Adders and Its Equations

Adder or addition is often located on the critical path of modern day digital signal processor (DSP). Since adders are often responsible for setting the minimum clock cycle time in a processor, they can be critical to any improvements seen at the VLSI level. Implementing fast adder has been an important subject since 1950, and Ripple Carry Adder (RCA) is the first and the most fundamental adder that is capable of performing binary number additions. Since its latency is proportional to the length of its input operands, it is not very useful. Weinberger and Smith (1958) have made the first major break-through in speeding-up additions by proposing Carry Look-ahead Adder (CLA). Rather than rippling carries throughout the adder. it uses parallelism to propagate carries much quickly. 

Half Adder (HA)

The most fundamental building block in the arithmetic system is Half Adder (HA). HA takes two bits

a_k and b_k and produces a sum bit s_k and a carry bit c_k+1. The logic equations for a HA are as follow:


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References:

• _{Digital Design and Implementation with Field Programmable Devices, Zainalabedin Navabi, Kluwer Academic Plubishers, 2005.}
• _{http://www.math-cs.gordon.edu/courses/cps111/Notes/Boolean%20logic/circuits.html (source of figure)}