\usepackage{amssymb}
\usepackage{amsfonts,amssymb}
\usepackage{amsmath}
+\usepackage{amscd}
\usepackage{graphicx}
\usepackage{subfigure}
\usepackage{makeidx}
\usepackage{pgfplots}
\usepackage{pgf}
\usepackage{tikz}
+\usepackage{url}
+\usepackage{dsfont}
+\usepackage{commath}
+\usepackage{stmaryrd}
+\usepackage{moreverb}
+\usepackage{commath}
%% \usepackage{listings}
%% \usepackage{subfigure}
\captionsetup[lstlisting]{format=listing,labelfont=white,textfont=white, singlelinecheck=false, margin=0pt, font={bf,footnotesize}}
+\DeclareGraphicsRule{.pdftex}{pdf}{*}{}
+
+\DeclareGraphicsRule{.pstex}{eps}{*}{}
+
+
\makeindex
\setcounter{page}{1}
\part{This is a Part}
-\include{Chapters/chapter1/ch1}
-\include{Chapters/chapter2/ch2}
-\include{Chapters/chapter3/ch3}
-\include{Chapters/chapter5/ch5}
-\include{Chapters/chapter6/ch6}
-\include{Chapters/chapter7/ch7}
-\include{Chapters/chapter8/ch8}
-\include{Chapters/chapter9/ch9}
-\include{Chapters/chapter11/ch11}
-\include{Chapters/chapter14/ch14}
-\include{Chapters/chapter15/ch15}
+%% \include{Chapters/chapter1/ch1}
+%% \include{Chapters/chapter2/ch2}
+%% \include{Chapters/chapter3/ch3}
+%% \include{Chapters/chapter5/ch5}
+%% \include{Chapters/chapter6/ch6}
+%% \include{Chapters/chapter7/ch7}
+%% \include{Chapters/chapter8/ch8}
+%% \include{Chapters/chapter9/ch9}
+%% \include{Chapters/chapter11/ch11}
+%% \include{Chapters/chapter14/ch14}
+%% \include{Chapters/chapter15/ch15}
+ \include{Chapters/chapter16/ch16}
+%% \include{Chapters/chapter18/ch18}
\bibliographystyle{hep}
%%%\bibliography{biblio}
--- /dev/null
+\subsection{Gear-2 based sensitivity calculation}
+\label{sec:gear}
+
+The Gear-2 integration method is a backward differentiation
+formula (BDF), which approximates the derivative of a function
+using information from past few steps. Gear-2 method has been
+shown to be more suitable for many practical problems such as
+stiff problems where circuit behavior is affected by different time constants
+(fast ones with large poles and slow ones with small poles)~\cite{Vlach:Book'94}.
+Switching power converters and RF systems are typically stiff systems
+as waveforms changing in two different time scales are involved.
+
+% Although on one time step, Gear-2
+% involves more computation than backward-Euler and trapezoidal method,
+% it has better convergence and stability, and usually allows larger
+% time step. As a result, fewer time points are needed for a given
+% transient sweep range, and the simulation is speeded up with the same
+% accuracy
+
+Given the DAE~(\ref{eq:dae}), at the $i$-th time step,
+Gear-2 integration\index{Gear-2 integration}
+approximates the derivative $\ud q(x(t))/\ud t$
+by a two-step backward finite difference,
+\[
+ \frac{\ud q(x(t_i))}{\ud t}
+ %\ud q(x(t_i)) / \ud t
+ = \frac{1}{h_i}
+ \left[ q(x(t_{i})) - \alpha_1^i q(x(t_{i-1})) - \alpha_2^i q(x(t_{i-2}))\right],
+\]
+where the coefficients $\alpha$'s are chosen to satisfy Gear's
+backward differentiation formula~\cite{Gear:book'71}.
+For uniformly discretised time steps,
+the index $i$ in $h_i$, $\alpha_1^i$ and $\alpha_2^i$ can be omitted.
+
+For the first step $t_1$,
+only the initial condition at $t_0$ is available.
+Therefore backward-Euler is used, i.e.,
+\begin{equation}\label{eq:BE}
+ \tfrac{1}{h_1}[q(x_1)-q(x_0)] + f(x_1) = b_1.
+\end{equation}
+Since $x_0$ directly affects the solution of $x_1$,
+the sensitivity matrix\index{sensitivity matrix} up to now is
+\begin{equation}\label{eq:sens1}
+ \frac{\ud x_1}{\ud x_0}
+ = \left[\frac{1}{h_1}C_1+G_1\right]^{-1} \frac{C_0}{h_1}
+ = J_1^{-1} \frac{C_0}{h_1}.
+\end{equation}
+%For the sake of simplicity,
+Let $J_i$ denote $(1/h_i)C_i + G_i$
+in the remaining part of this chapter.
+In addition, the LU\index{LU} factorizations of $J_i$ are already calculated
+since they are used to solve the circuit state at
+each time step before we calculate the sensitivity.
+
+Next, for the solution at $t_2$,
+with the previous two steps information available,
+Gear-2 integration can be applied,
+\begin{equation}\label{eq:Gear_t2}
+ \tfrac{1}{h_2}[q(x_2) - \alpha_1 q(x_1) - \alpha_2 q(x_0)] + f(x_2) = b_2.
+\end{equation}
+In view of the sensitivity of $x_2$ with respect to the changes of $x_0$,
+\eqref{eq:Gear_t2} indicates that $x_2$'s perturbation can be
+traced back to $x_0$ along two paths:
+$x_2$ is directly affected by $x_0$,
+and $x_2$ is also affected indirectly by $x_0$ via $x_1$.
+That is,
+\[ %begin{equation}
+ \frac{\ud x_2}{\ud x_0} = \frac{\partial x_2}{\partial x_1}\frac{\ud x_1}{\ud x_0}
+ + \frac{\partial x_2}{\partial x_0},
+\] %end{equation}
+where the two partial derivatives are
+\[
+ \frac{\partial x_2}{\partial x_1} = J_2^{-1} \frac{\alpha_1}{h_2} C_1, \qquad
+ \frac{\partial x_2}{\partial x_0} = J_2^{-1} \frac{\alpha_2}{h_2} C_0,
+\]
+and $\ud x_1/\ud x_0$ %, the sensitivity of $x_1$ with respect to $x_0$,
+is calculated previously in \eqref{eq:sens1}.
+Thus, the sensitivity matrix deduced from \eqref{eq:Gear_t2} is
+\begin{equation}\label{eq:sens2}
+ \frac{\ud x_2}{\ud x_0}
+ = J_2^{-1} \frac{\alpha_1}{h_2} C_1 \cdot J_1^{-1} \frac{C_0}{h_1}
+ + J_2^{-1} \frac{\alpha_2}{h_2} C_0.
+\end{equation}
+
+Likewise, for the third time step $t_3$,
+apply the Gear-2 integration formula to DAE~(\ref{eq:dae}),
+\begin{equation}\label{eq:Gear_t3}
+ \tfrac{1}{h_3} [q(x_3) - \alpha_1 q(x_2) - \alpha_2 q(x_1)] + f(x_3) = b_3,
+\end{equation}
+and the chain rule for sensitivity is
+\[ %begin{equation}\label{eq:sens3}
+% \begin{split}
+% \frac{\ud x_3}{\ud x_0} &= \frac{\partial x_3}{\partial x_2}\frac{\ud x_2}{\ud x_0}
+% + \frac{\partial x_3}{\partial x_1}\frac{\ud x_1}{\ud x_0} \\
+% &= J_3^{-1} \frac{\alpha_1}{h_3} C_2 \frac{\ud x_2}{\ud x_0}
+% + J_3^{-1} \frac{\alpha_2}{h_3} C_1 \frac{\ud x_1}{\ud x_0}.
+% \end{split}
+ \frac{\ud x_3}{\ud x_0} \!=\! \frac{\partial x_3}{\partial x_2}\frac{\ud x_2}{\ud x_0}
+ + \frac{\partial x_3}{\partial x_1}\frac{\ud x_1}{\ud x_0}
+ \!=\! J_3^{-1} \!\left[ \frac{\alpha_1}{h_3} C_2 \frac{\ud x_2}{\ud x_0}
+ \!+\!\frac{\alpha_2}{h_3} C_1 \frac{\ud x_1}{\ud x_0}\right],
+\] %end{equation}
+where both $\ud x_2/\ud x_0$ and $\ud x_1/\ud x_0$
+are ready from the previous two time steps, i.e., Eqs.~(\ref{eq:sens2}) and~(\ref{eq:sens1}).
+Follow this chain rule in the aforementioned recursive style,
+the sensitivity matrix for Gear-2 integration is computed
+along the remaining time steps up to the $M$-th step,
+\begin{equation}\label{eq:sensM}
+% \begin{split}
+% J = \frac{\ud x_M}{\ud x_0}
+% & = J_M^{-1} \frac{\alpha_1}{h_M} C_{M-1} \frac{\ud x_{M-1}}{\ud x_0} \\
+% &\quad + J_M^{-1} \frac{\alpha_2}{h_M} C_{M-2} \frac{\ud x_{M-2}}{\ud x_0}.
+% \end{split}
+J\! =\! \frac{\ud x_M}{\ud x_0}
+ \! =\! J_M^{-1} \! \left[ \frac{\alpha_1}{h_M} C_{M-1} \frac{\ud x_{M-1}}{\ud x_0}
+ \! + \! \frac{\alpha_2}{h_M} C_{M-2} \frac{\ud x_{M-2}}{\ud x_0}\right].
+\end{equation}
+
+%% \begin{algorithm}
+%% \caption{The matrix-free\index{matrix-free} method for
+%% Krylov subspace\index{Krylov subspace} construction.}
+%% \label{alg:mf_Gear}
+%% \begin{algorithmic}[1]
+%% \REQUIRE current Krylov subspace basis vector $v$,
+%% time step lengths $h_i$,
+%% saved $C_i$ matrices and LU\index{LU} factors of $J_i$, $i=0,\ldots,M$
+%% \ENSURE matrix-vector product $r$, such that $r = A v$
+%% \STATE solve $J_1 p_2 = h_1^{-1} C_0 v$ for $p_2$
+%% \STATE solve $J_2 p_1 = h_2^{-1} \left( \alpha_1 C_1 p_2 + \alpha_2 C_0 v \right)$ for $p_1$
+%% \FOR{$i=3 \ldots M$}
+%% \STATE solve
+%% $J_i p_0 = \alpha_1 h_i^{-1} C_{i-1} p_1 + \alpha_2 h_i^{-1} C_{i-2} p_2$
+%% for $p_0$ \label{line:mf_Gear_loop}
+%% \STATE $p_2 = p_1$
+%% \STATE $p_1 = p_0$
+%% \ENDFOR
+%% \STATE $r = (k-1) p_0 - k v$ \label{line:shift}
+%% \end{algorithmic}
+%% \end{algorithm}
+
+
+%It is easy to judge that, as $J$ is dense, explicitly
+%computing $J$ is expensive and impractical for large problems.
+%In addition, with a dense matrix,
+%both direct or iterative solvers will be inefficient.
+%Fortunately, one iterative solver, GMRES,
+%only requires $J x$ (actually $Ax$),
+%instead of an explicit $J$, to fulfill the solving process.
+%This virtue greatly saves computation cost.
+Note that as the matrix-free\index{matrix-free} GMRES method is applied,
+which only requires matrix-vector multiplication and no explicit $J$ is required,
+as explained in Algorithm~\ref{alg:mf_Gear}.
+% Next section will explain how to use this technique for
+% our proposed Gear-2 envelope-following.
+
+With matrix-free method, the matrix-vector multiplication
+in Line~\ref{line:mvp} of Algorithm~\ref{alg:GMRES}
+is replaced by the iteration shown in Algorithm~\ref{alg:mf_Gear},
+which calculates the multiplication product of the Gear-2 sensitivity we
+encounter in envelope-following and a basis vector in the Krylov
+subspace.
+Note that the scaling and shift of $J$ in $A=(k-1)J-kI$,
+as described in Eq.~\eqref{eq:A},
+is taken into consideration by Line~\ref{line:shift}
+of Algorithm~\ref{alg:mf_Gear}.
+
+For the matrix-free generation of new basis vectors, it is
+straightforward to apply CUBLAS and CUSPARSE routines, and some
+customized GPU kernel functions to implement
+Algorithm~\ref{alg:mf_Gear} with the stored LU matrices of $J_i$ and
+$C_i$ mentioned earlier. For example in Line~\ref{line:mf_Gear_loop},
+as the iteration index $i$ traverses all the $M$ time points'
+matrix information,
+sparse matrix-vector multiplication \verb|csrmv|
+is first called to calculate $C_{i-1} p_1$ and $C_{i-2} p_2$.
+And after the two resulted vectors are combined by \verb|axpy| of CUBLAS,
+$p_0$ is solved for by \verb|trsv|, since as we noted before,
+$J_i$ is already in LU factorization form when transient simulation
+at step $i$ finished.
--- /dev/null
+
+@Book{Trzynadlowski:Book'10,
+ author = "A. M. Trzynadlowski",
+ title = "Introduction to Modern Power Electronics",
+ publisher = "Wiley",
+ edition = "Second",
+ year = "2010"
+}
+
+@InProceedings{Kundert:ICCAD'88,
+ author = {K. Kundert and J. White and A. Sangiovanni-Vincentelli},
+ title = {An envelope following method for the efficient transient simulation
+ of switching power and filter circuits},
+ booktitle = {Proc. ICCAD},
+ pages = {446--449},
+ year = 1988,
+ month = {Oct.}
+}
+
+
+@Article{White:TPE'91,
+ author = {J. White and S. Leeb},
+ title = {An envelope-following approach to switching
+ power converter simulation},
+ journal = {IEEE Trans. Power Electron.},
+ year = 1991,
+ volume = 6,
+ number = 2,
+ pages = {303--307},
+ month = {Apr.}
+}
+
+@InProceedings{Feldmann:ICCAD'96,
+ author = {P. Feldmann and J. Roychowdhury},
+ title = {Computation of circuit waveform envelopes using an efficient,
+ matrix-decomposed harmonic balance algorithm},
+ booktitle = {Proc. ICCAD},
+ pages = {295--300},
+ year = 1996,
+ month = {Nov.}
+}
+
+@Book{Krein:book'97,
+ author = {P. Krein},
+ title = {Elements of Power Electronics},
+ publisher = {Oxford University Press},
+ year = 1997
+}
+
+@InProceedings{Telichevesky:DAC'95,
+ author = {R. Telichevesky and K. Kundert and J. White},
+ title = {Efficient steady-state analysis based on matrix-free {K}rylov-subspace methods},
+ booktitle = dac,
+ pages = {480--484},
+ year = 1995,
+}
+
+@Misc{IntelMC'06,
+ author = {{Intel Corporation}},
+ title = {Intel Multi-Core Processors, Making the move to quad-core and beyond ({W}hite {P}aper)},
+ year = 2006,
+ note = {\url{http://www.intel.com/multi-core}}
+}
+
+@Misc{AMDMC'06,
+ author = {{AMD Inc.}},
+ title = {Multi-Core Processors---The Next Evolution in Computing ({W}hite {P}aper)},
+ year = 2006,
+ note = {\url{http://multicore.amd.com}}
+}
+
+@MISC{nvidia,
+ author = {{NVIDIA Corporation}},
+ key = {openmp},
+ note = {{h}ttp://www.nvidia.com},
+ year = 2011
+}
+
+@Book{Kirk:Book'10,
+ Author="D. B. Kirk and W.-M. Hwu",
+ Title="Programming Massively Parallel Processors: A Hands-on Approach",
+ Year="2010",
+ Publisher="Morgan Kaufmann Publishers Inc.",
+ Address="San Francisco, CA",
+}
+
+@MISC{gpgpu,
+ author = {D. G\"{o}ddeke},
+ title = {General-Purpose Computation Using Graphics Harware},
+ howpublished = {\url{http://www.gpgpu.org/}},
+ year = 2011
+}
+
+@misc{cuda,
+ author = {{{NVIDIA} Corporation}},
+ title = {{CUDA} ({C}ompute {U}nified {D}evice {A}rchitecture)},
+ note = {\url{http://www.nvidia.com/object/cuda_home.html}},
+ year = 2011
+}
+
+@MISC{streamSDK,
+ author = {{AMD Inc.}},
+ title = {{AMD} {S}team {SDK}},
+ howpublished = {\url{http://developer.amd.com/gpu/ATIStreamSDK}},
+ year = 2011
+}
+
+@MISC{openCL,
+ author = {{Khronos Group}},
+ title = {Open {C}omputing {L}anguage ({O}pen{CL})},
+ howpublished = {\url{http://www.khronos.org/opencl}},
+ year = 2011
+}
+
+@Misc{CUDA_Zone,
+ title = {{CUDA} Community Showcase},
+ howpublished = {\url{http://www.nvidia.com/}}
+}
+
+@BOOK{Golub:Book'96,
+ AUTHOR = "G. H. Golub and C. Van Loan",
+ TITLE = "Matrix Computations",
+ PUBLISHER = "The Johns Hopkins University Press",
+ YEAR = "1996",
+ edition = "3rd",
+}
+
+@Misc{CUDASDK,
+ author = {{NVIDIA}},
+ year = 2011,
+ note = {\url{http://developer.nvidia.com/cuda-toolkit-40}}
+}
+
+@Book{Gear:book'71,
+ author = {C. W. Gear},
+ title = {Numerical Initial Value Problems in Ordinary Differential Equations},
+ publisher = {Prentice-Hall},
+ year = 1971,
+ address = {Englewood Cliffs, NJ}
+}
+
+@Misc{ngspice,
+ title = {{NGSPICE}},
+ howpublished = {\url{http://ngspice.sourceforge.net/}}
+}
+
+@INPROCEEDINGS{Kato:COMPEL'06,
+ author={Kato, T. and others},
+ booktitle={IEEE COMPEL'06},
+ title={Envelope Following Analysis of an Autonomous Power Electronic System},
+ year={2006},
+ month={July},
+ pages={29--33}
+}
+
+@Book{Vlach:Book'94,
+ author = {J. Vlach and K. Singhal},
+ title = {Computer Methods for Circuit Analysis and Design},
+ publisher = {Van Nostrand Reinhold},
+ year = {1995},
+ address = {New York, NY},
+}
+
+@conference{LiuTan1:DATE'12,
+author={X.-X. Liu and S. X.-D. Tan and H. Wang and H. Yu},
+title={A {GPU}-accelerated envelope-following method for switching power converter simulation”},
+booktitle= date,
+year={2012},
+pages = {1349-1354},
+month ={March},
+}
--- /dev/null
+\relax
+\@writefile{toc}{\author{X.-X. Liu}{}}
+\@writefile{toc}{\author{S. X.-D. Tan}{}}
+\@writefile{toc}{\author{H. Wang}{}}
+\@writefile{toc}{\author{H. Yu}{}}
+\@writefile{loa}{\addvspace {10\p@ }}
+\@writefile{toc}{\contentsline {chapter}{\numberline {1}GPU-Accelerated Envelope-Following Method}{3}}
+\@writefile{lof}{\addvspace {10\p@ }}
+\@writefile{lot}{\addvspace {10\p@ }}
+\@writefile{toc}{\contentsline {section}{\numberline {1.1}Introduction}{3}}
+\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
+\newlabel{fig:ef1}{{1.1(a)}{5}}
+\newlabel{sub@fig:ef1}{{(a)}{5}}
+\newlabel{fig:ef2}{{1.1(b)}{5}}
+\newlabel{sub@fig:ef2}{{(b)}{5}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.1}{\ignorespaces Transient envelope-following analysis. (Both two figures reflect backward-Euler style envelope-following.)\relax }}{5}}
+\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {Illustration of one envelope skip.}}}{5}}
+\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {The envelope changes in a slow time scale.}}}{5}}
+\newlabel{fig:ef_intro}{{1.1}{5}}
+\@writefile{toc}{\contentsline {section}{\numberline {1.2}The envelope-following method in a nutshell}{6}}
+\newlabel{sec:ef}{{1.2}{6}}
+\newlabel{eq:dae}{{1.1}{6}}
+\newlabel{eq:Newton}{{1.2}{7}}
+\newlabel{eq:A}{{1.3}{7}}
+\@writefile{toc}{\contentsline {section}{\numberline {1.3}New parallel envelope-following method}{8}}
+\newlabel{sec:gmres}{{1.3}{8}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.1}GMRES solver for Newton update equation}{8}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.2}{\ignorespaces The flow of envelope-following method.\relax }}{9}}
+\newlabel{fig:ef_flow}{{1.2}{9}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.2}Parallelization on GPU platforms}{10}}
+\newlabel{sec:gpu}{{1.3.2}{10}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.3}{\ignorespaces GPU parallel solver for envelope-following update.\relax }}{11}}
+\newlabel{fig:gmres}{{1.3}{11}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.3}Gear-2 based sensitivity calculation}{12}}
+\newlabel{sec:gear}{{1.3.3}{12}}
+\newlabel{eq:BE}{{1.4}{12}}
+\newlabel{eq:sens1}{{1.5}{12}}
+\newlabel{eq:Gear_t2}{{1.6}{13}}
+\newlabel{eq:sens2}{{1.7}{13}}
+\newlabel{eq:Gear_t3}{{1.8}{13}}
+\newlabel{eq:sensM}{{1.9}{13}}
+\@writefile{toc}{\contentsline {section}{\numberline {1.4}Numerical examples}{14}}
+\newlabel{sec:exp}{{1.4}{14}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.4}{\ignorespaces Diagram of a zero-voltage quasi-resonant flyback converter.\relax }}{15}}
+\newlabel{fig:flyback}{{1.4}{15}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.5}{\ignorespaces Illustration of power/ground network model.\relax }}{15}}
+\newlabel{fig:pg}{{1.5}{15}}
+\@writefile{lot}{\contentsline {table}{\numberline {1.1}{\ignorespaces CPU and GPU time comparisons (in seconds) for solving Newton update equation with the proposed Gear-2 sensitivity. \relax }}{15}}
+\newlabel{table:circuit}{{1.1}{15}}
+\newlabel{fig:flybackWhole}{{1.6(a)}{16}}
+\newlabel{sub@fig:flybackWhole}{{(a)}{16}}
+\newlabel{fig:flybackZoom}{{1.6(b)}{16}}
+\newlabel{sub@fig:flybackZoom}{{(b)}{16}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.6}{\ignorespaces Flyback converter solution calculated by envelope-following. The red curve is traditional SPICE simulation result, and the back curve is the envelope-following output with simulation points marked.\relax }}{16}}
+\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {The whole plot}}}{16}}
+\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {Detail of one EF simulation period}}}{16}}
+\newlabel{fig:flyback_wave}{{1.6}{16}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.7}{\ignorespaces Buck converter solution calculated by envelope-following.\relax }}{17}}
+\newlabel{fig:buck_wave}{{1.7}{17}}
+\@writefile{toc}{\contentsline {section}{\numberline {1.5}Summary}{17}}
+\newlabel{sec:summary}{{1.5}{17}}
+\@writefile{toc}{\contentsline {section}{\numberline {1.6}Glossary}{18}}
+\@writefile{toc}{\contentsline {section}{Bibliography}{18}}
+\@setckpt{Chapters/chapter16/ch16}{
+\setcounter{page}{20}
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--- /dev/null
+\chapterauthor{X.-X. Liu}{Dept. Electrical Engineering,
+ University of California, Riverside, CA 92521}
+\chapterauthor{S. X.-D. Tan}{Dept. Electrical Engineering,
+ University of California, Riverside, CA 92521}
+\chapterauthor{H. Wang}{Univ. of Electronics Science and Technology of China,
+ Chengdu, Sichuan, China}
+\chapterauthor{H. Yu}{School of Electrical \& Electronic Engineering,
+ Nanyang Technological University, Singapore}
+
+% \thanks{
+% This research was supported in part by NSF grants under
+% No.~CCF-1017090
+% No.~OISE-1051797, and
+% No.~OISE-0929699.
+% }
+
+\newcommand{\ud}{\,\mathrm{d}}
+
+%\renewcommand{\algorithmicrequire}{\textbf{Input:}}
+%\renewcommand{\algorithmicensure}{\textbf{Output:}}
+%\renewcommand{\algorithmiccomment}[1]{\% \textit{#1}}
+
+\chapter[GPU-Accelerated Envelope-Following Method]{A GPU-Accelerated Envelope-Following Method for Switching Power Converter Simulation}
+
+% \section{Abstract}
+% % Power converters have seen a surge of new trends and novel
+% % applications due to their widespread use in renewable energy
+% % systems and emerging hybrid and purely-electric vehicles. More
+% % efficient simulation techniques for power converters are urgently
+% % needed to meet more design constraints.
+% In this chapter, we propose a new envelope-following parallel transient analysis method for
+% the general switching power converters. The new method first exploits
+% the parallelisim in the envelope-following method
+% and parallelize the Newton update solving part,
+% which is the most computational expensive, in GPU platforms
+% to boost the simulation performance.
+% To further speed up the iterative GMRES
+% solving for Newton update equation in the envelope-following
+% method, we apply the matrix-free Krylov basis generation
+% technique, which was previously used for RF simulation.
+% Last, the new method also applies more robust
+% Gear-2 integration to compute the sensitivity matrix instead of
+% traditional integration methods.
+% %Furthermore, the resulted Gear-2 and matrix-free GMRES have been
+% Experimental results from several integrated on-chip power
+% converters show that the proposed GPU envelope-following algorithm leads to
+% about 10$\times$ speedup compared to its CPU counterpart,
+% and 100$\times$ faster than the traditional envelop-following methods
+% while still keeps the similar accuracy.
+
+\input{Chapters/chapter16/intro.tex}
+\input{Chapters/chapter16/ef.tex}
+%\input bdf.tex % now inside gpu.tex now
+\input{Chapters/chapter16/gpu.tex}
+\input{Chapters/chapter16/exp.tex}
+
+\section{Summary}
+\label{sec:summary}
+In this chapter, we present a new envelope-following method for
+transient analysis of switching power converters. First, the
+computationally expensive step, the solving of Newton update equation,
+has been parallelized on CUDA-enabled GPU platforms with iterative
+GMRES solver to boost performance of the analysis method. To further
+speed up the GMRES solving for Newton update equation, we have
+employed the matrix-free Krylov basis generation technique. The
+proposed method also applies the more robust Gear-2 integration to
+compute the sensitivity matrix. Experimental results from several
+integrated on-chip power converters have shown that the proposed GPU
+envelope-following algorithm can lead to about 10$\times$ speedup
+compared to its CPU counterpart, and 100$\times$ faster than the
+traditional envelope-following methods while still keeps the similar
+accuracy.
+
+
+\section{Glossary}
+\begin{Glossary}
+\item[Envelope-Following] In transient simulation of switching power circuits,
+nodal voltage waveforms in neighboring high frequency clock cycles are similar,
+but not exactly the duplicates. Envelope-following technique approximates
+the slowly changing transient trend over a lot of clock cycles
+without calculating waveforms in all cycles.
+\end{Glossary}
+
+\putbib[Chapters/chapter16/biblio16]
+%\bibliography{./envelope,../../bib/interconnect,../../bib/architecture,../../bib/simulation}
--- /dev/null
+\documentclass[conference]{IEEEtran}
+
+\usepackage{graphicx}
+\usepackage{algorithm,algorithmic}
+\usepackage{multirow}
+\usepackage{subfigure}
+\usepackage{booktabs}
+\usepackage{epstopdf}
+\usepackage{graphicx}
+\usepackage[dvips]{color}
+\usepackage{psfrag}
+\usepackage{pstricks}
+\usepackage{amssymb,amsmath}
+%\usepackage{authblk}
+\usepackage{cite}
+\usepackage{url}
+
+\DeclareGraphicsRule{.pdftex}{pdf}{*}{}
+\graphicspath{{matlab/}}
+
+
+\newtheorem{myformulation}{Formulation}
+\newtheorem{mytheorem}{Theorem}
+\newtheorem{mylemma}{Lemma}
+\newtheorem{mydefinition}{Definition}
+%\ninept
+
+% correct bad hyphenation here
+%\hyphenation{op-tical net-works semi-conduc-tor IEEEtran}
+%\renewcommand{\baselinestretch}{0.99}
+
+
+\renewcommand{\algorithmicrequire}{\textbf{Input:}}
+\renewcommand{\algorithmicensure}{\textbf{Output:}}
+\renewcommand{\algorithmiccomment}[1]{\% \textit{#1}}
+
+
+\newcommand{\setRn}{\mathbb{R}_{}^N}
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+\newcommand{\mtx}[1]{\mathbf{#1}}
+\newcommand{\mtxPhi}{\boldsymbol{\phi}}
+\newcommand{\mT}{\mathrm{T}}
+
+\IEEEoverridecommandlockouts
+
+
+\renewcommand{\baselinestretch}{0.97}
+
+\begin{document}
+\title{
+A GPU-Accelerated Envelope-Following Method for Switching Power Converter Simulation
+}
+\author{
+ \IEEEauthorblockN{
+ Xue-Xin Liu\IEEEauthorrefmark{1},
+ Sheldon X.-D. Tan\IEEEauthorrefmark{1},
+ Hai Wang\IEEEauthorrefmark{1}, and
+ Hao Yu\IEEEauthorrefmark{2}
+ }
+ \IEEEauthorblockA{
+ \IEEEauthorrefmark{1}Dept. Electrical Engineering,
+ University of California, Riverside, CA 92521
+ }
+ \IEEEauthorblockA{
+ \IEEEauthorrefmark{2}School of Electrical \& Electronic Engineering,
+ Nanyang Technological University
+ }
+ \thanks{
+ This research was supported in part by NSF grants under
+ No.~CCF-1017090
+ No.~OISE-1051797, and
+ No.~OISE-0929699.
+ }
+}
+
+%\IEEEpubid{978-3-9810801-8-6/DATE12/@2012 EDAA}
+
+\maketitle
+
+\begin{abstract}
+% Power converters have seen a surge of new trends and novel
+% applications due to their widespread use in renewable energy
+% systems and emerging hybrid and purely-electric vehicles. More
+% efficient simulation techniques for power converters are urgently
+% needed to meet more design constraints.
+In this paper, we propose a new envelope-following parallel transient analysis method for
+the general switching power converters. The new method first exploits
+the parallelisim in the envelope-following method
+and parallelize the Newton update solving part,
+which is the most computational expensive, in GPU platforms
+to boost the simulation performance.
+To further speed up the iterative GMRES
+solving for Newton update equation in the envelope-following
+method, we apply the matrix-free Krylov basis generation
+technique, which was previously used for RF simulation.
+Last, the new method also applies more robust
+Gear-2 integration to compute the sensitivity matrix instead of
+traditional integration methods.
+%Furthermore, the resulted Gear-2 and matrix-free GMRES have been
+Experimental results from several integrated on-chip power
+converters show that the proposed GPU envelope-following algorithm leads to
+about 10$\times$ speedup compared to its CPU counterpart,
+and 100$\times$ faster than the traditional envelop-following methods
+while still keeps the similar accuracy.
+\end{abstract}
+
+\input intro.tex
+\input ef.tex
+%\input bdf.tex % now inside gpu.tex now
+\input gpu.tex
+\input exp.tex
+
+\section{Conclusion} \label{sec:conclusion}
+A new envelope-following method for transient analysis of
+switching power converters has been introduced.
+First, the computationally expensive step,
+the solving of Newton update equation,
+has been parallelized on CUDA-enabled GPU platforms
+with iterative GMRES solver to boost performance of the analysis method.
+To further speed up the GMRES solving for Newton update equation,
+we have employed the matrix-free Krylov basis generation technique.
+The proposed method also applies the more robust Gear-2 integration
+to compute the sensitivity matrix.
+Experimental results from several integrated
+on-chip power converters have shown that the proposed GPU
+envelope-following algorithm can lead to about 10$\times$
+speedup compared to its CPU counterpart,
+and 100$\times$ faster than the traditional
+envelop-following methods while still keeps the similar accuracy.
+
+
+\bibliographystyle{unsrt} % IEEEtranS
+\bibliography{./envelope,../../bib/interconnect,../../bib/architecture,../../bib/simulation}
+% ./rfsim.bib
+
+\end{document}
--- /dev/null
+\section{The envelope-following method in a nutshell}
+\label{sec:ef}
+
+In transient analysis\index{transient analysis},
+a nonlinear circuit's behavior can be represented
+by a coupled set of nonlinear first-order
+differential algebraic equations\index{differential algebraic equations}
+(DAE) of the form
+\begin{equation}\label{eq:dae}
+ \frac{\ud}{\ud t} q(x(t)) + f(x(t)) = b(t),
+\end{equation}
+where $x(t)\in \mathbb{R}^N$ is the vector of circuit variables,
+usually comprising node voltages and possibly branch currents,
+$f(\cdot)$ is a nonlinear function that maps $x(t)$ to a vector
+of $N$ entries most of
+which are sums of resistive currents at a node,
+$q(\cdot)$ %is also a nonlinear function that
+maps $x(t)$ to a vector of $N$ entries of
+capacitor charges or inductor fluxes,
+and $b(t)$ contains the input source values.
+For many power electronics circuits, the input
+switching signal is known %as a circuit parameter
+and is periodic with clock period $T$.
+
+Assume the time inside one period $T$ has been discretised
+into $M$ time steps, $0=t_0 < t_1 < t_2 < \cdots < t_M=T$, and the
+$i$-th step length is $h_i = t_i-t_{i-1}$.
+In practice, the discretisation is nonuniform to
+control truncation error\index{truncation error} and convergence.
+Then, given an initial condition $x(0)$ at $t=0$,
+numerical integration is applied to find the time domain solution of
+circuit state $x(t)$ at each time step till $t=T$.
+%, in a consequent time series.
+
+For those circuits whose carrier waveforms\index{carrier waveforms}
+vary slowly in a large number of periods,
+the envelope-following\index{envelope-following} method only integrates the
+DAE in several selected periods and then jumps over to a new time point.
+By repeating this ``simulate and skip'' action,
+envelope-following method attains its efficiency
+compared to conventional transient analysis, but still can accurately
+obtain the slow varying envelope.
+
+For example, if the clock period is $T$, and the suitable jump
+interval for the envelope is $k$ periods, then the envelope step is
+$kT$. Suppose the state at time $t=mT$ is known as $x(mT)$, and the
+envelope-following wants to obtain the state at the next envelope
+step, $x((m+k)T)$. If the envelope step is much larger than the clock
+period ($k$ is much bigger than one), envelope-following will lead to
+significant savings in simulation time.
+
+% One simple way to estimate $x((m+k)T)$ is to use the information at
+% $mT$ and $(m-1)T$, which leads to the forward-Euler method as
+To estimate $x((m+k)T)$,
+a forward-Euler\index{forward-Euler} style jumping relies only on $x(mT)$ and $x((m-1)T)$,
+i.e.,
+\[
+ x((m+k)T)
+ =
+ x(mT) + k\left[ x(mT)-x((m-1)T) \right].
+\]
+However, this approach is inefficient due to its restriction on
+envelope step $k$, since larger $k$ usually causes instability.
+Instead, backward-Euler\index{backward-Euler} jumping,
+%and the equation becomes
+\[
+ x((m+k)T)-x(mT) = k\left[x((m+k)T)-x((m+k-1)T)\right],
+\]
+allows larger envelope steps.
+Here $x((m+k-1)T)$ is the unknown variable to be solved
+by Newton iteration\index{Newton iteration},
+and $x((m+k)T)$ is dependent on $x((m+k-1)T)$
+in each iteration.
+% Forward-Euler may be used to generate the initial guess.
+The Newton update equation in each iteration is thus expressed as
+% x^{j+1}((m+k-1)T) = x^{j}((m+k-1)T) - A^{-1} \left[(k-1)x^j((m+k)T) - kx^j((m+k-1)T)+ x(mT)\right],
+%\begin{equation} \label{eq:Newton}
+%\begin{split}
+\begin{multline} \label{eq:Newton}
+ \Delta x((m+k-1)T)= A^{-1} \left[(k-1)x^j((m+k)T) \right. \\
+ \left. - kx^j((m+k-1)T)+ x(mT)\right],
+\end{multline}
+%\end{split}
+%\end{equation}
+where the Jacobian matrix\index{Jacobian matrix} $A$ is computed as
+a combination of circuit sensitivity matrix\index{sensitivity matrix}
+and identity matrix, as
+\begin{equation} \label{eq:A}
+ A = (k-1)\frac{\ud x((m+k)T)}{\ud x((m+k-1)T)} -kI
+ = (k-1)J-kI.
+\end{equation}
+In each Newton iteration,
+$x((m+k-1)T)$ is used as initial condition to calculate
+$x((m+k)T)$ by integrating the DAE in one clock period.
+Meanwhile, %circuit matrices, for example,
+the conductance matrix $G_i=\ud f(x(t_i))/\ud x(t_i)$
+and the capacitance matrix $C_i=\ud q(x(t_i))/\ud x(t_i)$
+at each time step %in this period,
+are used to derive the %circuit
+sensitivity matrix $J= \ud x((m+k)T) / \ud x((m+k-1)T)$.
+% i.e.,
+% \[
+% x((m+k)T) = \Phi(x((m+k-1)T),(m+k-1)T,(m+k)T).
+% \]
+
+Different integration rules can be applied to
+the computation of sensitivity matrix.
+It can be easily derived that,
+%for one signal period with $M$ time steps,
+if the DAE is integrated with backward-Euler rule,
+the sensitivity is
+\[
+ J = \frac{\ud x_M}{\ud x_0}
+ = \prod_{i=1}^{M}
+ \left( \frac{1}{h_i} C_i + G_i \right)^{-1}
+ \frac{1}{h_i}C_{i-1}
+% \frac{\ud x_n}{\ud x_0}
+% = \left(\frac{1}{h_n} C_n + G_n\right)^{-1}
+% \frac{1}{h_n}C_{n-1} \frac{\ud x_{n-1}}{\ud x_0}
+\]
+%is the sensitivity matrix
+% In the next section, we will formulate the sensitivity in
+% Gear-2 scheme.
+% and for trapezoidal integration,
+% \[
+% \frac{\ud x_n}{\ud x_0}
+% = \left(\frac{1}{h_n} C_n + \frac{1}{2} G_n\right)^{-1}
+% \left(\frac{1}{h_n} C_{n-1} - \frac{1}{2} G_{n-1}\right)
+% \frac{\ud x_{n-1}}{\ud x_0}.
+% \]
+
+In summary, the envelope-following method is fundamentally
+a boosted version of traditional transient analysis,
+with certain skips over several periods and a Newton iteration
+to update or correct the errors brought by the skips,
+as is exhibited by Fig.~\ref{fig:ef_flow}.
+\begin{figure}[!tb]
+\centering
+\resizebox{.7\textwidth}{!}{\input{./Chapters/chapter16/figures/ef_flow.pdf_t}}
+\caption{The flow of envelope-following method.}
+\label{fig:ef_flow}
+%\end{center}
+\end{figure}
+
+
+
--- /dev/null
+\section{Numerical examples}
+\label{sec:exp}
+
+The presented algorithm has been prototyped and numerical experiments
+have been carried out on a server which has an Intel Xeon quad-core
+CPU with 2.0~GHz clock speed, and 24~GBytes memory. The GPU card
+mounted on this server is NVIDIA's Tesla~C2070 (Fermi), which contains
+448 cores (14 MPs $\times$ 32 cores per MP) running at a 1.30~GHz and
+has 4~GBytes on-chip memory. Some initial results have been published
+in~\cite{LiuTan1:DATE'12}.
+
+The envelope-following method with the proposed Gear-2
+sensitivity matrix computation is added to an open-source
+SPICE\index{SPICE}, implemented in C~\cite{ngspice}.
+Our envelope-following program is implemented by following
+the algorithm mentioned in~\cite{Kato:COMPEL'06}.
+To solve the Newton update equation,
+different methods are used to compare the computation time,
+such as direct LU, GMRES with explicitly formed matrix,
+and GMRES with implicit matrix-vector multiplication (matrix-free).
+Moreover, the matrix-free method is also incorporated to the same SPICE
+simulator using CUDA C programming interface, as described in
+Section~\ref{sec:gpu}.
+
+\begin{figure}
+ \centering
+ \resizebox{.8\textwidth}{!}{\input{./Chapters/chapter16/figures/resonant_flyback.pdf_t}}
+ \caption{Diagram of a zero-voltage quasi-resonant flyback converter.}
+ \label{fig:flyback}
+\end{figure}
+
+
+\begin{figure}%[hbfp]
+ \centering
+ \resizebox{.6\textwidth}{!}{\input{./Chapters/chapter16/figures/pgMesh.pdf_t}}
+ \caption{Illustration of power/ground network model.}
+ \label{fig:pg}
+\end{figure}
+
+\begin{figure}
+\centering
+\subfigure[The whole plot]{
+ \includegraphics[width=.6\textwidth]{./Chapters/chapter16/figures/flyback_wave_emb.eps}
+ \label{fig:flybackWhole}
+}
+\subfigure[Detail of one EF simulation period]{
+ \includegraphics[width=.6\textwidth]{./Chapters/chapter16/figures/flyback_zoomin_emb.eps}
+ \label{fig:flybackZoom}
+}
+\caption{Flyback converter solution calculated by envelope-following.
+The red curve is traditional SPICE simulation result, and
+the back curve is the envelope-following output with simulation points
+marked.}
+\label{fig:flyback_wave}
+\end{figure}
+
+
+\begin{figure}%[hbfp]
+ \centering
+ \includegraphics[width=.6\textwidth]{./Chapters/chapter16/figures/buck_wave_emb.eps}
+ \caption{Buck converter solution calculated by envelope-following.}
+ \label{fig:buck_wave}
+\end{figure}
+
+We use several integrated on-chip converters as simulation examples
+to measure running time and speedup. They include a Buck converter,
+a quasi-resonant flyback converter (shown in Fig.~\ref{fig:flyback}),
+and two boost converters.
+Each converter is directly integrated with on-chip power grid networks,
+since the performance of converters should be studied with their loads and
+we can easily observe the waveforms at different nodes in a power
+grid (see Fig.~\ref{fig:pg} for a simplified power grid structure).
+
+Fig.~\ref{fig:flyback_wave}
+and Fig.~\ref{fig:buck_wave}
+shows the waveform at output node of the resonant flyback converter
+and the Buck converter.
+Note that on the envelope curve, the darker
+dots in separated segments indicate the real simulation points were
+calculated in those cycles, and the segments without dots are the
+envelope jumps where no simulation were done.
+It can be verified that the proposed Gear-2 envelope-following method
+produces a envelope matching the original waveform well.
+
+\begin{table}
+\centering
+\caption{CPU and GPU time comparisons (in seconds) for solving Newton update equation
+ with the proposed Gear-2 sensitivity.
+}
+\vspace{0.1in}
+\label{table:circuit}
+{%\normalsize
+\begin{tabular}{@{}c|c|c|c|c|c|c@{}}
+\hline\hline
+Circuit & Nodes & Direct & Explicit & \multicolumn{3}{c}{Implicit GMRES}\\ \cline{5-7}
+ & & LU & GMRES & CPU & GPU & X \\
+\hline
+Buck & 910 & 423.8 & 420.3 & 36.8 & 3.9 & 9.4$\times$ \\
+Flyback & 941 & 462.4 & 459.6 & 64.5 & 7.4 & 8.7$\times$ \\
+Boost-1 & 976 & 695.1 & 687.7 & 73.2 & 6.2 & 11.8$\times$ \\
+Boost-2 & 1093 & 729.5 & 720.8 & 71.0 & 8.5 & 9.9$\times$ \\
+\hline\hline
+\end{tabular}
+}
+\end{table}
+
+For the comparison of running time spent in solving
+Newton update equation, Table~\ref{table:circuit} lists the time
+costed by direct method, explicit GMRES, matrix-free GMRES,
+and GPU matrix-free GMRES. All methods carry out the Gear-2 based
+envelope-following method, but they handle the sensitivity and
+equation solving in different implementation steps.
+It is obvious that as long as the sensitivity matrix is explicitly formed,
+such as the cases in direct method and explicitly GMRES,
+the cost is much higher than the implicit methods.
+When matrix-free technique is applied to generate matrix-vector
+products implicitly, the computation cost is greatly reduced.
+Thus, for the same example, implicit GMRES would be one order
+of magnitude faster than explicit GMRES. Furthermore, our GPU parallel
+implementation of implicit GMRES makes this method even faster,
+with a further 10$\times$ speedup.
--- /dev/null
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--- /dev/null
+#FIG 3.2
+Landscape
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+1200 2
+6 1950 2475 4200 3225
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+4 0 1 50 -1 0 9 0.0000 2 171 2048 2028 2700 save time steps $h_i$,\001
+4 0 1 50 -1 0 9 0.0000 2 156 1769 2028 2925 matrices $C_i$, and\001
+4 0 1 50 -1 0 9 0.0000 2 156 2141 2028 3150 the LU factors of $J_i$\001
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+4 0 0 50 -1 0 9 0.0000 2 180 2610 750 1350 make a guess of $x((m+k-1)T)$\001
+4 0 0 50 -1 0 9 0.0000 2 165 1800 1275 900 select jump size $k$\001
+4 0 0 50 -1 0 9 0.0000 2 165 3240 675 450 circuit state $x(mT)$ at time $t=mT$\001
+4 0 0 50 -1 0 9 0.0000 2 120 270 750 3450 yes\001
+4 0 0 50 -1 0 9 0.0000 2 165 2520 825 4500 solve Newton update equation\001
+4 0 0 50 -1 0 9 0.0000 2 90 180 1950 3750 no\001
+4 0 0 50 -1 0 9 0.0000 2 150 2340 975 4050 compute sensitivity matrix\001
+4 0 0 50 -1 0 9 0.0000 2 165 990 1425 5400 converged ?\001
+4 0 0 50 -1 0 9 0.0000 2 90 180 1125 5325 no\001
+4 0 0 50 -1 0 9 0.0000 2 120 270 1950 5625 yes\001
+4 0 0 50 -1 0 9 0.0000 2 135 630 1425 5850 $m=m+k$\001
+4 0 0 50 -1 0 9 0.0000 2 165 1080 1275 3525 $t<(m+k)T$ ?\001
+4 0 0 50 -1 0 9 0.0000 2 165 900 675 3000 for $x(t)$\001
+4 0 0 50 -1 0 9 0.0000 2 165 1170 675 2775 integrate DAE\001
--- /dev/null
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+\put(1426,-4561){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}converged ?}%
+}}}}
+\put(1126,-4486){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}no}%
+}}}}
+\put(1951,-4786){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}yes}%
+}}}}
+\put(1426,-5011){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$m=m+k$}%
+}}}}
+\put(1276,-2686){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$t<(m+k)T$ ?}%
+}}}}
+\put(676,-2161){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}for $x(t)$}%
+}}}}
+\put(676,-1936){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}integrate DAE}%
+}}}}
+\end{picture}%
--- /dev/null
+#FIG 3.2
+Landscape
+Center
+Inches
+Letter
+100.00
+Single
+-2
+1200 2
+6 2550 6450 4950 7650
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 4800 6600 75 75 4800 6600 4800 6675
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 4425 6600 75 75 4425 6600 4425 6675
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 4050 6675 75 75 4050 6675 4050 6750
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 3675 6750 75 75 3675 6750 3675 6825
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 2700 7500 75 75 2700 7500 2700 7575
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 3075 7350 75 75 3075 7350 3075 7425
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 3375 7050 75 75 3375 7050 3375 7125
+3 0 0 2 0 7 50 -1 -1 0.000 0 0 0 4
+ 2700 7500 3300 7350 3450 6675 4800 6600
+ 0.000 1.000 1.000 0.000
+-6
+6 450 7350 2850 8550
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 2700 7500 75 75 2700 7500 2700 7575
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 2325 7500 75 75 2325 7500 2325 7575
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 1950 7575 75 75 1950 7575 1950 7650
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 1575 7650 75 75 1575 7650 1575 7725
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 600 8400 75 75 600 8400 600 8475
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 975 8250 75 75 975 8250 975 8325
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 1275 7950 75 75 1275 7950 1275 8025
+3 0 0 2 0 7 50 -1 -1 0.000 0 0 0 4
+ 600 8400 1200 8250 1350 7575 2700 7500
+ 0.000 1.000 1.000 0.000
+-6
+6 8250 4050 10650 5250
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 10500 4200 75 75 10500 4200 10500 4275
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 10125 4200 75 75 10125 4200 10125 4275
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 9750 4275 75 75 9750 4275 9750 4350
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 9375 4350 75 75 9375 4350 9375 4425
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 8400 5100 75 75 8400 5100 8400 5175
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 8775 4950 75 75 8775 4950 8775 5025
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 9075 4650 75 75 9075 4650 9075 4725
+3 0 0 2 0 7 50 -1 -1 0.000 0 0 0 4
+ 8400 5100 9000 4950 9150 4275 10500 4200
+ 0.000 1.000 1.000 0.000
+-6
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+ 0 0 1.00 60.00 120.00
+ 0 0 1.00 60.00 120.00
+ 2700 5700 4800 5700
+2 1 1 2 0 7 50 -1 -1 4.000 0 0 7 0 0 2
+ 2700 7500 10500 4200
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 7 0 0 2
+ 2700 8100 2700 3900
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 7 0 0 2
+ 4800 8100 4800 3900
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 7 0 0 2
+ 8400 8100 8400 3900
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 7 0 0 2
+ 10500 8100 10500 3900
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+ 0 0 1.00 60.00 120.00
+ 0 0 1.00 60.00 120.00
+ 4800 3750 10500 3750
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+ 0 0 1.00 60.00 120.00
+ 1275 7350 1275 7725
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+ 0 0 1.00 60.00 120.00
+ 6600 6525 6300 6075
+2 4 0 1 0 7 50 -1 -1 4.000 0 0 7 0 0 5
+ 2175 7350 2175 6600 750 6600 750 7350 2175 7350
+4 0 0 50 -1 0 20 0.0000 2 165 1350 2850 5550 clock cycle $T$\001
+4 0 0 50 -1 0 20 0.0000 2 165 1440 6600 3600 jump $k$ periods\001
+4 0 0 50 -1 0 20 0.0000 2 165 720 6225 6825 envelope\001
+4 0 0 50 -1 0 20 0.0000 2 120 810 900 6900 transient\001
+4 0 0 50 -1 0 20 0.0000 2 150 540 900 7200 points\001
+4 0 0 50 -1 0 20 0.0000 2 135 540 4425 8400 $t=mT$\001
+4 0 0 50 -1 0 20 0.0000 2 165 1080 6525 8400 $t=(m+k-1)T$\001
+4 0 0 50 -1 0 20 0.0000 2 165 900 9450 8400 $t=(m+k)T$\001
--- /dev/null
+\begin{picture}(0,0)%
+\includegraphics{./Chapters/chapter16/figures/ef_intro.pdf}%
+\end{picture}%
+\setlength{\unitlength}{3947sp}%
+%
+\begingroup\makeatletter\ifx\SetFigFont\undefined%
+\gdef\SetFigFont#1#2#3#4#5{%
+ \reset@font\fontsize{#1}{#2pt}%
+ \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
+ \selectfont}%
+\fi\endgroup%
+\begin{picture}(10066,5172)(518,-7666)
+\put(2851,-4711){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}clock cycle $T$}%
+}}}}
+\put(6601,-2761){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}jump $k$ periods}%
+}}}}
+\put(6226,-5986){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}envelope}%
+}}}}
+\put(901,-6061){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}transient}%
+}}}}
+\put(901,-6361){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}points}%
+}}}}
+\put(4426,-7561){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$t=mT$}%
+}}}}
+\put(6526,-7561){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$t=(m+k-1)T$}%
+}}}}
+\put(9451,-7561){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$t=(m+k)T$}%
+}}}}
+\end{picture}%
--- /dev/null
+#FIG 3.2
+Landscape
+Center
+Inches
+Letter
+100.00
+Single
+-2
+1200 2
+0 32 #9c0000
+0 33 #8c8c8c
+0 34 #8c8c8c
+0 35 #424242
+0 36 #8c8c8c
+0 37 #424242
+0 38 #8c8c8c
+0 39 #424242
+0 40 #8c8c8c
+0 41 #424242
+0 42 #8c8c8c
+0 43 #424242
+0 44 #c2c2c2
+0 45 #6e6e6e
+0 46 #444444
+0 47 #8e8f8e
+0 48 #717171
+0 49 #aeaeae
+0 50 #333333
+0 51 #949395
+0 52 #747075
+0 53 #555555
+0 54 #b3b3b3
+0 55 #c3c3c3
+0 56 #6d6d6d
+0 57 #454545
+0 58 #8e8e8e
+0 59 #c6b797
+0 60 #eff8ff
+0 61 #dccba6
+0 62 #404040
+0 63 #808080
+0 64 #c0c0c0
+0 65 #e0e0e0
+0 66 #aaaaaa
+0 67 #d7d7d7
+0 68 #bebebe
+0 69 #515151
+0 70 #e7e3e7
+0 71 #000049
+0 72 #797979
+0 73 #303430
+0 74 #414141
+0 75 #c7b696
+0 76 #414541
+0 77 #dd9d93
+0 78 #f1ece0
+0 79 #e2c8a8
+0 80 #e1e1e1
+0 81 #d2d2d2
+0 82 #ededed
+0 83 #da7a1a
+0 84 #f1e41a
+0 85 #887dc2
+0 86 #d6d6d6
+0 87 #8c8ca5
+0 88 #4a4a4a
+0 89 #8c6b6b
+0 90 #5a5a5a
+0 91 #636363
+0 92 #b79b73
+0 93 #4193ff
+0 94 #bf703b
+0 95 #db7700
+0 96 #dab800
+0 97 #006400
+0 98 #5a6b3b
+0 99 #d3d3d3
+0 100 #8e8ea4
+0 101 #f3b95d
+0 102 #89996b
+0 103 #646464
+0 104 #b7e6ff
+0 105 #86c0ec
+0 106 #bdbdbd
+0 107 #d39552
+0 108 #98d2fe
+0 109 #8c9c6b
+0 110 #f76b00
+0 111 #5a6b39
+0 112 #8c9c6b
+0 113 #8c9c7b
+0 114 #184a18
+0 115 #adadad
+0 116 #f7bd5a
+0 117 #636b9c
+0 118 #f7f7f7
+0 119 #de0000
+0 120 #adadad
+0 121 #f7bd5a
+0 122 #adadad
+0 123 #f7bd5a
+0 124 #636b9c
+0 125 #526b29
+0 126 #949494
+0 127 #006300
+0 128 #00634a
+0 129 #7b844a
+0 130 #e7bd7b
+0 131 #a5b5c6
+0 132 #6b6b94
+0 133 #846b6b
+0 134 #529c4a
+0 135 #d6e7e7
+0 136 #526363
+0 137 #186b4a
+0 138 #9ca5b5
+0 139 #ff9400
+0 140 #ff9400
+0 141 #00634a
+0 142 #7b844a
+0 143 #63737b
+0 144 #e7bd7b
+0 145 #184a18
+0 146 #f7bd5a
+0 147 #dedede
+0 148 #f3eed3
+0 149 #f5ae5d
+0 150 #95ce99
+0 151 #b5157d
+0 152 #eeeeee
+0 153 #848484
+0 154 #7b7b7b
+0 155 #005a00
+0 156 #e77373
+0 157 #ffcb31
+0 158 #29794a
+0 159 #de2821
+0 160 #2159c6
+0 161 #f8f8f8
+0 162 #e6e6e6
+0 163 #21845a
+0 164 #c9c9c9
+0 165 #dfd8df
+0 166 #f7f3f7
+0 167 #cdcdcd
+0 168 #6c6c6c
+# Brace under
+6 1500 3975 2400 4425
+3 4 0 1 0 7 50 0 -1 0.000 0 0 0 7
+ 1573 4291 1598 4358 1980 4218 2004 4285 1980 4218 2363 4079
+ 2339 4012
+ 0.000 1.000 1.000 0.000 1.000 1.000 0.000
+-6
+# Brace under
+6 3000 3600 3900 3825
+3 4 0 1 0 7 50 0 -1 0.000 0 0 0 7
+ 3070 3747 3082 3818 3484 3747 3496 3816 3484 3747 3885 3676
+ 3873 3606
+ 0.000 1.000 1.000 0.000 1.000 1.000 0.000
+-6
+# Brace under
+6 4575 3525 5625 3900
+3 4 0 1 0 7 50 0 -1 0.000 0 0 0 7
+ 4639 3531 4622 3600 5089 3734 5072 3802 5089 3734 5556 3868
+ 5574 3799
+ 0.000 1.000 1.000 0.000 1.000 1.000 0.000
+-6
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+ 7500 4800 7500 5100
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+ 6225 4200 6225 5100
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+ 3000 3825 3000 5100
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+ 900 4575 900 5100
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+ 1500 4350 1500 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+ 0 0 1.00 60.00 120.00
+ 0 0 1.00 60.00 120.00
+ 900 5100 1500 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+ 0 0 1.00 60.00 120.00
+ 0 0 1.00 60.00 120.00
+ 1500 5100 3000 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+ 0 0 1.00 60.00 120.00
+ 0 0 1.00 60.00 120.00
+ 3075 5100 4575 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+ 0 0 1.00 60.00 120.00
+ 0 0 1.00 60.00 120.00
+ 4575 5100 6225 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+ 0 0 1.00 60.00 120.00
+ 0 0 1.00 60.00 120.00
+ 6300 5100 7500 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+ 0 0 1.00 60.00 120.00
+ 1500 3225 1050 4200
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+ 0 0 1.00 60.00 120.00
+ 3150 3075 3900 3225
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+ 0 0 1.00 60.00 120.00
+ 2250 3150 2475 3525
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+ 4575 3525 4575 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+ 0 0 1.00 60.00 120.00
+ 6225 3150 6675 4200
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+ 0 0 1.00 60.00 120.00
+ 3750 4200 3525 3825
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+ 0 0 1.00 60.00 120.00
+ 4200 4275 5025 3900
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+ 0 0 1.00 60.00 120.00
+ 3075 4275 2175 4275
+2 2 0 1 7 7 50 -1 -1 0.000 0 0 -1 0 0 5
+ 3300 2475 4500 2475 4500 2625 3300 2625 3300 2475
+3 0 0 3 4 7 50 -1 -1 0.000 0 0 0 3
+ 900 4500 1050 4200 1425 4275
+ 0.000 1.000 0.000
+3 0 0 3 4 7 50 -1 -1 0.000 0 0 0 3
+ 2400 3900 2550 3525 3000 3675
+ 0.000 1.000 0.000
+3 0 0 3 4 7 50 -1 -1 0.000 0 0 0 3
+ 3978 3436 4203 3136 4575 3450
+ 0.000 1.000 0.000
+3 0 0 3 4 7 50 -1 -1 0.000 0 0 0 3
+ 5622 3764 5847 3464 6225 4050
+ 0.000 1.000 0.000
+3 0 0 3 4 7 50 -1 -1 0.000 0 0 0 3
+ 7050 4425 7275 4125 7500 4725
+ 0.000 1.000 0.000
+3 0 1 2 1 7 50 -1 -1 6.000 0 0 0 5
+ 900 4500 3000 3600 4500 3300 6300 3975 7800 4950
+ 0.000 1.000 1.000 1.000 0.000
+4 0 0 50 -1 0 12 0.0000 2 150 540 2025 5025 $m_1T$\001
+4 0 0 50 -1 0 12 0.0000 2 150 540 3600 5025 $m_2T$\001
+4 0 0 50 -1 0 12 0.0000 2 150 540 5175 5025 $m_3T$\001
+4 0 0 50 -1 0 12 0.0000 2 150 540 6600 5025 $m_4T$\001
+4 0 4 50 -1 0 12 0.0000 2 165 1530 1425 3075 simulated periods\001
+4 0 1 50 -1 0 12 0.0000 2 165 720 5850 3075 envelope\001
+4 0 0 50 -1 0 12 0.0000 2 165 1350 3150 4425 skipped periods\001
+4 0 0 50 -1 0 12 0.0000 2 135 270 1125 5025 $T$\001
--- /dev/null
+\begin{picture}(0,0)%
+\includegraphics{./Chapters/chapter16/figures/envelope.pdf}%
+\end{picture}%
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+\put(5851,-2236){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,1}envelope}%
+}}}}
+\put(3151,-3586){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}skipped periods}%
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+\put(1126,-4186){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$T$}%
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+\end{picture}%
--- /dev/null
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+%%BeginProlog
+% MathWorks dictionary
+/MathWorks 160 dict begin
+% definition operators
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+/ldef {load def} bind def
+/xdef {exch def} bdef
+/xstore {exch store} bdef
+% operator abbreviations
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+/cc /concat ldef
+/cp /closepath ldef
+/gr /grestore ldef
+/gs /gsave ldef
+/mt /moveto ldef
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+/sm /setmatrix ldef
+/rm /rmoveto ldef
+/rl /rlineto ldef
+/s {show newpath} bdef
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+/sg /setgray ldef
+/w /setlinewidth ldef
+/j /setlinejoin ldef
+/cap /setlinecap ldef
+/rc {rectclip} bdef
+/rf {rectfill} bdef
+% page state control
+/pgsv () def
+/bpage {/pgsv save def} bdef
+/epage {pgsv restore} bdef
+/bplot /gsave ldef
+/eplot {stroke grestore} bdef
+% orientation switch
+/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def
+% coordinate system mappings
+/dpi2point 0 def
+% font control
+/FontSize 0 def
+/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0]
+ makefont setfont} bdef
+/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse
+ exch dup 3 1 roll findfont dup length dict begin
+ { 1 index /FID ne {def}{pop pop} ifelse } forall
+ /Encoding exch def currentdict end definefont pop} bdef
+/isroman {findfont /CharStrings get /Agrave known} bdef
+/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse
+ exch FMS} bdef
+/csm {1 dpi2point div -1 dpi2point div scale neg translate
+ dup landscapeMode eq {pop -90 rotate}
+ {rotateMode eq {90 rotate} if} ifelse} bdef
+% line types: solid, dotted, dashed, dotdash
+/SO { [] 0 setdash } bdef
+/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef
+/DA { [6 dpi2point mul] 0 setdash } bdef
+/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4
+ dpi2point mul] 0 setdash } bdef
+% macros for lines and objects
+/L {lineto stroke} bdef
+/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef
+/AP {{rlineto} repeat} bdef
+/PDlw -1 def
+/W {/PDlw currentlinewidth def setlinewidth} def
+/PP {closepath eofill} bdef
+/DP {closepath stroke} bdef
+/MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto
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+/FR {MR stroke} bdef
+/PR {MR fill} bdef
+/L1i {{currentfile picstr readhexstring pop} image} bdef
+/tMatrix matrix def
+/MakeOval {newpath tMatrix currentmatrix pop translate scale
+0 0 1 0 360 arc tMatrix setmatrix} bdef
+/FO {MakeOval stroke} bdef
+/PO {MakeOval fill} bdef
+/PD {currentlinewidth 2 div 0 360 arc fill
+ PDlw -1 eq not {PDlw w /PDlw -1 def} if} def
+/FA {newpath tMatrix currentmatrix pop translate scale
+ 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef
+/PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale
+ 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef
+/FAn {newpath tMatrix currentmatrix pop translate scale
+ 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef
+/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale
+ 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef
+/vradius 0 def /hradius 0 def /lry 0 def
+/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def
+/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef
+ /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly
+ vradius add translate hradius vradius scale 0 0 1 180 270 arc
+ tMatrix setmatrix lrx hradius sub uly vradius add translate
+ hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix
+ lrx hradius sub lry vradius sub translate hradius vradius scale
+ 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub
+ translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix
+ closepath} bdef
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+ newpath tMatrix currentmatrix pop ulx rad add uly rad add translate
+ rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad
+ sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix
+ closepath} bdef
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+/PlrRR {MlrRR fill } bdef
+/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def
+ newpath tMatrix currentmatrix pop ulx rad add uly rad add translate
+ rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad
+ sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix
+ closepath} bdef
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+/PtbRR {MtbRR fill } bdef
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+/tri_to_matrix {
+ 2 0 asub 3 1 asub 4 0 asub 5 1 asub
+ dup 0 get exch 1 get 7 -1 roll astore } bdef
+/compute_transform {
+ dmat dtri tri_to_matrix tmat1 invertmatrix
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+ currentfile
+ 3 index 0 eq {/ASCIIHexDecode filter}
+ {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if }
+ ifelse exch readstring pop
+ dup 0 3 index getinterval /rbmap xdef
+ dup 2 index dup getinterval /gbmap xdef
+ 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef
+/it {gs np dtri aload pop moveto lineto lineto cp c
+ cols rows 8 compute_transform
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+/il {newpath moveto lineto stroke}bdef
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+4 0 0 50 -1 0 12 0.0000 2 135 1620 1200 4650 calculate residual\001
+4 0 0 50 -1 0 12 0.0000 2 135 1890 1200 5175 residual below $tol$?\001
--- /dev/null
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--- /dev/null
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--- /dev/null
+\begin{picture}(0,0)%
+\includegraphics{./Chapters/chapter16/figures/resonant_flyback.pdf}%
+\end{picture}%
+\setlength{\unitlength}{3947sp}%
+%
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+\end{picture}%
--- /dev/null
+\section{New parallel envelope-following method}
+\label{sec:gmres}
+
+In this section, we explain how to efficiently
+use matrix-free GMRES to solve
+the Newton update problems with implicit sensitivity calculation,
+i.e., the steps enclosed by the double dashed block
+in Fig.~\ref{fig:ef_flow}.
+Then implementation issues of GPU acceleration
+will be discussed in detail.
+Finally, the Gear-2 integration is briefly introduced.
+
+\subsection{GMRES solver for Newton update equation}
+
+\underline{G}eneralized \underline{M}inimum \underline{Res}idual,
+or GMRES method is an iterative method for solving
+systems of linear equations ($A x=b$) with dense matrix $A$.
+The standard GMRES\index{GMRES} is given in Algorithm~\ref{alg:GMRES}.
+It constructs a Krylov subspace\index{Krylov subspace} with order $m$,
+\[ \mathcal{K}_m = \mathrm{span}( b, A^{} b, A^2 b,\ldots, A^{m-1} b ),\]
+where the approximate solution $x_m$ resides.
+In practice, an orthonormal basis $V_m$ that spans the
+subspace $\mathcal{K}_{m}$ can be generated by
+the Arnoldi iteration\index{Arnoldi iterations}.
+The goal of GMRES is to search for an optimal coefficient $y$
+such that the linear combination $x_m = V_m y$ will minimize
+its residual $\| b-Ax_m \|_2$.
+The Arnoldi iteration also creates a by-product,
+an upper Hessenberg matrix\index{Hessenberg matrix} $\tilde{H}_m \in R^{(m+1)\times m}$.
+Thus, the projection of $A$ on the orthonormal basis $V_m$ is described by
+the Arnoldi decomposition
+$
+ AV_m = V_{m+1} \tilde{H}_{m},
+$
+which is useful to check the residual at each iteration
+without forming $x_m$,
+and to solve for coefficient $y$ when residual is smaller than
+a preset tolerance~\cite{Golub:Book'96}.
+
+% Givens rotations are generated to triangularize
+% the Hessenberg matrix $\tilde{H}_m$ in order to solve the overdetermined system.
+% These same Givens rotations are also applied to the right-hand-side $h_{1,0}e_1$,
+% whose last element provides residual information on whether the tolerance
+% has been met in the current iteration.
+
+%$\left\| h_{1,0}e_1-\tilde{H}_m y_m \right\|_2$.
+
+
+%% \begin{algorithm}
+%% \caption{Standard GMRES\index{GMRES} algorithm.} \label{alg:GMRES}
+%% \begin{algorithmic}[1]
+%% \REQUIRE $ A \in \mathbb{R}^{N \times N}$, $b \in \mathbb{R}^N$,
+%% and initial guess $x_0 \in \mathbb{R}^N$
+%% \ENSURE $x \in \mathbb{R}^N$: $\| b - A x\|_2 < tol$
+
+%% \STATE $r = b - A x_0$
+%% \STATE $h_{1,0}=\left \| r \right \|_2$
+%% \STATE $m=0$
+
+%% \WHILE{$m < max\_iter$}
+%% \STATE $m = m+1$
+%% \STATE $v_{m} = r / h_{m,m-1}$
+%% \STATE \label{line:mvp} $r = A v_m$
+%% \FOR{$i = 1\ldots m$}
+%% \STATE $h_{i,m} = \langle v_i, r \rangle$
+%% \STATE $r = r - h_{i,m} v_i$
+%% \ENDFOR
+%% \STATE $h_{m+1,m} = \left\| r \right\|_2$\label{line:newnorm}
+%% %\STATE Generate Givens rotations to triangularize $\tilde{H}_m$
+%% %\STATE Apply Givens rotations on $h_{1,0}e_1$ to get residual $\epsilon$
+%% \STATE Compute the residual $\epsilon$
+%% \IF{$\epsilon < tol$}
+%% \STATE Solve the problem: minimize $\|b-Ax_m\|_2$
+%% \STATE Return $x_m = x_0 + V_m y_m$
+%% \ENDIF
+%% \ENDWHILE
+%% \end{algorithmic}
+%% \end{algorithm}
+
+At a first glance, the cost of using standard GMRES directly to
+solve the Newton update in Eq.~\eqref{eq:Newton}
+seems to come mainly from two parts: the
+formulation of the sensitivity matrix $J = \ud x_M/\ud x_0$
+in Eq.~\eqref{eq:sensM} in Section~\ref{sec:gear},
+and the iteration inside the standard GMRES,
+especially the matrix-vector multiplication and the orthonormal
+basis construction (Line~\ref{line:mvp} through
+Line~\ref{line:newnorm} in Algorithm~\ref{alg:GMRES}).
+Based on the observation that only the matrix-vector product is
+required in GMRES, the work in~\cite{Telichevesky:DAC'95} introduces
+an efficient matrix-free algorithm in the shooting-Newton method,
+where the equation solving part also involves with a sensitivity
+matrix.
+The matrix-free method does not take an explicit matrix as
+input, but directly passes the saved capacitance matrices $C_i$ and
+the LU factorizations of $J_i$, $i=0,\ldots,M$, into the Arnoldi
+iteration for Krylov subspace generation.
+Therefore, it avoids the
+cost of forming the dense sensitivity matrix and focuses on
+subspace construction.
+Briefly speaking, Line~\ref{line:mvp} will be replaced
+by a procedure without explict $A$, and we will talk
+about the flow of matrix-free generation of new basis vectors
+in later sections.
+
+
+\subsection{Parallelization on GPU platforms}
+\label{sec:gpu}
+% In this subsection, we describe how to parallelize
+% the proposed matrix-free GMRES solver with Gear-2 sensitivity
+% on GPU platforms.
+There exist many GPU-friendly computing operations
+in GMRES, such as the vector addition (\verb|axpy|),
+2-norm of vector (\verb|nrm2|), and
+sparse matrix-vector multiplication (\verb|csrmv|),
+which have been parallelized in the CUDA Basic Linear
+Algebra Subroutine (CUBLAS) Library and the CUDA Sparse Linear Algebra
+Library (CUSPARSE)~\cite{CUDASDK}.
+
+GPU programming is typically limited by the data transfer bandwidth as
+GPU favors computationally intensive algorithms~\cite{Kirk:Book'10}.
+So how to efficiently transfer the data and wisely partition
+the data between CPU memory and GPU memory
+is crucial for GPU programming.
+In the following, we discuss these two issues in our implementation.
+
+As noted in Section~\ref{sec:ef}, the envelope-following method
+requires the matrices gathered from all the time steps in a
+period in order to solve a Newton update.
+At each time step, SPICE\index{SPICE} has
+to linearize device models, stamp matrix elements
+into MNA (short for modified nodal analysis\index{modified nodal analysis, or MNA}) matrices,
+and solve circuit equations in its inner Newton iteration\index{Newton iteration}.
+When convergence is attained,
+circuit states are saved and then next time step begins.
+This is also the time when we store the needed matrices
+for the envelope-following computation.
+Since these data are involved in the calculation of Gear-2 sensitivity
+matrix in the generation of Krylov subspace vectors in
+Algorithm~\ref{alg:mf_Gear}, it is desirable that all of these
+matrices are transferred to GPU for its data parallel capability.
+
+To efficiently transfer those large data, we explore asynchronous
+memory copy between host and device in the recent GPUs (Fermi architecture),
+so that the copying overlaps with the host's computing of
+the next time step's circuit state. The implementation of asynchronous matrices
+copy includes two parts: allocating page-locked memory, also known as
+pinned memory, where we save matrices for one time step, and using
+asynchronous memory transfer to copy these data to GPU memory. While
+it is known that page-locked host memory is a scarce resource and
+should not be overused, the demand of memory size of the data
+generated at one time step can be generously accommodated by today's
+mainstream server configurations.
+
+\begin{figure}[htbp]
+ \centering
+ \resizebox{.9\textwidth}{!}{\input{./Chapters/chapter16/figures/gmres_flow.pdf_t}}
+ \caption{GPU parallel solver for envelope-following update.}
+ \label{fig:gmres}
+\end{figure}
+
+
+%When SPICE simulation reaches the end of the period, all data needed by
+%Newton update equation are ready in GPU memory, and the matrix-free
+%GMRES can be applied directly on them to solve for the update vector
+%in \eqref{eq:Newton}.
+
+% Knowing the dimensions or the sizes of the
+%matrices in a computation helps to boost GPU computing efficiency.
+The second issue is to decide the location of data between CPU and GPU memories.
+Therefore let us first make a rough sketch of the quantities in the
+GMRES Algorithm~\ref{alg:GMRES}. Although GMRES tends to converge
+quickly for most circuit examples, i.e., the iteration number $m \ll
+N$, the space for storing all the subspace basis $V_m$ of $N$-by-$m$,
+i.e., $m$ column vectors with $N$-length, is still big. In addition,
+every newly generated matrix-vector product needs to be orthogonalized
+with respect to all its previous basis vectors in
+the Arnoldi iterations\index{Arnoldi iterations}.
+Hence, keeping all the vectors of $V_m$ in GPU global
+memory allows GPU to handle those operations, such as inner-product of
+basis vectors (\verb|dot|) and vector subtraction (\verb|axpy|), in
+parallel.
+
+On the other hand, it is better to keep the Hessenberg matrix
+$\tilde{H}$, where intermediate results of
+the orthogonalization are stored, at the host side.
+This comes with the following reasons.
+First, its size is $(m+1)$-by-$m$ at most, rather small
+if compared to circuit matrices and Krylov basis.
+Besides, it is also necessary to triangularize $\tilde{H}$
+and check the residual regularly in
+each iteration so the GMRES can return the approximate solution as
+soon as the residual is below a preset tolerance.
+Hence, in consideration of the serial nature of the trianularization,
+the small size of Hessenberg matrix,
+and the frequent inspection of values by host, it is
+preferable to allocate $\tilde{H}$ in CPU (host) memory.
+As shown in Fig.~\ref{fig:gmres}, the memory copy from device to host
+is called each time when Arnoldi iteration generates a new vector
+and the orthogonalization produces the vector $h$.
+
+% \begin{figure}[!tb]
+% \centering
+% \resizebox{.45\textwidth}{!}{\input{./matlab/drawing.ps_tex}}
+% \caption{The flow of envelope-following method.}
+% \label{fig:gmres}
+% %\end{center}
+% \end{figure}
+
+% talk about gear 2 method here.
+\input{Chapters/chapter16/bdf.tex}
+
--- /dev/null
+\section{Introduction}
+
+
+Power converters\index{power converters} have seen a surge of new trends and novel
+applications due to their widespread use in renewable energy systems
+and emerging hybrid and purely-electric vehicles. More efficient
+simulation techniques for power converters are urgently needed to meet
+more design constraints. In this chapter, we present a novel
+envelope-following parallel transient analysis method for the general
+switching power converters. The new method first exploits the
+parallelisim in the envelope-following method and parallelize the
+Newton update solving part, which is the most computational expensive,
+in GPU platforms to boost the simulation performance. To further
+speed up the iterative GMRES\index{GMRES} solving for Newton update equation in the
+envelope-following method, we apply the matrix-free\index{matrix-free}
+Krylov subspace\index{Krylov subspace} basis
+generation technique, which was previously used for RF simulation.
+Last, the new method also applies more robust Gear-2 integration to
+compute the sensitivity matrix instead of traditional integration
+methods.
+
+Over the past decades, power electronics\index{power electronics} and especially switching
+power converters have seen a surge of new trends and novel applications
+--- from the growing significance of PWM (pulse-width modulation)
+rectifiers and multilevel inverters to their widespread use in
+renewable (like solar and wind) energy systems, smart grids and emerging
+electric and hybrid vehicles~\cite{Trzynadlowski:Book'10}.
+
+This requires more efficient simulation techniques for power
+electronics to meet the new application and demanding design
+constraints. To facilitate the design of typical power electronics
+circuits, many special-purpose simulation algorithms and
+tools were developed. Among them is the envelope-following
+method~\cite{Kundert:ICCAD'88,White:TPE'91,Feldmann:ICCAD'96}, %,Bose:TIE'09
+which is able to calculate the slowly changing contour, or
+envelope, of a carrier waveform with a much higher
+switching frequency\index{switching frequency}.
+This is the case for switching power converters,
+which have fast switching currents to convert powers
+from one level to another level.
+In those switching power converters, it is the envelope,
+which is the power voltage delivered,
+not the fast switching waves in every cycle,
+that is of interest to the designers.
+As shown in Fig.~\ref{fig:ef1}, the solid line is
+the waveform of the output node in a Buck
+converter~\cite{Krein:book'97}, the dots are the simulation points
+of SPICE\index{SPICE}, and the appended dash line is the envelope.
+
+Obviously, traditional SPICE will incur an extraordinarily high
+simulation time for this task, since it has to integrate the
+circuit's differential equation with many time points in every
+clock cycle to get the accurate details of the carrier. For
+switching power converters, the waveform of the carrier in
+consequent cycles does not change much, envelope-following method
+is an approximation analysis method, which skips over several
+cycles (the dash line in Fig.~\ref{fig:ef2}), the so called
+envelope step, without simulating them, and then carries out a
+correction, which usually contains a sensitivity-based Newton
+iteration or shooting until convergence, in order to begin the
+next envelope step.
+
+% \begin{figure}[!b]
+% \resizebox{.5\textwidth}{!}{\input{./matlab/ef_intro.pstex_t}}
+% %\includegraphics[width=.45\textwidth]{./matlab/ef_intro_ppt.eps}
+% \caption{Transient envelope-following analysis.
+% (This figure reflects a backward-Euler style envelope-following.)}
+% \label{fig:ef_intro}
+% \end{figure}
+\begin{figure}
+ \centering
+ \subfigure[Illustration of one envelope skip.]
+ {\resizebox{.8\textwidth}{!}{\input{./Chapters/chapter16/figures/ef_intro.pdf_t}}
+ \label{fig:ef1} }
+ \subfigure[The envelope changes in a slow time scale.]
+ {\resizebox{.9\textwidth}{!}{\input{./Chapters/chapter16/figures/envelope.pdf_t}}
+ \label{fig:ef2} }
+ \caption{Transient envelope-following\index{envelope-following} analysis.
+ (Both two figures reflect backward-Euler\index{backward-Euler} style envelope-following.)}
+ \label{fig:ef_intro}
+\end{figure}
+
+%\IEEEpubidadjcol
+
+Also, iterative GMRES\index{GMRES} solver is typically used in the
+envelope-following method to compute the solution of Newton update
+due to its efficiency compared to direct LU\index{LU} method.
+However, as the Jacobian matrix\index{Jacobian matrix}
+or the sensitivity matrix\index{sensitivity matrix}
+in the equation to be solved is dense,
+explicit computing of the Jacobian is a very expense process.
+Recently, the matrix-free\index{matrix-free} GMRES
+was proposed~\cite{Telichevesky:DAC'95}
+for RF shooting based simulation.
+The new method leads to significant
+savings due to its implicit calculation of new basis vectors
+without the explicit formulation of the sensitivity matrix.
+
+
+Modern computer architecture has shifted towards designs that employ
+so called multi-core processor or chip-multiprocessors (CMP)~\cite{IntelMC'06,AMDMC'06}.
+The family of graphic processing units (GPU) are among the most
+powerful many-core computing systems in mass-market use~\cite{nvidia}.
+For instance, the state-of-the-art NVIDIA Tesla T10 chip has a peak
+performance of over 1~TFLOPS versus about 80--100~GFLOPS of Intel i5
+series Quad-core CPUs~\cite{Kirk:Book'10}.
+In addition to the primary
+use of GPUs in accelerating graphics rendering operations, there has
+been considerable interest in exploiting GPUs for general purpose
+computation (GPGPU)~\cite{gpgpu}. The introduction of new parallel
+programming interfaces for general purpose computation, such as
+Computer Unified Device Architecture (CUDA),
+Stream SDK, and OpenCL~\cite{cuda,streamSDK,openCL},
+has made GPUs an attractive choice
+for developing high-performance scientific computation tools
+and solving practical engineering problems.
+Hence, the applications with GPGPUs are rapidly growing in a broad
+variety of parallel numerical computation works~\cite{CUDA_Zone}.
+
+%For this correction to be efficient and precise, it is critical
+%for the Newton iteration to converge fast and yield accurate
+%result.
+
+%Switching power converters and RF (radio-frequency) circuits are
+%typically stiff systems where
+%the fast signals (switching signals or carrier signals) and
+%the slow moving signals (the output voltage waveforms and user signals)
+%coexist.
+%However, it is challenging to simulate the stiff systems
+%using the traditional integration method such as
+%Backward Euler and Trapezoidal method~\cite{Iserles:Book'96}.
+%It has been shown that Gear-2 is more suitable
+%for many practical problems such as stiff problems
+%where one have different time constants
+%(fast ones with large poles and slow ones with small poles)~\cite{Vlach:Book'94}.
+
+%GMRES is a Krylov-subspace based method for solving linear
+%equation, which guarantees convergence and is efficient if the
+%eigenvalues are tightly clustered together~\cite{Golub:Book'96}.
+%Fortunately, most problems arising from the shooting method and
+%the envelope-following method satisfy naturally the convergence
+%criterion of GMRES~\cite{Nastov:ProcIEEE'07} (or after
+%preconditioners applied if not).
+
+
+%\item Newton update equation for envelope-following is formed with
+% the sensitivity matrix derived from the Gear-2 integration rule for
+% better stability.
+% \item Detailed mathematical derivations to show
+% the relationship between the sensitivity matrix and
+% the information extracted from transient steps.
+
+After the basic algorithm of envelope-following is briefly reviewed
+and Newton update equation is derived in Section~\ref{sec:ef}. In
+Section~\ref{sec:gmres} presents the new parallel envelope method
+where the CPU parallelization, matrix-free GMRES and Gear-2
+integration will be discussed. We will show the parallelization of the
+Newton update, which is the most time consuming step, in the
+envelop-follow method, in GPU platforms. We will also present the new
+GMRES solver using matrix-free Jacobian-vector multiplication and the
+Gear-2 integration for sensitivity based Newton update equation.
+Numerical examples are shown in Section~\ref{sec:exp}, and finally,
+this chapter is summarized in Section~\ref{sec:summary}.
pdflatex ${BOOK}
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+# bibtex bu7
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+# #don't put chapter 14, refs are included
+# bibtex bu11
+# # bibtex bu12
+
makeindex ${BOOK}.idx
pdflatex ${BOOK}
pdflatex ${BOOK}
