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<p>A novel hybrid number representation is proposed. It includes the two's complement representation and the signed-digit representation as special cases. The hybrid number representations proposed are capable of bounding the maximum length of carry propagation chains during addition to any desired value between 1 and the entire word length. The framework reveals a continuum of number representations between the two extremes of two's complement and signed-digit number systems and allows a unified performance analysis of the entire spectrum of implementations of adders, multipliers and alike. We present several static CMOS implementations of a two-operand adder which employ the proposed representations. We then derive quantitative estimates of area (in terms of the required number of transistors) and the maximum carry propagation delay for such an adder. The analysis clearly illustrates the trade-offs between area and execution time associated with each of the possible representations. We also discuss adder trees for parallel multipliers and show that the proposed representations lead to compact adder trees with fast execution times. In practice, the area available to a designer is often limited. In such cases, the designer can select the particular hybrid representation that yields the most suitable implementation (fastest, lowest power consumption, etc.) while satisfying the area constraint. Similarly, if the worst case delay is predetermined, the designer can select a hybrid representation that minimizes area or power under the delay constraint.</p>
adders; digital arithmetic; multiplying circuits; carry logic; logic design; hybrid signed-digit number systems; redundant number representations; bounded carry propagation chains; 2s complement representation; signed-digit representation; carry propagation chains; performance analysis; CMOS; execution time; parallel multipliers; adder trees; bounded carry propagation; carry-free addition.

I. Koren and D. Phatak, "Hybrid Signed-Digit Number Systems: A Unified Framework for Redundant Number Representations with Bounded Carry Propagation Chains," in IEEE Transactions on Computers, vol. 43, no. , pp. 880-891, 1994.
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