diff --git a/ifcs2018_journal.tex b/ifcs2018_journal.tex index 216f688..a44b624 100644 --- a/ifcs2018_journal.tex +++ b/ifcs2018_journal.tex @@ -142,9 +142,9 @@ We want create a new methodology to develop any Digital Signal Processing (DSP) and for any hardware platform (Altera, Xilinx...). To do this we have defined an abstract model to represent some basic operations of DSP. -For the moment, we are focused on only two operations: the filtering and the shift of data. +For the moment, we are focused on only two operations: the filtering and the shifting of data. We have chosen this basic operation because the shifting and the filtering have already be studied in -lot of works {\color{red} mettre les nouvelles référence ici} hence it will be easier +lot of works \cite{lim_1996, lim_1988, young_1992, smith_1998} hence it will be easier to check and validate our results. However having only two operations is insufficient to work with complex DSP but @@ -283,170 +283,569 @@ better rejection than expected. \label{fig:sum_rejection} \end{figure} +Finally we can describe our abstract model with following expressions : +\begin{align} +\text{Maximize } & \sum_{i=1}^n r_i \notag \\ +\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} \\ +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} \\ +\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} \\ +\pi_1^- &= \Pi^I \label{eq:init} +\end{align} + +{\color{red} Je sais que l'idée est de ne pas parler du programme linéaire mais +ça me semble quand même indispensable. Au pire, j'essaierai de revoir ça si on +est vraiment en manque de place.} + +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 +the area for a filter. More precisely, it is the area of the FIR as the Shifter +does not need any circuitry. We consider that the FIR needs $C_i$ registers of size +$\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the +input data and the coefficients. Equation~\ref{eq:rejectiondef} gives the +definition of the rejection of the filter thanks to function~$F$ that we defined +previously. The Shifter does not introduce negative rejection as we explain later, +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 +$\pi_i^C$ 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 +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 +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 +increases the maximum rejection level of 6~dB. We add one to take the sign bit +into account. If equation~\ref{eq:maxshift} was not present, the Shifter could +shift too much and introduce some noise in the output data. Each supplementary +shift bit would cause 6~dB of noise. A totally equivalent equation is: +$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right) $. +Finally, equation~\ref{eq:init} gives the global input's number of bits. + +This model is non-linear and even non-quadratic, as $F$ does not have a known +linear or quadratic expression. We introduce $p$ FIR configurations + $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. We define binary + variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ + and 0 otherwise. The new equations are as follows: + +\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} \\ +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} \\ +\sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config} +\end{align} + +Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace +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. + +The next section shows the results for this quadratic program but the section~\ref{sec:fixed_rej} +presents the results for the complementary problem. In this case we want +minimize the occupied area for a targeted rejection level. Hence we have replace +the objective function with: +\begin{align} +\text{Minimize } & \sum_{i=1}^n a_i \notag +\end{align} +We adapt our constraints of quadratic program to replace the equation \ref{eq:area} +by the equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal +rejection required. + +\begin{align} +\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min} +\end{align} + +\section{Design workflow} +\label{sec:workflow} + +In this section, we describe the workflow to compute all the results presented in section~\ref{sec:fixed_area}. +Figure~\ref{fig:workflow} shows the global workflow and the different steps involved in the computations of the results. + +\begin{figure} + \centering + \begin{tikzpicture}[node distance=0.75cm and 2cm] + \node[draw,minimum size=1cm] (Solver) { Filter Solver } ; + \node (Start) [left= 3cm of Solver] { } ; + \node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ; + \node (Input) [above= of TCL] { } ; + \node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ; + \node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ; + \node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ; + \node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ; + \node (Results) [left= of Postproc] { } ; + + \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) ; + \draw[->] (Solver) edge node [below] { (1a) } (TCL) ; + \draw[->] (Solver) edge node [right] { (1b) } (Deploy) ; + \draw[->] (TCL) edge node [left] { (2) } (Bitstream) ; + \draw[->,dashed] (Bitstream) -- (Deploy) ; + \draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ; + \draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ; + \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ; + \draw[->] (Postproc) -- (Results) ; + \end{tikzpicture} + \caption{Design workflow from the input parameters to the results} + \label{fig:workflow} +\end{figure} + +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 $\Pi^I$, +the FIR configurations $(C_{ij}, \pi_{ij}^C)$ and the function $F$. It creates +the quadratic programs and uses the Gurobi solver to get the optimal results. +Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow}) +and a deploy script ((1b) on figure~\ref{fig:workflow}). + +The TCL script describes the whole digital processing chain from the beginning +(the raw signal data) to the end (the filtered data). +The raw input data generated from a Pseudo Random Number (PRN) +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 +comes from a skeleton library. The generic FIR is highly configurable +with the number of coefficients and the size of the coefficients. The coefficients +themselves are not stored in the script. +Whereas the signal is processed in real-time, the output signal is stored as +consecutive bursts of data. + +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 +bitstream on a Redpitaya board fitted with a Xilinx Zynq-7010 series +FPGA (xc7z010clg400-1) and two 125~MS/s ADC. +The board works with a Buildroot Linux image. We have developed some tools and +drivers to flash and communicate with the FPGA. They are used to automatize all +the workflow inside the board: load the filter coefficients and retrieve the +computed data. + +The deploy script uploads the bitstream to the board ((3) on +figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers, +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}). + +Finally, an Octave post-processing script computes the final results thanks to +the output data ((5) on figure~\ref{fig:workflow}). +The results are normalized so that the Power Spectrum Density (PSD) starts at zero +and the different configurations can be compared. + +The workflow used to compute the results in section~\ref{sec:fixed_rej}, we +have just adapted the quadratic program but the rest of the workflow is unchanged. + \section{Experiments with fixed area space} +\label{sec:fixed_area} +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. +This is interesting to understand the choices made by the solver to compute its solutions. + +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 $\Pi^I$. +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. +The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$ +ranging from 2 to 22. In each case, the quadratic program has been able to give a +result up to five stages ($n = 5$) in the cascaded filter. + +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. +Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500. + +\renewcommand{\arraystretch}{1.4} + +\begin{table} + \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500} + \label{tbl:gurobi_max_500} + \centering + {\scalefont{0.77} + \begin{tabular}{|c|ccccc|c|c|} + \hline + $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ + \hline + 1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\ + 2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\ + 3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\ + 4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\ + 5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\ + \hline + \end{tabular} + } +\end{table} + +\begin{table} + \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000} + \label{tbl:gurobi_max_1000} + \centering + {\scalefont{0.77} + \begin{tabular}{|c|ccccc|c|c|} + \hline + $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ + \hline + 1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\ + 2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\ + 3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\ + 4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\ + 5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\ + \hline + \end{tabular} + } +\end{table} + +\begin{table} + \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500} + \label{tbl:gurobi_max_1500} + \centering + {\scalefont{0.77} + \begin{tabular}{|c|ccccc|c|c|} + \hline + $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ + \hline + 1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\ + 2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\ + 3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\ + 4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\ + 5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\ + \hline + \end{tabular} + } +\end{table} + +\renewcommand{\arraystretch}{1} + +From these tables, we can first state that the more stages are used to define +the cascaded FIR filters, the better the rejection. It was an expected result as it has +been previously observed that many small filters are better than +a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusion +being hardly used in practice due to the lack of tools for identifying individual filter +coefficients in the cascaded approach. + +Second, the larger the silicon area, the better the rejection. This was also an +expected result as more area means a filter of better quality (more coefficients +or more bits per coefficient). + +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 +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} +gives the relation between both values. + +Finally, we note that the solver consumes all the given silicon area. + +The following graphs present the rejection for real data on the FPGA. In all following +figures, the solid line represents the actual rejection of the filtered +data on the FPGA as measured experimentally and the dashed line are the noise level +given by the quadratic solver. The configurations are those computed in the previous section. + +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. +Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500. \begin{figure} \centering -\includegraphics[width=\linewidth]{images/max_rejection/prn_500} -\caption{Experimental results for design with PRN as data input and 500 a.u. as max arbitrary space} -\label{fig:prn_500} +\includegraphics[width=\linewidth]{images/max_500} +\caption{Signal spectrum for MAX/500} +\label{fig:max_500_result} \end{figure} \begin{figure} \centering -\includegraphics[width=\linewidth]{images/max_rejection/prn_1000} -\caption{Experimental results for design with PRN as data input and 1000 a.u. as max arbitrary space} -\label{fig:prn_1000} +\includegraphics[width=\linewidth]{images/max_1000} +\caption{Signal spectrum for MAX/1000} +\label{fig:max_1000_result} \end{figure} \begin{figure} \centering -\includegraphics[width=\linewidth]{images/max_rejection/prn_2000} -\caption{Experimental results for design with PRN as data input and 2000 a.u. as max arbitrary space} -\label{fig:prn_2000} +\includegraphics[width=\linewidth]{images/max_1500} +\caption{Signal spectrum for MAX/1500} +\label{fig:max_1500_result} \end{figure} -\begin{table} -\centering -\begin{tabular}{|c|c|ccc|c|c|} -\hline -\multicolumn{2}{|c|}{\multirow{2}{*}{Stage}} & \multicolumn{3}{c|}{Stage} & \multirow{2}{*}{Rejection} & \multirow{2}{*}{Area} \\ \cline{3-5} -\multicolumn{2}{|c|}{} & i = 1 & i = 2 & i = 3 & & \\ \hline - & C & 19 & - & - & & \\ -n = 1 & $pi^C$ & 7 & - & - & 33 dB & 437 a.u. \\ - & $pi^S$ & 0 & - & - & & \\ \hline - & C & 11 & 19 & - & & \\ -n = 2 & $pi^C$ & 5 & 7 & - & 53 dB & 478 a.u. \\ - & $pi^S$ & 16 & 0 & - & & \\ \hline - & C & 9 & 15 & 11 & & \\ -n = 3 & $pi^C$ & 4 & 6 & 5 & 57 dB & 499 a.u. \\ - & $pi^S$ & 16 & 3 & 0 & & \\ \hline -\end{tabular} -\caption{Solver results for design with PRN as data input and 500 a.u. as max arbitrary space} -\label{tbl:prn_500} -\end{table} +In all cases, we observe that the actual rejection is close to the rejection computed by the solver. + +We compare the actual silicon resources given by Vivado to the +resources in arbitrary units. +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 +number of arbitrary units, the actual silicon resources do not depend on the +number of stages $n$. Most significantly, our approach aims +at remaining far enough from the practical logic gate implementation used by +various vendors to remain platform independent and be portable from one +architecture to another. + +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 +and 1500 arbitrary units. We have taken care to extract solely the resources used by +the FIR filters and remove additional processing blocks including FIFO and PL to +PS communication. \begin{table} -\centering -{\scalefont{0.85} -\begin{tabular}{|c|c|ccccc|c|c|} -\hline -\multicolumn{2}{|c|}{\multirow{2}{*}{Stage}} & \multicolumn{5}{c|}{Stage} & \multirow{2}{*}{Rejection} & \multirow{2}{*}{Area} \\ \cline{3-7} -\multicolumn{2}{|c|}{} & i = 1 & i = 2 & i = 3 & i = 4 & i = 5 & & \\ \hline - & C & 37 & - & - & - & - & & \\ -n = 1 & $pi^C$ & 11 & - & - & - & - & 56 dB & 999 a.u. \\ - & $pi^S$ & 0 & - & - & - & - & & \\ \hline - & C & 11 & 39 & - & - & - & & \\ -n = 2 & $pi^C$ & 5 & 13 & - & - & - & 82 dB & 972 a.u. \\ - & $pi^S$ & 16 & 0 & - & - & - & & \\ \hline - & C & 9 & 31 & 19 & - & - & & \\ -n = 3 & $pi^C$ & 7 & 8 & 7 & - & - & 93 dB & 990 a.u. \\ - & $pi^S$ & 19 & 2 & 0 & - & - & & \\ \hline - & C & 9 & 19 & 17 & 11 & - & & \\ -n = 4 & $pi^C$ & 4 & 7 & 7 & 5 & - & 99 dB & 992 a.u. \\ - & $pi^S$ & 16 & 3 & 3 & 0 & - & & \\ \hline - & C & 9 & 15 & 11 & 11 & 11 & & \\ -n = 5 & $pi^C$ & 4 & 7 & 5 & 5 & 5 & 99 dB & 998 a.u. \\ - & $pi^S$ & 16 & 3 & 2 & 1 & 1 & & \\ \hline -\end{tabular} -} -\caption{Solver results for design with PRN as data input and 1000 a.u. as max arbitrary space} -\label{tbl:prn_1000} + \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} + \label{tbl:resources_usage} + \centering + \begin{tabular}{|c|c|ccc|c|} + \hline + $n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline + & LUT & 249 & 453 & 627 & \emph{17600} \\ + 1 & BRAM & 1 & 1 & 1 & \emph{120} \\ + & DSP & 21 & 37 & 47 & \emph{80} \\ \hline + & LUT & 2374 & 5494 & 691 & \emph{17600} \\ + 2 & BRAM & 2 & 2 & 2 & \emph{120} \\ + & DSP & 0 & 0 & 70 & \emph{80} \\ \hline + & LUT & 2443 & 3304 & 3521 & \emph{17600} \\ + 3 & BRAM & 3 & 3 & 3 & \emph{120} \\ + & DSP & 0 & 19 & 35 & \emph{80} \\ \hline + & LUT & 2634 & 3753 & 2557 & \emph{17600} \\ + 4 & BRAM & 4 & 4 & 4 & \emph{120} \\ + & DPS & 0 & 19 & 46 & \emph{80} \\ \hline + & LUT & 2423 & 3047 & 2847 & \emph{17600} \\ + 5 & BRAM & 5 & 5 & 5 & \emph{120} \\ + & DPS & 0 & 22 & 46 & \emph{80} \\ \hline + \end{tabular} \end{table} +In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that, +when the filters coefficients are small enough, or when the input size is small +enough, Vivado optimized resource consumption by selecting multiplexers to +implement the multiplications instead of a DSP. In this case, it is quite difficult +to compare the whole silicon budget. + +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 $n \geq 2$, +we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon +area use. With this equivalence, our 500 arbitraty units corresponds to 2500 LUTs, +1000 arbitrary units corresponds to 5000 LUTs and 1500 arbitrary units corresponds +to 7300 LUTs. The conclusion is that the orders of magnitude of our arbitrary +unit are quite good. The relatively small differences can probably be explained +by the optimizations done by Vivado based on the detailed map of available processing resources. + +We present the computation time to solve the quadratic problem. +For each case, the filter solver software are executed with a Intel(R) Xeon(R) CPU E5606 +cadenced 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 +problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units. + \begin{table} +\caption{Time to solve the quadratic program with Gurobi} +\label{tbl:area_time} \centering -{\scalefont{0.85} -\begin{tabular}{|c|c|ccccc|c|c|} -\hline -\multicolumn{2}{|c|}{\multirow{2}{*}{Stage}} & \multicolumn{5}{c|}{Stage} & \multirow{2}{*}{Rejection} & \multirow{2}{*}{Area} \\ \cline{3-7} -\multicolumn{2}{|c|}{} & i = 1 & i = 2 & i = 3 & i = 4 & i = 5 & & \\ \hline - & C & 39 & - & - & - & - & & \\ -n = 1 & $pi^C$ & 13 & - & - & - & - & 61 dB & 1131 a.u. \\ - & $pi^S$ & 0 & - & - & - & - & & \\ \hline - & C & 37 & 39 & - & - & - & & \\ -n = 2 & $pi^C$ & 11 & 13 & - & - & - & 117 dB & 1974 a.u. \\ - & $pi^S$ & 17 & 0 & - & - & - & & \\ \hline - & C & 15 & 35 & 35 & - & - & & \\ -n = 3 & $pi^C$ & 9 & 11 & 11 & - & - & 138 dB & 1985 a.u. \\ - & $pi^S$ & 19 & 3 & 0 & - & - & & \\ \hline - & C & 11 & 27 & 27 & 23 & - & & \\ -n = 4 & $pi^C$ & 5 & 9 & 9 & 9 & - & 148 dB & 1993 a.u. \\ - & $pi^S$ & 16 & 3 & 2 & 0 & - & & \\ \hline - & C & 11 & 27 & 31 & 11 & 11 & & \\ -n = 5 & $pi^C$ & 5 & 9 & 8 & 5 & 5 & 153 dB & 2000 a.u. \\ - & $pi^S$ & 16 & 3 & 1 & 0 & 1 & & \\ \hline +\begin{tabular}{|c|c|c|c|}\hline +$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline +1 & 0.1~s & 0.1~s & 0.3~s \\ +2 & 1.1~s & 2.2~s & 12~s \\ +3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\ +4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\ +5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline \end{tabular} -} -\caption{Solver results for design with PRN as data input and 2000 a.u. as max arbitrary space} -\label{tbl:prn_2000} \end{table} +As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ? +When the area is limited, the design exploration space is more limited and the solver is able to +find an optimal solution faster. On the contrary, in the case of MAX/1500 with +5~stages, we were not able to obtain a result after 40~hours of computation so we decided to stop. + +\section{Experiments with fixed rejection target} +\label{sec:fixed_rej} +This section presents the results of complementary quadratic program which we +minimize the area occupation for a targeted noise level. + +The experimental setup is also composed of three cases. The raw input is the same +as previous section, a PRN generator, which fixes the input data size $\Pi^I$. +Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60 or 80~dB. +Hence, the three cases have been named: MIN/40, MIN/60, MIN/80. +The number of configurations $p$ is the same as previous section. + +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. +Table~\ref{tbl:gurobi_min_80} shows the results obtained by the filter solver for MIN/80. + +\renewcommand{\arraystretch}{1.4} + \begin{table} -\centering -\begin{tabular}{|c|c|c|c|c|}\hline -Input & Stages & Computation time & Vivado time & Redpitaya time \\\hline\hline - & 1 & 0.02~s & $\approx$ 20 min & $\approx$ 1 min \\ -PRN & 2 & 1.70~s & $\approx$ 20 min & $\approx$ 1 min \\ - & 3 & 19~s & $\approx$ 20 min & $\approx$ 1 min \\\hline -\end{tabular} -\caption{Time to compute and deploy the designs for PRN 500} -\label{tbl:time_prn_500} + \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40} + \label{tbl:gurobi_min_40} + \centering + {\scalefont{0.77} + \begin{tabular}{|c|ccccc|c|c|} + \hline + $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ + \hline + 1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\ + 2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\ + 3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\ + 4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\ + \hline + \end{tabular} + } \end{table} \begin{table} -\centering -\begin{tabular}{|c|c|c|c|c|}\hline -Input & Stages & Computation time & Vivado time & Redpitaya time \\\hline\hline - & 1 & 0.07~s & $\approx$ 20 min & $\approx$ 1 min \\ - & 2 & 1.31~s & $\approx$ 20 min & $\approx$ 1 min \\ -PRN & 3 & 119~s ($\approx$ 2~min) & $\approx$ 20 min & $\approx$ 1 min \\ - & 4 & 270~s ($\approx$ 5~min) & $\approx$ 20 min & $\approx$ 1 min \\ - & 5 & 5998~s ($\approx$ 2~h) & $\approx$ 20 min & $\approx$ 1 min \\\hline -\end{tabular} -\caption{Time to compute and deploy the designs for PRN 1000} -\label{tbl:time_prn_1000} + \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60} + \label{tbl:gurobi_min_60} + \centering + {\scalefont{0.77} + \begin{tabular}{|c|ccccc|c|c|} + \hline + $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ + \hline + 1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\ + 2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\ + 3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\ + 4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\ + 5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\ + \hline + \end{tabular} + } \end{table} \begin{table} -\centering -\begin{tabular}{|c|c|c|c|c|}\hline -Input & Stages & Computation time & Vivado time & Redpitaya time \\\hline\hline - & 1 & 0.07~s & $\approx$ 20 min & $\approx$ 1 min \\ - & 2 & 0.75~s & $\approx$ 20 min & $\approx$ 1 min \\ -PRN & 3 & 36~s & - & - \\ - & 4 & 14500~s ($\approx$ 4~h) & $\approx$ 20 min & $\approx$ 1 min \\ - & 5 & 74237~s ($\approx$ 20~h) & $\approx$ 20 min & $\approx$ 1 min \\\hline -\end{tabular} -\caption{Time to compute and deploy the designs for PRN 2000} -\label{tbl:time_prn_2000} + \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80} + \label{tbl:gurobi_min_80} + \centering + {\scalefont{0.77} + \begin{tabular}{|c|ccccc|c|c|} + \hline + $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ + \hline + 1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\ + 2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\ + 3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\ + 4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\ + 5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\ + \hline + \end{tabular} + } \end{table} +\renewcommand{\arraystretch}{1} -\section{Experiments with fixed rejection target} +From these tables, we can first state that all configuration reach the target rejection +level and more we have stages lesser is the area occupied in arbitrary unit. +Futhermore, the area of the monolithic filter is twice bigger than the two cascaded. +More generally, more there is filters lower is the occupied area. + +Like in previous section, the solver choose always a little filter as first +filter stage and the second one is often the biggest filter. this choice can be explain +as the previous section. The solver uses just enough bits to not degrade the input +signal and in second filter it can choose a better filter to improve rejection without +have too bits in the output data. + +For the specific case in MIN/40 for $n = 5$ the solver has determined that the optimal +number of filter is 4 so it not chose any configuration in last filter. Hence this +solution is equivalent to the result for $n = 4$. + +The following graphs present the rejection for real data on the FPGA. In all following +figures, the solid line represents the actual rejection of the filtered +data on the FPGA as measured experimentally and the dashed line are the noise level +given by the quadratic solver. + +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. +Figure~\ref{fig:min_80} shows the rejection of the different configurations in the case of MIN/80. \begin{figure} \centering -\includegraphics[width=\linewidth]{images/min_area/prn_50} -\caption{Results for design with PRN as data input and 50 dB as aimed rejection level} -\label{fig:prn_500} +\includegraphics[width=\linewidth]{images/min_40} +\caption{Signal spectrum for MIN/40} +\label{fig:min_40} \end{figure} \begin{figure} \centering -\includegraphics[width=\linewidth]{images/min_area/prn_100} -\caption{Results for design with PRN as data input and 50 dB as aimed rejection level} -\label{fig:prn_100} +\includegraphics[width=\linewidth]{images/min_60} +\caption{Signal spectrum for MIN/60} +\label{fig:min_60} \end{figure} \begin{figure} \centering -\includegraphics[width=\linewidth]{images/min_area/prn_150} -\caption{Results for design with PRN as data input and 2000 a.u. as max arbitrary space} -\label{fig:prn_150} +\includegraphics[width=\linewidth]{images/min_80} +\caption{Signal spectrum for MIN/80} +\label{fig:min_80} \end{figure} +We observe that all rejections given by the quadratic solver are close to the real +rejection. All curves prove that the constraint to reach the target rejection is +respected both monolithic filter or cascaded filters. + +Table~\ref{tbl:resources_usage} shows the resources usage in the case of MIN/40, MIN/60 and +MIN/80 \emph{i.e.} when the target rejection is fixed to 40, 60 and 80~dB. We +have taken care to extract solely the resources used by +the FIR filters and remove additional processing blocks including FIFO and PL to +PS communication. + +\begin{table} + \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} + \label{tbl:resources_usage_comp} + \centering + \begin{tabular}{|c|c|ccc|c|} + \hline + $n$ & & MIN/40 & MIN/60 & MIN/80 & \emph{Zynq 7010} \\ \hline\hline + & LUT & 343 & 334 & 772 & \emph{17600} \\ + 1 & BRAM & 1 & 1 & 1 & \emph{120} \\ + & DSP & 27 & 39 & 55 & \emph{80} \\ \hline + & LUT & 1252 & 2862 & 5099 & \emph{17600} \\ + 2 & BRAM & 2 & 2 & 2 & \emph{120} \\ + & DSP & 0 & 0 & 0 & \emph{80} \\ \hline + & LUT & 891 & 2148 & 2023 & \emph{17600} \\ + 3 & BRAM & 3 & 3 & 3 & \emph{120} \\ + & DSP & 0 & 0 & 19 & \emph{80} \\ \hline + & LUT & 662 & 1729 & 2451 & \emph{17600} \\ + 4 & BRAM & 4 & 4 & 4 & \emph{120} \\ + & DPS & 0 & 0 & 7 & \emph{80} \\ \hline + & LUT & - & 1259 & 2602 & \emph{17600} \\ + 5 & BRAM & - & 5 & 5 & \emph{120} \\ + & DPS & - & 0 & 0 & \emph{80} \\ \hline + \end{tabular} +\end{table} + +If we keep the previous estimation of cost of one DSP in term of LUT (1 DSP $\approx$ 100 LUT) +the real resource consumption decrease in function of number of stage filter according +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 +filters is lesser than expected. + +Finally, the table~\ref{tbl:area_time_comp} shows the computation time to solve +the quadratic program. + +\begin{table} +\caption{Time to solve the quadratic program with Gurobi} +\label{tbl:area_time_comp} +\centering +\begin{tabular}{|c|c|c|c|}\hline +$n$ & Time (MIN/40) & Time (MIN/60) & Time (MIN/80) \\\hline\hline +1 & 0.07~s & 0.02~s & 0.01~s \\ +2 & 7.8~s & 16~s & 14~s \\ +3 & 4.7~s & 14~s & 28~s \\ +4 & 39~s & 20~s & 193~s \\ +5 & 126~s & 12~s & 170~s \\\hline +\end{tabular} +\end{table} + +The time needed to solve this configuration are substantially faster than time +needed in the previous section. Indeed the worst time in this case is only 3~minutes +in balance of 3~days on previous section. We are able to solve more easily this +problem than the previous one. + \section{Conclusion} +In this paper, we have proposed a new approach to work with a cascade of FIR filter inside a FPGA. +This method aims to be hardware independent and focus an high-level of abstraction. +We have modeled the FIR filter operation and the data shift impact. With this model +we have created a quadratic program to select the optimal FIR coefficient set to reject a +maximum of noise. In our experiments we have chosen deliberately some common tools +to design the filter coefficients but we can use any other method. + +Our experimental results are very promising in providing a rational approach to selecting +the coefficients of each FIR filter in the context of a performance target for a chain of +such filters. The FPGA design that is produced automatically by our +workflow is able to filter an input signal as expected which validates our model and our approach. +We can easily change the quadratic program to adapt it to an other problem. + +A perspective is to model and add the decimators to the processing chain to have a classical +FIR filter and decimator. The impact of the decimator is not so trivial, especially in terms of silicon +area for the subsequent stages since some hardware optimization can be applied in +this case. + +The software used to demonstrate the concepts developed in this paper is based on the +CPU-FPGA co-design framework available at \url{https://github.com/oscimp/oscimpDigital}. + \section*{Acknowledgement} This work is supported by the ANR Programme d'Investissement d'Avenir in diff --git a/images/max_1000.pdf b/images/max_1000.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e8226b1b31fd8fbe21fa311d308fcb7d7dca4538 GIT binary patch literal 35582 zcmafZcT|%>(=RGWl`g%bLMYNpLX(aNp#&8{r1wrjZ%Xe)5kV3{5keIZlp=ya=p_WC zcMuYg-a9w%d%t_n{qN>HC$nXCXZE)y!R&o_#I)S3VfGsKww`wO6w1mJP?(RswL3+?wD4a^ z6LBInbrK5*kF2Dj&d}<}$$*TBS1*p@?`zjt*g>^dBlw6d4T_EBtS%X{x6`?5)xAXUo7ECSN&f_eq}y%IL1IW*MP~U_RSwG zZbcB?5_)1xG?w;N0RQ{CubN{8Q_d%Jt79+`?7(&5P^Bo9mU<3-c)q_oZj%`K9xBm2g7C*$jc{*)`40 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b/references.bib index 844afc1..39b5f40 100644 --- a/references.bib +++ b/references.bib @@ -10,7 +10,7 @@ } @article{kodek1980design, - title={Design of optimal finite wordlength {FIR} digital filters using integer + title={Design of optimal finite wordlength {FIR} digital filters using integer programming techniques}, author={Kodek, Dusan}, journal={IEEE Transactions on Acoustics, Speech, and Signal Processing}, @@ -44,3 +44,54 @@ note={availble online, accessed May 2018} publisher={AIP Publishing} } +@inproceedings{lim_1996, +author={Y.-C. Lim and R. Yang and B. Liu}, +booktitle={1996 IEEE International Symposium on Circuits and Systems. Circuits and Systems Connecting the World. ISCAS 96}, +title={The design of cascaded FIR filters}, +year={1996}, +volume={2}, +number={}, +pages={181-184 vol.2}, +keywords={cascade networks;digital filters;FIR filters;filtering theory;linear programming;frequency response;cascaded FIR filters;stopband response;minimum attenuation requirement;passband ripple magnitude;linear-programming technique;FIR filter design;filter optimisation;Finite impulse response filter;IIR filters;Passband;Frequency;Signal sampling;Band pass filters;Digital filters;Attenuation;Image sampling;Linear programming}, +doi={10.1109/ISCAS.1996.540382}, +ISSN={}, +month={May},} + +@article{lim_1988, +author={Y. C. {Lim} and B. {Liu}}, +journal={IEEE Transactions on Acoustics, Speech, and Signal Processing}, +title={Design of cascade form FIR filters with discrete valued coefficients}, +year={1988}, +volume={36}, +number={11}, +pages={1735-1739}, +keywords={cascade networks;digital filters;filtering and prediction theory;iterative equalisation strategy;cascade form FIR filters;discrete valued coefficients;peak ripple;prototype filter;roundoff noise property;Finite impulse response filter;Low pass filters;Band pass filters;Passband;Prototypes;Frequency;Digital filters;Digital arithmetic;Design optimization;Sampling methods}, +doi={10.1109/29.9010}, +ISSN={0096-3518}, +month={Nov},} + +@inproceedings{young_1992, +author={C. {Young} and D. L. {Jones}}, +booktitle={[Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing}, +title={Improvement in finite wordlength FIR digital filter design by cascading}, +year={1992}, +volume={5}, +number={}, +pages={109-112 vol.5}, +keywords={approximation theory;digital filters;integer programming;series (mathematics);finite wordlength filter;quantization;FIR digital filter design;finite impulse response;digital systems;finite wordlength coefficients;cascaded subfilters;stopband suppression;Taylor series approximation;linear integer program;passband deviation;Finite impulse response filter;Digital filters;Linear programming;Passband;Quantization;Frequency response;Digital systems;Taylor series;Minimax techniques;Design optimization}, +doi={10.1109/ICASSP.1992.226646}, +ISSN={1520-6149}, +month={March},} + +@article{smith_1998, +author={L. M. {Smith}}, +journal={IEEE Transactions on Signal Processing}, +title={Decomposition of FIR digital filters for realization via the cascade connection of subfilters}, +year={1998}, +volume={46}, +number={6}, +pages={1681-1684}, +keywords={FIR filters;digital filters;cascade networks;Z transforms;transfer functions;frequency response;Newton-Raphson method;convergence of numerical methods;search problems;poles and zeros;FIR digital filters;subfilters cascade connection;even-order linear-phase FIR filters;filter decomposition;fourth-order subfilters;second-order subfilters;roots;z-domain filter transfer function;complex z plane;impulse response symmetry;unit circle;perimeter;complex values;real values;impulse response coefficients;root-finding algorithm;Newton-Raphson method;2D search;Cauchy-Riemann relations;convergence speed;frequency response characteristics;Finite impulse response filter;Digital filters;Polynomials;Programmable logic arrays;Transfer functions;Testing;Frequency response;Application specific integrated circuits;Nonlinear filters;Passband}, +doi={10.1109/78.678490}, +ISSN={1053-587X}, +month={June},}