By Martin Kumm

ISBN-10: 3658133228

ISBN-13: 9783658133221

ISBN-10: 3658133236

ISBN-13: 9783658133238

This paintings covers box programmable gate array (FPGA)-specific optimizations of circuits computing the multiplication of a variable by means of numerous constants, regularly denoted as a number of consistent multiplication (MCM). those optimizations specialise in low source utilization yet excessive functionality. They contain using speedy carry-chains in adder-based consistent multiplications together with ternary (3-input) adders in addition to the mixing of look-up table-based consistent multipliers and embedded multipliers to get the optimum mapping to fashionable FPGAs. The proposed equipment can be utilized for the effective implementation of electronic filters, discrete transforms and lots of different circuits within the area of electronic sign processing, communique and picture processing.

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**Extra info for Multiple Constant Multiplication Optimizations for Field Programmable Gate Arrays**

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Here, each node with a rectangular shape includes a pipeline register, i. , two-input nodes marked with ‘+’ or ‘–’ are pipelined adders and subtractors, respectively, while single input nodes are pure registers. Each node in the PAG corresponds to similar BLE costs. 10(c), there obviously exists a solution with less nodes. This PAG will typically use less BLEs although more adders are used. Note that optimizing PAGs is complementary to methods that increase the speed of a single adder, e. , deeply pipelined adders [67].

The lowest possible adder depth is limited by the node(s) with maximum adder depth Dmax = max ADmin (t) . 19) Limiting the adder depth to Dmax leads to the lowest possible delay and to the corresponding optimization problem: Deﬁnition 5 (MCM with Bounded Adder Depth (MCMBAD ) Problem). Given a set of positive target constants T = {t1 , . . , tM }, ﬁnd an adder graph with minimum cost such that AD(t) ≤ Dmax for all t ∈ T . The adder depth has an impact on the power consumption [34, 50–55]. It is based on the insight that a transition which is generated at the input or an adder output produces more transitions (glitches) in the following stages.

Y1 y0 ), with xi , yi ∈ {0, 1} can be written as B−1 x·y =x· B−1 2i y i i=0 2i x · yi = i=0 . 1) partial product In a generic multiplier, the partial products x · yi can be obtained by a bitwise AND-operation. The ﬁnal product is then obtained by adding the bit-shifted partial products. Now, if y is a constant bit vector, all bits yi which are zero lead to zero partial products and the corresponding adders can be removed. Thus, the number of required adders for the constant multiplication using this representation is equal to the number of non-zero elements (Hamming weight) in the binary representation of y minus one.

### Multiple Constant Multiplication Optimizations for Field Programmable Gate Arrays by Martin Kumm

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