183 183 controlling clocks, e.g. in a quartz-stabilized high performance clock whose long term behavior
is controlled by non-piezoelectric resonator (sapphire resonator, microwave or optical 184 184 is controlled by non-piezoelectric resonator (sapphire resonator, microwave or optical
atomic transition). 185 185 atomic transition).
186 186