From: Christiaan Baaij Date: Tue, 2 Mar 2010 15:40:27 +0000 (+0100) Subject: Update FIR example to include jan's comments, and do general cleanup of that section X-Git-Url: https://git.stderr.nl/gitweb?a=commitdiff_plain;h=b67d2a163cc5e8d55cecc17cc0ca2d7dcfb827e8;p=matthijs%2Fmaster-project%2Fdsd-paper.git Update FIR example to include jan's comments, and do general cleanup of that section --- diff --git a/clash.bib b/clash.bib index 5bec913..e0ef497 100644 --- a/clash.bib +++ b/clash.bib @@ -106,6 +106,18 @@ timestamp = {2010.01.21} } +@BOOK{lambdacalculus, + title = {{The Lambda Calculus: its Syntax and Semantics}}, + publisher = {{Elsevier Science}}, + year = {1984}, + author = {{H.P. Barendregt}}, + volume = {103}, + series = {{Studies in Logic and the Foundations of Mathematics}}, + edition = {{Revised}}, + owner = {baaijcpr}, + timestamp = {2010.03.02} +} + @INPROCEEDINGS{DAISY, author = {Johnson, Steven D.}, title = {Applicative programming and digital design}, diff --git "a/c\316\273ash.lhs" "b/c\316\273ash.lhs" index d0ab48b..7eaf97f 100644 --- "a/c\316\273ash.lhs" +++ "b/c\316\273ash.lhs" @@ -1082,10 +1082,10 @@ translation. The second stage of the compiler, the \emph{normalization} phase exhaustively applies a set of \emph{meaning-preserving} transformations on the \emph{Core} description until this description is in a \emph{normal form}. This set of transformations includes transformations typically found in -reduction systems for lambda calculus, such a $\beta$-reduction and -$\eta$-expansion, but also includes self-defined transformations that are -responsible for the reduction of higher-order functions to `regular' -first-order functions. +reduction systems for lambda calculus~\cite{lambdacalculus}, such a +$\beta$-reduction and $\eta$-expansion, but also includes self-defined +transformations that are responsible for the reduction of higher-order +functions to `regular' first-order functions. The final step in the compiler pipeline is the translation to a \VHDL\ \emph{netlist}, which is a straightforward process due to resemblance of a @@ -1094,6 +1094,8 @@ end-product of the \CLaSH\ compiler a \VHDL\ \emph{netlist} as the resulting \VHDL\ resembles an actual netlist description and not idiomatic \VHDL. \section{Use cases} + +\subsection{FIR Filter} \label{sec:usecases} As an example of a common hardware design where the use of higher-order functions leads to a very natural description is a FIR filter, which is @@ -1123,48 +1125,48 @@ xs *+* ys = foldl1 (+) (zipWith (*) xs hs) The \hs{zipWith} function is very similar to the \hs{map} function seen earlier: It takes a function, two vectors, and then applies the function to each of the elements in the two vectors pairwise (\emph{e.g.}, \hs{zipWith (*) -[1, 2] [3, 4]} becomes \hs{[1 * 3, 2 * 4]} $\equiv$ \hs{[3,8]}). +[1, 2] [3, 4]} becomes \hs{[1 * 3, 2 * 4]}). -The \hs{foldl1} function takes a function, a single vector, and applies +The \hs{foldl1} function takes a binary function, a single vector, and applies the function to the first two elements of the vector. It then applies the -function to the result of the first application and the next element from -the vector. This continues until the end of the vector is reached. The -result of the \hs{foldl1} function is the result of the last application. -As you can see, the \hs{zipWith (*)} function is just pairwise -multiplication and the \hs{foldl1 (+)} function is just summation. - -Returning to the actual FIR filter, we will slightly change the -equation belong to it, so as to make the translation to code more obvious. -What we will do is change the definition of the vector of input samples. -So, instead of having the input sample received at time -$t$ stored in $x_t$, $x_0$ now always stores the current sample, and $x_i$ -stores the $ith$ previous sample. This changes the equation to the -following (Note that this is completely equivalent to the original -equation, just with a different definition of $x$ that will better suit -the transformation to code): +function to the result of the first application and the next element in the +vector. This continues until the end of the vector is reached. The result of +the \hs{foldl1} function is the result of the last application. It is obvious +that the \hs{zipWith (*)} function is basically pairwise multiplication and +that the \hs{foldl1 (+)} function is just summation. + +Returning to the actual FIR filter, we will slightly change the equation +describing it, so as to make the translation to code more obvious and concise. +What we do is change the definition of the vector of input samples and delay +the computation by one sample. Instead of having the input sample received at +time $t$ stored in $x_t$, $x_0$ now always stores the newest sample, and $x_i$ +stores the $ith$ previous sample. This changes the equation to the following +(note that this is completely equivalent to the original equation, just with a +different definition of $x$ that will better suit the transformation to code): \begin{equation} y_t = \sum\nolimits_{i = 0}^{n - 1} {x_i \cdot h_i } \end{equation} -Consider that the vector \hs{hs} contains the FIR coefficients and the -vector \hs{xs} contains the current input sample in front and older -samples behind. The function that shifts the input samples is shown below: +The complete definition of the FIR filter in code then becomes: \begin{code} -x >> xs = x +> tail xs +fir (State (xs,hs)) x = (State (x >> xs,hs), xs *+* hs) \end{code} -Where the \hs{tail} function returns all but the first element of a -vector, and the concatenate operator ($\succ$) adds a new element to the -left of a vector. The complete definition of the FIR filter then becomes: +Where the vector \hs{hs} contains the FIR coefficients and the vector \hs{xs} +contains the latest input sample in front and older samples behind. The code +for the shift (\hs{>>}) operator that adds the new input sample (\hs{x}) to +the list of previous input samples (\hs{xs}) and removes the oldest sample is +shown below: \begin{code} -fir (State (xs,hs)) x = (State (x >> xs,hs), xs *+* hs) +x >> xs = x +> init xs \end{code} -The resulting netlist of a 4-taps FIR filter based on the above definition -is depicted in \Cref{img:4tapfir}. +The \hs{init} function returns all but the last element of a vector, and the +concatenate operator ($\succ$) adds a new element to the left of a vector. The +resulting netlist of a 4-taps FIR filter, created by specializing the vectors of the above definition to a length of 4, is depicted in \Cref{img:4tapfir}. \begin{figure} \centerline{\includegraphics{4tapfir.svg}} @@ -1172,10 +1174,8 @@ is depicted in \Cref{img:4tapfir}. \label{img:4tapfir} \end{figure} - \subsection{Higher order CPU} - \begin{code} type FuState = State Word fu :: (a -> a -> a)