jfriedt / IFCS2018 article

% fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee

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% demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de

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% rejection par bit et perte si moins de bits que rejection/6

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% developper programme lineaire en incluant le decalage de bits

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% insister que avant on etait synthetisable mais pas implementable, alors que maintenant on

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% implemente et on demontre que ca tourne

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% gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ?

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% Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer

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% (zedboard ou redpit)

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% label schema : verifier que "argumenter de la cascade de FIR" est fait

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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}

\pagestyle{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. The

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required and keeping timing constraints within the targeted measurement bandwidths. The

presented technique is applicable to scheduling any sequence of processing blocks characterized

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presented technique is applicable to scheduling any sequence of processing blocks characterized

by a throughput, resource occupation and performance tabulated as a function of configuration

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by a throughput, resource occupation and performance tabulated as a function of configuration

characateristics, as is the case for filters with their coefficients and resolution yielding

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characateristics, as is the case for filters with their coefficients and resolution yielding

rejection and number of multipliers.

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rejection and number of multipliers.

\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 filters for their unconditional stability and ease of design. A FIR filter is defined

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We select FIR filters 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 offers more degrees of freedom since

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processor architecture, implementing such a filter on an FPGA offers 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

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the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language

(VHDL) level.

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(VHDL) level.

Since latency is not an issue in a openloop phase noise characterization instrument,

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Since latency is not an issue in a openloop phase noise characterization instrument,

the large

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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/zero_values}

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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: processing

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taps become null, making the large number of coefficients irrelevant: processing

resources

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resources

are hence saved by shrinking the filter length. This tradeoff aimed at minimizing resources

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are hence saved 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}

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Our objective is to develop a new methodology applicable to any Digital Signal Processing (DSP)

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Our objective is to develop a new methodology applicable to any Digital Signal Processing (DSP)

chain obtained by assembling basic processing blocks, with hardware and manufacturer independence.

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chain obtained by assembling basic processing blocks, with hardware and manufacturer independence.

Achieving such a target requires defining an abstract model to represent some basic properties

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Achieving such a target requires defining an abstract model to represent some basic properties

of DSP blocks such as performance (i.e. rejection or ripples in the bandpass for filters) and

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of DSP blocks such as performance (i.e. rejection or ripples in the bandpass for filters) and

resource occupation. These abstract properties, not necessarily related to the detailed hardware

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resource occupation. These abstract properties, not necessarily related to the detailed hardware

implementation of a given platform, will feed a scheduler solver aimed at assembling the optimum

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implementation of a given platform, will feed a scheduler solver aimed at assembling the optimum

target, whether in terms of maximizing performance for a given arbitrary resource occupation, or

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target, whether in terms of maximizing performance for a given arbitrary resource occupation, or

minimizing resource occupation for a given performance. In our approach, the solution of the

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minimizing resource occupation for a given performance. In our approach, the solution of the

solver is then synthesized using the dedicated tool provided by each platform manufacturer

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solver is then synthesized using the dedicated tool provided by each platform manufacturer

to assess the validity of our abstract resource occupation indicator, and the result of running

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to assess the validity of our abstract resource occupation indicator, and the result of running

the DSP chain on the FPGA allows for assessing the performance of the scheduler. We emphasize

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the DSP chain on the FPGA allows for assessing the performance of the scheduler. We emphasize

that all solutions found by the solver are synthesized and executed on hardware at the end

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that all solutions found by the solver are synthesized and executed on hardware at the end

of the analysis.

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of the analysis.

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In this demonstration, we focus on only two operations: filtering and shifting the number of

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In this demonstration, we focus on only two operations: filtering and shifting the number of

bits needed to represent the data along the processing chain.

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bits needed to represent the data along the processing chain.

We have chosen these basic operations because shifting and the filtering have already been studied

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We have chosen these basic operations because shifting and the filtering have already been studied

in the literature \cite{lim_1996, lim_1988, young_1992, smith_1998} providing a framework for

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in the literature \cite{lim_1996, lim_1988, young_1992, smith_1998} providing a framework for

assessing our results. Furthermore, filtering is a core step in any radiofrequency frontend

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assessing our results. Furthermore, filtering is a core step in any radiofrequency frontend

requiring pipelined processing at full bandwidth for the earliest steps, including for

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requiring pipelined processing at full bandwidth for the earliest steps, including for

time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}.

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time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}.

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Addressing only two operations allows for demonstrating the methodology but should not be

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Addressing only two operations allows for demonstrating the methodology but should not be

considered as a limitation of the framework which can be extended to assembling any number

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considered as a limitation of the framework which can be extended to assembling any number

of skeleton blocks as long as performance and resource occupation can be determined.

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of skeleton blocks as long as performance and resource occupation can be determined.

Hence,

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Hence,

in this paper we will apply our methodology on simple DSP chains: a white noise input signal

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in this paper we will apply our methodology on simple DSP chains: a white noise input signal

is generated using a Pseudo-Random Number (PRN) generator or by sampling a wideband (125~MS/s)

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is generated using a Pseudo-Random Number (PRN) generator or by sampling a wideband (125~MS/s)

14-bit Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor. Once samples have been

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14-bit Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor. Once samples have been

digitized at a rate of 125~MS/s, filtering is applied to qualify the processing block performance --

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digitized at a rate of 125~MS/s, filtering is applied to qualify the processing block performance --

practically meeting the radiofrequency frontend requirement of noise and bandwidth reduction

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practically meeting the radiofrequency frontend requirement of noise and bandwidth reduction

by filtering and decimating. Finally, bursts of filtered samples are stored for post-processing,

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by filtering and decimating. Finally, bursts of filtered samples are stored for post-processing,

allowing to assess either filter rejection for a given resource usage, or validating the rejection

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allowing to assess either filter rejection for a given resource usage, or validating the rejection

when implementing a solution minimizing resource occupation.

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when implementing a solution minimizing resource occupation.

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The first step of our approach is to model the DSP chain. Since we aim at only optimizing

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The first step of our approach is to model the DSP chain. Since we aim at only optimizing

the filtering part of the signal processing chain, we have not included the PRN generator or the

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the filtering part of the signal processing chain, we have not included the PRN generator or the

ADC in the model: the input data size and rate are considered fixed and defined by the hardware.

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ADC in the model: the input data size and rate are considered fixed and defined by the hardware.

The filtering can be done in two ways, either by considering a single monolithic FIR filter

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The filtering can be done in two ways, either by considering a single monolithic FIR filter

requiring many coefficients to reach the targeted noise rejection ratio, or by

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requiring many coefficients to reach the targeted noise rejection ratio, or by

cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.

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cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.

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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}

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A cascade of filters is composed of $n$ FIR stages. In stage $i$ ($1 \leq i \leq n$)

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A cascade of filters is composed of $n$ FIR stages. In stage $i$ ($1 \leq i \leq n$)

the FIR has $C_i$ coefficients and each coefficient is an integer value with $\pi^C_i$

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the FIR has $C_i$ coefficients and each coefficient is an integer value with $\pi^C_i$

bits while the filtered data are shifted by $\pi^S_i$ bits. We define also $\pi^-_i$ as

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bits while the filtered data are shifted by $\pi^S_i$ 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 $i$ has been characterized through numerical simulation as able to reject $F(C_i, \pi_i^C)$ dB.

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FIR $i$ has been characterized through numerical simulation as able to reject $F(C_i, \pi_i^C)$ dB.

This rejection has been computed using GNU Octave software FIR coefficient design functions

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This rejection has been computed using GNU Octave software FIR coefficient design functions

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

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(\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 others are coded on much 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 others are coded on much 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

transfer function.

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transfer function.

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

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

the FIR magnitude exhibits two parts: we focus here on the transitions width and the rejection rather than on the

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the FIR magnitude exhibits two parts: we focus here on the transitions width and the rejection rather than on the

bandpass ripples as emphasized in \cite{lim_1988,lim_1996}. Throughout this demonstration,

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bandpass ripples as emphasized in \cite{lim_1988,lim_1996}. Throughout this demonstration,

we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\%

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we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\%

of the Nyquist frequency to the end of the band, as would be typically selected to prevent

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of the Nyquist frequency to the end of the band, as would be typically selected to prevent

aliasing before decimating the dataflow by 2. The method is however generalized to any filter

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aliasing before decimating the dataflow by 2. The method is however generalized to any filter

shape as long as it is defined from the initial modeling steps: Fig. \ref{fig:rejection_pyramid}

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shape as long as it is defined from the initial modeling steps: Fig. \ref{fig:rejection_pyramid}

as described below is indeed unique for each filter shape.

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as described below is indeed unique for each filter shape.

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\begin{figure}

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\begin{figure}

\begin{center}

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\begin{center}

\scalebox{0.8}{

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\scalebox{0.8}{

\centering

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245

\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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}

\end{center}

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\end{center}

\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 define the passband and the stopband characteristics.

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

% r2.7

293

Initial considered criteria include the mean value of the stopband rejection which yields unacceptable results since notches

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Initial considered criteria include the mean value of the stopband rejection which yields unacceptable results since notches

overestimate the rejection capability of the filter.

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overestimate the rejection capability of the filter.

% Furthermore, the losses within

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An intermediate criterion considered the maximal rejection within the stopband, to which the sum of the absolute values

% the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients.

297

{\color{red} An intermediate criterion considered the maximal rejection within the stopband, to which the sum of the absolute values

298

% JMF : je fais le choix de remplacer minimal par maximal rejection pour etre coherent avec caption de Fig custom_criterion mais surtout parceque

299

% rejection me semble plus convaincant si on la maximise (il me semble que -120 dB de S21 signifie 120 dB de rejection donc on veut maximiser)

300

within the passband is subtracted to avoid filters with excessive ripples, normalized to the

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within the passband is subtracted to avoid filters with excessive ripples, normalized to the

bin width to remain consistent with the passband criterion (dBc/Hz units in all cases).

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bin width to remain consistent with the passband criterion (dBc/Hz units in all cases).

In this case, cascading too many filters with individual excessive ($>$ 1~dB) passband ripples

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In this case, cascading too many filters with individual excessive ($>$ 1~dB) passband ripples

led to unacceptable ($>$ 10~dB) final ripple levels, especially close to the transition band.

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led to unacceptable ($>$ 10~dB) final ripple levels, especially close to the transition band.

Hence, the final criterion considers the minimal rejection in the stopband to which the

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Hence, the final criterion considers the minimal rejection in the stopband to which the

the maximal amplitude in the passband (maximum value minus the minimum value) is substracted, with

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the maximal amplitude in the passband (maximum value minus the minimum value) is substracted, with

a 1~dB threshold on the latter quantity over which the filter is discarded.}

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a 1~dB threshold on the latter quantity over which the filter is discarded.

% Our final criterion to compute the filter rejection considers

308

% % r2.8 et r2.2 r2.3

309

% the minimal rejection within the stopband, to which the sum of the absolute values

310

% within the passband is subtracted to avoid filters with excessive ripples, normalized to the

311

% bin width to remain consistent with the passband criterion (dBc/Hz units in all cases).

312

With this

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With this

criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.

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criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.

{\color{red} The best filter has a correct rejection estimation and the worst filter

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The best filter has a correct rejection estimation and the worst filter

is discarded based on the excessive passband ripple criterion.}

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is discarded based on the excessive passband ripple criterion.

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295

% \begin{figure}

318

% \centering

319

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

320

% \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters}

321

% \label{fig:mean_criterion}

322

% \end{figure}

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324

\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/custom_criterion}

\caption{\color{red}Selected filter qualification criterion computed as the maximum rejection in the stopband

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\caption{Selected filter qualification criterion computed as the maximum rejection in the stopband

minus the maximal ripple amplitude in the passband with a $>$ 1~dB threshold above which the filter is discarded:

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minus the maximal ripple amplitude in the passband with a $>$ 1~dB threshold above which the filter is discarded:

comparison between monolithic filter (blue, rejected in this case) and cascaded filters (red).}

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comparison between monolithic filter (blue, rejected in this case) and cascaded filters (red).}

\label{fig:custom_criterion}

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\label{fig:custom_criterion}

\end{figure}

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303

\end{figure}

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Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps

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Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps

and estimate their rejection. Figure~\ref{fig:rejection_pyramid} exhibits the

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and estimate their rejection. Figure~\ref{fig:rejection_pyramid} exhibits the

rejection as a function of the number of coefficients and the number of bits representing these coefficients.

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rejection as a function of the number of coefficients and the number of bits representing these coefficients.

The curve shaped as a pyramid exhibits optimum configurations sets at the vertex where both edges meet.

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The curve shaped as a pyramid exhibits optimum configurations sets at the vertex where both edges meet.

Indeed for a given number of coefficients, increasing the number of bits over the edge will not improve the rejection.

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Indeed for a given number of coefficients, increasing the number of bits over the edge will not improve the rejection.

Conversely when setting the a given number of bits, increasing the number of coefficients will not improve

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Conversely when setting the a given number of bits, increasing the number of coefficients will not improve

the rejection. Hence the best coefficient set are on the vertex of the pyramid. {\color{red} Notice that the word length

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the rejection. Hence the best coefficient set are on the vertex of the pyramid. Notice that the word length

and number of coefficients do not start at 1: filters with too few coefficients or too little tap word size are rejected

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and number of coefficients do not start at 1: filters with too few coefficients or too little tap word size are rejected

by the excessive ripple constraint of the criterion. Hence, the size of the pyramid is significantly reduced by discarding

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by the excessive ripple constraint of the criterion. Hence, the size of the pyramid is significantly reduced by discarding

these filters and so is the solution search space.} % ajout JMF

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these filters and so is the solution search space.

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\begin{figure}

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\begin{figure}

\centering

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317

\centering

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

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

\caption{\color{red}Filter rejection as a function of number of coefficients and number of bits

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\caption{Filter rejection as a function of number of coefficients and number of bits

: this lookup table will be used to identify which filter parameters -- number of bits

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: this lookup table will be used to identify which filter parameters -- number of bits

representing coefficients and number of coefficients -- best match the targeted transfer function. {\color{red}Filters

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representing coefficients and number of coefficients -- best match the targeted transfer function. Filters

with fewer than 10~taps or with coefficients coded on fewer than 5~bits are discarded due to excessive

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with fewer than 10~taps or with coefficients coded on fewer than 5~bits are discarded due to excessive

ripples in the passband.}} % ajout JMF

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ripples in the passband.}

\label{fig:rejection_pyramid}

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\label{fig:rejection_pyramid}

\end{figure}

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\end{figure}

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Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps),

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Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps),

we have a problem when we cascade filters and estimate the criterion as a sum two or more individual criteria.

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we have a problem when we cascade filters and estimate the criterion as a sum two or more individual criteria.

If the FIR filter coefficients are the same between the stages, we have:

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If the FIR filter coefficients are the same between the stages, we have:

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

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

But selecting two different sets of coefficient will yield a more complex situation in which

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But selecting two different sets of coefficient will yield a more complex situation in which

the previous relation is no longer valid as illustrated on figure~\ref{fig:sum_rejection}. The red and blue curves

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the previous relation is no longer valid as illustrated on figure~\ref{fig:sum_rejection}. The red and blue curves

are two different filters with maximums and notches not located at the same frequency offsets.

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are two different filters with maximums and notches not located at the same frequency offsets.

Hence when summing the transfer functions, the resulting rejection shown as the dashed yellow line is improved

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Hence when summing the transfer functions, the resulting rejection shown as the dashed yellow line is improved

with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.

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with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.

% r2.9

365

Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection

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Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection

criteria of each filter. However since the individual filter rejection sum underestimates the rejection capability of the cascade,

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criteria of each filter. However since the individual filter rejection sum underestimates the rejection capability of the cascade,

% r2.10

368

this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability

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this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability

of the filter cascade to meet design criteria.

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of the filter cascade to meet design criteria.

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\begin{figure}

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\begin{figure}

\centering

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\centering

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

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

\caption{Transfer function of individual filters and after cascading the two filters,

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\caption{Transfer function of individual filters and after cascading the two filters,

demonstrating that the selected criterion of maximum rejection in the bandstop (horizontal

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demonstrating that the selected criterion of maximum rejection in the bandstop (horizontal

lines) is met. Notice that the cascaded filter has better rejection than summing the bandstop

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lines) is met. Notice that the cascaded filter has better rejection than summing the bandstop

maximum of each individual filter.

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maximum of each individual filter.

}

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}

\label{fig:sum_rejection}

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\label{fig:sum_rejection}

\end{figure}

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\end{figure}

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Finally in our case, we consider that the input signal are fully known. The

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Finally in our case, we consider that the input signal are fully known. The

resolution of the input data stream are fixed and still the same for all experiments

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resolution of the input data stream are fixed and still the same for all experiments

in this paper.

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in this paper.

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Based on this analysis, we address the estimate of resource consumption (called

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Based on this analysis, we address the estimate of resource consumption (called

% r2.11

388

silicon area -- in the case of FPGAs this means processing cells) as a function of

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silicon area -- in the case of FPGAs this means processing cells) as a function of

filter characteristics. As a reminder, we do not aim at matching actual hardware

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filter characteristics. As a reminder, we do not aim at matching actual hardware

configuration but consider an arbitrary silicon area occupied by each processing function,

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configuration but consider an arbitrary silicon area occupied by each processing function,

and will assess after synthesis the adequation of this arbitrary unit with actual

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and will assess after synthesis the adequation of this arbitrary unit with actual

hardware resources provided by FPGA manufacturers. The sum of individual processing

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hardware resources provided by FPGA manufacturers. The sum of individual processing

unit areas is constrained by a total silicon area representative of FPGA global resources.

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unit areas is constrained by a total silicon area representative of FPGA global resources.

Formally, variable $a_i$ is the area taken by filter~$i$

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Formally, variable $a_i$ is the area taken by filter~$i$

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

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

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

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

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\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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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 used by a filter, considered as the area of the FIR since the Shifter is

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the area used by a filter, considered as the area of the FIR since the Shifter is

assumed not to require significant resources. We consider that the FIR needs $C_i$ registers of size

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assumed not to require significant resources. 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 with the coefficients. Equation~\ref{eq:rejectiondef} gives the

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

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

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

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

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previously. The Shifter does not introduce negative rejection as we will 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 by 6~dB. We add one to take the sign bit

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increases the maximum rejection level by 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 an additional 6~dB rejection rise. A totally equivalent equation is:

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shift bit would cause an additional 6~dB rejection rise. 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 number of bits of the global input.

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

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This model is non-linear since we multiply some variable with another variable

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This model is non-linear since we multiply some variable with another variable

and it is even non-quadratic, as the cost function $F$ does not have a known

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and it is even non-quadratic, as the cost function $F$ does not have a known

linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.

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linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.

% AH: conflit merge

436

% This variable must be defined by the user, it represent the number of different

437

% set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}

438

% functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid}

439

% to restrict the number of configurations. Indeed, it is useless to have too many coefficients or

440

% too many bits, hence we take the configurations close to edge of pyramid. Thank to theses

441

% configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant

442

% and the function $F$ can be estimate for each configurations

443

% thanks our rejection criterion. We also defined binary

444

This variable $p$ is defined by the user, and represents the number of different

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404

This variable $p$ is defined by the user, and represents the number of different

set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1}

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set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1}

functions from GNU Octave) based on the targeted filter characteristics and implementation

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functions from GNU Octave) based on the targeted filter characteristics and implementation

assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and

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assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and

$\pi_{ij}^C$ become constants and

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$\pi_{ij}^C$ become constants and

we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table)

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we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table)

for each configurations thanks to the rejection criterion. We also define the binary

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for each configurations thanks to the rejection criterion. We also define the 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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413

\begin{align}

455

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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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415

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}

460

419

\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.

465

424

% JM: conflict merge

466

% However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}

467

% we multiply

468

% $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can

469

% linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size,

470

% we define $0 < \pi_i^- \leq 128$ which is the maximum data size whose estimation is

471

% assumed on hardware characteristics.

472

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

473

% model is able to linearize the model provided as is. This model

474

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

475

The problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}

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The problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}

we multiply

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we multiply

$\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can

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427

$\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can

linearize this multiplication. The following formula shows how to linearize

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linearize this multiplication. The following formula shows how to linearize

this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$):

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429

this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$):

\begin{equation*}

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\begin{equation*}

m = x \times y \implies

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m = x \times y \implies

\left \{

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432

\left \{

\begin{split}

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433

\begin{split}

m & \geq 0 \\

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m & \geq 0 \\

m & \leq y \times X^{max} \\

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m & \leq y \times X^{max} \\

m & \leq x \\

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m & \leq x \\

m & \geq x - (1 - y) \times X^{max} \\

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m & \geq x - (1 - y) \times X^{max} \\

\end{split}

489

438

\end{split}

\right .

490

439

\right .

\end{equation*}

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\end{equation*}

So if we bound up $\pi_i^-$ by 128~bits which is the maximum data size whose estimation is

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So if we bound up $\pi_i^-$ by 128~bits which is the maximum data size whose estimation is

assumed on hardware characteristics,

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assumed on hardware characteristics,

the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize

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the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize

for us the quadratic problem so the model is left as is. This model

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for us the quadratic problem so the model is left as is. This model

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

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

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

498

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

499

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

500

% % r2.12

501

% This variable must be defined by the user, it represent the number of different

502

% set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}

503

% functions from GNU Octave).

504

% We define binary

505

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

506

% and 0 otherwise. The new equations are as follows:

507

%

508

% \begin{align}

509

% 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} \\

510

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

511

% \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} \\

512

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

513

% \end{align}

514

%

515

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

516

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

517

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

518

%

519

% % r2.13

520

% This modified model is quadratic since we multiply two variables in the

521

% equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.

522

% The Gurobi

523

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

524

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

525

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

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527

Two problems will be addressed using the workflow described in the next section: on the one

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Two problems will be addressed using the workflow described in the next section: on the one

hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary

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hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary

silicon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area

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silicon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area

for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the

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for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the

objective function is replaced with:

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451

objective function is replaced with:

\begin{align}

533

452

\begin{align}

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

534

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

\end{align}

535

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\end{align}

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

536

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

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

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

rejection required.

538

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rejection required.

539

458

\begin{align}

540

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\begin{align}

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

541

460

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

\end{align}

542

461

\end{align}

543

462

\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 sections~\ref{sec:fixed_area}

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

and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved

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and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved

in the computation of the results.

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in the computation of the results.

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469

\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 allowing for

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\caption{Design workflow from the input parameters to the results allowing for

a fully automated optimal solution search.}

576

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a fully automated optimal solution search.}

\label{fig:workflow}

577

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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 estimate the optimal results.

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the quadratic programs and uses the Gurobi solver to estimate 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

587

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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) in a language compatible

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(the raw signal data) to the end (the filtered data) in a language compatible

with proprietary synthesis software, namely Vivado for Xilinx and Quartus for

589

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with proprietary synthesis software, namely Vivado for Xilinx and Quartus for

Intel/Altera. The raw input data generated from a 20-bit Pseudo Random Number (PRN)

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Intel/Altera. The raw input data generated from a 20-bit 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.

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

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

consecutive bursts of data for post-processing, mainly assessing the consistency of the

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consecutive bursts of data for post-processing, mainly assessing the consistency of the

implemented FIR cascade transfer function with the design criteria and the expected

598

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implemented FIR cascade transfer function with the design criteria and the expected

transfer function.

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transfer function.

600

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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 LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to

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FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to

provide a broadband noise source.

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provide a broadband noise source.

The board runs the Linux kernel and surrounding environment produced from the

606

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The board runs the Linux kernel and surrounding environment produced from the

Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring

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Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring

the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and

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the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and

fetching the results is automated.

609

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fetching the results is automated.

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529

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

611

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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,

612

531

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

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

613

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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}).

614

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

615

534

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

616

535

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

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

617

536

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

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

618

537

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

and the different configurations can be compared.

619

538

and the different configurations can be compared.

620

539

\section{Maximizing the rejection at fixed silicon area}

621

540

\section{Maximizing the rejection at fixed silicon area}

\label{sec:fixed_area}

622

541

\label{sec:fixed_area}

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

623

542

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.

Such results allow for understanding the choices made by the solver to compute its solutions.

625

544

Such results allow for understanding the choices made by the solver to compute its solutions.

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545

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

627

546

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 $\Pi^I$.

628

547

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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549

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

The number of configurations $p$ is {\color{red}1133}, with $C_i$ ranging from 3 to 60 and $\pi^C$

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550

The number of configurations $p$ is 1133, 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

632

551

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

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553

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

635

554

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.

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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.

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

638

557

\renewcommand{\arraystretch}{1.4}

639

558

\renewcommand{\arraystretch}{1.4}

640

559

\begin{table}

641

560

\begin{table}

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

642

561

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

\label{tbl:gurobi_max_500}

643

562

\label{tbl:gurobi_max_500}

\centering

644

563

\centering

{\color{red}

645

564

{\scalefont{0.77}

\scalefont{0.77}

646

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

647

565

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

\hline

648

566

\hline

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

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567

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

\hline

650

568

\hline

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

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

2 & (3, 5, 18) & (33, 10, 0) & - & - & - & 48~dB & 492 \\

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570

2 & (3, 5, 18) & (33, 10, 0) & - & - & - & 48~dB & 492 \\

3 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\

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571

3 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\

4 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\

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572

4 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\

5 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\

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573

5 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\

\hline

656

574

\hline

\end{tabular}

657

575

\end{tabular}

}

658

576

}

\end{table}

659

577

\end{table}

660

578

\begin{table}

661

579

\begin{table}

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

662

580

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

\label{tbl:gurobi_max_1000}

663

581

\label{tbl:gurobi_max_1000}

\centering

664

582

\centering

{\color{red}\scalefont{0.77}

665

583

{\scalefont{0.77}

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

666

584

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

\hline

667

585

\hline

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

668

586

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

\hline

669

587

\hline

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

670

588

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

2 & (15, 8, 17) & (35, 11, 0) & - & - & - & 80~dB & 990 \\

671

589

2 & (15, 8, 17) & (35, 11, 0) & - & - & - & 80~dB & 990 \\

3 & (3, 13, 26) & (31, 9, 1) & (27, 9, 0) & - & - & 92~dB & 999 \\

672

590

3 & (3, 13, 26) & (31, 9, 1) & (27, 9, 0) & - & - & 92~dB & 999 \\

4 & (3, 5, 18) & (19, 7, 1) & (19, 7, 0) & (19, 7, 0) & - & 98~dB & 994 \\

673

591

4 & (3, 5, 18) & (19, 7, 1) & (19, 7, 0) & (19, 7, 0) & - & 98~dB & 994 \\

5 & (3, 5, 18) & (19, 7, 1) & (19, 7, 0) & (19, 7, 0) & - & 98~dB & 994 \\

674

592

5 & (3, 5, 18) & (19, 7, 1) & (19, 7, 0) & (19, 7, 0) & - & 98~dB & 994 \\

\hline

675

593

\hline

\end{tabular}

676

594

\end{tabular}

}

677

595

}

\end{table}

678

596

\end{table}

679

597

\begin{table}

680

598

\begin{table}

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

681

599

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

\label{tbl:gurobi_max_1500}

682

600

\label{tbl:gurobi_max_1500}

\centering

683

601

\centering

{\color{red}\scalefont{0.77}

684

602

{\scalefont{0.77}

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

685

603

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

\hline

686

604

\hline

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

687

605

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

\hline

688

606

\hline

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

689

607

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

2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 102~dB & 1489 \\

690

608

2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 102~dB & 1489 \\

3 & (15, 9, 18) & (31, 8, 0) & (27, 9, 0) & - & - & 116~dB & 1488 \\

691

609

3 & (15, 9, 18) & (31, 8, 0) & (27, 9, 0) & - & - & 116~dB & 1488 \\

4 & (3, 9, 22) & (31, 9, 1) & (27, 9, 0) & (19, 7, 0) & - & 125~dB & 1500 \\

692

610

4 & (3, 9, 22) & (31, 9, 1) & (27, 9, 0) & (19, 7, 0) & - & 125~dB & 1500 \\

5 & (3, 9, 22) & (31, 9, 1) & (27, 9, 0) & (19, 7, 0) & - & 125~dB & 1500 \\

693

611

5 & (3, 9, 22) & (31, 9, 1) & (27, 9, 0) & (19, 7, 0) & - & 125~dB & 1500 \\

\hline

694

612

\hline

\end{tabular}

695

613

\end{tabular}

}

696

614

}

\end{table}

697

615

\end{table}

698

616

\renewcommand{\arraystretch}{1}

699

617

\renewcommand{\arraystretch}{1}

700

618

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

701

619

By analyzing these tables, we can first state that we reach an optimal solution

% the cascaded FIR filters, the better the rejection.

702

{\color{red} By analyzing these tables, we can first state that we reach an optimal solution

703

for each case : $n = 3$ for MAX/500, and $n = 4$ for MAX/1000 and MAX/1500. Moreover

704

620

for each case : $n = 3$ for MAX/500, and $n = 4$ for MAX/1000 and MAX/1500. Moreover

the cascaded filters always exhibit better performance than the monolithic solution.}

705

621

the cascaded filters always exhibit better performance than the monolithic solution.

It was an expected result as it has

706

622

It was an expected result as it has

been previously observed that many small filters are better than

707

623

been previously observed that many small filters are better than

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

708

624

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

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

709

625

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

coefficients in the cascaded approach.

710

626

coefficients in the cascaded approach.

711

627

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

712

628

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 with more coefficients

713

629

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

or more bits per coefficient.

714

630

or more bits per coefficient.

715

631

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

716

632

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

717

633

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

718

634

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}

719

635

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

gives the relation between both values.

720

636

gives the relation between both values.

721

637

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

722

638

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

723

639

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

724

640

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

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

725

641

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 levels

726

642

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

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

727

643

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

728

644

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

729

645

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.

730

646

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.

731

647

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

732

648

% \begin{figure}

733

% \centering

734

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

735

% \caption{Signal spectrum for MAX/500}

736

% \label{fig:max_500_result}

737

% \end{figure}

738

%

739

% \begin{figure}

740

% \centering

741

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

742

% \caption{Signal spectrum for MAX/1000}

743

% \label{fig:max_1000_result}

744

% \end{figure}

745

%

746

% \begin{figure}

747

% \centering

748

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

749

% \caption{Signal spectrum for MAX/1500}

750

% \label{fig:max_1500_result}

751

% \end{figure}

752

753

% r2.14 et r2.15 et r2.16

754

\begin{figure}

755

649

\begin{figure}

\centering

756

650

\centering

\begin{subfigure}{\linewidth}

757

651

\begin{subfigure}{\linewidth}

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

758

652

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

\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving

759

653

\caption{Filter transfer functions for varying number of cascaded filters solving

the MAX/500 problem of maximizing rejection for a given resource allocation (500~arbitrary units).}

760

654

the MAX/500 problem of maximizing rejection for a given resource allocation (500~arbitrary units).}

\label{fig:max_500_result}

761

655

\label{fig:max_500_result}

\end{subfigure}

762

656

\end{subfigure}

763

657

\begin{subfigure}{\linewidth}

764

658

\begin{subfigure}{\linewidth}

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

765

659

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

\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving

766

660

\caption{Filter transfer functions for varying number of cascaded filters solving

the MAX/1000 problem of maximizing rejection for a given resource allocation (1000~arbitrary units).}

767

661

the MAX/1000 problem of maximizing rejection for a given resource allocation (1000~arbitrary units).}

\label{fig:max_1000_result}

768

662

\label{fig:max_1000_result}

\end{subfigure}

769

663

\end{subfigure}

770

664

\begin{subfigure}{\linewidth}

771

665

\begin{subfigure}{\linewidth}

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

772

666

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

\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving

773

667

\caption{Filter transfer functions for varying number of cascaded filters solving

the MAX/1500 problem of maximizing rejection for a given resource allocation (1500~arbitrary units).}

774

668

the MAX/1500 problem of maximizing rejection for a given resource allocation (1500~arbitrary units).}

\label{fig:max_1500_result}

775

669

\label{fig:max_1500_result}

\end{subfigure}

776

670

\end{subfigure}

\caption{\color{red}Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing

777

671

\caption{Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing

rejection for a given resource allocation.

778

672

rejection for a given resource allocation.

The filter shape constraint (bandpass and bandstop) is shown as thick

779

673

The filter shape constraint (bandpass and bandstop) is shown as thick

horizontal lines on each chart.}

780

674

horizontal lines on each chart.}

\end{figure}

781

675

\end{figure}

782

676

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

783

677

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

784

678

We compare the actual silicon resources given by Vivado to the

785

679

We compare the actual silicon resources given by Vivado to the

resources in arbitrary units.

786

680

resources in arbitrary units.

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

787

681

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

788

682

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

789

683

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

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

790

684

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

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

791

685

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

various vendors to remain platform independent and be portable from one

792

686

various vendors to remain platform independent and be portable from one

architecture to another.

793

687

architecture to another.

794

688

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

795

689

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

796

690

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

797

691

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 Programmable

798

692

the FIR filters and remove additional processing blocks including FIFO and Programmable

Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.

799

693

Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.

800

694

\begin{table}[h!tb]

801

695

\begin{table}[h!tb]

\caption{Resource occupation following synthesis of the solutions found for

802

696

\caption{Resource occupation following synthesis of the solutions found for

the problem of maximizing rejection for a given resource allocation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}

803

697

the problem of maximizing rejection for a given resource allocation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}

\label{tbl:resources_usage}

804

698

\label{tbl:resources_usage}

\color{red}

805

\centering

806

699

\centering

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

807

700

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

\hline

808

701

\hline

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

809

702

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

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

810

703

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

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

811

704

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

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

812

705

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

& LUT & 2253 & 474 & 691 & \emph{17600} \\

813

706

& LUT & 2253 & 474 & 691 & \emph{17600} \\

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

814

707

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

& DSP & 0 & 50 & 70 & \emph{80} \\ \hline

815

708

& DSP & 0 & 50 & 70 & \emph{80} \\ \hline

& LUT & 1329 & 2006 & 3158 & \emph{17600} \\

816

709

& LUT & 1329 & 2006 & 3158 & \emph{17600} \\

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

817

710

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

& DSP & 15 & 30 & 42 & \emph{80} \\ \hline

818

711

& DSP & 15 & 30 & 42 & \emph{80} \\ \hline

& LUT & 1329 & 1600 & 2260 & \emph{17600} \\

819

712

& LUT & 1329 & 1600 & 2260 & \emph{17600} \\

4 & BRAM & 3 & 4 & 4 & \emph{120} \\

820

713

4 & BRAM & 3 & 4 & 4 & \emph{120} \\

& DPS & 15 & 38 & 49 & \emph{80} \\ \hline

821

714

& DPS & 15 & 38 & 49 & \emph{80} \\ \hline

& LUT & 1329 & 1600 & 2260 & \emph{17600} \\

822

715

& LUT & 1329 & 1600 & 2260 & \emph{17600} \\

5 & BRAM & 3 & 4 & 4 & \emph{120} \\

823

716

5 & BRAM & 3 & 4 & 4 & \emph{120} \\

& DPS & 15 & 38 & 49 & \emph{80} \\ \hline

824

717

& DPS & 15 & 38 & 49 & \emph{80} \\ \hline

\end{tabular}

825

718

\end{tabular}

\end{table}

826

719

\end{table}

827

720

{\color{red} In case $n = 2$ for MAX/500}, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that,

828

721

In case $n = 2$ for MAX/500, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that,

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

829

722

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

enough, Vivado optimizes resource consumption by selecting multiplexers to

830

723

enough, Vivado optimizes resource consumption by selecting multiplexers to

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

831

724

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

to compare the whole silicon budget.

832

725

to compare the whole silicon budget.

833

726

However, a rough estimation can be made with a simple equivalence: looking at

834

727

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 $n \geq 2$,

835

728

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

836

729

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

area use. With this equivalence, our 500 arbitrary units correspond to 2500 LUTs,

837

730

area use. With this equivalence, our 500 arbitrary units correspond to 2500 LUTs,

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

838

731

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

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

839

732

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

unit map well to actual hardware resources. The relatively small differences can probably be explained

840

733

unit map well to actual hardware resources. The relatively small differences can probably be explained

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

841

734

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

842

735

We now present the computation time needed to solve the quadratic problem.

843

736

We now present the computation time needed to solve the quadratic problem.

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

844

737

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

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

845

738

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

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

846

739

the quadratic problem. 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.

847

740

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

848

741

\begin{table}[h!tb]

849

742

\begin{table}[h!tb]

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

850

743

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

\label{tbl:area_time}

851

744

\label{tbl:area_time}

\centering

852

745

\centering

\color{red}

853

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

854

746

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

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

855

747

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

1 & 0.01~s & 0.02~s & 0.03~s \\

856

748

1 & 0.01~s & 0.02~s & 0.03~s \\

2 & 0.1~s & 1~s & 2~s \\

857

749

2 & 0.1~s & 1~s & 2~s \\

3 & 5~s & 27~s & 351~s ($\approx$ 6~min) \\

858

750

3 & 5~s & 27~s & 351~s ($\approx$ 6~min) \\

4 & 4~s & 141~s ($\approx$ 3~min) & 1134~s ($\approx$ 18~min) \\

859

751

4 & 4~s & 141~s ($\approx$ 3~min) & 1134~s ($\approx$ 18~min) \\

5 & 6~s & 630~s ($\approx$ 10~min) & 49400~s ($\approx$ 13~h) \\\hline

860

752

5 & 6~s & 630~s ($\approx$ 10~min) & 49400~s ($\approx$ 13~h) \\\hline

\end{tabular}

861

753

\end{tabular}

\end{table}

862

754

\end{table}

863

755

As expected, the computation time seems to rise exponentially with the number of stages.

864

756

As expected, the computation time seems to rise exponentially with the number of stages.

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

865

757

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

find an optimal solution faster.

866

758

find an optimal solution faster.

{\color{red} We also notice that the solution with $n$ greater than the optimal value

867

759

We also notice that the solution with $n$ greater than the optimal value

takes more time to be found than the optimal one. This can be explained since the search space is

868

760

takes more time to be found than the optimal one. This can be explained since the search space is

larger and we need more time to ensure that the previous solution (from the

869

761

larger and we need more time to ensure that the previous solution (from the

smaller value of $n$) still remains the optimal solution.}

870

762

smaller value of $n$) still remains the optimal solution.

871

763

\subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej}

872

764

\subsection{Minimizing resource occupation at fixed rejection}

765

\label{sec:fixed_rej}

873

766

This section presents the results of the complementary quadratic program aimed at

874

767

This section presents the results of the complementary quadratic program aimed at

minimizing the area occupation for a targeted rejection level.

875

768

minimizing the area occupation for a targeted rejection level.

876

769

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

877

770

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

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

878

771

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

Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60, 80 or 100~dB.

879

772

Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60, 80 or 100~dB.

Hence, the three cases have been named: MIN/40, MIN/60, MIN/80 and MIN/100.

880

773

Hence, the three cases have been named: MIN/40, MIN/60, MIN/80 and MIN/100.

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

881

774

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

882

775

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

883

776

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.

884

777

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.

885

778

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

Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.

886

779

Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.

887

780

\renewcommand{\arraystretch}{1.4}

888

781

\renewcommand{\arraystretch}{1.4}

889

782

\begin{table}[h!tb]

890

783

\begin{table}[h!tb]

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

891

784

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

\label{tbl:gurobi_min_40}

892

785

\label{tbl:gurobi_min_40}

\centering

893

786

\centering

{\scalefont{0.77}\color{red}

894

787

{\scalefont{0.77}

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

895

788

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

\hline

896

789

\hline

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

897

790

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

\hline

898

791

\hline

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

899

792

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

2 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\

900

793

2 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\

3 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\

901

794

3 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\

4 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\

902

795

4 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\

5 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\

903

796

5 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\

\hline

904

797

\hline

\end{tabular}

905

798

\end{tabular}

}

906

799

}

\end{table}

907

800

\end{table}

908

801

\begin{table}[h!tb]

909

802

\begin{table}[h!tb]

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

910

803

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

\label{tbl:gurobi_min_60}

911

804

\label{tbl:gurobi_min_60}

\centering

912

805

\centering

{\scalefont{0.77}\color{red}

913

806

{\scalefont{0.77}

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

914

807

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

\hline

915

808

\hline

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

916

809

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

\hline

917

810

\hline

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

918

811

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

2 & (15, 6, 16) & (23, 9, 0) & - & - & - & 60~dB & 675 \\

919

812

2 & (15, 6, 16) & (23, 9, 0) & - & - & - & 60~dB & 675 \\

3 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\

920

813

3 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\

4 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\

921

814

4 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\

5 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\

922

815

5 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\

\hline

923

816

\hline

\end{tabular}

924

817

\end{tabular}

}

925

818

}

\end{table}

926

819

\end{table}

927

820

\begin{table}[h!tb]

928

821

\begin{table}[h!tb]

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

929

822

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

\label{tbl:gurobi_min_80}

930

823

\label{tbl:gurobi_min_80}

\centering

931

824

\centering

{\scalefont{0.77}\color{red}

932

825

{\scalefont{0.77}

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

933

826

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

\hline

934

827

\hline

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

935

828

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

\hline

936

829

\hline

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

937

830

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

2 & (15, 8, 17) & (35, 11, 0) & - & - & - & 80~dB & 990 \\

938

831

2 & (15, 8, 17) & (35, 11, 0) & - & - & - & 80~dB & 990 \\

3 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\

939

832

3 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\

4 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\

940

833

4 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\

5 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\

941

834

5 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\

\hline

942

835

\hline

\end{tabular}

943

836

\end{tabular}

}

944

837

}

\end{table}

945

838

\end{table}

946

839

\begin{table}[h!tb]

947

840

\begin{table}[h!tb]

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}

948

841

\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}

\label{tbl:gurobi_min_100}

949

842

\label{tbl:gurobi_min_100}

\centering

950

843

\centering

{\scalefont{0.77}\color{red}

951

844

{\scalefont{0.77}

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

952

845

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

\hline

953

846

\hline

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

954

847

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

\hline

955

848

\hline

1 & - & - & - & - & - & - & - \\

956

849

1 & - & - & - & - & - & - & - \\

2 & (27, 9, 15) & (35, 11, 0) & - & - & - & 100~dB & 1410 \\

957

850

2 & (27, 9, 15) & (35, 11, 0) & - & - & - & 100~dB & 1410 \\

3 & (3, 5, 18) & (35, 11, 1) & (27, 9, 0) & - & - & 100~dB & 1147 \\

958

851

3 & (3, 5, 18) & (35, 11, 1) & (27, 9, 0) & - & - & 100~dB & 1147 \\

4 & (3, 5, 18) & (15, 6, 2) & (27, 9, 0) & (19, 7, 0) & - & 100~dB & 1067 \\

959

852

4 & (3, 5, 18) & (15, 6, 2) & (27, 9, 0) & (19, 7, 0) & - & 100~dB & 1067 \\

5 & (3, 5, 18) & (15, 6, 2) & (27, 9, 0) & (19, 7, 0) & - & 100~dB & 1067 \\

960

853

5 & (3, 5, 18) & (15, 6, 2) & (27, 9, 0) & (19, 7, 0) & - & 100~dB & 1067 \\

\hline

961

854

\hline

\end{tabular}

962

855

\end{tabular}

}

963

856

}

\end{table}

964

857

\end{table}

\renewcommand{\arraystretch}{1}

965

858

\renewcommand{\arraystretch}{1}

966

859

From these tables, we can first state that almost all configurations reach the targeted rejection

967

860

From these tables, we can first state that almost all configurations reach the targeted rejection

level or even better thanks to our underestimate of the cascade rejection as the sum of the

968

861

level or even better thanks to our underestimate of the cascade rejection as the sum of the

individual filter rejection. The only exception is for the monolithic case ($n = 1$) in

969

862

individual filter rejection. The only exception is for the monolithic case ($n = 1$) in

MIN/100: no solution is found for a single monolithic filter reach a 100~dB rejection.

970

863

MIN/100: no solution is found for a single monolithic filter reach a 100~dB rejection.

Furthermore, the area of the monolithic filter is twice as big as the two cascaded filters

971

864

Furthermore, the area of the monolithic filter is twice as big as the two cascaded filters

{\color{red}(675 and 1131 arbitrary units v.s 990 and 1760 arbitrary units for 60 and 80~dB rejection}

972

865

(675 and 1131 arbitrary units v.s 990 and 1760 arbitrary units for 60 and 80~dB rejection

respectively). More generally, the more filters are cascaded, the lower the occupied area.

973

866

respectively). More generally, the more filters are cascaded, the lower the occupied area.

974

867

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

975

868

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

filter stage and the second one is often the biggest filter. This choice can be explained

976

869

filter stage and the second one is often the biggest filter. This choice can be explained

as in the previous section, with the solver using just enough bits not to degrade the input

977

870

as in the previous section, with the solver using just enough bits not to degrade the input

signal and in the second filter selecting a better filter to improve rejection without

978

871

signal and in the second filter selecting a better filter to improve rejection without

having too many bits in the output data.

979

872

having too many bits in the output data.

980

873

{\color{red} For each case, we found an optimal solution with $n < 5$: for MIN/40 $n=2$,

981

874

For each case, we found an optimal solution with $n < 5$: for MIN/40 $n=2$,

for MIN/60 and MIN/80 $n = 3$ and for MIN/100 $n = 4$. In all cases, the solutions

982

875

for MIN/60 and MIN/80 $n = 3$ and for MIN/100 $n = 4$. In all cases, the solutions

when $n$ is greater than this optimal $n$ remain identical to the optimal one.}

983

876

when $n$ is greater than this optimal $n$ remain identical to the optimal one.

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

984

% number of filters is 4 so it did not chose any configuration for the last filter. Hence this

985

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

986

987

877

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

988

878

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

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

989

879

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

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

990

880

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

given by the quadratic solver.

991

881

given by the quadratic solver.

992

882

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

993

883

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.

994

884

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.

995

885

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

Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.

996

886

Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.

997

887

% \begin{figure}

998

% \centering

999

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

1000

% \caption{Signal spectrum for MIN/40}

1001

% \label{fig:min_40}

1002

% \end{figure}

1003

%

1004

% \begin{figure}

1005

% \centering

1006

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

1007

% \caption{Signal spectrum for MIN/60}

1008

% \label{fig:min_60}

1009

% \end{figure}

1010

%

1011

% \begin{figure}

1012

% \centering

1013

% \includegraphics[width=\linewidth]{images/min_80}

1014

% \caption{Signal spectrum for MIN/80}

1015

% \label{fig:min_80}

1016

% \end{figure}

1017

%

1018

% \begin{figure}

1019

% \centering

1020

% \includegraphics[width=\linewidth]{images/min_100}

1021

% \caption{Signal spectrum for MIN/100}

1022

% \label{fig:min_100}

1023

% \end{figure}

1024

1025

% r2.14 et r2.15 et r2.16

1026

\begin{figure}

1027

888

\begin{figure}

\centering

1028

889

\centering

\begin{subfigure}{\linewidth}

1029

890

\begin{subfigure}{\linewidth}

\includegraphics[width=.91\linewidth]{images/min_40}

1030

891

\includegraphics[width=.91\linewidth]{images/min_40}

\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving

1031

892

\caption{Filter transfer functions for varying number of cascaded filters solving

the MIN/40 problem of minimizing resource allocation for reaching a 40~dB rejection.}

1032

893

the MIN/40 problem of minimizing resource allocation for reaching a 40~dB rejection.}

\label{fig:min_40}

1033

894

\label{fig:min_40}

\end{subfigure}

1034

895

\end{subfigure}

1035

896

\begin{subfigure}{\linewidth}

1036

897

\begin{subfigure}{\linewidth}

\includegraphics[width=.91\linewidth]{images/min_60}

1037

898

\includegraphics[width=.91\linewidth]{images/min_60}

\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving

1038

899

\caption{Filter transfer functions for varying number of cascaded filters solving

the MIN/60 problem of minimizing resource allocation for reaching a 60~dB rejection.}

1039

900

the MIN/60 problem of minimizing resource allocation for reaching a 60~dB rejection.}

\label{fig:min_60}

1040

901

\label{fig:min_60}

\end{subfigure}

1041

902

\end{subfigure}

1042

903

\begin{subfigure}{\linewidth}

1043

904

\begin{subfigure}{\linewidth}

\includegraphics[width=.91\linewidth]{images/min_80}

1044

905

\includegraphics[width=.91\linewidth]{images/min_80}

\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving

1045

906

\caption{Filter transfer functions for varying number of cascaded filters solving

the MIN/80 problem of minimizing resource allocation for reaching a 80~dB rejection.}

1046

907

the MIN/80 problem of minimizing resource allocation for reaching a 80~dB rejection.}

\label{fig:min_80}

1047

908

\label{fig:min_80}

\end{subfigure}

1048

909

\end{subfigure}

1049

910

\begin{subfigure}{\linewidth}

1050

911

\begin{subfigure}{\linewidth}

\includegraphics[width=.91\linewidth]{images/min_100}

1051

912

\includegraphics[width=.91\linewidth]{images/min_100}

\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving

1052

913

\caption{Filter transfer functions for varying number of cascaded filters solving

the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.}

1053

914

the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.}

\label{fig:min_100}

1054

915

\label{fig:min_100}

\end{subfigure}

1055

916

\end{subfigure}

\caption{\color{red}Solutions for the MIN/40, MIN/60, MIN/80 and MIN/100 problems of reaching a

1056

917

\caption{Solutions for the MIN/40, MIN/60, MIN/80 and MIN/100 problems of reaching a

given rejection while minimizing resource allocation. The filter shape constraint (bandpass and

1057

918

given rejection while minimizing resource allocation. The filter shape constraint (bandpass and

bandstop) is shown as thick

1058

919

bandstop) is shown as thick

horizontal lines on each chart.}

1059

920

horizontal lines on each chart.}

\end{figure}

1060

921

\end{figure}

1061

922

We observe that all rejections given by the quadratic solver are close to the experimentally

1062

923

We observe that all rejections given by the quadratic solver are close to the experimentally

measured rejection. All curves prove that the constraint to reach the target rejection is

1063

924

measured rejection. All curves prove that the constraint to reach the target rejection is

respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters.

1064

925

respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters.

1065

926

Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60;

1066

927

Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60;

MIN/80 and MIN/100 \emph{i.e.} when the target rejection is fixed to 40, 60, 80 and 100~dB. We

1067

928

MIN/80 and MIN/100 \emph{i.e.} when the target rejection is fixed to 40, 60, 80 and 100~dB. We

have taken care to extract solely the resources used by

1068

929

have taken care to extract solely the resources used by

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

1069

930

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

PS communication.

1070

931

PS communication.

1071

932

\renewcommand{\arraystretch}{1.2}

1072

933

\renewcommand{\arraystretch}{1.2}

\begin{table}

1073

934

\begin{table}

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

1074

935

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

\label{tbl:resources_usage_comp}

1075

936

\label{tbl:resources_usage_comp}

\centering

1076

937

\centering

{\scalefont{0.90}\color{red}

1077

938

{\scalefont{0.90}

\begin{tabular}{|c|c|cccc|c|}

1078

939

\begin{tabular}{|c|c|cccc|c|}

\hline

1079

940

\hline

$n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline

1080

941

$n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline

& LUT & 343 & 334 & 772 & - & \emph{17600} \\

1081

942

& LUT & 343 & 334 & 772 & - & \emph{17600} \\

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

1082

943

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

& DSP & 27 & 39 & 55 & - & \emph{80} \\ \hline

1083

944

& DSP & 27 & 39 & 55 & - & \emph{80} \\ \hline

& LUT & 1664 & 2329 & 474 & 620 & \emph{17600} \\

1084

945

& LUT & 1664 & 2329 & 474 & 620 & \emph{17600} \\

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

1085

946

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

& DSP & 0 & 15 & 50 & 62 & \emph{80} \\ \hline

1086

947

& DSP & 0 & 15 & 50 & 62 & \emph{80} \\ \hline

& LUT & 1664 & 3114 & 1884 & 2873 & \emph{17600} \\

1087

948

& LUT & 1664 & 3114 & 1884 & 2873 & \emph{17600} \\

3 & BRAM & 2 & 3 & 3 & 3 & \emph{120} \\

1088

949

3 & BRAM & 2 & 3 & 3 & 3 & \emph{120} \\

& DSP & 0 & 0 & 22 & 27 & \emph{80} \\ \hline

1089

950

& DSP & 0 & 0 & 22 & 27 & \emph{80} \\ \hline

& LUT & 1664 & 3114 & 2570 & 4318 & \emph{17600} \\

1090

951

& LUT & 1664 & 3114 & 2570 & 4318 & \emph{17600} \\

4 & BRAM & 2 & 3 & 4 & 4 & \emph{120} \\

1091

952

4 & BRAM & 2 & 3 & 4 & 4 & \emph{120} \\

& DPS & 0 & 15 & 19 & 19 & \emph{80} \\ \hline

1092

953

& DPS & 0 & 15 & 19 & 19 & \emph{80} \\ \hline

& LUT & 1664 & 3114 & 2570 & 4318 & \emph{17600} \\

1093

954

& LUT & 1664 & 3114 & 2570 & 4318 & \emph{17600} \\

5 & BRAM & 2 & 3 & 4 & 4 & \emph{120} \\

1094

955

5 & BRAM & 2 & 3 & 4 & 4 & \emph{120} \\

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

1095

956

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

\end{tabular}

1096

957

\end{tabular}

}

1097

958

}

\end{table}

1098

959

\end{table}

\renewcommand{\arraystretch}{1}

1099

960

\renewcommand{\arraystretch}{1}

1100

961

If we keep the previous estimation of cost of one DSP in terms of LUT (1 DSP $\approx$ 100 LUT)

1101

962

If we keep the previous estimation of cost of one DSP in terms of LUT (1 DSP $\approx$ 100 LUT)

the real resource consumption decreases as a function of the number of stages in the cascaded

1102

963

the real resource consumption decreases as a function of the number of stages in the cascaded

filter according

1103

964

filter according

to the solution given by the quadratic solver. Indeed, we have always a decreasing

1104

965

to the solution given by the quadratic solver. Indeed, we have always a decreasing

consumption even if the difference between the monolithic and the two cascaded

1105

966

consumption even if the difference between the monolithic and the two cascaded

filters is less than expected.

1106

967

filters is less than expected.

1107

968

Finally, table~\ref{tbl:area_time_comp} shows the computation time to solve

1108

969

Finally, table~\ref{tbl:area_time_comp} shows the computation time to solve

the quadratic program.

1109

970

the quadratic program.

1110

971

\renewcommand{\arraystretch}{1.2}

1111

972

\renewcommand{\arraystretch}{1.2}

\begin{table}[h!tb]

1112

973

\begin{table}[h!tb]

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

1113

974

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

\label{tbl:area_time_comp}

1114

975

\label{tbl:area_time_comp}

\centering

1115

976

\centering

{\scalefont{0.90}\color{red}

1116

977

{\scalefont{0.90}

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

1117

978

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

$n$ & Time (MIN/40) & Time (MIN/60) & Time (MIN/80) & Time (MIN/100) \\\hline\hline

1118

979

$n$ & Time (MIN/40) & Time (MIN/60) & Time (MIN/80) & Time (MIN/100) \\\hline\hline

1 & 0.04~s & 0.01~s & 0.01~s & - \\

1119

980

1 & 0.04~s & 0.01~s & 0.01~s & - \\

2 & 2.7~s & 2.4~s & 2.4~s & 0.8~s \\

1120

981

2 & 2.7~s & 2.4~s & 2.4~s & 0.8~s \\

3 & 4.6~s & 7~s & 7~s & 18~s \\

1121

982

3 & 4.6~s & 7~s & 7~s & 18~s \\

4 & 3~s & 22~s & 70~s & 220~s ($\approx$ 3~min) \\

1122

983

4 & 3~s & 22~s & 70~s & 220~s ($\approx$ 3~min) \\

5 & 5~s & 122~s & 200~s & 384~s ($\approx$ 5~min) \\\hline

1123

984

5 & 5~s & 122~s & 200~s & 384~s ($\approx$ 5~min) \\\hline

\end{tabular}

1124

985

\end{tabular}

}

1125

986

}

GITLAB

jfriedt / IFCS2018 article

Dernière correction.