ajout qlq ref et qlq corrections

[dmems12.git] / dmems12.tex

diff --git a/dmems12.tex b/dmems12.tex
index 572feb269713bf7861174e775175b282d9542b57..ad693f6d8121197a6f18a58235b7bda47d07445c 100644 (file)
--- a/dmems12.tex
+++ b/dmems12.tex
@@ -26,7 +26,6 @@
 \newcommand{\tab}{\ \ \ }
 
 
 \newcommand{\tab}{\ \ \ }
 
 

-

 \begin{document}
 
 
 \begin{document}
 
 


@@ -188,7 +187,7 @@ the cantilevers. They are called base profile and tip profile in the
 following. Furthermore, a reference profile is taken on the base of
 the cantilever array.
 
 following. Furthermore, a reference profile is taken on the base of
 the cantilever array.
 

-The pixels intensity $I$ (in gray level) of each profile is modelized by :
+The pixels intensity $I$ (in gray level) of each profile is modelized by:

 
 \begin{equation}
 \label{equ:profile}
 
 \begin{equation}
 \label{equ:profile}


@@ -217,7 +216,7 @@ images coming from the camera. The accuracy of results must be close
 to the maximum precision ever obtained experimentally on the
 architecture, i.e. 0.3nm. Finally, the latency between an image
 entering in the unit and the deflections must be as small as possible
 to the maximum precision ever obtained experimentally on the
 architecture, i.e. 0.3nm. Finally, the latency between an image
 entering in the unit and the deflections must be as small as possible

-(NB : future works plan to add some control on the cantilevers).\\
+(NB: future works plan to add some control on the cantilevers).\\

 
 If we put aside some hardware issues like the speed of the link
 between the camera and the computation unit, the time to deserialize
 
 If we put aside some hardware issues like the speed of the link
 between the camera and the computation unit, the time to deserialize


@@ -240,7 +239,7 @@ E6650 at 2.33GHz, this program reaches an average of 155Mflops.
 
 Obviously, some cache effects and optimizations on
 huge amount of computations can drastically increase these
 
 Obviously, some cache effects and optimizations on
 huge amount of computations can drastically increase these

-performances : peak efficiency is about 2.5Gflops for the considered
+performances: peak efficiency is about 2.5Gflops for the considered

 CPU. But this is not the case for phase computation that used only few
 tenth of values.\\
 
 CPU. But this is not the case for phase computation that used only few
 tenth of values.\\
 


@@ -256,7 +255,7 @@ case of an occasional load of the system, it could be largely
 overtaken. A solution would be to use a real-time operating system but
 another one to search for a more efficient algorithm.
 
 overtaken. A solution would be to use a real-time operating system but
 another one to search for a more efficient algorithm.
 

-But the main drawback is the latency of such a solution : since each
+But the main drawback is the latency of such a solution: since each

 profile must be treated one after another, the deflection of 100
 cantilevers takes about $200\times 10.5 = 2.1$ms, which is inadequate
 for an efficient control. An obvious solution is to parallelize the
 profile must be treated one after another, the deflection of 100
 cantilevers takes about $200\times 10.5 = 2.1$ms, which is inadequate
 for an efficient control. An obvious solution is to parallelize the


@@ -290,25 +289,23 @@ computation, we give some general information about FPGAs and the board we use.
 
 \subsection{FPGAs}
 
 
 \subsection{FPGAs}
 

-A field-programmable gate array (FPGA) is an integrated circuit
-designed to be configured by the customer. FGPAs are composed of
-programmable logic components, called configurable logic blocks
-(CLB). These blocks mainly contains look-up tables (LUT), flip/flops
-(F/F) and latches, organized in one or more slices connected
-together. Each CLB can be configured to perform simple (AND, XOR, ...)
-or complex combinational functions. They are interconnected by
-reconfigurable links. Modern FPGAs contain memory elements and
-multipliers which enable to simplify the design and to increase the
-performance. Nevertheless, all other complex operations, like
-division, trigonometric functions, $\ldots$ are not available and must
-be done by configuring a set of CLBs. Since this configuration is not
-obvious at all, it can be done via a framework, like ISE. Such a
-software can synthetize a design written in an hardware description
-language (HDL), map it onto CLBs, place/route them for a specific
-FPGA, and finally produce a bitstream that is used to configre the
-FPGA. Thus, from the developper point of view, the main difficulty is
-to translate an algorithm in HDL code, taking account FPGA resources
-and constraints like clock signals and I/O values that drive the FPGA.
+A field-programmable gate  array (FPGA) is an integrated  circuit designed to be
+configured by the customer. FGPAs are composed of programmable logic components,
+called  configurable logic blocks  (CLB). These  blocks mainly  contains look-up
+tables  (LUT), flip/flops (F/F)  and latches,  organized in  one or  more slices
+connected together. Each CLB can be configured to perform simple (AND, XOR, ...)
+or complex  combinational functions.  They are interconnected  by reconfigurable
+links.  Modern FPGAs  contain memory  elements and  multipliers which  enable to
+simplify the  design and  to increase the  performance. Nevertheless,  all other
+complex  operations, like  division, trigonometric  functions, $\ldots$  are not
+available  and  must  be  done  by   configuring  a  set  of  CLBs.  Since  this
+configuration  is not  obvious at  all, it  can be  done via  a  framework, like
+ISE~\cite{ISE}. Such  a software  can synthetize a  design written in  a hardware
+description language  (HDL), map it onto  CLBs, place/route them  for a specific
+FPGA, and finally  produce a bitstream that is used to  configre the FPGA. Thus,
+from  the developper  point of  view,  the main  difficulty is  to translate  an
+algorithm in HDL code, taking  account FPGA resources and constraints like clock
+signals and I/O values that drive the FPGA.

 
 Indeed, HDL programming is very different from classic languages like
 C. A program can be seen as a state-machine, manipulating signals that
 
 Indeed, HDL programming is very different from classic languages like
 C. A program can be seen as a state-machine, manipulating signals that


@@ -337,7 +334,7 @@ pipelines in order to handle multiple data streams.
 The board we use is designed by the Armadeus compagny, under the name
 SP Vision. It consists in a development board hosting a i.MX27 ARM
 processor (from Freescale). The board includes all classical
 The board we use is designed by the Armadeus compagny, under the name
 SP Vision. It consists in a development board hosting a i.MX27 ARM
 processor (from Freescale). The board includes all classical

-connectors : USB, Ethernet, ... A Flash memory contains a Linux kernel
+connectors: USB, Ethernet, ... A Flash memory contains a Linux kernel

 that can be launched after booting the board via u-Boot.
 
 The processor is directly connected to a Spartan3A FPGA (from Xilinx)
 that can be launched after booting the board via u-Boot.
 
 The processor is directly connected to a Spartan3A FPGA (from Xilinx)


@@ -371,23 +368,23 @@ Let consider a profile $P$, that is a segment of $M$ pixels with an
 intensity in gray levels. Let call $I(x)$ the intensity of profile in $x
 \in [0,M[$. 
 
 intensity in gray levels. Let call $I(x)$ the intensity of profile in $x
 \in [0,M[$. 
 

-At first, only $M$ values of $I$ are known, for $x = 0, 1,
-\ldots,M-1$. A normalisation allows to scale known intensities into
-$[-1,1]$. We compute splines that fit at best these normalised
-intensities. Splines are used to interpolate $N = k\times M$ points
-(typically $k=4$ is sufficient), within $[0,M[$. Let call $x^s$ the
-coordinates of these $N$ points and $I^s$ their intensities.
+At  first, only $M$  values of  $I$ are  known, for  $x =  0, 1,  \ldots,M-1$. A
+normalisation  allows  to scale  known  intensities  into  $[-1,1]$. We  compute
+splines  that fit  at best  these normalised  intensities. Splines  (SPL  in the
+following) are  used to interpolate $N  = k\times M$ points  (typically $k=4$ is
+sufficient), within $[0,M[$. Let call  $x^s$ the coordinates of these $N$ points
+    and $I^s$ their intensities.

 
 In order to have the frequency, the mean line $a.x+b$ (see equation \ref{equ:profile}) of $I^s$ is
 computed. Finding intersections of $I^s$ and this line allow to obtain
 the period thus the frequency.
 
 
 In order to have the frequency, the mean line $a.x+b$ (see equation \ref{equ:profile}) of $I^s$ is
 computed. Finding intersections of $I^s$ and this line allow to obtain
 the period thus the frequency.
 

-The phase is computed via the equation :
+The phase is computed via the equation:

 \begin{equation}
 \theta = atan \left[ \frac{\sum_{i=0}^{N-1} sin(2\pi f x^s_i) \times I^s(x^s_i)}{\sum_{i=0}^{N-1} cos(2\pi f x^s_i) \times I^s(x^s_i)} \right]
 \end{equation}
 
 \begin{equation}
 \theta = atan \left[ \frac{\sum_{i=0}^{N-1} sin(2\pi f x^s_i) \times I^s(x^s_i)}{\sum_{i=0}^{N-1} cos(2\pi f x^s_i) \times I^s(x^s_i)} \right]
 \end{equation}
 

-Two things can be noticed :
+Two things can be noticed:

 \begin{itemize}
 \item the frequency could also be obtained using the derivates of
   spline equations, which only implies to solve quadratic equations.
 \begin{itemize}
 \item the frequency could also be obtained using the derivates of
   spline equations, which only implies to solve quadratic equations.


@@ -400,9 +397,9 @@ Two things can be noticed :
 \subsubsection{Least square algorithm}
 
 Assuming that we compute the phase during the acquisition loop,
 \subsubsection{Least square algorithm}
 
 Assuming that we compute the phase during the acquisition loop,

-equation \ref{equ:profile} has only 4 parameters :$a, b, A$, and
+equation \ref{equ:profile} has only 4 parameters: $a, b, A$, and

 $\theta$, $f$ and $x$ being already known. Since $I$ is non-linear, a
 $\theta$, $f$ and $x$ being already known. Since $I$ is non-linear, a

-least square method based an Gauss-newton algorithm must be used to
+least square method based on a Gauss-newton algorithm can be used to

 determine these four parameters. Since it is an iterative process
 ending with a convergence criterion, it is obvious that it is not
 particularly adapted to our design goals.
 determine these four parameters. Since it is an iterative process
 ending with a convergence criterion, it is obvious that it is not
 particularly adapted to our design goals.


@@ -410,16 +407,16 @@ particularly adapted to our design goals.
 Fortunatly, it is quite simple to reduce the number of parameters to
 only $\theta$. Let $x^p$ be the coordinates of pixels in a segment of
 size $M$. Thus, $x^p = 0, 1, \ldots, M-1$. Let $I(x^p)$ be their
 Fortunatly, it is quite simple to reduce the number of parameters to
 only $\theta$. Let $x^p$ be the coordinates of pixels in a segment of
 size $M$. Thus, $x^p = 0, 1, \ldots, M-1$. Let $I(x^p)$ be their

-intensity. Firstly, we "remove" the slope by computing :
+intensity. Firstly, we "remove" the slope by computing:

 
 \[I^{corr}(x^p) = I(x^p) - a.x^p - b\]
 
 Since linear equation coefficients are searched, a classical least
 
 \[I^{corr}(x^p) = I(x^p) - a.x^p - b\]
 
 Since linear equation coefficients are searched, a classical least

-square method can be used to determine $a$ and $b$ :
+square method can be used to determine $a$ and $b$:

 
 \[a = \frac{covar(x^p,I(x^p))}{var(x^p)} \]
 
 
 \[a = \frac{covar(x^p,I(x^p))}{var(x^p)} \]
 

-Assuming an overlined symbol means an average, then :
+Assuming an overlined symbol means an average, then:

 
 \[b = \overline{I(x^p)} - a.\overline{{x^p}}\]
 
 
 \[b = \overline{I(x^p)} - a.\overline{{x^p}}\]
 


@@ -427,22 +424,22 @@ Let $A$ be the amplitude of $I^{corr}$, i.e.
 
 \[A = \frac{max(I^{corr}) - min(I^{corr})}{2}\]
 
 
 \[A = \frac{max(I^{corr}) - min(I^{corr})}{2}\]
 

-Then, the least square method to find $\theta$ is reduced to search the minimum of :
+Then, the least square method to find $\theta$ is reduced to search the minimum of:

 
 \[\sum_{i=0}^{M-1} \left[ cos(2\pi f.i + \theta) - \frac{I^{corr}(i)}{A} \right]^2\]
 
 
 \[\sum_{i=0}^{M-1} \left[ cos(2\pi f.i + \theta) - \frac{I^{corr}(i)}{A} \right]^2\]
 

-It is equivalent to derivate this expression and to solve the following equation :
+It is equivalent to derivate this expression and to solve the following equation:

 
 \begin{eqnarray*}
 2\left[ cos\theta \sum_{i=0}^{M-1} I^{corr}(i).sin(2\pi f.i) + sin\theta \sum_{i=0}^{M-1} I^{corr}(i).cos(2\pi f.i)\right] \\
 - A\left[ cos2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right]   = 0
 \end{eqnarray*}
 
 
 \begin{eqnarray*}
 2\left[ cos\theta \sum_{i=0}^{M-1} I^{corr}(i).sin(2\pi f.i) + sin\theta \sum_{i=0}^{M-1} I^{corr}(i).cos(2\pi f.i)\right] \\
 - A\left[ cos2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right]   = 0
 \end{eqnarray*}
 

-Several points can be noticed :
+Several points can be noticed:

 \begin{itemize}
 \item As in the spline method, some parts of this equation can be
   computed before the acquisition loop. It is the case of sums that do
 \begin{itemize}
 \item As in the spline method, some parts of this equation can be
   computed before the acquisition loop. It is the case of sums that do

-  not depend on $\theta$ :
+  not depend on $\theta$:

 
 \[ \sum_{i=0}^{M-1} sin(4\pi f.i), \sum_{i=0}^{M-1} cos(4\pi f.i) \] 
 
 
 \[ \sum_{i=0}^{M-1} sin(4\pi f.i), \sum_{i=0}^{M-1} cos(4\pi f.i) \] 
 


@@ -452,7 +449,7 @@ computed.
 \item The simplest method to find the good $\theta$ is to discretize
   $[-\pi,\pi]$ in $nb_s$ steps, and to search which step leads to the
   result closest to zero. By the way, three other lookup tables can
 \item The simplest method to find the good $\theta$ is to discretize
   $[-\pi,\pi]$ in $nb_s$ steps, and to search which step leads to the
   result closest to zero. By the way, three other lookup tables can

-  also be computed before the loop :
+  also be computed before the loop:

 
 \[ sin \theta, cos \theta, \]
 
 
 \[ sin \theta, cos \theta, \]
 


@@ -462,7 +459,7 @@ computed.
 
 \end{itemize}
 
 
 \end{itemize}
 

-Finally, the whole summarizes in an algorithm (called LSQ in the following) in two parts, one before and one during the acquisition loop :
+Finally, the whole summarizes in an algorithm (called LSQ in the following) in two parts, one before and one during the acquisition loop:

 \begin{algorithm}[h]
 \caption{LSQ algorithm - before acquisition loop.}
 \label{alg:lsq-before}
 \begin{algorithm}[h]
 \caption{LSQ algorithm - before acquisition loop.}
 \label{alg:lsq-before}


@@ -549,7 +546,7 @@ Finally, the whole summarizes in an algorithm (called LSQ in the following) in t
 
 \subsubsection{Comparison}
 
 
 \subsubsection{Comparison}
 

-We compared the two algorithms on the base of three criterions :
+We compared the two algorithms on the base of three criteria:

 \begin{itemize}
 \item precision of results on a cosinus profile, distorted with noise,
 \item number of operations,
 \begin{itemize}
 \item precision of results on a cosinus profile, distorted with noise,
 \item number of operations,


@@ -568,12 +565,12 @@ discretization correspond to an error of 0.15nm on the lever
 deflection, which is smaller than the best precision they achieved,
 i.e. 0.3nm.
 
 deflection, which is smaller than the best precision they achieved,
 i.e. 0.3nm.
 

-For each test, we add some noise to the profile : each group of two
+For each test, we add some noise to the profile: each group of two

 pixels has its intensity added to a random number picked in $[-N,N]$
 (NB: it should be noticed that picking a new value for each pixel does
 not distort enough the profile). The absolute error on the result is
 evaluated by comparing the difference between the reference and
 pixels has its intensity added to a random number picked in $[-N,N]$
 (NB: it should be noticed that picking a new value for each pixel does
 not distort enough the profile). The absolute error on the result is
 evaluated by comparing the difference between the reference and

-computed phase, out of $2\pi$, expressed in percents. That is : $err =
+computed phase, out of $2\pi$, expressed in percents. That is: $err =

 100\times \frac{|\theta_{ref} - \theta_{comp}|}{2\pi}$.
 
 Table \ref{tab:algo_prec} gives the maximum and average error for the two algorithms and increasing values of $N$.
 100\times \frac{|\theta_{ref} - \theta_{comp}|}{2\pi}$.
 
 Table \ref{tab:algo_prec} gives the maximum and average error for the two algorithms and increasing values of $N$.


@@ -615,7 +612,7 @@ largely beyond the worst experimental ones.
 
 \begin{figure}[ht]
 \begin{center}
 
 \begin{figure}[ht]
 \begin{center}

-  \includegraphics[width=9cm]{intens-noise20}
+  \includegraphics[width=\columnwidth]{intens-noise20}

 \end{center}
 \caption{Sample of worst profile for N=10}
 \label{fig:noise20}
 \end{center}
 \caption{Sample of worst profile for N=10}
 \label{fig:noise20}


@@ -623,7 +620,7 @@ largely beyond the worst experimental ones.
 
 \begin{figure}[ht]
 \begin{center}
 
 \begin{figure}[ht]
 \begin{center}

-  \includegraphics[width=9cm]{intens-noise60}
+  \includegraphics[width=\columnwidth]{intens-noise60}

 \end{center}
 \caption{Sample of worst profile for N=30}
 \label{fig:noise60}
 \end{center}
 \caption{Sample of worst profile for N=30}
 \label{fig:noise60}


@@ -640,48 +637,42 @@ $<$, $>$) is taken account. Translating the two algorithms in C code, we
 obtain about 430 operations for LSQ and 1550 (plus few tenth for
 $atan$) for SPL. This result is largely in favor of LSQ. Nevertheless,
 considering the total number of operations is not really pertinent for
 obtain about 430 operations for LSQ and 1550 (plus few tenth for
 $atan$) for SPL. This result is largely in favor of LSQ. Nevertheless,
 considering the total number of operations is not really pertinent for

-an FPGA implementation : it mainly depends on the type of operations
+an FPGA implementation: it mainly depends on the type of operations

 and their
 ordering. The final decision is thus driven by the third criterion.\\
 
 and their
 ordering. The final decision is thus driven by the third criterion.\\
 

-The Spartan 6 used in our architecture has hard constraint : it has no
-built-in floating point units. Obviously, it is possible to use some
-existing "black-boxes" for double precision operations. But they have
-a quite long latency. It is much simpler to exclusively use integers,
-with a quantization of all double precision values. Obviously, this
-quantization should not decrease too much the precision of
-results. Furthermore, it should not lead to a design with a huge
-latency because of operations that could not complete during a single
-or few clock cycles. Divisions are in this case and, moreover, they
-need an varying number of clock cycles to complete. Even
-multiplications can be a problem : DSP48 take inputs of 18 bits
-maximum. For larger multiplications, several DSP must be combined,
-increasing the latency.
-
-Nevertheless, the hardest constraint does not come from the FPGA
-characteristics but from the algorithms. Their VHDL implentation will
-be efficient only if they can be fully (or near) pipelined. By the
-way, the choice is quickly done : only a small part of SPL can be.
-Indeed, the computation of spline coefficients implies to solve a
-tridiagonal system $A.m = b$. Values in $A$ and $b$ can be computed
-from incoming pixels intensity but after, the back-solve starts with
-the lastest values, which breaks the pipeline. Moreover, SPL relies on
-interpolating far more points than profile size. Thus, the end
-of SPL works on a larger amount of data than the beginning, which
-also breaks the pipeline.
-
-LSQ has not this problem : all parts except the dichotomial search
-work on the same amount of data, i.e. the profile size. Furthermore,
-LSQ needs less operations than SPL, implying a smaller output
-latency. Consequently, it is the best candidate for phase
-computation. Nevertheless, obtaining a fully pipelined version
-supposes that operations of different parts complete in a single clock
-cycle. It is the case for simulations but it completely fails when
-mapping and routing the design on the Spartan6. By the way,
-extra-latency is generated and there must be idle times between two
-profiles entering into the pipeline.
-
-%%Before obtaining the least bitstream, the crucial question is : how to
+The Spartan 6 used in our architecture has a hard constraint: it has no built-in
+floating  point  units.   Obviously,  it  is  possible  to   use  some  existing
+"black-boxes"  for double  precision  operations.  But they  have  a quite  long
+latency. It is much simpler to  exclusively use integers, with a quantization of
+all double  precision values. Obviously,  this quantization should  not decrease
+too much the  precision of results. Furthermore, it should not  lead to a design
+with  a huge  latency because  of operations  that could  not complete  during a
+single or few clock cycles. Divisions  are in this case and, moreover, they need
+a varying  number of  clock cycles  to complete. Even  multiplications can  be a
+problem:  DSP48 take  inputs of  18  bits maximum.  For larger  multiplications,
+several DSP must be combined, increasing the latency.
+
+Nevertheless, the hardest constraint does not come from the FPGA characteristics
+but from the algorithms. Their VHDL  implentation will be efficient only if they
+can be fully (or near) pipelined. By the way, the choice is quickly done: only a
+small  part of  SPL  can be.   Indeed,  the computation  of spline  coefficients
+implies to solve  a tridiagonal system $A.m =  b$. Values in $A$ and  $b$ can be
+computed from  incoming pixels intensity  but after, the back-solve  starts with
+the  lastest  values,  which  breaks  the  pipeline.  Moreover,  SPL  relies  on
+interpolating far more points than profile size. Thus, the end of SPL works on a
+larger amount of data than the beginning, which also breaks the pipeline.
+
+LSQ has  not this problem: all parts  except the dichotomial search  work on the
+same  amount  of  data, i.e.  the  profile  size.  Furthermore, LSQ  needs  less
+operations than SPL, implying a  smaller output latency. Consequently, it is the
+best candidate for phase  computation. Nevertheless, obtaining a fully pipelined
+version supposes that  operations of different parts complete  in a single clock
+cycle. It is  the case for simulations but it completely  fails when mapping and
+routing the design  on the Spartan6. By the way,  extra-latency is generated and
+there must be idle times between two profiles entering into the pipeline.
+
+%%Before obtaining the least bitstream, the crucial question is: how to

 %%translate the C code the LSQ into VHDL ?
 
 
 %%translate the C code the LSQ into VHDL ?
 
 


@@ -689,20 +680,58 @@ profiles entering into the pipeline.
 
 \section{Experimental tests}
 
 
 \section{Experimental tests}
 

+In this section we explain what  we have done yet. Until now, we could not perform
+real experiments  since we just have  received the FGPA  board. Nevertheless, we
+will include real experiments in the final version of this paper.
+

 \subsection{VHDL implementation}
 
 \subsection{VHDL implementation}
 

+
+

 % - ecriture d'un code en C avec integer
 % - calcul de la taille max en bit de chaque variable en fonction de la quantization.
 % - tests de quantization : équilibre entre précision et contraintes FPGA
 % - en parallèle : simulink et VHDL à la main
 % - ecriture d'un code en C avec integer
 % - calcul de la taille max en bit de chaque variable en fonction de la quantization.
 % - tests de quantization : équilibre entre précision et contraintes FPGA
 % - en parallèle : simulink et VHDL à la main

-%
+
+
+From the  LSQ algorithm,  we have written  a C  program which uses  only integer
+values  that have  been  previously  scaled. The  quantization  of doubles  into
+integers has  been performed  in order  to obtain a  good trade-off  between the
+number of bits  used and the precision. We have  compared the result of
+the LSQ version  using integers and doubles. We have observed  that the results of
+both versions were similar.
+
+Then we have built  two versions of VHDL codes: one directly  by hand coding and
+the     other     with    Matlab     using     the     Simulink    HDL     coder
+feature~\cite{HDLCoder}. Although  the approach is completely  different we have
+obtain VHDL  codes that are quite  comparable. Each approach  has advantages and
+drawbacks.   Roughly speaking, hand  coding provides  beautiful and  much better
+structured code  while HDL  coder provides  code faster.  In  terms of  speed of
+code, we  think that both  approaches will be  quite comparable with  a slightly
+advantage for hand coding.  We hope that real experiments will confirm that.  In
+the  LSQ algorithm, we  have replaced  all the  divisions by  multiplications by
+constants since divisions  are performed with constants depending  of the number
+of pixels in the profile (i.e. $M$).
+

 \subsection{Simulation}
 
 \subsection{Simulation}
 

+Currently, we have only simulated our VHDL codes with GHDL and GTKWave (two free
+tools with linux).  Both approaches led to correct results.  At the beginning of
+our simulations, our  pipiline could compute a new phase each  33 cycles and the
+length of the  pipeline was equal to  95 cycles.  When we tried  to generate the
+corresponding bitsream  with ISE environment  we had many problems  because many
+stages required  more than the  10$n$s required by  the clock frequency.   So we
+needed to decompose  some part of the  pipeline in order to add  some cycles and
+simplify some parts between a clock top.

 % ghdl + gtkwave
 % au mieux : une phase tous les 33 cycles, latence de 95 cycles.
 % mais routage/placement impossible.
 \subsection{Bitstream creation}
 
 % ghdl + gtkwave
 % au mieux : une phase tous les 33 cycles, latence de 95 cycles.
 % mais routage/placement impossible.
 \subsection{Bitstream creation}
 

+Currently both  approaches provide synthesable  bitstreams with ISE.   We expect
+that the  pipeline will  have a latency  of 112  cycles, i.e. 1.12$\mu$s  and it
+could accept new profiles of pixel each 48 cycles, i.e. 480$n$s.
+

 % pas fait mais prévision d'une sortie tous les 480ns avec une latence de 1120
 
 \label{sec:results}
 % pas fait mais prévision d'une sortie tous les 480ns avec une latence de 1120
 
 \label{sec:results}