# Low Power and Low complexity Constant Multiplication using Serial Arithmetic

Abstract: The main issue in this thesis is to minimize the energy consumption per operation for the arithmetic parts of DSP circuits, such as digital filters. More specific, the focus is on single- and multiple-constant multiplication using serial arithmetic. The possibility to reduce the complexity and energy consumption is investigated. The main difference between serial and parallel arithmetic, which is of interest here, is that a shift operation in serial arithmetic require a flip-flop, while it can be hardwired in parallel arithmetic.The possible ways to connect a certain number of adders is limited, i.e., for single-constant multiplication, the number of possible structures is limited for a given number of adders. Furthermore, for each structure there is a limited number of ways to place the shift operations. Hence, it is possible to find the best solution for each constant, in terms of complexity, by an exhaustive search. Methods to bound the search space are discussed. We show that it is possible to save both adders and shifts compared to CSD serial/parallel multipliers. Besides complexity, throughput is also considered by defining structures where the critical path, for bit-serial arithmetic, is no longer than one full adder.Two algorithms for the design of multiple-constant multiplication using serial arithmetic are proposed. The difference between the proposed design algorithms is the trade-offs between adders and shifts. For both algorithms, the total complexity is decreased compared to an algorithm for parallel arithmetic.The impact of the digit-size, i.e., the number of bits to be processed in parallel, in FIR filters is studied. Two proposed multiple-constant multiplication algorithms are compared to an algorithm for parallel arithmetic and separate realization of the multipliers. The results provide some guidelines for designing low power multiple-constant multiplication algorithms for FIR filters implemented using digit-serial arithmetic.A method for computing the number of logic switchings in bit-serial constant multipliers is proposed. The average switching activity in all possible multiplier structures with up to four adders is determined. Hence, it is possible to reduce the switching activity by selecting the best structure for any given constant. In addition, a simplified method for computing the switching activity in constant serial/parallel multipliers is presented. Here it is possible to reduce the energy consumption by selecting the best signed-digit representation of the constant.Finally, a data dependent switching activity model is proposed for ripple-carry adders. For most applications, the input data is correlated, while previous estimations assumed un-correlated data. Hence, the proposed method may be included in high-level power estimation to obtain more accurate estimates. In addition, the model can be used as cost function in multiple-constant multiplication algorithms. A modified model based on word-level statistics, which is accurate in estimating the switching activity when real world signals are applied, is also presented.

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