jfriedt / IFCS2018 article

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\hyphenation{op-tical net-works semi-conduc-tor}

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\begin{document}

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\begin{document}

\title{Filter optimization for real time digital processing of radiofrequency signals: application

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\title{Filter optimization for real time digital processing of radiofrequency signals: application

to oscillator metrology}

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to oscillator metrology}

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\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},

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\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},

G. Goavec-M\'erou\IEEEauthorrefmark{1},

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G. Goavec-M\'erou\IEEEauthorrefmark{1},

P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}

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P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}

\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }

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\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 \\

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\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\

Email: \{pyb2,jmfriedt\}@femto-st.fr}

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Email: \{pyb2,jmfriedt\}@femto-st.fr}

}

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}

\maketitle

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\maketitle

\thispagestyle{plain}

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\thispagestyle{plain}

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\pagestyle{plain}

\newtheorem{definition}{Definition}

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\newtheorem{definition}{Definition}

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\begin{abstract}

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\begin{abstract}

Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to

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Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to

radiofrequency signal processing. Applied to oscillator characterization in the context

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radiofrequency signal processing. Applied to oscillator characterization in the context

of ultrastable clocks, stringent filtering requirements are defined by spurious signal or

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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

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noise rejection needs. Since real time radiofrequency processing must be performed in a

Field Programmable Array to meet timing constraints, we investigate optimization strategies

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Field Programmable Array to meet timing constraints, we investigate optimization strategies

to design filters meeting rejection characteristics while limiting the hardware resources

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to design filters meeting rejection characteristics while limiting the hardware resources

required and keeping timing constraints within the targeted measurement bandwidths.

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required and keeping timing constraints within the targeted measurement bandwidths.

\end{abstract}

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\end{abstract}

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\begin{IEEEkeywords}

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\begin{IEEEkeywords}

Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter

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Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter

\end{IEEEkeywords}

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\end{IEEEkeywords}

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\section{Digital signal processing of ultrastable clock signals}

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\section{Digital signal processing of ultrastable clock signals}

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Analog oscillator phase noise characteristics are classically performed by downconverting

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Analog oscillator phase noise characteristics are classically performed by downconverting

the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband,

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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

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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

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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}.

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multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.

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\begin{figure}[h!tb]

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\begin{figure}[h!tb]

\begin{center}

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\begin{center}

\includegraphics[width=.8\linewidth]{images/schema}

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\includegraphics[width=.8\linewidth]{images/schema}

\end{center}

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\end{center}

\caption{Fully digital oscillator phase noise characterization: the Device Under Test

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\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

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(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

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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

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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

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Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays

the spectral characteristics of the phase fluctuations.}

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the spectral characteristics of the phase fluctuations.}

\label{schema}

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\label{schema}

\end{figure}

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\end{figure}

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As with the analog mixer,

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As with the analog mixer,

the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as

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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.

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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

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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

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for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the

downconverter

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downconverter

and the decimation processing blocks are core characteristics of an oscillator characterization

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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

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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

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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

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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

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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

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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

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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

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tunable number of coefficients and tunable number of bits representing the coefficients and the

data being processed.

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data being processed.

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\section{Finite impulse response filter}

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\section{Finite impulse response filter}

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We select FIR filter for their unconditional stability and ease of design. A FIR filter is defined

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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

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by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the

outputs $y_k$

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outputs $y_k$

\begin{align}

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\begin{align}

y_n=\sum_{k=0}^N b_k x_{n-k}

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y_n=\sum_{k=0}^N b_k x_{n-k}

\label{eq:fir_equation}

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\label{eq:fir_equation}

\end{align}

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\end{align}

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As opposed to an implementation on a general purpose processor in which word size is defined by the

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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

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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

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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

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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

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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

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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.

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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

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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,

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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.

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is not considered as an issue as would be in a closed loop system.

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The coefficients are classically expressed as floating point values. However, this binary

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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,

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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

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we select to quantify these floating point values into integer values. This quantization

will result in some precision loss.

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will result in some precision loss.

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\begin{figure}[h!tb]

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\begin{figure}[h!tb]

\includegraphics[width=\linewidth]{images/demo_filtre}

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\includegraphics[width=\linewidth]{images/zero_values}

\caption{Impact of the quantization resolution of the coefficients: the quantization is

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\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

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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

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the 30~first and 30~last coefficients out of the initial 128~band-pass

filter coefficients to 0 (red dots).}

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filter coefficients to 0 (red dots).}

\label{float_vs_int}

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\label{float_vs_int}

\end{figure}

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\end{figure}

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The tradeoff between quantization resolution and number of coefficients when considering

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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

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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

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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

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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

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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

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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

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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

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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

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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

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and tap length, as will be shown in the next section. Indeed, our development strategy closely

follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}

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follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}

in which basic blocks are defined and characterized before being assembled \cite{hide}

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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

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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

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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

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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

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current implementation: the decimation is assumed to be located after the FIR cascade at the

moment.

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moment.

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\section{Methodology description}

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\section{Methodology description}

We want create a new methodology to develop any Digital Signal Processing (DSP) chain

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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

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and for any hardware platform (Altera, Xilinx...). To do this we have defined an

abstract model to represent some basic operations of DSP.

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abstract model to represent some basic operations of DSP.

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For the moment, we are focused on only two operations: the filtering and the shifting of data.

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For the moment, we are focused on only two operations: the filtering and the shifting of data.

We have chosen this basic operation because the shifting and the filtering have already be studied in

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We have chosen this basic operation because the shifting and the filtering have already be studied in

lot of works \cite{lim_1996, lim_1988, young_1992, smith_1998} hence it will be easier

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lot of works \cite{lim_1996, lim_1988, young_1992, smith_1998} hence it will be easier

to check and validate our results.

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to check and validate our results.

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However having only two operations is insufficient to work with complex DSP but

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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.

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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

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In future work it will be possible to add more operations and we are able to

model any DSP chain.

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model any DSP chain.

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We will apply our methodology on very simple DSP chain. We generate a digital signal

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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

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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.

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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.

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Finally we stored some burst of filtered samples before post-processing it.

% TODO: faire un schéma

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% TODO: faire un schéma

In this particular case, we want optimize the filtering step to have the best noise

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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

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rejection for constrain number of resource or to have the minimal resources

consumption for a given rejection objective.

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consumption for a given rejection objective.

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The first step of our approach is to model the DSP chain and since we just optimize

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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

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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

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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

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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

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we select different FIR filters with less coefficients the monolithic filter and we cascaded

it to filtering the signal.

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it to filtering the signal.

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After each filter we leave the possibility of shifting the filtered data to consume

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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

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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).

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and a shifter (the shift could be omitted if we do not need to divide the filtered data).

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\subsection{Model of a FIR filter}

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\subsection{Model of a FIR filter}

A cascade of filter are composed of $n$ stage. In stage $i$ ($1 \leq i \leq n$)

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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 $\pi^C_i$

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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

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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}

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the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage}

shows a filtering stage.

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shows a filtering stage.

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\begin{figure}

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\begin{figure}

\centering

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\centering

\begin{tikzpicture}[node distance=2cm]

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\begin{tikzpicture}[node distance=2cm]

\node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ;

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\node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ;

\node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ;

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\node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ;

\node (Start) [left of=FIR] { } ;

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\node (Start) [left of=FIR] { } ;

\node (End) [right of=Shift] { } ;

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\node (End) [right of=Shift] { } ;

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\node[draw,fit=(FIR) (Shift)] (Filter) { } ;

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\node[draw,fit=(FIR) (Shift)] (Filter) { } ;

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\draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ;

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\draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ;

\draw[->] (FIR) -- (Shift) ;

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\draw[->] (FIR) -- (Shift) ;

\draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ;

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\draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ;

\end{tikzpicture}

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\end{tikzpicture}

\caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)}

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\caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)}

\label{fig:fir_stage}

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\label{fig:fir_stage}

\end{figure}

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\end{figure}

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FIR $i$ can reject $F(C_i, \pi_i^C)$ dB. $F$ is determined numerically.

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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

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To measure this rejection, we use GNU Octave software to design FIR filter coefficients thanks to two

algorithms (\texttt{firls} and \texttt{fir1}).

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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.

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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,

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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.

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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.

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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.

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With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter.

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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},

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Comparing the performance between FIRs requires however a unique criterion. As shown in figure~\ref{fig:fir_mag},

the FIR magnitude exhibits two parts.

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the FIR magnitude exhibits two parts.

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\begin{figure}

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\begin{figure}

\centering

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\centering

\begin{tikzpicture}[scale=0.3]

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\begin{tikzpicture}[scale=0.3]

\draw[<->] (0,15) -- (0,0) -- (21,0) ;

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\draw[<->] (0,15) -- (0,0) -- (21,0) ;

\draw[thick] (0,12) -- (8,12) -- (20,0) ;

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\draw[thick] (0,12) -- (8,12) -- (20,0) ;

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\draw (0,14) node [left] { $P$ } ;

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\draw (0,14) node [left] { $P$ } ;

\draw (20,0) node [below] { $f$ } ;

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\draw (20,0) node [below] { $f$ } ;

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\draw[>=latex,<->] (0,14) -- (8,14) ;

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\draw[>=latex,<->] (0,14) -- (8,14) ;

\draw (4,14) node [above] { passband } node [below] { $40\%$ } ;

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\draw (4,14) node [above] { passband } node [below] { $40\%$ } ;

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\draw[>=latex,<->] (8,14) -- (12,14) ;

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\draw[>=latex,<->] (8,14) -- (12,14) ;

\draw (10,14) node [above] { transition } node [below] { $20\%$ } ;

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\draw (10,14) node [above] { transition } node [below] { $20\%$ } ;

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\draw[>=latex,<->] (12,14) -- (20,14) ;

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\draw[>=latex,<->] (12,14) -- (20,14) ;

\draw (16,14) node [above] { stopband } node [below] { $40\%$ } ;

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\draw (16,14) node [above] { stopband } node [below] { $40\%$ } ;

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\draw[>=latex,<->] (16,12) -- (16,8) ;

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\draw[>=latex,<->] (16,12) -- (16,8) ;

\draw (16,10) node [right] { rejection } ;

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\draw (16,10) node [right] { rejection } ;

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\draw[dashed] (8,-1) -- (8,14) ;

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\draw[dashed] (8,-1) -- (8,14) ;

\draw[dashed] (12,-1) -- (12,14) ;

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\draw[dashed] (12,-1) -- (12,14) ;

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\draw[dashed] (8,12) -- (16,12) ;

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\draw[dashed] (8,12) -- (16,12) ;

\draw[dashed] (12,8) -- (16,8) ;

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\draw[dashed] (12,8) -- (16,8) ;

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\end{tikzpicture}

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\end{tikzpicture}

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% \includegraphics[width=.5\linewidth]{images/fir_magnitude}

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% \includegraphics[width=.5\linewidth]{images/fir_magnitude}

\caption{Shape of the filter transmitted power $P$ as a function of frequency $f$:

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\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,

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the passband is considered to occupy the initial 40\% of the Nyquist frequency range,

the stopband the last 40\%, allowing 20\% transition width.}

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the stopband the last 40\%, allowing 20\% transition width.}

\label{fig:fir_mag}

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\label{fig:fir_mag}

\end{figure}

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\end{figure}

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In the transition band, the behavior of the filter is left free, we only care about the passband and the stopband.

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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.

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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}.

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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}.

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\begin{figure}

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\begin{figure}

\centering

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\centering

\includegraphics[width=\linewidth]{images/mean_criterion}

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\includegraphics[width=\linewidth]{images/colored_mean_criterion}

\caption{Mean criterion comparison between monolithic filter and cascade filters}

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\caption{Mean criterion comparison between monolithic filter and cascade filters}

\label{fig:mean_criterion}

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\label{fig:mean_criterion}

\end{figure}

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\end{figure}

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\begin{figure}

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\begin{figure}

\centering

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\centering

\includegraphics[width=\linewidth]{images/custom_criterion}

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\includegraphics[width=\linewidth]{images/colored_custom_criterion}

\caption{Custom criterion comparison between monolithic filter and cascade filters}

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\caption{Custom criterion comparison between monolithic filter and cascade filters}

\label{fig:custom_criterion}

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\label{fig:custom_criterion}

\end{figure}

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\end{figure}

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Although we have a efficient criterion to estimate the rejection of one set of coefficient

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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

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we have a problem when we sum two or more criterion. If the FIR filter coefficients are the same

between the stage, we have:

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between the stage, we have:

$$F_{total} = F_1 + F_2$$

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$$F_{total} = F_1 + F_2$$

But when we choose two different set of coefficient, the previous equality are not

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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

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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

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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)

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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

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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.

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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

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However this summation gives us an upper bound for rejection although in fact we obtain

better rejection than expected.

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better rejection than expected.

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\begin{figure}

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\begin{figure}

\centering

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\centering

\includegraphics[width=\linewidth]{images/sum_rejection}

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\includegraphics[width=\linewidth]{images/cascaded_criterion}

\caption{Rejection of two cascaded filters}

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\caption{Rejection of two cascaded filters}

\label{fig:sum_rejection}

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\label{fig:sum_rejection}

\end{figure}

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\end{figure}

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The first problem we address is to maximize the rejection under bounded silicon area

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and feasibility constraints. Variable $a_i$ is the area taken by filter~$i$

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(in arbitrary unit). Variable $r_i$ is the rejection of filter~$i$ (in dB).

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Constant $\mathcal{A}$ is the total available area. We model our problem as follows:

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Finally we can describe our abstract model with following expressions :

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Finally we can describe our abstract model with following expressions :

\begin{align}

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\begin{align}

\text{Maximize } & \sum_{i=1}^n r_i \notag \\

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\text{Maximize } & \sum_{i=1}^n r_i \notag \\

\sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\

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\sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\

a_i & = C_i \times (\pi_i^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef} \\

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a_i & = C_i \times (\pi_i^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef} \\

r_i & = F(C_i, \pi_i^C), & \forall i \in [1, n] \label{eq:rejectiondef} \\

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r_i & = F(C_i, \pi_i^C), & \forall i \in [1, n] \label{eq:rejectiondef} \\

\pi_i^+ & = \pi_i^- + \pi_i^C - \pi_i^S, & \forall i \in [1, n] \label{eq:bits} \\

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\pi_i^+ & = \pi_i^- + \pi_i^C - \pi_i^S, & \forall i \in [1, n] \label{eq:bits} \\

\pi_{i - 1}^+ & = \pi_i^-, & \forall i \in [2, n] \label{eq:inout} \\

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\pi_{i - 1}^+ & = \pi_i^-, & \forall i \in [2, n] \label{eq:inout} \\

\pi_i^+ & \geq 1 + \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right), & \forall i \in [1, n] \label{eq:maxshift} \\

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\pi_i^+ & \geq 1 + \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right), & \forall i \in [1, n] \label{eq:maxshift} \\

\pi_1^- &= \Pi^I \label{eq:init}

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\pi_1^- &= \Pi^I \label{eq:init}

\end{align}

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\end{align}

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{\color{red} Je sais que l'idée est de ne pas parler du programme linéaire mais

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ça me semble quand même indispensable. Au pire, j'essaierai de revoir ça si on

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est vraiment en manque de place.}

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Equation~\ref{eq:area} states that the total area taken by the filters must be

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Equation~\ref{eq:area} states that the total area taken by the filters must be

less than the available area. Equation~\ref{eq:areadef} gives the definition of

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less than the available area. Equation~\ref{eq:areadef} gives the definition of

the area for a filter. More precisely, it is the area of the FIR as the Shifter

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the area for a filter. More precisely, it is the area of the FIR as the Shifter

does not need any circuitry. We consider that the FIR needs $C_i$ registers of size

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does not need any circuitry. We consider that the FIR needs $C_i$ registers of size

$\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the

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$\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the

input data and the coefficients. Equation~\ref{eq:rejectiondef} gives the

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input data and the coefficients. Equation~\ref{eq:rejectiondef} gives the

definition of the rejection of the filter thanks to function~$F$ that we defined

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definition of the rejection of the filter thanks to function~$F$ that we defined

previously. The Shifter does not introduce negative rejection as we explain later,

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previously. The Shifter does not introduce negative rejection as we explain later,

so the rejection only comes from the FIR. Equation~\ref{eq:bits} states the

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so the rejection only comes from the FIR. Equation~\ref{eq:bits} states the

relation between $\pi_i^+$ and $\pi_i^-$. The multiplications in the FIR add

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relation between $\pi_i^+$ and $\pi_i^-$. The multiplications in the FIR add

$\pi_i^C$ bits as most coefficients are close to zero, and the Shifter removes

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$\pi_i^C$ bits as most coefficients are close to zero, and the Shifter removes

$\pi_i^S$ bits. Equation~\ref{eq:inout} states that the output number of bits of

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$\pi_i^S$ bits. Equation~\ref{eq:inout} states that the output number of bits of

a filter is the same as the input number of bits of the next filter.

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a filter is the same as the input number of bits of the next filter.

Equation~\ref{eq:maxshift} ensures that the Shifter does not introduce negative

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Equation~\ref{eq:maxshift} ensures that the Shifter does not introduce negative

rejection. Indeed, the results of the FIR can be right shifted without compromising

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rejection. Indeed, the results of the FIR can be right shifted without compromising

the quality of the rejection until a threshold. Each bit of the output data

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the quality of the rejection until a threshold. Each bit of the output data

increases the maximum rejection level of 6~dB. We add one to take the sign bit

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increases the maximum rejection level of 6~dB. We add one to take the sign bit

into account. If equation~\ref{eq:maxshift} was not present, the Shifter could

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into account. If equation~\ref{eq:maxshift} was not present, the Shifter could

shift too much and introduce some noise in the output data. Each supplementary

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shift too much and introduce some noise in the output data. Each supplementary

shift bit would cause 6~dB of noise. A totally equivalent equation is:

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shift bit would cause 6~dB of noise. A totally equivalent equation is:

$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right) $.

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$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right) $.

Finally, equation~\ref{eq:init} gives the global input's number of bits.

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Finally, equation~\ref{eq:init} gives the global input's number of bits.

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This model is non-linear and even non-quadratic, as $F$ does not have a known

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This model is non-linear and even non-quadratic, as $F$ does not have a known

linear or quadratic expression. We introduce $p$ FIR configurations

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linear or quadratic expression. We introduce $p$ FIR configurations

$(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. We define binary

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$(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. We define binary

variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$

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variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$

and 0 otherwise. The new equations are as follows:

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and 0 otherwise. The new equations are as follows:

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\begin{align}

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\begin{align}

a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\

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a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\

r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\

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r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\

\pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\

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\pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\

\sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}

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\sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}

\end{align}

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\end{align}

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Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace

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Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace

respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.

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respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.

Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.

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Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.

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The next section shows the results for this quadratic program but the section~\ref{sec:fixed_rej}

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This modified model is quadratic, and it can be linearised if necessary. The Gurobi

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(\url{www.gurobi.com}) optimization software is used to solve this quadratic

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model, and since Gurobi is able to linearize, the model is left as is. This model

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has $O(np)$ variables and $O(n)$ constraints.

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The section~\ref{sec:fixed_area} shows the results for the first version of quadratic program but the section~\ref{sec:fixed_rej}

presents the results for the complementary problem. In this case we want

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presents the results for the complementary problem. In this case we want

minimize the occupied area for a targeted rejection level. Hence we have replace

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minimize the occupied area for a targeted rejection level. Hence we have replace

the objective function with:

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the objective function with:

\begin{align}

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\begin{align}

\text{Minimize } & \sum_{i=1}^n a_i \notag

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\text{Minimize } & \sum_{i=1}^n a_i \notag

\end{align}

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\end{align}

We adapt our constraints of quadratic program to replace the equation \ref{eq:area}

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We adapt our constraints of quadratic program to replace the equation \ref{eq:area}

by the equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal

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by the equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal

rejection required.

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rejection required.

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\begin{align}

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\begin{align}

\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}

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\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}

\end{align}

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\end{align}

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\section{Design workflow}

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\section{Design workflow}

\label{sec:workflow}

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\label{sec:workflow}

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In this section, we describe the workflow to compute all the results presented in section~\ref{sec:fixed_area}.

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In this section, we describe the workflow to compute all the results presented in section~\ref{sec:fixed_area}.

Figure~\ref{fig:workflow} shows the global workflow and the different steps involved in the computations of the results.

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Figure~\ref{fig:workflow} shows the global workflow and the different steps involved in the computations of the results.

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\begin{figure}

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\begin{figure}

\centering

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\centering

\begin{tikzpicture}[node distance=0.75cm and 2cm]

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\begin{tikzpicture}[node distance=0.75cm and 2cm]

\node[draw,minimum size=1cm] (Solver) { Filter Solver } ;

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\node[draw,minimum size=1cm] (Solver) { Filter Solver } ;

\node (Start) [left= 3cm of Solver] { } ;

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\node (Start) [left= 3cm of Solver] { } ;

\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;

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\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;

\node (Input) [above= of TCL] { } ;

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\node (Input) [above= of TCL] { } ;

\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;

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\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;

\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;

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\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;

\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;

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\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;

\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;

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\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;

\node (Results) [left= of Postproc] { } ;

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\node (Results) [left= of Postproc] { } ;

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\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ;

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\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ;

\draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;

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\draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;

\draw[->] (Solver) edge node [below] { (1a) } (TCL) ;

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\draw[->] (Solver) edge node [below] { (1a) } (TCL) ;

\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;

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\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;

\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;

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\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;

\draw[->,dashed] (Bitstream) -- (Deploy) ;

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\draw[->,dashed] (Bitstream) -- (Deploy) ;

\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;

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\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;

\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;

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\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;

\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;

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\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;

\draw[->] (Postproc) -- (Results) ;

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\draw[->] (Postproc) -- (Results) ;

\end{tikzpicture}

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\end{tikzpicture}

\caption{Design workflow from the input parameters to the results}

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\caption{Design workflow from the input parameters to the results}

\label{fig:workflow}

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\label{fig:workflow}

\end{figure}

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\end{figure}

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The filter solver is a C++ program that takes as input the maximum area

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The filter solver is a C++ program that takes as input the maximum area

$\mathcal{A}$, the number of stages $n$, the size of the input signal $\Pi^I$,

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$\mathcal{A}$, the number of stages $n$, the size of the input signal $\Pi^I$,

the FIR configurations $(C_{ij}, \pi_{ij}^C)$ and the function $F$. It creates

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the FIR configurations $(C_{ij}, \pi_{ij}^C)$ and the function $F$. It creates

the quadratic programs and uses the Gurobi solver to get the optimal results.

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the quadratic programs and uses the Gurobi solver to get the optimal results.

Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})

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Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})

and a deploy script ((1b) on figure~\ref{fig:workflow}).

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and a deploy script ((1b) on figure~\ref{fig:workflow}).

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The TCL script describes the whole digital processing chain from the beginning

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The TCL script describes the whole digital processing chain from the beginning

(the raw signal data) to the end (the filtered data).

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(the raw signal data) to the end (the filtered data).

The raw input data generated from a Pseudo Random Number (PRN)

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The raw input data generated from a Pseudo Random Number (PRN)

generator inside the FPGA and $\Pi^I$ is fixed at 16~bits.

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generator inside the FPGA and $\Pi^I$ is fixed at 16~bits.

Then the script builds each stage of the chain with a generic FIR task that

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Then the script builds each stage of the chain with a generic FIR task that

comes from a skeleton library. The generic FIR is highly configurable

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comes from a skeleton library. The generic FIR is highly configurable

with the number of coefficients and the size of the coefficients. The coefficients

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with the number of coefficients and the size of the coefficients. The coefficients

themselves are not stored in the script.

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themselves are not stored in the script.

Whereas the signal is processed in real-time, the output signal is stored as

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Whereas the signal is processed in real-time, the output signal is stored as

consecutive bursts of data.

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consecutive bursts of data.

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The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}).

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The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}).

We use the 2018.2 version of Xilinx Vivado and we execute the synthesized

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We use the 2018.2 version of Xilinx Vivado and we execute the synthesized

bitstream on a Redpitaya board fitted with a Xilinx Zynq-7010 series

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bitstream on a Redpitaya board fitted with a Xilinx Zynq-7010 series

FPGA (xc7z010clg400-1) and two 125~MS/s ADC.

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FPGA (xc7z010clg400-1) and two 125~MS/s ADC.

The board works with a Buildroot Linux image. We have developed some tools and

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The board works with a Buildroot Linux image. We have developed some tools and

drivers to flash and communicate with the FPGA. They are used to automatize all

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drivers to flash and communicate with the FPGA. They are used to automatize all

the workflow inside the board: load the filter coefficients and retrieve the

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the workflow inside the board: load the filter coefficients and retrieve the

computed data.

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computed data.

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The deploy script uploads the bitstream to the board ((3) on

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The deploy script uploads the bitstream to the board ((3) on

figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,

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figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,

configures the coefficients of the FIR filters. It then waits for the results

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configures the coefficients of the FIR filters. It then waits for the results

and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).

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and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).

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Finally, an Octave post-processing script computes the final results thanks to

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Finally, an Octave post-processing script computes the final results thanks to

the output data ((5) on figure~\ref{fig:workflow}).

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the output data ((5) on figure~\ref{fig:workflow}).

The results are normalized so that the Power Spectrum Density (PSD) starts at zero

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The results are normalized so that the Power Spectrum Density (PSD) starts at zero

and the different configurations can be compared.

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and the different configurations can be compared.

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The workflow used to compute the results in section~\ref{sec:fixed_rej}, we

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The workflow used to compute the results in section~\ref{sec:fixed_rej}, we

have just adapted the quadratic program but the rest of the workflow is unchanged.

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have just adapted the quadratic program but the rest of the workflow is unchanged.

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\section{Experiments with fixed area space}

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\section{Experiments with fixed area space}

\label{sec:fixed_area}

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\label{sec:fixed_area}

This section presents the output of the filter solver {\em i.e.} the computed

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This section presents the output of the filter solver {\em i.e.} the computed

configurations for each stage, the computed rejection and the computed silicon area.

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configurations for each stage, the computed rejection and the computed silicon area.

This is interesting to understand the choices made by the solver to compute its solutions.

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This is interesting to understand the choices made by the solver to compute its solutions.

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The experimental setup is composed of three cases. The raw input is generated

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The experimental setup is composed of three cases. The raw input is generated

by a Pseudo Random Number (PRN) generator, which fixes the input data size $\Pi^I$.

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by a Pseudo Random Number (PRN) generator, which fixes the input data size $\Pi^I$.

Then the total silicon area $\mathcal{A}$ has been fixed to either 500, 1000 or 1500

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Then the total silicon area $\mathcal{A}$ has been fixed to either 500, 1000 or 1500

arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500.

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arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500.

The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$

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The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$

ranging from 2 to 22. In each case, the quadratic program has been able to give a

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ranging from 2 to 22. In each case, the quadratic program has been able to give a

result up to five stages ($n = 5$) in the cascaded filter.

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449

result up to five stages ($n = 5$) in the cascaded filter.

444

450

Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500.

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451

Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500.

Table~\ref{tbl:gurobi_max_1000} shows the results obtained by the filter solver for MAX/1000.

446

452

Table~\ref{tbl:gurobi_max_1000} shows the results obtained by the filter solver for MAX/1000.

Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.

447

453

Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.

448

454

\renewcommand{\arraystretch}{1.4}

449

455

\renewcommand{\arraystretch}{1.4}

450

456

\begin{table}

451

457

\begin{table}

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}

452

458

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}

\label{tbl:gurobi_max_500}

453

459

\label{tbl:gurobi_max_500}

\centering

454

460

\centering

{\scalefont{0.77}

455

461

{\scalefont{0.77}

\begin{tabular}{|c|ccccc|c|c|}

456

462

\begin{tabular}{|c|ccccc|c|c|}

\hline

457

463

\hline

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

458

464

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

\hline

459

465

\hline

1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\

460

466

1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\

2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\

461

467

2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\

3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\

462

468

3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\

4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\

463

469

4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\

5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\

464

470

5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\

\hline

465

471

\hline

\end{tabular}

466

472

\end{tabular}

}

467

473

}

\end{table}

468

474

\end{table}

469

475

\begin{table}

470

476

\begin{table}

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}

471

477

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}

\label{tbl:gurobi_max_1000}

472

478

\label{tbl:gurobi_max_1000}

\centering

473

479

\centering

{\scalefont{0.77}

474

480

{\scalefont{0.77}

\begin{tabular}{|c|ccccc|c|c|}

475

481

\begin{tabular}{|c|ccccc|c|c|}

\hline

476

482

\hline

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

477

483

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

\hline

478

484

\hline

1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\

479

485

1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\

2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\

480

486

2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\

3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\

481

487

3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\

4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\

482

488

4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\

5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\

483

489

5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\

\hline

484

490

\hline

\end{tabular}

485

491

\end{tabular}

}

486

492

}

\end{table}

487

493

\end{table}

488

494

\begin{table}

489

495

\begin{table}

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}

490

496

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}

\label{tbl:gurobi_max_1500}

491

497

\label{tbl:gurobi_max_1500}

\centering

492

498

\centering

{\scalefont{0.77}

493

499

{\scalefont{0.77}

\begin{tabular}{|c|ccccc|c|c|}

494

500

\begin{tabular}{|c|ccccc|c|c|}

\hline

495

501

\hline

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

496

502

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

\hline

497

503

\hline

1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\

498

504

1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\

2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\

499

505

2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\

3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\

500

506

3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\

4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\

501

507

4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\

5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\

502

508

5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\

\hline

503

509

\hline

\end{tabular}

504

510

\end{tabular}

}

505

511

}

\end{table}

506

512

\end{table}

507

513

\renewcommand{\arraystretch}{1}

508

514

\renewcommand{\arraystretch}{1}

509

515

From these tables, we can first state that the more stages are used to define

510

516

From these tables, we can first state that the more stages are used to define

the cascaded FIR filters, the better the rejection. It was an expected result as it has

511

517

the cascaded FIR filters, the better the rejection. It was an expected result as it has

been previously observed that many small filters are better than

512

518

been previously observed that many small filters are better than

a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusion

513

519

a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusion

being hardly used in practice due to the lack of tools for identifying individual filter

514

520

being hardly used in practice due to the lack of tools for identifying individual filter

coefficients in the cascaded approach.

515

521

coefficients in the cascaded approach.

516

522

Second, the larger the silicon area, the better the rejection. This was also an

517

523

Second, the larger the silicon area, the better the rejection. This was also an

expected result as more area means a filter of better quality (more coefficients

518

524

expected result as more area means a filter of better quality (more coefficients

or more bits per coefficient).

519

525

or more bits per coefficient).

520

526

Then, we also observe that the first stage can have a larger shift than the other

521

527

Then, we also observe that the first stage can have a larger shift than the other

stages. This is explained by the fact that the solver tries to use just enough

522

528

stages. This is explained by the fact that the solver tries to use just enough

bits for the computed rejection after each stage. In the first stage, a

523

529

bits for the computed rejection after each stage. In the first stage, a

balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}

524

530

balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}

gives the relation between both values.

525

531

gives the relation between both values.

526

532

Finally, we note that the solver consumes all the given silicon area.

527

533

Finally, we note that the solver consumes all the given silicon area.

528

534

The following graphs present the rejection for real data on the FPGA. In all following

529

535

The following graphs present the rejection for real data on the FPGA. In all following

figures, the solid line represents the actual rejection of the filtered

530

536

figures, the solid line represents the actual rejection of the filtered

data on the FPGA as measured experimentally and the dashed line are the noise level

531

537

data on the FPGA as measured experimentally and the dashed line are the noise level

given by the quadratic solver. The configurations are those computed in the previous section.

532

538

given by the quadratic solver. The configurations are those computed in the previous section.

533

539

Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500.

534

540

Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500.

Figure~\ref{fig:max_1000_result} shows the rejection of the different configurations in the case of MAX/1000.

535

541

Figure~\ref{fig:max_1000_result} shows the rejection of the different configurations in the case of MAX/1000.

Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.

536

542

Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.

537

543

\begin{figure}

538

544

\begin{figure}

\centering

539

545

\centering

\includegraphics[width=\linewidth]{images/max_500}

540

546

\includegraphics[width=\linewidth]{images/max_500}

\caption{Signal spectrum for MAX/500}

541

547

\caption{Signal spectrum for MAX/500}

\label{fig:max_500_result}

542

548

\label{fig:max_500_result}

\end{figure}

543

549

\end{figure}

544

550

\begin{figure}

545

551

\begin{figure}

\centering

546

552

\centering

\includegraphics[width=\linewidth]{images/max_1000}

547

553

\includegraphics[width=\linewidth]{images/max_1000}

\caption{Signal spectrum for MAX/1000}

548

554

\caption{Signal spectrum for MAX/1000}

\label{fig:max_1000_result}

549

555

\label{fig:max_1000_result}

\end{figure}

550

556

\end{figure}

551

557

\begin{figure}

552

558

\begin{figure}

\centering

553

559

\centering

\includegraphics[width=\linewidth]{images/max_1500}

554

560

\includegraphics[width=\linewidth]{images/max_1500}

\caption{Signal spectrum for MAX/1500}

555

561

\caption{Signal spectrum for MAX/1500}

\label{fig:max_1500_result}

556

562

\label{fig:max_1500_result}

\end{figure}

557

563

\end{figure}

558

564

In all cases, we observe that the actual rejection is close to the rejection computed by the solver.

559

565

In all cases, we observe that the actual rejection is close to the rejection computed by the solver.

560

566

We compare the actual silicon resources given by Vivado to the

561

567

We compare the actual silicon resources given by Vivado to the

resources in arbitrary units.

562

568

resources in arbitrary units.

The goal is to check that our arbitrary units of silicon area models well enough

563

569

The goal is to check that our arbitrary units of silicon area models well enough

the real resources on the FPGA. Especially we want to verify that, for a given

564

570

the real resources on the FPGA. Especially we want to verify that, for a given

number of arbitrary units, the actual silicon resources do not depend on the

565

571

number of arbitrary units, the actual silicon resources do not depend on the

number of stages $n$. Most significantly, our approach aims

566

572

number of stages $n$. Most significantly, our approach aims

at remaining far enough from the practical logic gate implementation used by

567

573

at remaining far enough from the practical logic gate implementation used by

various vendors to remain platform independent and be portable from one

568

574

various vendors to remain platform independent and be portable from one

architecture to another.

569

575

architecture to another.

570

576

Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and

571

577

Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and

MAX/1500 \emph{i.e.} when the maximum allowed silicon area is fixed to 500, 1000

572

578

MAX/1500 \emph{i.e.} when the maximum allowed silicon area is fixed to 500, 1000

and 1500 arbitrary units. We have taken care to extract solely the resources used by

573

579

and 1500 arbitrary units. We have taken care to extract solely the resources used by

the FIR filters and remove additional processing blocks including FIFO and PL to

574

580

the FIR filters and remove additional processing blocks including FIFO and PL to

PS communication.

575

581

PS communication.

576

582

\begin{table}

577

583

\begin{table}

\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}

578

584

\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}

\label{tbl:resources_usage}

579

585

\label{tbl:resources_usage}

\centering

580

586

\centering

\begin{tabular}{|c|c|ccc|c|}

581

587

\begin{tabular}{|c|c|ccc|c|}

\hline

582

588

\hline

$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline

583

589

$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline

& LUT & 249 & 453 & 627 & \emph{17600} \\

584

590

& LUT & 249 & 453 & 627 & \emph{17600} \\

1 & BRAM & 1 & 1 & 1 & \emph{120} \\

585

591

1 & BRAM & 1 & 1 & 1 & \emph{120} \\

& DSP & 21 & 37 & 47 & \emph{80} \\ \hline

586

592

& DSP & 21 & 37 & 47 & \emph{80} \\ \hline

& LUT & 2374 & 5494 & 691 & \emph{17600} \\

587

593

& LUT & 2374 & 5494 & 691 & \emph{17600} \\

2 & BRAM & 2 & 2 & 2 & \emph{120} \\

588

594

2 & BRAM & 2 & 2 & 2 & \emph{120} \\

& DSP & 0 & 0 & 70 & \emph{80} \\ \hline

589

595

& DSP & 0 & 0 & 70 & \emph{80} \\ \hline

& LUT & 2443 & 3304 & 3521 & \emph{17600} \\

590

596

& LUT & 2443 & 3304 & 3521 & \emph{17600} \\

3 & BRAM & 3 & 3 & 3 & \emph{120} \\

591

597

3 & BRAM & 3 & 3 & 3 & \emph{120} \\

& DSP & 0 & 19 & 35 & \emph{80} \\ \hline

592

598

& DSP & 0 & 19 & 35 & \emph{80} \\ \hline

& LUT & 2634 & 3753 & 2557 & \emph{17600} \\

593

599

& LUT & 2634 & 3753 & 2557 & \emph{17600} \\

4 & BRAM & 4 & 4 & 4 & \emph{120} \\

594

600

4 & BRAM & 4 & 4 & 4 & \emph{120} \\

& DPS & 0 & 19 & 46 & \emph{80} \\ \hline

595

601

& DPS & 0 & 19 & 46 & \emph{80} \\ \hline

& LUT & 2423 & 3047 & 2847 & \emph{17600} \\

596

602

& LUT & 2423 & 3047 & 2847 & \emph{17600} \\

5 & BRAM & 5 & 5 & 5 & \emph{120} \\

597

603

5 & BRAM & 5 & 5 & 5 & \emph{120} \\

& DPS & 0 & 22 & 46 & \emph{80} \\ \hline

598

604

& DPS & 0 & 22 & 46 & \emph{80} \\ \hline

\end{tabular}

599

605

\end{tabular}

\end{table}

600

606

\end{table}

601

607

In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that,

602

608

In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that,

when the filters coefficients are small enough, or when the input size is small

603

609

when the filters coefficients are small enough, or when the input size is small

enough, Vivado optimized resource consumption by selecting multiplexers to

604

610

enough, Vivado optimized resource consumption by selecting multiplexers to

implement the multiplications instead of a DSP. In this case, it is quite difficult

605

611

implement the multiplications instead of a DSP. In this case, it is quite difficult

to compare the whole silicon budget.

606

612

to compare the whole silicon budget.

607

613

However, a rough estimation can be made with a simple equivalence. Looking at

608

614

However, a rough estimation can be made with a simple equivalence. Looking at

the first column (MAX/500), where the number of LUTs is quite stable for $n \geq 2$,

609

615

the first column (MAX/500), where the number of LUTs is quite stable for $n \geq 2$,

we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon

610

616

we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon

area use. With this equivalence, our 500 arbitraty units corresponds to 2500 LUTs,

611

617

area use. With this equivalence, our 500 arbitraty units corresponds to 2500 LUTs,

1000 arbitrary units corresponds to 5000 LUTs and 1500 arbitrary units corresponds

612

618

1000 arbitrary units corresponds to 5000 LUTs and 1500 arbitrary units corresponds

to 7300 LUTs. The conclusion is that the orders of magnitude of our arbitrary

613

619

to 7300 LUTs. The conclusion is that the orders of magnitude of our arbitrary

unit are quite good. The relatively small differences can probably be explained

614

620

unit are quite good. The relatively small differences can probably be explained

by the optimizations done by Vivado based on the detailed map of available processing resources.

615

621

by the optimizations done by Vivado based on the detailed map of available processing resources.

616

622

We present the computation time to solve the quadratic problem.

617

623

We present the computation time to solve the quadratic problem.

For each case, the filter solver software are executed with a Intel(R) Xeon(R) CPU E5606

618

624

For each case, the filter solver software are executed with a Intel(R) Xeon(R) CPU E5606

cadenced at 2.13~GHz. The CPU has 8 cores that are used by Gurobi to solve

619

625

cadenced at 2.13~GHz. The CPU has 8 cores that are used by Gurobi to solve

the quadratic problem.

620

626

the quadratic problem.

621

627

Table~\ref{tbl:area_time} shows the time needed to solve the quadratic

622

628

Table~\ref{tbl:area_time} shows the time needed to solve the quadratic

problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.

623

629

problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.

624

630

\begin{table}

625

631

\begin{table}

\caption{Time to solve the quadratic program with Gurobi}

626

632

\caption{Time to solve the quadratic program with Gurobi}

\label{tbl:area_time}

627

633

\label{tbl:area_time}

\centering

628

634

\centering

\begin{tabular}{|c|c|c|c|}\hline

629

635

\begin{tabular}{|c|c|c|c|}\hline

$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline

630

636

$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline

1 & 0.1~s & 0.1~s & 0.3~s \\

631

637

1 & 0.1~s & 0.1~s & 0.3~s \\

2 & 1.1~s & 2.2~s & 12~s \\

632

638

2 & 1.1~s & 2.2~s & 12~s \\

3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\

633

639

3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\

4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\

634

640

4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\

5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline

635

641

5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline

\end{tabular}

636

642

\end{tabular}

\end{table}

637

643

\end{table}

638

644

As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ?

639

645

As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ?

When the area is limited, the design exploration space is more limited and the solver is able to

640

646

When the area is limited, the design exploration space is more limited and the solver is able to

find an optimal solution faster. On the contrary, in the case of MAX/1500 with

641

647

find an optimal solution faster. On the contrary, in the case of MAX/1500 with

5~stages, we were not able to obtain a result after 40~hours of computation so we decided to stop.

642

648

5~stages, we were not able to obtain a result after 40~hours of computation so we decided to stop.

643

649

\section{Experiments with fixed rejection target}

644

650

\section{Experiments with fixed rejection target}

\label{sec:fixed_rej}

645

651

\label{sec:fixed_rej}

This section presents the results of complementary quadratic program which we

646

652

This section presents the results of complementary quadratic program which we

minimize the area occupation for a targeted noise level.

647

653

minimize the area occupation for a targeted noise level.

648

654

The experimental setup is also composed of three cases. The raw input is the same

649

655

The experimental setup is also composed of three cases. The raw input is the same

as previous section, a PRN generator, which fixes the input data size $\Pi^I$.

650

656

as previous section, a PRN generator, which fixes the input data size $\Pi^I$.

Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60 or 80~dB.

651

657

Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60 or 80~dB.

Hence, the three cases have been named: MIN/40, MIN/60, MIN/80.

652

658

Hence, the three cases have been named: MIN/40, MIN/60, MIN/80.

The number of configurations $p$ is the same as previous section.

653

659

The number of configurations $p$ is the same as previous section.

654

660

Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40.

655

661

Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40.

Table~\ref{tbl:gurobi_min_60} shows the results obtained by the filter solver for MIN/60.

656

662

Table~\ref{tbl:gurobi_min_60} shows the results obtained by the filter solver for MIN/60.

Table~\ref{tbl:gurobi_min_80} shows the results obtained by the filter solver for MIN/80.

657

663

Table~\ref{tbl:gurobi_min_80} shows the results obtained by the filter solver for MIN/80.

658

664

\renewcommand{\arraystretch}{1.4}

659

665

\renewcommand{\arraystretch}{1.4}

660

666

\begin{table}

661

667

\begin{table}

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}

662

668

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}

\label{tbl:gurobi_min_40}

663

669

\label{tbl:gurobi_min_40}

\centering

664

670

\centering

{\scalefont{0.77}

665

671

{\scalefont{0.77}

\begin{tabular}{|c|ccccc|c|c|}

666

672

\begin{tabular}{|c|ccccc|c|c|}

\hline

667

673

\hline

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

668

674

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

\hline

669

675

\hline

1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\

670

676

1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\

2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\

671

677

2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\

3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\

672

678

3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\

4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\

673

679

4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\

\hline

674

680

\hline

\end{tabular}

675

681

\end{tabular}

}

676

682

}

\end{table}

677

683

\end{table}

678

684

\begin{table}

679

685

\begin{table}

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}

680

686

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}

\label{tbl:gurobi_min_60}

681

687

\label{tbl:gurobi_min_60}

\centering

682

688

\centering

{\scalefont{0.77}

683

689

{\scalefont{0.77}

\begin{tabular}{|c|ccccc|c|c|}

684

690

\begin{tabular}{|c|ccccc|c|c|}

\hline

685

691

\hline

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

686

692

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

\hline

687

693

\hline

1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\

688

694

1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\

2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\

689

695

2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\

3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\

690

696

3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\

4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\

691

697

4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\

5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\

692

698

5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\

\hline

693

699

\hline

\end{tabular}

694

700

\end{tabular}

}

695

701

}

\end{table}

696

702

\end{table}

697

703

\begin{table}

698

704

\begin{table}

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}

699

705

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}

\label{tbl:gurobi_min_80}

700

706

\label{tbl:gurobi_min_80}

\centering

701

707

\centering

{\scalefont{0.77}

702

708

{\scalefont{0.77}

\begin{tabular}{|c|ccccc|c|c|}

703

709

\begin{tabular}{|c|ccccc|c|c|}

\hline

704

710

\hline

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

705

711

$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\

\hline

706

712

\hline

1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\

707

713

1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\

2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\

708

714

2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\

3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\

709

715

3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\

4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\

710

716

4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\

5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\

711

717

5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\

\hline

712

718

\hline

\end{tabular}

713

719

\end{tabular}

}

714

720

}

\end{table}

715

721

\end{table}

\renewcommand{\arraystretch}{1}

716

722

\renewcommand{\arraystretch}{1}

717

723

From these tables, we can first state that all configuration reach the target rejection

718

724

From these tables, we can first state that all configuration reach the target rejection

level and more we have stages lesser is the area occupied in arbitrary unit.

719

725

level and more we have stages lesser is the area occupied in arbitrary unit.

Futhermore, the area of the monolithic filter is twice bigger than the two cascaded.

720

726

Futhermore, the area of the monolithic filter is twice bigger than the two cascaded.

More generally, more there is filters lower is the occupied area.

721

727

More generally, more there is filters lower is the occupied area.

722

728

Like in previous section, the solver choose always a little filter as first

723

729

Like in previous section, the solver choose always a little filter as first

filter stage and the second one is often the biggest filter. this choice can be explain

724

730

filter stage and the second one is often the biggest filter. this choice can be explain

as the previous section. The solver uses just enough bits to not degrade the input

725

731

as the previous section. The solver uses just enough bits to not degrade the input

signal and in second filter it can choose a better filter to improve rejection without

726

732

signal and in second filter it can choose a better filter to improve rejection without

have too bits in the output data.

727

733

have too bits in the output data.

728

734

For the specific case in MIN/40 for $n = 5$ the solver has determined that the optimal

729

735

For the specific case in MIN/40 for $n = 5$ the solver has determined that the optimal

number of filter is 4 so it not chose any configuration in last filter. Hence this

730

736

number of filter is 4 so it not chose any configuration in last filter. Hence this

solution is equivalent to the result for $n = 4$.

731

737

solution is equivalent to the result for $n = 4$.

732

738

The following graphs present the rejection for real data on the FPGA. In all following

733

739

The following graphs present the rejection for real data on the FPGA. In all following

figures, the solid line represents the actual rejection of the filtered

734

740

figures, the solid line represents the actual rejection of the filtered

data on the FPGA as measured experimentally and the dashed line are the noise level

735

741

data on the FPGA as measured experimentally and the dashed line are the noise level

given by the quadratic solver.

736

742

given by the quadratic solver.

737

743

Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40.

738

744

Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40.

Figure~\ref{fig:min_60} shows the rejection of the different configurations in the case of MIN/60.

739

745

Figure~\ref{fig:min_60} shows the rejection of the different configurations in the case of MIN/60.

Figure~\ref{fig:min_80} shows the rejection of the different configurations in the case of MIN/80.

740

746

Figure~\ref{fig:min_80} shows the rejection of the different configurations in the case of MIN/80.

741

747

\begin{figure}

742

748

\begin{figure}

\centering

743

749

\centering

\includegraphics[width=\linewidth]{images/min_40}

744

750

\includegraphics[width=\linewidth]{images/min_40}

\caption{Signal spectrum for MIN/40}

745

751

\caption{Signal spectrum for MIN/40}

\label{fig:min_40}

746

752

\label{fig:min_40}

\end{figure}

747

753

\end{figure}

748

754

\begin{figure}

749

755

\begin{figure}

\centering

750

756

\centering

\includegraphics[width=\linewidth]{images/min_60}

751

757

\includegraphics[width=\linewidth]{images/min_60}

\caption{Signal spectrum for MIN/60}

752

758

\caption{Signal spectrum for MIN/60}

GITLAB

jfriedt / IFCS2018 article

Final draft.