# Low Power and Low Complexity Shift-and-Add Based Computations

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 multiplications, which are realized using shift-and-add based computations. The possibilities to reduce the complexity, i.e., the chip area, and the energy consumption are investigated. Both serial and parallel arithmetic are considered. The main difference, which is of interest here, is that shift operations in serial arithmetic require flip-flops, while shifts can be hardwired in parallel arithmetic.The possible ways to connect a given number of adders is limited. Thus, for single-constant multiplication, the number of shift-and-add structures is finite. We show that it is possible to save both adders and shifts compared to traditional multipliers. Two algorithms for multiple-constant multiplication using serial arithmetic are proposed. For both algorithms, the total complexity is decreased compared to one of the best-known algorithms designed for parallel arithmetic. Furthermore, the impact of the digit-size, i.e., the number of bits to be processed in parallel, is studied for FIR filters implemented using serial arithmetic. Case studies indicate that the minimum energy consumption per sample is often obtained for a digit-size of around four bits.The energy consumption is proportional to the switching activity, i.e., the average number of transitions between the two logic levels per clock cycle. To achieve low power designs, it is necessary to develop accurate high-level models that can be used to estimate the switching activity. A method for computing the switching activity in bit-serial constant multipliers is proposed.For parallel arithmetic, a detailed complexity model for constant multiplication is introduced. The model counts the required number of full and half adder cells. It is shown that the complexity can be significantly reduced by considering the interconnection between the adders. A main factor for energy consumption in constant multipliers is the adder depth, i.e., the number of cascaded adders. The reason for this is that the switching activity will increase when glitches are propagated to subsequent adders. We propose an algorithm, where all multiplier coefficients are guaranteed to be realized at the theoretically lowest depth possible. Implementation examples show that the energy consumption is significantly reduced using this algorithm compared to solutions with fewer word level adders.For most applications, the input data are correlated since real world signals are processed. A data dependent switching activity model is derived for ripple-carry adders. Furthermore, a switching activity model for the single adder multiplier is proposed. This is a good starting point for accurate modeling of shift-and-add based computations using more adders.Finally, a method to rewrite an arbitrary function as a sum of weighted bit-products is presented. It is shown that for many elementary functions, a majority of the bit-products can be neglected while still maintaining reasonable high accuracy, since the weights are significantly smaller than the allowed error. The function approximation algorithms can be implemented using a low complexity architecture, which can easily be pipelined to an arbitrary degree for increased throughput.

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