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\documentclass[a4paper,conference]{IEEEtran/IEEEtran} 1 1 \documentclass[a4paper,conference]{IEEEtran/IEEEtran}
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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{Finite impulse response filter} 77 77 \section{Finite impulse response filter}
78 78
We select FIR filter for their unconditional stability and ease of design. A FIR filter is defined 79 79 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 $y_k$ 80 80 by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the outputs $y_k$
$$y_n=\sum_{k=0}^N b_k x_{n-k}$$ 81 81 $$y_n=\sum_{k=0}^N b_k x_{n-k}$$
82 82
As opposed to an implementation on a general purpose processor in which word size is defined by the 83 83 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 84 84 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 85 85 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. 86 86 the coefficients and the sample size.
87 87
Ideally the coefficient are expressed as floating point value but this notation isn't a efficient way to 88 88 The coefficients are classically expressed as floating point values. However, this binary
work with FPGA. Instead we prefer convert this floating point values into integer values. However this 89 89 number representation is not efficient for fast arithmetic computation by an FPGA. Instead,
conversion result in some precision loss. Actually as show figure \ref{float_vs_int}, we see that we aren't 90 90 we select to quantify these floating point values into integer values. This quantization
91 will result in some precision loss. As illustrated in Fig. \ref{float_vs_int}, we see that we aren't
need too coefficients or too sample size. If we have lot of coefficients but a small sample size, 91 92 need too coefficients or too sample size. If we have lot of coefficients but a small sample size,
the first and last are equal to zero. But if we have too sample size for few coefficients that not improve the quality. 92 93 the first and last are equal to zero. But if we have too sample size for few coefficients that not improve the quality.
93 94
95 % JMF je ne comprends pas la derniere phrase ci-dessus ni la figure ci dessous
\begin{figure}[h!tb] 94 96 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/float-vs-integer.pdf} 95 97 \includegraphics[width=\linewidth]{images/float-vs-integer.pdf}
\caption{Illistration of coefficients choice impact} 96 98 \caption{Impact of the quantization resolution of the coefficients}
\label{float_vs_int} 97 99 \label{float_vs_int}
\end{figure} 98 100 \end{figure}
99 101
\section{Filter optimization} 100 102 \section{Filter optimization}
101 103
A basic approach for implementing the FIR filter is to compute the transfer function of 102 104 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 103 105 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 104 106 (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 105 107 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. 106 108 this filter must process data stream at the radiofrequency sampling rate after the mixer.
107 109
An optimization problem \cite{leung2004handbook} aims at improving one or many 108 110 An optimization problem \cite{leung2004handbook} aims at improving one or many
performance criteria within a constrained resource environment. Amongst the tools 109 111 performance criteria within a constrained resource environment. Amongst the tools
developed to meet this aim, Mixed-Integer Linear Programming (MILP) provides the framework to 110 112 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 111 113 provide a formal definition of the stated problem and search for an optimal use of available
resources \cite{yu2007design, kodek1980design}. 112 114 resources \cite{yu2007design, kodek1980design}.
113 115
The degrees of freedom when addressing the problem of replacing the single monolithic 114 116 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 $i$, 115 117 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 116 118 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, 117 119 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 118 120 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 119 121 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. 120 122 the number of bits needed in a worst case condition to represent the output of the FIR.
121 123
122 124
\begin{figure}[h!tb] 123 125 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/noise-rejection.pdf} 124 126 \includegraphics[width=\linewidth]{images/noise-rejection.pdf}
\caption{Rejection as a function of number of coefficients and number of bits} 125 127 \caption{Rejection as a function of number of coefficients and number of bits}
\label{noise-rejection} 126 128 \label{noise-rejection}
\end{figure} 127 129 \end{figure}
128 130
The objective function maximizes the noise rejection while keeping resource occupation below 129 131 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 130 132 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 131 133 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 132 134 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 133 135 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 134 136 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 135 137 half the Nyquist frequency and the Nyquist frequency is stored as computed by the frequency response
of the digital filter (Fig. \ref{noise-rejection}). 136 138 of the digital filter (Fig. \ref{noise-rejection}).
137 139
Linear program formalism for solving the problem is well documented: an objective function is 138 140 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 139 141 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}. 140 142 as linear equation and solved using one of the available solvers, in our case GLPK\cite{glpk}.
141 143
The MILP solver provides a solution to the problem by selecting a series of small FIR with 142 144 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 143 145 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 144 146 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 145 147 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 146 148 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 147 149 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 148 150 rejection than the monolithic FIR at the expense of a lower cutoff frequency which remains to
be tuned or compensated for. 149 151 be tuned or compensated for.
150 152
\begin{figure}[h!tb] 151 153 \begin{figure}[h!tb]
% \includegraphics[width=\linewidth]{images/compare-fir.pdf} 152 154 % \includegraphics[width=\linewidth]{images/compare-fir.pdf}
\includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-200dB.pdf} 153 155 \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 154 156 \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.} 155 157 with a cutoff frequency set at half the Nyquist frequency.}
\label{compare-fir} 156 158 \label{compare-fir}
\end{figure} 157 159 \end{figure}
158 160
The resource occupation when synthesizing such FIR on a Xilinx FPGA is summarized as Tab. \ref{t1}. 159 161 The resource occupation when synthesizing such FIR on a Xilinx FPGA is summarized as Tab. \ref{t1}.
160 162
\begin{table}[h!tb] 161 163 \begin{table}[h!tb]
\caption{Resource occupation on a Xilinx Zynq-7000 series FPGA when synthesizing the FIR cascade 162 164 \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 163 165 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 164 166 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.} 165 167 value from 0.6 to 1 Nyquist frequency.}
\begin{center} 166 168 \begin{center}
\begin{tabular}{|c|cccc|}\hline 167 169 \begin{tabular}{|c|cccc|}\hline
FIR & BlockRAM & LookUpTables & DSP & rejection (dB)\\\hline\hline 168 170 FIR & BlockRAM & LookUpTables & DSP & rejection (dB)\\\hline\hline
1 (monolithic) & 1 & 4064 & 40 & -72 \\ 169 171 1 (monolithic) & 1 & 4064 & 40 & -72 \\
5 & 5 & 12332 & 0 & -217 \\ 170 172 5 & 5 & 12332 & 0 & -217 \\
10 & 10 & 12717 & 0 & -251 \\\hline\hline 171 173 10 & 10 & 12717 & 0 & -251 \\\hline\hline
Zynq 7010 & 60 & 17600 & 80 & \\\hline 172 174 Zynq 7010 & 60 & 17600 & 80 & \\\hline
\end{tabular} 173 175 \end{tabular}
\end{center} 174 176 \end{center}
%\vspace{-0.7cm} 175 177 %\vspace{-0.7cm}
\label{t1} 176 178 \label{t1}
\end{table} 177 179 \end{table}
178 180
\section{Filter coefficient selection} 179 181 \section{Filter coefficient selection}
180 182
The coefficients of a single monolithic filter are computed as the impulse response 181 183 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 182 184 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 183 185 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 184 186 (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} 185 187 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 186 188 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 187 189 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 188 190 the processing chain, the type of selected filter also changes depending on the number of coefficients
and evolves along the processing chain. 189 191 and evolves along the processing chain.
190 192
\begin{figure}[h!tb] 191 193 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/fir1-vs-firls} 192 194 \includegraphics[width=\linewidth]{images/fir1-vs-firls}