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Mathematics and Its Applications

Systolic processing elements before it returns to memory much system is easy to implement because of its regularity as blood circulates to and from the heart. This can be and easy to reconfigure.. This architecture can result in rectangular, triangular or hexagonal to make use of cost-effective, high-performance performance special-purpose special higher degrees of parallelism. Moreover to implement systems for a wide range of problems.

Some simple a variety of computations, mputations, data flow in a systolic examples of systolic array models are shown in the system may be at multiple speeds in multiple Fig. The Fig. Each cell Systolic Array. The systolic array is often The restt of the work in the paper is arranged as; rectangular where data flows across the array between Section 2 briefs out the basic knowledge of systolic neighbor DPUs, often with different data flowing in array and its architecture.

Section 3 explains the differentt directions. Systolic architectures, an implementation of systolic array using evolutionary architectural concept originally proposed for VLSI computation; also define the meaning of evolutionary implementation of some matrix operations.

A systolic computation utation and how we can use it in our design. As array, there we have taken example of 3 tap FIR filter simple, regular egular communication and controlled and 3x3 matrix multiplication to understand the structures have substantial advantages over design. Section 5 we have discussed results which are complicated ones in design and implementation; cells obtained btained after implementing the systolic array for FIR in a systolic system are typically interconnected to filter and matrix multiplication ltiplication using MATLAB tool.

Information in systolic systemss flows between cells in a pipelined 2. Each processor at each be achieved without increasing memory bandwidth step takes in data from one or more neighbor neighbors, Being able to use each input data item a number of processes it and, in the next step, outputs results in the times and thus achieving high computation opposite direction.

Kung and Charles Leiserson throughput with only modest memory bandwidth is were the first to publish a paper on systolic arrays in one of the many advantages of systolic approach. Systolic array is a specialized form of parallel computing with multiple 3. Each unit is an Evolutionary computation uses computational models independent processor. Cells processors compute of evolutionary processes as key elements in the data and store it independently of each other.

Every design and implementation of computer-based computer processor has some registers and an ALU [3]. There are a variety of evolutionary computational models that have been proposed and studied which we will refer to as a evolutionary algorithms. They share a common conceptual base of simulating the evolution of individual structures via processes of selection and reproduction. These processes depend on the perceived performance fitness of o the individual structures as defined by an environment.

More b c precisely, evolutionary algorithms maintain a Fig. Selection focuses attention on high generation. The process of generating new trials and selecting fitness information. Recombination and mutation those with least error continues until a sufficient perturb those individuals, providing general heuristics solution is reached or the available computation is for exploration. Systolic array architectures are designed by using linear mapping techniques on regular dependence graphs DG.

DECOMPOSER: A synthesizer for systolic systems. — Penn State

A DG is said to be regular, if a presence of an edge in a certain direction at any node in the DG represents presence of an edge in the same direction at all nodes in the DG[4]. Basic vectors involved in the systolic array design are: 1. A population of individual structures is initialized and then evolved from generation to 2. The executed by processor in space time representation population size N is generally constant in an by: evolutionary algorithm, although there is no a priori i reason other than convenience to make this assumption.

P is selected at random from a feasible range in 4. Hardware Utilization Efficiency each dimension. This is because two tasks executed by the same 4. These vectors must mapping is done. If points A and B differ by the projection vector, then an edge PT e is introduced in a systolic array i.


Once edge mapping is done, then executed by the same processor. In other words: we construct low-level implementation of the particular design. The 2. If A and B are mapped to the same processor, then low level architecture for the systolic array is shown they cannot be executed at the same time. The Low level architecture for the above design II is Fig. The coordinates for the FIR filter has been calculated from the basic vectors in the The Low level architecture for the above FIR filter is systolic array and finally the Fig.

The coordinates for the FIR representation for the considered design I have been filter has been calculated from the basic vectors in the presented. Low level architecture for systolic array for Fig. The Design 2 deals with Fan-in results, Move inputs 6. The general scheme of the computation can be viewed as follows. The y i , which are initially zero, are pumped to the left while the x i are pumped to the right and the a ij are marching down. All the moves are synchronized. Note that when y 1 and y 2 are output they have the correct values. Observe also that at any given time alternate processors are idle.

Specifying the operation of the systolic array more precisely, we assume that the processors are numbered by integers, 1, 2,. Each processor has three registers, R A , R x and R y , which will hold entries in A, x and y, respectively. Initially, all registers contain zeros. Each pulsation of the systolic array consists of the following operations, but for odd numbered pulses only odd numbered processors are activated and for even numbered pulses only even numbered processors are activated. The R x in processor 1 gets a new component of x.

Processor 1 outputs its R y contents and the R y in processor w gets zero. Using the square type inner product step processor illustrated in FIG. A second problem which is ideally solved by our invention is that of multiplying two n x n matrices. Let A and B be nxn band matrices of band width w 1 and w 2 , respectively.

We will show how the recurrences above can be evaluated by pipelining the a ij , b ij and c ij through a systolic array having w 1 w 2 hex-connected inner product step processors. We illustrate the general scheme by considering the matrix multiplication problem depicted in FIG. The diamond shaped systolic array for this case is shown in FIG. The elements in the bands of A, B and C are pumped through the systolic network in three directions synchronously.

Each c ij is initialized to zero as it enters the network through the bottom boundaries. One can easily see that with the hexagonal inner product processors illustrated at FIGS. Note that in any row or column of the network, out of every three consecutive processors, only one is active at given time. Another problem ideally solved by the systolic array system of our invention is that of factoring a matrix A into lower and upper triangular matrices L and U. We deal with the latter problem hereafter. In FIG. We assume that matrix A has the property that its LU-decomposition can be done by Gaussian elimination without pivoting.

This is true, for example, when A is a symmetric positive-definite, or an irreducible, diagonally dominant matrix. The evaluation of these recurrences can be pipelined on a hex-connected systolic array. A global view of this pipelined computation is shown in FIG.

The systolic array in FIG. The processors below the upper boundaries are the standard inner product step processors in FIG. The processor at the top, denoted by a circle, is a special processor. It computes the reciprocal of its input and passes the result southwest, and also passes the same input northward unchanged.

The other processors on the upper boundaries are again hexagonal inner product step processors, as illustrated in FIG. The flow of data on the systolic array is indicated by arrows in the figure. Similar to matrix multiplication, each processor only operates every third time pulse.

Thus, a 52 , for example, can be viewed as a 52 2 when it enters the network. There are several equivalent systolic arrays that reflect only minor changes to the network presented in this section. For example, the elements of L and U can be retrieved as output in a number of different ways. The fact that the matrix multiplication network forms a part of the LU-decomposition network is due to the similarity of the defining recurrences.

In any row or column of the LU-decomposition systolic array, only one out of every three consecutive processors is active at a given time. Still another problem which the present invention can uniquely handle is that of solving a triangular linear system. After having finished the LU-decomposition of A e. This section concerns itself with the solution of triangular linear systems. An upper triangular linear system can always be rewritten as a lower triangular linear system.

Without loss of generality, this section deals exclusively with lower triangular linear systems. The vector x can be computed by the following recurrences. Then the above recurrences can be evaluated by a systolic array similar to that used for band matrix-vector multiplication.

A Systolic Array Optimizing Compiler

The similarity of the defining recurrences for these two problems should be noted. We illustrate our result by considering the linear system problem in FIG. For this case, the systolic array is described in FIG. The y i , which are initially zero, move leftward through the systolic array while the x i , a ij and b i are moving as indicated in FIG. In fact, the special reciprocal processor introduced to solve the LU-decomposition problem is a special case of this more general processor.

Each y i accumulates inner product terms in the rest of the processors as it moves to the left. From the figure one can check that the final values of x 1 , x 2 , x 3 and x 4 are all correct. Although most of our illustrations are done for band matrices all the systolic arrays work for the regular nxn dense matrix. If the band width of a matrix is so large that it requires more processors than a given network provides, then one should decompose the matrix and solve each subproblem on the network.

Understanding Systolic vs Diastolic dysfunction mathematically

One can often reduce the number of processors required by a systolic array, if the matrix is known to be sparse or symmetric. For example, the matrices arising from a set of finite differences or finite elements approximations to differential equations are usually "sparse band matrices.

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In this case by introducing proper delays to each processor for shifting its data to its neighbors, the number of processors required by the systolic array illustrated in FIG. This variant is useful for performing iterative methods involving sparse band matrices.

If matrix A is symmetric positive definite, then it is possible to use only the left portion of the hex-connected network, since in this case U is simply DL T where D is the diagonal matrix a kk k. The optimal choice of the size of the systolic network to solve a particular problem depends upon not only the problem but also the memory band width to the host computer.

For achieving high performance, it is desirable to have as many processors as possible in the network, provided they can all be kept busy doing useful computations. For example, some pattern matching problems can be viewed as matrix problems with comparison and Boolean operations. It is possible to store a dynamically changing data structure in a systolic array so that an order statistic can always be executed in constant time.

There are a number of important problems which can be formulated as matrix-vector multiplication problems and thus can be solved rapidly by the systolic array in FIG.

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The problems of computing convolutions, finite impulse response FIR filters, and discrete Fourier transforms are such examples. If a matrix has the property that the entries on any line parallel to the diagonal are all the same, then the matrix is a Toeplitz matrix. The convolution problem of vectors a and x is simply the matrix-vector multiplication where the matrix is a triangular Toeplitz matrix as follows: EQU2.

The computation of a 4-tap filter with coefficients a 1 , a 2 , a 3 and a 4 may be represented as: EQU3.

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The discrete Fourier transformation of vector x may be represented as: EQU4. In such convolution and filter problems each processor needs to receive an entry of the matrix only once, and this entry can be shipped to the processor through horizontal connections and stay in the processor during the rest of the computation.

As a result, for these three problems it is not necessary for each processor in the network to have the external input connection on the top of the processor, as depicted in FIG. This requirement can be fulfilled by using the algorithm below. We assume that each processor has one additional register R t. All processors except the middle one perform the following operations in each step, but for odd respectively, even numbered time steps only processors which are odd even units apart from the middle processor are activated.

For all processors except the middle one the contents of both R A and R t are initially zero. If the processor is in the left respectively, right hand side of the middle processor then. The middle processor is special; it performs the following operations at every even numbered time step. For this processor the contents of both R A and R t are initially one. Note that all the systolic arrays described above store and retrieve elements of the matrix in the same order. See FIGS. Therefore, we recommend that matrices be always arranged in memory according to this particular ordering so that they can be accessed efficiently by any of the systolic arrays.

One of the most important features of our systolic arrays are that their data paths are very simple and regular, and they support pipeline computations. Simple and regular data paths imply lost cost and efficient implementations of the arrays in VLSI or even printed circuit technology. Since loading of data into the network occurs naturally as computation proceeds, no extra control logic is required. Nor is initialization logic needed. We have discovered that some data flow patterns are fundamental in matrix computations.

For example, the two-way flow on the linearly connected network is common to both matrix-vector multiplication and solution of triangular linear systems FIGS. A practical implication of this fact is that one systolic device may be used for solving many different problems. Moreover, we note that almost all the processors needed in any of these devices are the inner product step processor illustrated in FIGS.

Using the system of this invention with a linearly connected network of size O n both the convolution of two n-vectors and the n-point discrete Fourier transform can be computed in O n units of time, rather than O n log n as required by the sequential FFT algorithm. In the foregoing specification we have set out certain preferred practices and embodiments of this invention; however, it will be understood that this invention may be otherwise embodied within the scope of the following claims.

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