\input{caption} \makeatletter \long\def\unmarkedfootnote#1{{\long\def\@makefntext##1{##1}\footnotetext{#1}}} \makeatother \newcommand{\ignore}[1]{} \documentstyle[popl,picture]{article} \begin{document} \bibliographystyle{acm} \pagestyle{myheadings} \markboth {\mbox{Whelchel, Withers, O'Hallaron, and Lieu - DRAFT - \today\ }} {\mbox{Whelchel, Withers, O'Hallaron, and Lieu - DRAFT - \today\ }} \title{Computational Aspects of the Pipelined Phase-Rotation FFT} \author{ \begin{tabular}[c]{ c@{\extracolsep{2em}} c} Lang P. Withers, Jr., John E. Whelchel & David R. O'Hallaron, Peter J. Lieu \\ \\ {E-Systems, Inc., Melpar Div.} & {School of Computer Science}\\ {7700 Arlington Boulevard} & {Carnegie Mellon University}\\ {Falls Church, VA 22046} & {Pittsburgh, PA 15213} \end{tabular} } \date{} \maketitle{} \thispagestyle{empty} \unmarkedfootnote{ Supported in part by the Advanced Research Projects Agency, Information Science and Technology Office, under the title "Research on Parallel Computing," ARPA Order No. 7330. Work furnished in connection with this research is provided under prime contract MDA972--90--C--0035 issued by ARPA/CMO to Carnegie Mellon University, and in part by the Air Force Office of Scientific Research under Contract F49620--92--J--0131. %\vspace {1in} %ACM copyright notice } \begin{abstract} The phase-rotation FFT has a simple, constant-geometry, parallel-pipeline architecture. This paper considers computational aspects of implementing the FFT on a parallel computer system. Shuffle address and twiddle recipes, described directly in terms of the pipeline, are provided, and a recent full-bandwidth implementation on the iWarp parallel computer system is described. A new, parallel-pipeline equivalent of the index-reversing shuffle is added to complete the original phase-rotation FFT design. \end{abstract} \section {Introduction} The Fast Fourier Transform (FFT) is an important algorithm with many applications in signal processing and scientific computing. The Whelchel phase-rotation FFT \cite{whelchel90} is a pipelined version of the Pease constant-geometry FFT \cite{pease68}, which itself is derived from the original Cooley-Tukey FFT \cite{cooley65}. The phase-rotation FFT of radix $r$ is designed for a pipeline of $r$ parallel data channels. At each time step, in each stage, the pipeline carries the next $r$ data points, one from each channel, into a DFT kernel. Unlike earlier pipelined FFT's \cite{mcclellan78,corinthios71}, the phase-rotation FFT has the key property that {\em no data is switched across channels, except within the DFT kernel and at the input and output}. The phase-rotation FFT extends easily to higher radices, reducing memory and latency while preserving high throughput and parallel shuffling simplicity of lower radix versions. The phase-rotation FFT has also been extended to a vector-radix, multidimensional parallel-pipeline FFT with the same qualities of the one-dimensional algorithm, and without transposes \cite{withers91}. This paper reports some results from a project to implement the 1D phase-rotation FFT on a parallel computer system. There are three main results: \begin{itemize} \item The bit reversing shuffle step in the original version of the phase-rotation FFT \cite{whelchel90} is a potential pipeline bottleneck. We describe a new version that corrects this problem by using a parallel-pipeline bit-reversing step. \item While the structure of the phase-rotation FFT is extremely simple, we soon learned that generating the appropriate twiddles and shuffle indices from the original tensor algebraic formulation \cite{whelchel90} is quite difficult, even for the designers of the algorithm! To try to help the implementer, we have reformulated the phase-rotation FFT. Following \cite{vanloan92}, we present a new set of recipes, in a {\sc Matlab}-like format, for generating the twiddles and shuffle indices directly in terms of the parallel pipeline. \item Finally, we describe an implementation of the phase-rotation FFT on the iWarp, a private-memory multicomputer system developed by Intel and Carnegie Mellon \cite{iwarp,iwarpcomm}. In particular, we demonstrate that the 1D radix-2 phase-rotation FFT can run at the full 40 Mbytes/sec rate of the iWarp physical links. \end{itemize} Section~\ref{sec:basic} introduces the phase-rotation concept. Section~\ref{sec:phasedef} formally defines the improved FFT algorithm. Section~\ref{sec:recipes} gives the recipes for generating the twiddles and shuffle indices in terms of the parallel pipeline. Finally, Section~\ref{sec:implementation} describes the full-bandwidth implementation on iWarp. \section{The basic idea} \label{sec:basic} This section introduces the concept of the phase-rotation FFT. Starting with the Pease constant-geometry FFT, we informally derive the pipelined phase-rotation FFT, identifying the key insights along the way. \subsection{Constant-geometry FFT} Figure~\ref{fig:phase}(a) shows the flowgraph for a radix $r$ $N$-point decimation-in-frequency (DIF) constant-geometry FFT, with $r=2$ and $N=r^n=8$. There are $n$ stages. Each stage consists of $N/r$ {\em kernels}, each of which is an operator that inputs two complex numbers and outputs two complex numbers. \begin{figure} \centering \makebox{ \scaledpict 50mm by 29mm (phase scaled 1000) } \caption{Derivation of the phase-rotation FFT. {\rm (a) Initial constant-geometry FFT. (b) Pipelined constant geometry FFF. (c) Pipelined FFT based on cyclic rotations. (d) Phase-rotation FFT.} } \label{fig:phase} \end{figure} Each stage in the constant-geometry performs an identical perfect stride-by-$s$ shuffle of its data vector, where $N=rs$. If the data vector is regarded as an $s \times r$ array, then the perfect shuffle is a transpose operation. For example, the following transpose is a stride-by-4 perfect shuffle, for $N=8$ points and radix $r=2$: \begin{center} \begin{tabular} {l c c c} $\left[ \begin{array} {c c} 0 & 4 \\ 1 & 5 \\ 2 & 6 \\ 3 & 7 \end{array} \right]$ & $\stackrel{T}{\longrightarrow}$ & $\left[ \begin{array} {c c c c} 0 & 1 & 2 & 3 \\ 4 & 5 & 6 & 7 \end{array} \right]$\\ \end{tabular} \end{center} The data items, labeled by their indices in the original column vector, are regarded as equivalent to a $4 \times 2$ array composed by a stride-by-4 unstacking of the 8-point column vector. After the transpose, the $2 \times 4$ array is equivalent to a new 8-point column vector composed by a stride-by-2 stacking. As we shall see, this transpose creates difficulties when we try to pipeline the constant-geometry FFT. And it is precisely these difficulties that the phase-rotation FFT addresses. \subsection{Pipelining the FFT} The constant-geometry FFT can be pipelined by computing each stage on a single processor. For example, Figure~\ref{fig:phase}(b) shows the pipeline for a single stage. The pipeline consists of a sequence of operators connected by {\em pipeline segments}. Each pipeline segment consists of $r$ parallel {\em channels}. Each channel consists of a {\em stream} of $N/r$ data points, which are labeled in this example by their indices from the original column vectors in Figure~\ref{fig:phase}(a). For each pipeline segment, the $r$ data points in the same position in each stream are known as an $r$-{\em frame}, or simply, a {\em frame}. For example, in Figure~\ref{fig:phase}(b), the first frame in the pipeline segment between $S$ and $F$ is (0,4), the second frame is (1,5), and so on. Each time step, the twiddle operators ($W$) collectively read a frame, perform an element-wise complex multiplication, and write the resulting frame. Notice that each stream is operated on independently. Similarly, the kernel operator ($F$) reads a frame, computes the radix-$r$ kernel, and writes the resulting frame. However, in this case, the streams are not independent; each data item in the output frame is a function of every data point in the input frame. The twiddle and kernel operators pipeline nicely because during each time step they read and write a single frame. However, the pipelined shuffle operator ($S$) is less well behaved. It must read multiple input frames before it can begin to produce the first output frame. For example, for the radix-2 FFT in Figure~\ref{fig:phase}(b), the shuffle operator must read half the frames before it can begin producing any output frames. For the radix-4 FFT, the situation is worse, the shuffle operator must read {\em all} of the frames before it can produce a single output frame. The pipelined shuffle operator creates a number of problems for the implementer. In general the shuffle operator must read and store the entire pipeline segment before it can begin to produce any outputs. Since double buffering must be used to guarantee full bandwidth, each shuffle operator must have enough RAM to store $2N$ data points. Further, this memory must be $r$-ported. As the FFT radix increases, building such memories become increasingly expensive. Notice that the kernel operator suffers from a similar problem, but to a much lesser extent because it operates only on single frames. \subsection{The phase-rotation concept} The phase-rotation FFT attacks the problems caused by the perfect shuffle in the pipelined constant-geometry FFT. The idea is based on a simple factorization of the perfect shuffle operator. Once in this form, the DFT cyclic shift theorem can be applied to replace cyclic rotations with phasor multipliers, which are then merged into the twiddles. The result is a pipelined FFT where the only communication between streams occurs at each kernel and during the bit reversal at the end. Further, the communications that do exist only involve data points contained in the same frame. The rest of this section provides the intuition for these central ideas. When represented in matrix form, the shuffle operator $S$ in Figure~\ref{fig:phase}(b) can be factored into the double matrix product of the form \begin{equation} {\bf S} = {\bf C}_{fast} {\overline{\overline{\bf S}}} {\bf C}_{slow} \label{eqn:decomp} \end{equation} where ${\overline{\overline{\bf S}}}$ is a {\em parallel-pipeline shuffle} that shuffles independently along each input stream, and ${\bf C}_{slow}$ and ${\bf C}_{fast}$ are frame-wise (i.e., involving only data items in a single frame) cyclic rotations. We will formally define ${\overline{\overline{\bf S}}}$, ${\bf C}_{slow}$, and ${\bf C}_{fast}$ in Section~\ref{sec:phasedef}. For now, we will rely on the example in Figure~\ref{fig:decomp} to get the idea across. \begin{figure} \centering \makebox{ \begin{tabular} {l c c c} & $\left[ \begin{array} {c c c c} 0 & 2 & 4 & 6 \\ 1 & 3 & 5 & 7 \end{array} \right]$ & $\stackrel{{\bf S}}{\longrightarrow}$ & $\left[ \begin{array} {c c c c} 0 & 1 & 2 & 3 \\ 4 & 5 & 6 & 7 \end{array} \right]$ \\ & \\ & $\downarrow \; {\bf C}_{slow}$ & & $\uparrow \; {\bf C}_{fast}$ \\ & \\ & $\left[ \begin{array} {c c c c} 0 & 2 & 5 & 7 \\ 1 & 3 & 4 & 6 %% needs hand-drawn circles, squares \end{array} \right]$ & $\stackrel{ \overline{\overline{\bf S}} }{\longrightarrow}$ & $\left[ \begin{array} {c c c c} 0 & 5 & 2 & 7 \\ 4 & 1 & 6 & 3 %% needs hand-drawn circles, squares \end{array} \right]$ \end{tabular} } \caption{Interpretation of ${\bf S} = {\bf C}_{fast} {\overline{\overline{\bf S}}} {\bf C}_{slow}$} \label{fig:decomp} \end{figure} In this example, a pipeline segment is represented as a matrix. Each row in the matrix is a stream and each column is a frame. Notice that the operators in (\ref{eqn:decomp}) are applied from right to left, and not left to right. The first step in Figure~\ref{fig:decomp} is a cyclic rotation ${\bf C}_{slow}$, which rotates each frame. This rotation is frame-wise, in the sense that only data points contained in the same frame are rotated across the streams. Next, a parallel-pipeline shuffle permutes the data in each stream. Notice that no data points need to be transferred between streams in this step. The last step is a cyclic rotation ${\bf C}_{fast}$, which also performs a frame-wise rotation. If we apply (\ref{eqn:decomp}) to the pipelined FFT in Figure~\ref{fig:phase}(b), we get a pipelined FFT based on cyclic rotations, as shown in Figure~\ref{fig:phase}(c). The basic rotation in Figure~\ref{fig:phase}(c) that is applied at slowly varying, then fast-varying rates, is represented by the $r \times r$ cyclic (circular) shift permutation matrix ${\bf C}_{r}$, made by permuting the rows of the identity matrix down by one row, and moving the bottom row up to the top. \ignore{ For example, \begin{eqnarray*} {\bf C}_{4} = \left( \begin{array}{ c c c c } 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0 \end{array} \right)\;\;. \end{eqnarray*} }%end ignore The key insight of the phase-rotation FFT is that the cyclic shift theorem for the DFT can be applied to the cyclic shift operators in Figure~\ref{fig:phase}(c). In matrix form, the cyclic shift theorem for a DFT is the relation \begin{equation} {\bf F}_{r}{\bf C}_{r} = {\bf D}_{r}{\bf F}_{r}, \label{eqn:shifttheorem} \end{equation} where ${\bf D}_{r}= diag( 1, \omega, \omega^{2}, ..., \omega^{r-1} )$ is a set of twiddles, and the DFT matrix of size $r$ is \begin{displaymath} {\bf F}_{r} = \frac {1}{\sqrt{r}} ( \omega_{r}^{jk})_{j,k=0}^{r-1} , \end{displaymath} where $\omega = e^{-\frac{2 \pi i}{r}}$. For the pipelined FFT, (\ref{eqn:shifttheorem}) says that phasor multipliers after a DFT kernel give the same effect as physical data rotations before the DFT kernel. Likewise, physical rotations after the kernel are equivalent to phasor multipliers before it. This meaning of (\ref{eqn:shifttheorem}) is shown graphically in Figure~\ref{fig:theorem} for a pipelined radix-2 kernel. \begin{figure} \centering \makebox{ \scaledpict 50mm by 29mm (theorem scaled 1000) } \caption{Interpretation of ${\bf F}_{r}{\bf C}_{r} = {\bf D}_{r}{\bf F}_{r}$} \label{fig:theorem} \end{figure} The key point is that the cyclic rotations in Figure~\ref{fig:phase}(c) can be replaced by constant phasor multipliers. These phasors can then be absorbed by the twiddle factors on either side of the kernel, leaving only a parallel-pipeline shuffle. The result is the pipelined phase-rotation FFT, which is shown in Figure~\ref{fig:phase}(d). This completes our informal derivation of the phase-rotation FFT. The structure of the phase-rotation FFT in Figure~\ref{fig:phase}(d) is quite similar to the original pipelined FFT in Figure~\ref{fig:phase}(b). The twiddle operators ($W^\prime$) are identical to the original twiddle operators ($W$), except now the twiddles incorporate the original twiddles, phasors for the ${\bf C}_{fast}$ operator from the previous stage, and phasors for the ${\bf C}_{slow}$ operator from the next stage. The kernel operators are identical as well. The important difference is that the troublesome perfect shuffle operator has now been replaced by a parallel-pipeline shuffle that requires no communication across the streams. There are several other important things to note about the phase-rotation FFT in Figure~\ref{fig:phase}(d). First, there are no additional multiplications or additions, compared to the original pipelined FFT. Second, the only communication across streams occurs at the kernel, and this communication is constrained, in the sense that only data points within a single frame need to be switched across channels. \section{Improved phase-rotation FFT} \label{sec:phasedef} In this section we give a formal definition of an improved version of the original phase-rotation FFT \cite{whelchel90}. The new version replaces the bit reversing permutation in the original phase rotation FFT with a parallel-pipeline shuffle, followed by a frame-wise cyclic rotation. The advantage of this new approach is that during the bit reversal step at the end, all communication between streams is limited to data points within a single frame. For radix $r$ and $N=r^n$ points ($n>1$), the 1-dimensional phase-rotation FFT is a matrix factorization of the $N$-point DFT matrix ${\bf F}_{N}$. Starting with the Pease constant-geometry factorization, we replace its perfect shuffles ${\bf S}$ by ${\bf S} = {\bf C}_{fast} \overline{\overline{{\bf S}}} {\bf C}_{slow}$. Similarly, at the left end we replace the radix-$r$ index-digit-reversing permutation ${\bf Q} = {\bf Q}_{N,r}$ of $N$ data points by ${\bf Q} = {\bf C}_{slow}^T \overline{\overline{\bf Q}}{\bf C}_{slow}$, where $\overline{\overline{\bf Q}}$ is another parallel-pipeline shuffle that will be defined formally in Section~\ref{sec:recipes}. The phase-rotation FFT is then defined by: \begin{eqnarray} {\bf F}_{N} &=& {\bf Q} \cdot \prod_{j=1}^{n} \left( {\bf F}{\bf S}{\bf T}_{j} \right) = \cdots \left( \mbox{ \small \begin{tabular}[c]{c} \small vigorous\\ \small algebraic\\ \small shuffling \end{tabular} } \right) \cdots \nonumber\\ %&=& %\left( \prod_{j=1}^{n-1} {\bf F} %{\bf C}_{fast} \overline{\overline{{\bf S}}} % {\bf C}_{slow} {\bf T}_{j} \right) {\bf F}{\bf S}{\bf Q} % = \cdots \nonumber\\ &=& {\bf C}_{slow}^T \cdot \overline{\overline{\bf Q}}{\bf D}_{fast}^{\prime} \left[ \prod_{j=1}^{n} \left( {\bf F} \overline{\overline{{\bf S}}}{\bf D}_{j}^{\prime} \right) \right] \cdot {\bf C}_{slow} . \label{eqn:phase} \end{eqnarray} % As before, we let $s = N/r$ and $r^{\prime} = N/r^2$. ${\bf F}$ is a direct (tensor, Kronecker) product ${\bf I}_{s} \otimes {\bf F}_{r} = \mbox{diag}({\bf F}_{r}, {\bf F}_{r}, ...,{\bf F}_{r})$. We interpret this as a kernel DFT ${\bf F}_{r}$ operating on $s$ successive frames of $r$ points placed in the pipeline. For $j=1:n$, the other parts of (\ref{eqn:phase}) are defined by \begin{eqnarray} {\bf C}_{slow} &=& \bigoplus_{k=0}^{r-1} \left( {\bf I}_{r^{\prime}} \otimes {\bf C}_{r}^{k} \right ) \nonumber\\ {\bf C}_{fast} &=& {\bf I}_{r^{\prime}} \otimes \left( \bigoplus_{k=0}^{r-1} ({\bf C}_{r}^{T})^{k} \right ) \nonumber\\ \omega_j &=& exp \left({-\frac{2 \pi i}{r^{j}}} \right) \nonumber\\ {\bf D}_{r} &=& diag( 1, \omega_{1}, \omega_{1}^{2}, ..., \omega_{1}^{r-1} ) \nonumber\\ {\bf D}_{r^{j+1}} &=& diag( 1, \omega_{j+1}, \omega_{j+1}^{2}, ..., \omega_{j+1}^{r^{j}-1} ) \nonumber\\ {\bf D}_{slow}^{-1} &=& \bigoplus_{k=0}^{r-1} \left( {\bf I}_{r^{\prime}} \otimes {\bf D}_{r}^{-k} \right) \nonumber\\ {\bf D}_{slow}^{\prime} &=& {\bf C}_{fast}^T {\bf D}_{slow}^{-1} {\bf C}_{fast} \nonumber\\ {\bf D}_{slow}^{\prime \prime} &=& \overline{\overline{{\bf S}}}^T {\bf D}_{slow}^{\prime} \overline{\overline{{\bf S}}} \nonumber\\ {\bf D}_{fast}^{-1} &=& {\bf I}_{r^{\prime}} \otimes \left( \bigoplus_{k=0}^{r-1} {\bf D}_{r}^{-k} \right) \nonumber\\ {\bf D}_{fast}^{\prime} &=& {\bf C}_{slow}^T {\bf D}_{fast}^{-1} {\bf C}_{slow} \nonumber\\ {\tilde{\bf T}}_{j} &=& {\bf I}_{ \frac{N}{r^{(j+1)}} } \otimes \left( \bigoplus_{k=0}^{r-1}{\bf D}_{r^{j+1}}^{k} \right) \nonumber\\ % {\bf T}_{j} &=& {\bf S}^{j\;T}{\tilde{\bf T}}_{j}{\bf S}^{j} \nonumber\\ {\bf T}_{j}^{\prime} &=& {\bf C}_{slow} {\bf T}_{j} {\bf C}_{slow}^T \nonumber\\ {\bf D}_{1}^{\prime} &=& \left( \overline{\overline{{\bf S}}}^T {\bf D}_{slow}^{-1} \overline{\overline{{\bf S}}} \right) \cdot {\bf T}_{1}^{\prime}{\bf D}_{fast}^{-1} \nonumber\\ {\bf D}_{j}^{\prime} &=& {\bf D}_{slow}^{\prime \prime} {\bf T}_{j}^{\prime} {\bf D}_{fast}^{-1} \;\;, \;\;\;\; j=2:n-1 \nonumber\\ {\bf D}_{n}^{\prime} &=& {\bf D}_{slow}^{\prime \prime} {\bf T}_{n}^{\prime} = {\bf D}_{slow}^{\prime \prime} \;\;. \label{eqn:phasederivation} \end{eqnarray} The direct sums are of the form \begin{displaymath} \bigoplus_{k=0}^{r-1} {\bf A}_{k} = \mbox{diag}( {\bf A}_{0}, {\bf A}_{1},..., {\bf A}_{r-1} ). \end{displaymath} See \cite{withers91} for more on the basic definitions and relations used to derive (\ref{eqn:phase}), as well as the generalization to higher dimension FFT's. Note that the stages in (\ref{eqn:phase}) are counted in reverse time order by the index $j$. This is in keeping with the fact that (\ref{eqn:phase}) is a decimation-in-frequency (DIF) version of the FFT. The transpose of (\ref{eqn:phase}), with the product $\prod_{j=n}^{1}$, is the decimation-in-time (DIT) version of the phase-rotation FFT. A ${\bf C}_{slow}$ shuffle and its inverse remain at the input and output ends of the pipeline, respectively. As we have seen, ${\bf C}_{slow}$ is a completely frame-wise rotation. It rotates (commutates) the data within each successive frame (column $r$-vector) of the $r \times s$ pipeline segment for a stage. There is also an implicit frame-wise broadcast within each FFT kernel engine, when an $r$-point DFT is somehow computed. So in the phase-rotation FFT, data motion is all parallel, except for frame-wise motions at I/O and at every FFT kernel. The simplicity of the phase-rotation FFT is that no data point ever moves both down and across the pipeline in one time-step. \section{Pipeline recipes} \label{sec:recipes} While the structure of the pipelined phase-rotation FFT is extremely simple, experience has taught us that generating the appropriate twiddles and shuffle indices from the matrix formulations of (\ref{eqn:phase}) and (\ref{eqn:phasederivation}) is difficult and confusing. To address this problem, we have developed a collection of recipes, in a {\sc matlab}-like format, for generating the phase-rotation twiddles and shuffle indices off-line. The recipes are defined for any 1D phase-rotation FFT with a power of two radix. As we saw in (\ref{eqn:phase}), the pipelined phase-rotation FFT performs a typical ``twiddle, shuffle, kernel'' cycle at each stage. Only the twiddles vary from stage to stage, and there is a digit-reversing shuffle at the end. To implement this FFT using parallel $r \times s$ pipeline segments (one per stage), we insert the $N$-vector of input data ${\bf x}$ into the pipeline as an $r \times s$ array $X$: the first $r$ points of ${\bf x}$ go into the first frame (column) $X$, the second $r$ points go into the second frame, and so on. We must also have a shuffle address and a twiddle factor ready for each point in the pipeline. In other words, we would like to fill one $r \times s$ copy $A$ of the pipeline segment with addresses, and another copy $D$ with twiddles. With this information in hand, the behavior of the processors that execute each stage, ${\bf F} \overline{\overline{{\bf S}}}{\bf D}_{j}^{\prime}$, in the pipeline is completely determined. Using the current frame of addresses, each processor will fetch the current $r$-frame of data $X(0$:$r-1, A(0$:$r-1,t))$, and the current $r$-frame of twiddles $D(0$:$r-1, A(0$:$r-1,t))$ (pointwise in parallel), multiply these two frames pointwise, and then do an $r$-point DFT ${\bf F}_r$ of the twiddled data frame. The twiddle and shuffle in this section are recipes are ``in place'' in the sense that they work inside the $r \times s$ pipeline segments that will contain the desired addresses and twiddles. They are not ``in place'' in the usual sense, as we will freely use an input and an output copy of a pipeline segment. This approach avoids constructing and operating with large $N \times N$ matrices (each containing only $N$ non-zero elements). Each parallel-pipeline function recipe is given a name similar to that of the $N \times N$ matrix factor in the FFT (\ref{eqn:phase}) that it effectively implements. \subsection{Shuffle recipes} As a convention, pipeline addresses (pipeline array row and column indices) run 0:$r-1$ and 0:$s-1$, respectively. To do parallel-pipeline shuffles, we only need the horizontal (column) addresses, since the data inside each pipe will only jump within that stream (row). The cross-stream shuffles, Cslow and Cfast, are implemented using $\pi_{r}$, a cyclic rotation of a frame (a vertical slice of the parallel pipeline) that has the effect of ${\bf y}_r = {\bf C}_r{\bf x}_r$. $\pi_{r}$ takes a column $r$-vector ${\bf x}_r = (x_0, x_1, x_2, ..., x_{r-1})^T \mapsto {\bf y}_r = (x_{r-1}, x_0, x_1, ..., x_{r-2})^T$. \begin{tabbing} ttttabbb \=tab\=tabbb \=tabbb1234512345123451234512\= \kill \> {\bf function} $Y$ = Cslow$( X )$ \\ \> $col = 0$\\ \> for $k = 1:r$\\ \> \> for $j = 1:r^{\prime}$\\ \> \> \> $Y(:,col) = \pi_{r}^{k} (X(:,col))$\\ \> \> \> $col = col + 1$\\ \> \> end \\ \> end \end{tabbing} \begin{tabbing} ttttabbb \=tab\=tabbb \=tabbb1234512345123451234512\= \kill \> {\bf function} $Y$ = Cfast$( X )$ \\ \> $col = 0$\\ \> for $j = 1:r$\\ \> \> for $k = 1:r^{\prime}$\\ \> \> \> $Y(:,col) = \pi_{r}^{k} (X(:,col))$\\ \> \> \> $col = col + 1$\\ \> \> end \\ \> end \end{tabbing} The inverses of Cslow and Cfast are formed by simply reversing $\pi_{r}$. Next, we define some perfect shuffles. \begin{tabbing} ttttabbb \=tab\=tabbb \=tabbb1234512345123451234512\= \kill \> {\bf function} $Y$ = S$( X ) \;\;$ !stride by $s$\\ \> $col = 0$\\ \> for $row = 0:r-1$\\ \> \> for $k1 = 0:r:s-r$\\ \> \> \> $k2 = k1+r-1$\\ \> \> \> $Y(row,k1:k2) = X(:,col)$\\ \> \> \> $col = col + 1$\\ \> \> end \\ \> end \end{tabbing} \begin{tabbing} ttttabbb \=tab\=tabbb \=tabbb1234512345123451234512\= \kill \> {\bf function} $Y$ = S$^{-1}( X )\;\;$ !stride by $r$\\ \> $col = 0$\\ \> for $row = 0:r-1$\\ \> \> for $k1 = 0:r:s-r$\\ \> \> \> $k2 = k1+r-1$\\ \> \> \> $Y(:,col) = X(row,k1:k2)$\\ \> \> \> $col = col + 1$\\ \> \> end \\ \> end \end{tabbing} To implement the parallel-pipeline shuffles, $\overline{\overline{{\mbox{S}}}}$, $\overline{\overline{{\mbox{S}}}}^{-1}$, and $\overline{\overline{{\mbox{Q}}}}$, we will use the parallel-pipeline addresses $A$, which are computed by the following function: \begin{tabbing} ttttabbb \=tab\=tabbb \=tabbb1234512345123451234512\= \kill \> {\bf function} $A$ = $\overline{\overline{\mbox{S}}}$\_addresses$(r,s)$ \\ \> ${\bf a} = ( 0, r^{\prime}, ..., (r-1)r^{\prime})^T$ \> \> \>\\ \> $col = 0$\\ \> for $j = 1:r^{\prime}$\\ \> \> for $k = 1:r$\\ \> \> \> $A(:,col) = {\bf a}$\\ \> \> \> $col = col + 1$\\ \> \> \> ${\bf a} = \pi_{r}({\bf a})$ \> \\ \> \> end \\ \> \> ${\bf a} = {\bf a} + {\bf 1}_{r}$ \\ %% \;\;\;(mod\;\;\; r)$ \> \> \\ \> end \end{tabbing} Looking closely, one can see Cfast$^{-1}$ at work producing the addresses $A$ in the last function. The addresses $A$ can also be generated by loading a pipeline segment with simple $r \times r$ address blocks $B_{rr}$, and then applying Cfast$^{-1}$ to the pipeline segment. The first block to load is $B_{rr} =$ diag$(0$:$r^{\prime}$:$s-1)*{\bf 1}_{rr}$, where ${\bf 1}_{rr}$ is the $r \times r$ matrix whose entries are all 1's. The next block is always $B_{rr} = B_{rr} + {\bf 1}_{rr}$, until the pipeline segment contains $r^{\prime}$ blocks and is full. \begin{tabbing} ttttabbb \=tab\=tabbb \=tabbb1234512345123451234512\= \kill \> {\bf function} $Y = \overline{\overline{{\mbox{S}}}}( X )$ \\ \> $A$ = $\overline{\overline{{\mbox{S}}}}$\_addresses$(r,s)$ \\ \> for $row = 0:r-1$\\ \> \> $Y(row,:) = X(row,A(row,:))$ \\ \> end \end{tabbing} \begin{tabbing} ttttabbb \=tab\=tabbb \=tabbb1234512345123451234512\= \kill \> {\bf function} $Y = \overline{\overline{\mbox{S}}}^{-1}( X )$ \\ \> $A$ = $\overline{\overline{{\mbox{S}}}}$\_addresses$(r,s)$ \\ \> $[AA,I] =$ sort($A$) \\ \> for $row = 0:r-1$\\ \> \> $Y(row,:) = X(row,I(row,:))$ \\ \> end \end{tabbing} In the above functions, sort($A$) sorts each row of an array $A$ in ascending order. It returns the row-sorted array $AA$ and the corresponding array of addresses $I$ where the successive row elements were found in $A$. After we have sorted the addresses $A$ for $\overline{\overline{{\mbox{S}}}}$, $I$ has the addresses for $\overline{\overline{\mbox{S}}}^{-1}$. The pipeline addresses for $\overline{\overline{\mbox{Q}}}$ are obtained by block-perfect shuffles (along the length of the pipeline) of the addresses for $\overline{\overline{\mbox{S}}}$: \begin{tabbing} ttttabbb \=tab\=tab\=tab\=tab\=tab\=12345123451234512\= \kill \> {\bf function} $Y = \overline{\overline{\mbox{Q}}}( X, n )$ \\ \> $A$ = $\overline{\overline{{\mbox{S}}}}$\_addresses$(r,s)$ \\ \\ \> if $n > 2$ \\ \> \> for $ns = (n-2):-1:1 \;$ \\ %% !perfect shuffle index exponent \> \> \> $stride= r^ns$\\ \> \> \> $block = r^{n-2-ns}\;\;\;$ ! block length \\ \> \> \> $col2=0$\\ \> \> \> for $k_1=1:stride$\\ \> \> \> \> $col1 = (k_1-1)*block$\\ \> \> \> \> for $k=1:r$\\ \> \> \> \> \> for $j=1:block$\\ \> \> \> \> \> \> $B(:,col2)=A(:,col1)$\\ \> \> \> \> \> \> $col1 = col1 + 1$\\ \> \> \> \> \> \> $col2 = col2 + 1$\\ \> \> \> \> \> end\\ \> \> \> \> \> $col1 = col1 + (stride-1)*block$\\ \> \> \> \> end\\ \> \> \> end\\ \> \> \> $A=B$\\ \> \> end\\ \> end \\ \\ \> for $row = 0:r-1$\\ \> \> $Y(row,:) = X(row,A(row,:))$ \\ \> end \end{tabbing} \subsection{Twiddle recipes} Every twiddle matrix ${\bf D}$ is diagonal, so it operates on a data vector as a point-to-point vector multiply. Given some permutation matrix ${\bf P}$, a new twiddle matrix ${\bf PDP}^T$ is equivalent to a rediagonalizing of the vector shuffle of the diagonal of ${\bf D}$, that is, ${\bf PDP}^T$ = diag$({\bf P}*$diag$({\bf D}))$. (This is a {\sc Matlab} notation: diag() puts the diagonal of a matrix in a vector, and puts a vector in the diagonal of a matrix.) Since we want to perform shuffles within pipeline arrays, we reshape the twiddle $N$-vector diag$({\bf D})$ as an $r \times s$ pipeline array $D$, just as we originally reshaped the data vector. Then we shuffle the pipelined twiddles, to effect the equivalent of the vector shuffle ${\bf P}*$diag$({\bf D})$. So we interpret the ${\bf PDP}^T$ operator as an in-pipeline shuffle of the pipelined twiddles $D$, which are then in position to operate on the pipelined data $X$ directly by point-to-point multiplication, $Y = D*X$. (As mentioned, the data will actually be twiddled frame-by-frame in the pipelined implementation.) We will interpret the twiddles expressed in (\ref{eqn:phasederivation}) this way. Each twiddle function below returns an $r \times s$ array $D$ of twiddle factors (the actual twiddling of the data is not included): \begin{tabbing} ttttabbb \=tab\=tabbb \=tabbb1234512345123451234512\= \kill \> {\bf function} $D_{slow}$ = Dslow\_twiddles$( r, s )$ \\ \> $\omega_{j} = exp(-2\pi i /r)$\\ \> $t = 0$\\ \> for $j = 0:(r-1)$\\ \> \> for $k = 0:(r^{\prime}-1)$\\ \> \> \> $D_{slow}(:,t) = (1, \omega_r^k, \omega_r^{2k}, ..., \omega_r^{(r-1)k})^T$ \\ \> \> \> $t = t + 1$\\ \> \> end \\ \> end \end{tabbing} \begin{tabbing} ttttabbb \=tab\=tabbb \=tabbb1234512345123451234512\= \kill \> {\bf function} $D_{fast}$ = Dfast\_twiddles$( r, s )$ \\ \> $\omega_{j} = exp(-2\pi i /r)$\\ \> $t = 0$\\ \> for $k = 0:(r^{\prime}-1)$\\ \> \> for $j = 0:(r-1)$\\ \> \> \> $D_{fast}(:,t) = (1, \omega_r^j, \omega_r^{2j}, ..., \omega_r^{(r-1)j})^T$ \\ \> \> \> $t = t + 1$\\ \> \> end \\ \> end \end{tabbing} The inverses of $D_{slow}$ and $D_{fast}$ are just their complex conjugates, and are generated simply by replacing $\omega_{j}$ by $\omega_{j}^{-1}$. For stages $j = 1$:$n$ (counted down from $n$), we generate pipelined twiddles $\tilde{T}_j$ by \begin{tabbing} ttttabbb \=tab\=tab\=tab\=1234512345123451234512\= \kill \> {\bf function} $\tilde{T}_j$ = $\tilde{\mbox{T}}$\_twiddles$( r, s, j )$ \\ \> $\omega_{j} = exp(2\pi i /r^{j+1})$\\ \> $\omega_{j}^{\prime} = \omega_{r^{j+1}}^r$\\ \> for $k = 0:(r-1) \;\;$ ! direct sum loop\\ \> \> $t_1 = k \cdot r^{j-1}$\\ \> \> for $p = 0:(r-1)$\\ \> \> \> $\tilde{T}_j(p,t_1) = \omega_{j}^{kp} $\\ \> \> end \\ \> \> $t_1 = t_1 + 1$\\ \> \> $t_2 = t_1 + r^{j-1}$\\ \> \> for $t = t_1:t_2\;\;$ ! fill next column from last \\ \> \> \> $\tilde{T}_j(:,t) = \omega_{j}^{\prime k} \cdot \tilde{T}_j(:,t-1)$ \\ \> \> end \\ \> end\\ \\ \> if $j < n$ \\ \> \> $t_2 = r^j$\\ \> \> for $k = 0:(N/r^{j+1}) \;\;$ \\ \> \> \> $t_1 = t_2$\\ \> \> \> $t_2 = k \cdot r^j$\\ \> \> \> $t = 0$\\ \> \> \> for $t_0 = t_1:t_2$\\ \> \> \> \>$\tilde{T}_j(:,t_0) = \tilde{T}_j(:,t)\;\;$ ! copy columns\\ \> \> \> \>$t = t + 1$\\ \> \> \> end \\ \> \> end\\ \> end \end{tabbing} The rest of the twiddle arrays can now be defined in terms of the shuffles: \begin{tabbing} ttttabbb \=tab\=tabbbb\=tabbb \=tabbb1234512345123451234512\= \kill \> $D_{slow}^{\prime}$ \> \> = S$^{-1}( D_{slow}^{-1} )$ \\ \> $D_{slow}^{\prime \prime}$ \> \> = Cslow$( D_{slow}^{\prime} )$ \\ \\ \> $D_{fast}^{\prime}$ \> \> = Cslow$( D_{fast}^{-1} )$ \\ \\ \> $T_j$ \> \> = S$^{-1}( \tilde{T}_j )$ \\ \> $T_j^{\prime}$ \> \> = Cslow$( T_j )$ \\ \\ \> $D_1^{\prime}$ \> = $\overline{\overline{\mbox{S}}}(D_{slow}^{-1}) .* T_1^{\prime}.*D_{fast}^{-1}$\\ \> if $1 \> $D_j^{\prime}$ \> = $D_{slow}^{\prime \prime}.* T_j^{\prime}.*D_{fast}^{-1}$\\ \> end \\ \> $D_n^{\prime}$ \> = $D_{slow}^{\prime \prime}$ \end{tabbing} \section{Implementation issues} \label{sec:implementation} In this section we describe the issues that arise when the phase-rotation FFT is implemented on a parallel system. In particular, we describe implementation approaches for the radix-2 FFT on the iWarp system. The main result is a scalable implementation of the phase-rotation FFT that runs at the full 40 Mbytes/second rate of the iWarp physical links. \subsection{iWarp} The iWarp is a private-memory multicomputer developed jointly by Carnegie Mellon and Intel Corporation \cite{iwarp,iwarpcomm}. iWarp systems are 2-dimensional tori of iWarp nodes, ranging in size from 4 to 1024 node. Each node consists of an iWarp {\em component}, up to 16 Mbytes of off-chip local memory, and a set of 8 unidirectional communication {\em links} that physically connect the node to four neighboring nodes. The iWarp component is a VLSI chip that contains a {\em processing agent} and a {\em communication agent}. The processing agent is a general-purpose load-store microprocessor, centered around a $32 \times 128$-bit register file, that runs at a maximum rate of 20 MFLOPS. The local memory is accessed at a rate of 160 Mbytes/sec. Each link runs at 40 Mbytes/sec, for a maximum aggregate bandwidth of 320 Mbytes/sec per node. The key feature of the iWarp is its communication system, which is summarized in Figure~\ref{fig:pathways}. Each communication agent contains a set of 20 hardware FIFO {\em queues}. Each queue can hold up to 8 32-bit words. iWarp nodes communicate with other nodes using unidirectional point-to-point structures called {\em pathways}. Each pathway is a sequence of queues that can be defined dynamically at runtime. Figure~\ref{fig:pathways} shows a pair of such pathways. \begin{figure} \centering \makebox{ \psfig{figure=pathways.eps} } \caption{iWarp communication structures.} \label{fig:pathways} \end{figure} Data traveling along a pathway passes from queue to queue {\em automatically}, without disturbing the computations on intermediate nodes. For example, in Figure~\ref{fig:pathways}, data traveling over the pair of pathways does not disturb the computation on node 1. The latency from queue to queue is small, ranging from 100-300 nanoseconds. Multiple pathways can share the same link. For example, in Figure~\ref{fig:pathways}, two pathways share the link from node 1 to node 2. In this case, the pathways share the link bandwidth in a round-robin fashion, one word at a time. If only one pathway is sending data over a link link, then it gets the entire link bandwidth. If multiple pathways are sending data over a link, then the link is utilized at the full 40 Mbytes/sec, and each pathway is guaranteed a proportional fraction of the bandwidth. User programs can directly access the queues, one word at a time, by reading and writing special registers in the register file called {\em gates}. To an iWarp instruction, a gate is just another register in the register file. The important point is that a program can read or write a word in a queue with the latency of a register access. A single instruction can read and write up to 4 words from queues, with a maximum aggregate bandwidth of 160 Mbytes/sec. Gates are accessed from user-level C programs using \verb.send_word(). and \verb.recv_word(). primitives. \subsection{Mapping strategies on iWarp} The problem is to develop a mapping of the flowgraph in Figure~\ref{fig:phase}(d) to an iWarp array. The objective is a pipelined radix-2 FFT that runs at the maximum 40 Mbytes/sec rate of an iWarp communication link. We will also consider a mapping that uses fewer nodes per stage, at the price of lower bandwidth. In the final paper, we plan to treat the radix-4 case as well. The simplest strategy is to assign each flowgraph node to a unique processor node of a linear array, route the flowgraph arcs through this array, and then embed the resulting linear array in the iWarp torus. This approach, called the PHASE5 mapping because it uses 5 iWarp nodes for each FFT stage, is shown in Figure~\ref{fig:mappings}(a). \begin{figure} \centering \makebox{ \psfig{figure=mappings.eps} } \caption{Mapping strategies.} \label{fig:mappings} \end{figure} Each iWarp node in PHASE5 executes a small program that implements its flowgraph operator. Each twiddle node ($W^\prime$) repeatedly reads a complex number from its input pathway, multiplies by the appropriate twiddle (precomputed off-line using the recipes in Section~\ref{sec:recipes}), and sends the result to its output pathway. Each shuffle operator ($\overline{\overline{S}}$) repeatedly reads a complex data item from its input pathway, stores it in memory, and uses the appropriate shuffle index (again precomputed off-line using the recipes in Section~\ref{sec:recipes}) to send an appropriate double-buffered data point to the output pathway. The kernel node ($F$) repeatedly reads two complex numbers from its input pathways, performs the radix-2 DFT kernel operation, and outputs two complex numbers to its output pathways. While the details are beyond the scope of this paper, each node program in the PHASE5 mapping performs each of its iterations in at most 8 clocks. At its peak rate of 40 Mbytes/sec, each link can produce and consume a 32-bit floating point number every 2 clocks. Further, each data point in the pipeline is a complex number consisting of 2 32-bit floating point words. {\em As a result, each pathway in the PHASE5 mapping requires exactly half of the available link bandwidth}. Since each link is shared by two pathways, and since the iWarp communication agent gives each pathway an equal share of the link bandwidth, without disturbing the computations on intermediate nodes, each link is fully utilized. The result is a radix-2 FFT that runs at the full 40 Mbytes/sec rate of an iWarp link, regardless of the number of points in the FFT! The computational throughput is 8 MFLOPS per node, again independent of the size of the FFT. A potential disadvantage of the PHASE5 mapping is that it is node-inefficient, requiring 5 iWarp nodes for each FFT stage, plus an additional 5 nodes for the bit reversal at the end. Another approach, the PHASE3 mapping, combines the twiddle and shuffle operators on a single node, as shown in Figure~\ref{fig:mappings}(b), so that each stage requires 3 nodes instead of 5 nodes. Each node program in the PHASE3 mapping executes in at most 11 clocks per iteration, for a communication throughput of 30 Mbytes/sec, and a computational throughput of 10 MFLOPS per node. So on the iWarp, the PHASE3 mapping trades off higher computational throughput for lower communication throughput. \subsection{Example FFT on iWarp} Figure~\ref{fig:example} shows a working implementation of a 2K-point radix-2 phase-rotation FFT running on a 64-node iWarp array at Carnegie Mellon. The large squares are iWarp nodes, labeled with the corresponding operator and stage number. The bit reversal at the end is treated as another stage. The small squares are queues. The arrows are iWarp pathways. The implementation uses a PHASE5 mapping. Each of the 11 FFT stages uses 5 nodes, with an additional 5 nodes for the bit-reversing step at the end, for a total of 60 nodes. The measured communication throughput of this FFT on the iWarp hardware is 40 Mbytes/second (5 Msamples/sec). The average computational throughput for each node is 8 MFLOPS, for an aggregate computational throughput of $8 \cdot 60 = 480$ MFLOPS. \begin{figure} \centering \makebox{ \psfig{figure=example.eps} } \caption{2K phase-rotation FFT on iWarp} \label{fig:example} \end{figure} \section{Concluding remarks} \section*{Acknowledgements} Thanks to thank Tom Warfel and LeeAnn Tzeng for their help with the iWarp implementation, and Doug Noll and Doug Smith for discussions that led to the node-efficient mapping. \small \bibliography{/afs/cs/project/iwarp/member/droh/bib/compiler,/afs/cs/project/iwarp/member/droh/bib/defs,/afs/cs/project/iwarp/member/droh/bib/me,/afs/cs/project/iwarp/member/droh/bib/refs} \end{document}