7 %% http://www.michaelshell.org/
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42 %% File list of work: IEEEtran.cls, IEEEtran_HOWTO.pdf, bare_adv.tex,
43 %% bare_conf.tex, bare_jrnl.tex, bare_jrnl_compsoc.tex
44 %%*************************************************************************
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63 % should be used if it is desired that the figures are to be displayed in
66 \documentclass[conference,pdf,a4paper,10pt,final,twoside,twocolumn]{IEEEtran}
67 % Add the compsoc option for Computer Society conferences.
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71 % \documentclass[conference]{../sty/IEEEtran}
73 % Some very useful LaTeX packages include:
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76 % *** MISC UTILITY PACKAGES ***
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234 % IEEEtran contains the IEEEeqnarray family of commands that can be used to
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239 %\usepackage{eqparbox}
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243 % http://www.ctan.org/tex-archive/macros/latex/contrib/eqparbox/
249 % *** SUBFIGURE PACKAGES ***
250 %\usepackage[tight,footnotesize]{subfigure}
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252 % easy to put subfigures in your figures. e.g., "Figure 1a and 1b". For IEEE
253 % work, it is a good idea to load it with the tight package option to reduce
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262 %\usepackage[caption=false]{caption}
263 %\usepackage[font=footnotesize]{subfig}
264 % subfig.sty, also written by Steven Douglas Cochran, is the modern
265 % replacement for subfigure.sty. However, subfig.sty requires and
266 % automatically loads Axel Sommerfeldt's caption.sty which will override
267 % IEEEtran.cls handling of captions and this will result in nonIEEE style
268 % figure/table captions. To prevent this problem, be sure and preload
269 % caption.sty with its "caption=false" package option. This is will preserve
270 % IEEEtran.cls handing of captions. Version 1.3 (2005/06/28) and later
271 % (recommended due to many improvements over 1.2) of subfig.sty supports
272 % the caption=false option directly:
273 %\usepackage[caption=false,font=footnotesize]{subfig}
275 % The latest version and documentation can be obtained at:
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277 % The latest version and documentation of caption.sty can be obtained at:
278 % http://www.ctan.org/tex-archive/macros/latex/contrib/caption/
283 % *** FLOAT PACKAGES ***
285 %\usepackage{fixltx2e}
286 % fixltx2e, the successor to the earlier fix2col.sty, was written by
287 % Frank Mittelbach and David Carlisle. This package corrects a few problems
288 % in the LaTeX2e kernel, the most notable of which is that in current
289 % LaTeX2e releases, the ordering of single and double column floats is not
290 % guaranteed to be preserved. Thus, an unpatched LaTeX2e can allow a
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297 %\usepackage{stfloats}
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300 % as the top. (e.g., "\begin{figure*}[!b]" is not normally possible in
301 % LaTeX2e). It also provides a command:
303 % to enable the placement of footnotes below bottom floats (the standard
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312 % IEEE should note that IEEE rarely uses double column equations and
313 % that authors should try to avoid such use. Do not be tempted to use the
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315 % not format its papers in such ways.
321 % *** PDF, URL AND HYPERLINK PACKAGES ***
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328 % Read the url.sty source comments for usage information. Basically,
335 % *** Do not adjust lengths that control margins, column widths, etc. ***
336 % *** Do not use packages that alter fonts (such as pslatex). ***
337 % There should be no need to do such things with IEEEtran.cls V1.6 and later.
338 % (Unless specifically asked to do so by the journal or conference you plan
339 % to submit to, of course. )
341 % correct bad hyphenation here
342 \hyphenation{op-tical net-works semi-conduc-tor}
344 % Macro for certain acronyms in small caps. Doesn't work with the
345 % default font, though (it contains no smallcaps it seems).
346 \def\acro#1{{\small{#1}}}
347 \def\VHDL{\acro{VHDL}}
349 \def\CLaSH{{\small{C}}$\lambda$a{\small{SH}}}
351 % Macro for pretty printing haskell snippets. Just monospaced for now, perhaps
352 % we'll get something more complex later on.
353 \def\hs#1{\texttt{#1}}
354 \def\quote#1{``{#1}"}
356 \newenvironment{xlist}[1][\rule{0em}{0em}]{%
358 \settowidth{\labelwidth}{#1:}
359 \setlength{\labelsep}{0.5cm}
360 \setlength{\leftmargin}{\labelwidth}
361 \addtolength{\leftmargin}{\labelsep}
362 \setlength{\rightmargin}{0pt}
363 \setlength{\listparindent}{\parindent}
364 \setlength{\itemsep}{0 ex plus 0.2ex}
365 \renewcommand{\makelabel}[1]{##1:\hfil}
370 \usepackage{paralist}
372 \def\comment#1{{\color[rgb]{1.0,0.0,0.0}{#1}}}
374 %include polycode.fmt
380 % can use linebreaks \\ within to get better formatting as desired
381 \title{C$\lambda$aSH: Structural Descriptions \\ of Synchronous Hardware using Haskell}
384 % author names and affiliations
385 % use a multiple column layout for up to three different
387 \author{\IEEEauthorblockN{Christiaan P.R. Baaij, Matthijs Kooijman, Jan Kuper, Marco E.T. Gerards, Bert Molenkamp, Sabih H. Gerez}
388 \IEEEauthorblockA{University of Twente, Department of EEMCS\\
389 P.O. Box 217, 7500 AE, Enschede, The Netherlands\\
390 c.p.r.baaij@@utwente.nl, matthijs@@stdin.nl}}
392 % \IEEEauthorblockN{Homer Simpson}
393 % \IEEEauthorblockA{Twentieth Century Fox\\
395 % Email: homer@thesimpsons.com}
397 % \IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
398 % \IEEEauthorblockA{Starfleet Academy\\
399 % San Francisco, California 96678-2391\\
400 % Telephone: (800) 555--1212\\
401 % Fax: (888) 555--1212}}
403 % conference papers do not typically use \thanks and this command
404 % is locked out in conference mode. If really needed, such as for
405 % the acknowledgment of grants, issue a \IEEEoverridecommandlockouts
406 % after \documentclass
408 % for over three affiliations, or if they all won't fit within the width
409 % of the page, use this alternative format:
411 %\author{\IEEEauthorblockN{Michael Shell\IEEEauthorrefmark{1},
412 %Homer Simpson\IEEEauthorrefmark{2},
413 %James Kirk\IEEEauthorrefmark{3},
414 %Montgomery Scott\IEEEauthorrefmark{3} and
415 %Eldon Tyrell\IEEEauthorrefmark{4}}
416 %\IEEEauthorblockA{\IEEEauthorrefmark{1}School of Electrical and Computer Engineering\\
417 %Georgia Institute of Technology,
418 %Atlanta, Georgia 30332--0250\\ Email: see http://www.michaelshell.org/contact.html}
419 %\IEEEauthorblockA{\IEEEauthorrefmark{2}Twentieth Century Fox, Springfield, USA\\
420 %Email: homer@thesimpsons.com}
421 %\IEEEauthorblockA{\IEEEauthorrefmark{3}Starfleet Academy, San Francisco, California 96678-2391\\
422 %Telephone: (800) 555--1212, Fax: (888) 555--1212}
423 %\IEEEauthorblockA{\IEEEauthorrefmark{4}Tyrell Inc., 123 Replicant Street, Los Angeles, California 90210--4321}}
428 % use for special paper notices
429 %\IEEEspecialpapernotice{(Invited Paper)}
434 % make the title area
440 The abstract goes here.
442 % IEEEtran.cls defaults to using nonbold math in the Abstract.
443 % This preserves the distinction between vectors and scalars. However,
444 % if the conference you are submitting to favors bold math in the abstract,
445 % then you can use LaTeX's standard command \boldmath at the very start
446 % of the abstract to achieve this. Many IEEE journals/conferences frown on
447 % math in the abstract anyway.
454 % For peer review papers, you can put extra information on the cover
456 % \ifCLASSOPTIONpeerreview
457 % \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
460 % For peerreview papers, this IEEEtran command inserts a page break and
461 % creates the second title. It will be ignored for other modes.
462 \IEEEpeerreviewmaketitle
465 \section{Introduction}
466 Hardware description languages has allowed the productivity of hardware
467 engineers to keep pace with the development of chip technology. Standard
468 Hardware description languages, like \VHDL~\cite{VHDL2008} and
469 Verilog~\cite{Verilog}, allowed an engineer to describe circuits using a
470 programming language. These standard languages are very good at describing
471 detailed hardware properties such as timing behavior, but are generally
472 cumbersome in expressing higher-level abstractions. In an attempt to raise the
473 abstraction level of the descriptions, a great number of approaches based on
474 functional languages has been proposed \cite{T-Ruby,Hydra,HML2,Hawk1,Lava,
475 ForSyDe1,Wired,reFLect}. The idea of using functional languages for hardware
476 descriptions started in the early 1980s \cite{Cardelli1981, muFP,DAISY,FHDL},
477 a time which also saw the birth of the currently popular hardware description
478 languages such as \VHDL. The merit of using a functional language to describe
479 hardware comes from the fact that basic combinatorial circuits are equivalent
480 to mathematical functions and that functional languages are very good at
481 describing and composing mathematical functions.
483 In an attempt to decrease the amount of work involved with creating all the
484 required tooling, such as parsers and type-checkers, many functional hardware
485 description languages are embedded as a domain specific language inside the
486 functional language Haskell \cite{Hydra,Hawk1,Lava,ForSyDe1,Wired}. This
487 means that a developer is given a library of Haskell~\cite{Haskell} functions
488 and types that together form the language primitives of the domain specific
489 language. As a result of how the signals are modeled and abstracted, the
490 functions used to describe a circuit also build a large domain-specific
491 datatype (hidden from the designer) which can be further processed by an
492 embedded compiler. This compiler actually runs in the same environment as the
493 description; as a result compile-time and run-time become hard to define, as
494 the embedded compiler is usually compiled by the same Haskell compiler as the
495 circuit description itself.
497 The approach taken in this research is not to make another domain specific
498 language embedded in Haskell, but to use (a subset of) the Haskell language
499 itself for the purpose of describing hardware. By taking this approach, we can
500 capture certain language constructs, such as Haskell's choice elements
501 (if-constructs, case-constructs, pattern matching, etc.), which are not
502 available in the functional hardware description languages that are embedded
503 in Haskell as a domain specific languages. As far as the authors know, such
504 extensive support for choice-elements is new in the domain of functional
505 hardware description language. As the hardware descriptions are plain Haskell
506 functions, these descriptions can be compiled for simulation using using the
507 optimizing Haskell compiler \GHC.
509 Where descriptions in a conventional hardware description language have an
510 explicit clock for the purpose state and synchronicity, the clock is implied
511 in this research. The functions describe the behavior of the hardware between
512 clock cycles, as such, only synchronous systems can be described. Many
513 functional hardware description models signals as a stream of all values over
514 time; state is then modeled as a delay on this stream of values. The approach
515 taken in this research is to make the current state of a circuit part of the
516 input of the function and the updated state part of the output.
518 Like the standard hardware description languages, descriptions made in a
519 functional hardware description language must eventually be converted into a
520 netlist. This research also features a prototype translator called \CLaSH\
521 (pronounced: clash), which converts the Haskell code to equivalently behaving
522 synthesizable \VHDL\ code, ready to be converted to an actual netlist format
523 by optimizing \VHDL\ synthesis tools.
525 \section{Hardware description in Haskell}
527 \subsection{Function application}
528 The basic syntactic elements of a functional program are functions
529 and function application. These have a single obvious \VHDL\
530 translation: each top level function becomes a hardware component,
531 where each argument is an input port and the result value is the
532 (single) output port. This output port can have a complex type (such
533 as a tuple), so having just a single output port does not create a
536 Each function application in turn becomes component instantiation.
537 Here, the result of each argument expression is assigned to a
538 signal, which is mapped to the corresponding input port. The output
539 port of the function is also mapped to a signal, which is used as
540 the result of the application itself.
542 Since every top level function generates its own component, the
543 hierarchy of of function calls is reflected in the final \VHDL\
544 output as well, creating a hierarchical \VHDL\ description of the
545 hardware. This separation in different components makes the
546 resulting \VHDL\ output easier to read and debug.
548 Example that defines the \texttt{mac} function by applying the
549 \texttt{add} and \texttt{mul} functions to calculate $a * b + c$:
552 mac a b c = add (mul a b) c
555 \comment{TODO: Pretty picture}
558 Although describing components and connections allows describing a
559 lot of hardware designs already, there is an obvious thing missing:
560 choice. We need some way to be able to choose between values based
561 on another value. In Haskell, choice is achieved by \hs{case}
562 expressions, \hs{if} expressions, pattern matching and guards.
564 The easiest of these are of course case expressions (and \hs{if}
565 expressions, which can be very directly translated to \hs{case}
566 expressions). A \hs{case} expression can in turn simply be
567 translated to a conditional assignment in \VHDL, where the
568 conditions use equality comparisons against the constructors in the
569 \hs{case} expressions.
571 A slightly more complex (but very powerful) form of choice is
572 pattern matching. A function can be defined in multiple clauses,
573 where each clause specifies a pattern. When the arguments match the
574 pattern, the corresponding clause will be used.
576 A pattern match (with optional guards) can also be implemented using
577 conditional assignments in \VHDL, where the condition is the logical
578 and of comparison results of each part of the pattern as well as the
581 Contrived example that sums two values when they are equal or
582 non-equal (depending on the predicate given) and returns 0
583 otherwise. This shows three implementations, one using and if
584 expression, one using only case expressions and one using pattern
588 sumif pred a b = if pred == Eq && a == b ||
589 pred == Neq && a != b
593 sumif pred a b = case pred of
597 Neq -> case a != b of
601 sumif Eq a b | a == b = a + b
602 sumif Neq a b | a != b = a + b
606 \comment{TODO: Pretty picture}
609 Translation of two most basic functional concepts has been
610 discussed: function application and choice. Before looking further
611 into less obvious concepts like higher-order expressions and
612 polymorphism, the possible types that can be used in hardware
613 descriptions will be discussed.
615 Some way is needed to translate every value used to its hardware
616 equivalents. In particular, this means a hardware equivalent for
617 every \emph{type} used in a hardware description is needed.
619 The following types are \emph{built-in}, meaning that their hardware
620 translation is fixed into the \CLaSH\ compiler. A designer can also
621 define his own types, which will be translated into hardware types
622 using translation rules that are discussed later on.
624 \subsection{Built-in types}
627 This is the most basic type available. It can have two values:
628 \hs{Low} and \hs{High}. It is mapped directly onto the
629 \texttt{std\_logic} \VHDL\ type.
631 This is a basic logic type. It can have two values: \hs{True}
632 and \hs{False}. It is translated to \texttt{std\_logic} exactly
633 like the \hs{Bit} type (where a value of \hs{True} corresponds
634 to a value of \hs{High}). Supporting the Bool type is
635 particularly useful to support \hs{if ... then ... else ...}
636 expressions, which always have a \hs{Bool} value for the
638 \item[\hs{SizedWord}, \hs{SizedInt}]
639 These are types to represent integers. A \hs{SizedWord} is unsigned,
640 while a \hs{SizedInt} is signed. These types are parametrized by a
641 length type, so you can define an unsigned word of 32 bits wide as
645 type Word32 = SizedWord D32
648 Here, a type synonym \hs{Word32} is defined that is equal to the
649 \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
650 is the \emph{type level representation} of the decimal number 32,
651 making the \hs{Word32} type a 32-bit unsigned word. These types are
652 translated to the \VHDL\ \texttt{unsigned} and \texttt{signed}
655 This is a vector type, that can contain elements of any other type and
656 has a fixed length. The \hs{Vector} type constructor takes two type
657 arguments: the length of the vector and the type of the elements
658 contained in it. The state type of an 8 element register bank would
662 type RegisterState = Vector D8 Word32
665 Here, a type synonym \hs{RegisterState} is defined that is equal to
666 the \hs{Vector} type constructor applied to the types \hs{D8} (The
667 type level representation of the decimal number 8) and \hs{Word32}
668 (The 32 bit word type as defined above). In other words, the
669 \hs{RegisterState} type is a vector of 8 32-bit words. A fixed size
670 vector is translated to a \VHDL\ array type.
671 \item[\hs{RangedWord}]
672 This is another type to describe integers, but unlike the previous
673 two it has no specific bit-width, but an upper bound. This means that
674 its range is not limited to powers of two, but can be any number.
675 A \hs{RangedWord} only has an upper bound, its lower bound is
676 implicitly zero. The main purpose of the \hs{RangedWord} type is to be
677 used as an index to a \hs{Vector}.
679 \comment{TODO: Perhaps remove this example?} To define an index for
680 the 8 element vector above, we would do:
683 type RegisterIndex = RangedWord D7
686 Here, a type synonym \hs{RegisterIndex} is defined that is equal to
687 the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
688 other words, this defines an unsigned word with values from
689 0 to 7 (inclusive). This word can be be used to index the
690 8 element vector \hs{RegisterState} above. This type is translated to
691 the \texttt{unsigned} \VHDL type.
694 \subsection{User-defined types}
695 There are three ways to define new types in Haskell: algebraic
696 data-types with the \hs{data} keyword, type synonyms with the \hs{type}
697 keyword and type renamings with the \hs{newtype} keyword. \GHC\
698 offers a few more advanced ways to introduce types (type families,
699 existential typing, {\small{GADT}}s, etc.) which are not standard
700 Haskell. These are not currently supported.
702 Only an algebraic datatype declaration actually introduces a
703 completely new type, for which we provide the \VHDL\ translation
704 below. Type synonyms and renamings only define new names for
705 existing types (where synonyms are completely interchangeable and
706 renamings need explicit conversion). Therefore, these do not need
707 any particular \VHDL\ translation, a synonym or renamed type will
708 just use the same representation as the original type. The
709 distinction between a renaming and a synonym does no longer matter
710 in hardware and can be disregarded in the generated \VHDL.
712 For algebraic types, we can make the following distinction:
715 \item[\bf{Single constructor}]
716 Algebraic datatypes with a single constructor with one or more
717 fields, are essentially a way to pack a few values together in a
718 record-like structure. An example of such a type is the following pair
722 data IntPair = IntPair Int Int
725 Haskell's builtin tuple types are also defined as single
726 constructor algebraic types and are translated according to this
727 rule by the \CLaSH\ compiler. These types are translated to \VHDL\
728 record types, with one field for every field in the constructor.
729 \item[\bf{No fields}]
730 Algebraic datatypes with multiple constructors, but without any
731 fields are essentially a way to get an enumeration-like type
732 containing alternatives. Note that Haskell's \hs{Bool} type is also
733 defined as an enumeration type, but we have a fixed translation for
734 that. These types are translated to \VHDL\ enumerations, with one
735 value for each constructor. This allows references to these
736 constructors to be translated to the corresponding enumeration value.
737 \item[\bf{Multiple constructors with fields}]
738 Algebraic datatypes with multiple constructors, where at least
739 one of these constructors has one or more fields are not
744 A very important concept in hardware it the concept of state. In a
745 stateful design, the outputs depend on the history of the inputs, or the
746 state. State is usually stored in registers, which retain their value
747 during a clock cycle. As we want to describe more than simple
748 combinatorial designs, \CLaSH\ needs an abstraction mechanism for state.
750 An important property in Haskell, and in most other functional languages,
751 is \emph{purity}. A function is said to be \emph{pure} if it satisfies two
754 \item given the same arguments twice, it should return the same value in
756 \item when the function is called, it should not have observable
759 This purity property is important for functional languages, since it
760 enables all kinds of mathematical reasoning that could not be guaranteed
761 correct for impure functions. Pure functions are as such a perfect match
762 for a combinatorial circuit, where the output solely depends on the
763 inputs. When a circuit has state however, it can no longer be simply
764 described by a pure function. Simply removing the purity property is not a
765 valid option, as the language would then lose many of it mathematical
766 properties. In an effort to include the concept of state in pure
767 functions, the current value of the state is made an argument of the
768 function; the updated state becomes part of the result.
770 A simple example is the description of an accumulator circuit:
772 acc :: Word -> State Word -> (State Word, Word)
773 acc inp (State s) = (State s', outp)
778 This approach makes the state of a function very explicit: which variables
779 are part of the state is completely determined by the type signature. This
780 approach to state is well suited to be used in combination with the
781 existing code and language features, such as all the choice constructs, as
782 state values are just normal values.
783 \section{\CLaSH\ prototype}
787 \section{Related work}
788 Many functional hardware description languages have been developed over the
789 years. Early work includes such languages as $\mu$\acro{FP}~\cite{muFP}, an
790 extension of Backus' \acro{FP} language to synchronous streams, designed
791 particularly for describing and reasoning about regular circuits. The
792 Ruby~\cite{Ruby} language uses relations, instead of functions, to describe
793 circuits, and has a particular focus on layout. \acro{HML}~\cite{HML2} is a
794 hardware modeling language based on the strict functional language
795 \acro{ML}, and has support for polymorphic types and higher-order functions.
796 Published work suggests that there is no direct simulation support for
797 \acro{HML}, and that the translation to \VHDL\ is only partial.
799 Like this work, many functional hardware description languages have some sort
800 of foundation in the functional programming language Haskell.
801 Hawk~\cite{Hawk1} uses Haskell to describe system-level executable
802 specifications used to model the behavior of superscalar microprocessors. Hawk
803 specifications can be simulated, but there seems to be no support for
804 automated circuit synthesis. The ForSyDe~\cite{ForSyDe2} system uses Haskell
805 to specify abstract system models, which can (manually) be transformed into an
806 implementation model using semantic preserving transformations. ForSyDe has
807 several simulation and synthesis backends, though synthesis is restricted to
808 the synchronous subset of the ForSyDe language.
810 Lava~\cite{Lava} is a hardware description language that focuses on the
811 structural representation of hardware. Besides support for simulation and
812 circuit synthesis, Lava descriptions can be interfaced with formal method
813 tools for formal verification. Lava descriptions are actually circuit
814 generators when viewed from a synthesis viewpoint, in that the language
815 elements of Haskell, such as choice, can be used to guide the circuit
816 generation. If a developer wants to insert a choice element inside an actual
817 circuit he will have to specify this explicitly as a component. In this
818 respect \CLaSH\ differs from Lava, in that all the choice elements, such as
819 case-statements and pattern matching, are synthesized to choice elements in the
820 eventual circuit. As such, richer control structures can both be specified and
821 synthesized in \CLaSH\ compared to any of the languages mentioned in this
824 The merits of polymorphic typing, combined with higher-order functions, are
825 now also recognized in the `main-stream' hardware description languages,
826 exemplified by the new \VHDL-2008 standard~\cite{VHDL2008}. \VHDL-2008 has
827 support to specify types as generics, thus allowing a developer to describe
828 polymorphic components. Note that those types still require an explicit
829 generic map, whereas type-inference and type-specialization are implicit in
832 % Wired~\cite{Wired},, T-Ruby~\cite{T-Ruby}, Hydra~\cite{Hydra}.
834 % A functional language designed specifically for hardware design is
835 % $re{\mathit{FL}}^{ect}$~\cite{reFLect}, which draws experience from earlier
836 % language called \acro{FL}~\cite{FL} to la
838 % An example of a floating figure using the graphicx package.
839 % Note that \label must occur AFTER (or within) \caption.
840 % For figures, \caption should occur after the \includegraphics.
841 % Note that IEEEtran v1.7 and later has special internal code that
842 % is designed to preserve the operation of \label within \caption
843 % even when the captionsoff option is in effect. However, because
844 % of issues like this, it may be the safest practice to put all your
845 % \label just after \caption rather than within \caption{}.
847 % Reminder: the "draftcls" or "draftclsnofoot", not "draft", class
848 % option should be used if it is desired that the figures are to be
849 % displayed while in draft mode.
853 %\includegraphics[width=2.5in]{myfigure}
854 % where an .eps filename suffix will be assumed under latex,
855 % and a .pdf suffix will be assumed for pdflatex; or what has been declared
856 % via \DeclareGraphicsExtensions.
857 %\caption{Simulation Results}
861 % Note that IEEE typically puts floats only at the top, even when this
862 % results in a large percentage of a column being occupied by floats.
865 % An example of a double column floating figure using two subfigures.
866 % (The subfig.sty package must be loaded for this to work.)
867 % The subfigure \label commands are set within each subfloat command, the
868 % \label for the overall figure must come after \caption.
869 % \hfil must be used as a separator to get equal spacing.
870 % The subfigure.sty package works much the same way, except \subfigure is
871 % used instead of \subfloat.
874 %\centerline{\subfloat[Case I]\includegraphics[width=2.5in]{subfigcase1}%
875 %\label{fig_first_case}}
877 %\subfloat[Case II]{\includegraphics[width=2.5in]{subfigcase2}%
878 %\label{fig_second_case}}}
879 %\caption{Simulation results}
883 % Note that often IEEE papers with subfigures do not employ subfigure
884 % captions (using the optional argument to \subfloat), but instead will
885 % reference/describe all of them (a), (b), etc., within the main caption.
888 % An example of a floating table. Note that, for IEEE style tables, the
889 % \caption command should come BEFORE the table. Table text will default to
890 % \footnotesize as IEEE normally uses this smaller font for tables.
891 % The \label must come after \caption as always.
894 %% increase table row spacing, adjust to taste
895 %\renewcommand{\arraystretch}{1.3}
896 % if using array.sty, it might be a good idea to tweak the value of
897 % \extrarowheight as needed to properly center the text within the cells
898 %\caption{An Example of a Table}
899 %\label{table_example}
901 %% Some packages, such as MDW tools, offer better commands for making tables
902 %% than the plain LaTeX2e tabular which is used here.
903 %\begin{tabular}{|c||c|}
913 % Note that IEEE does not put floats in the very first column - or typically
914 % anywhere on the first page for that matter. Also, in-text middle ("here")
915 % positioning is not used. Most IEEE journals/conferences use top floats
916 % exclusively. Note that, LaTeX2e, unlike IEEE journals/conferences, places
917 % footnotes above bottom floats. This can be corrected via the \fnbelowfloat
918 % command of the stfloats package.
923 The conclusion goes here.
928 % conference papers do not normally have an appendix
931 % use section* for acknowledgement
932 \section*{Acknowledgment}
935 The authors would like to thank...
941 % trigger a \newpage just before the given reference
942 % number - used to balance the columns on the last page
943 % adjust value as needed - may need to be readjusted if
944 % the document is modified later
945 %\IEEEtriggeratref{8}
946 % The "triggered" command can be changed if desired:
947 %\IEEEtriggercmd{\enlargethispage{-5in}}
951 % can use a bibliography generated by BibTeX as a .bbl file
952 % BibTeX documentation can be easily obtained at:
953 % http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
954 % The IEEEtran BibTeX style support page is at:
955 % http://www.michaelshell.org/tex/ieeetran/bibtex/
956 \bibliographystyle{IEEEtran}
957 % argument is your BibTeX string definitions and bibliography database(s)
958 \bibliography{IEEEabrv,clash.bib}
960 % <OR> manually copy in the resultant .bbl file
961 % set second argument of \begin to the number of references
962 % (used to reserve space for the reference number labels box)
963 % \begin{thebibliography}{1}
965 % \bibitem{IEEEhowto:kopka}
966 % H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
967 % 0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
969 % \end{thebibliography}
977 % vim: set ai sw=2 sts=2 expandtab: