Let N + (k) = 2 k/2 k 3/2 f (k) and N − (k) = 2 k/2 k 1/2 g(k) where f (k) → ∞ and g(k) → 0 arbitrarily slowly as k → ∞. We show that the probability of a random 2-coloring of {1, 2, . . . , N + (k)} containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of {1, 2, . . . , N − (k)} containing a monochromatic kterm arithmetic progression approaches 0, as k → ∞. This improves an upper bound due to Brown [3], who had established an analogous result for N + (k) = 2 k log kf (k).
The number of adders (subtractors) needed to implement the coefficient multipliers determines the complexity of the finite impulse response (FIR) filters. A greedy common subexpression elimination (CSE) algorithm based on the canonic signed digit representation of filter coefficients for implementing low complexity FIR filters is proposed in this paper. Our look-ahead algorithm chooses the maximum number of frequently occurring subexpressions to eliminate redundant computations in coefficient multiplication and hence reduces the number of adders required to implement the filter. When compared to existing CSE algorithms, our method results in a very low number of unpaired bits. Design examples of FIR filters show that the proposed method offers an average adder reduction of about 20% over the best known CSE method.the identification and elimination of two nonzero bit subexpressions (2-bit CSs) was proposed. A method to eliminate the most commonly occurring 2-bit CSs was proposed in [3]. As an additional criterion in the subexpression identification process, an estimation of a latch count improvement was also used in [3]. A modification of the method in [2] for identifying and eliminating the best subexpressions to maximize the optimization impact is proposed in [4]. In [5], a nonrecursive signed CSE (NRSCSE) algorithm has been proposed as a modification of the technique in [3] that minimizes the logic depth into the digital structure.The main idea in [6] is reordering computations and identifying common computations that maximize computation sharing between different multipliers. However the method in [6] offers only a slight improvement in reduction of adders (11%) over the CSE method [3]. Moreover, this method results in an increase in delay, corresponding to the delay of one adder-step on average. Instead of exploring optimizations over the original filter coefficients, differential coefficients were considered in [7], where differences between absolute values of filter coefficients were employed to reduce the dynamic range of computation. However the DCM suffers from overheads since it requires extra adders to compute the sum of the stored partial product of previous computation in order to compensate the effect of differential coefficients. In [8], the idea of using differential coefficient was applied to the multiplierless
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.