Block Size Optimization in Carry-Skip Adder

Block Size Optimization in Carry-Skip Adder By Lee Hathcock

Carry-Skip Adders • Carry-Skip a quick and dirty method to increase speed over ripple-carry • Small hardware / power cost • Comparatively large speed increase • My approach • Use a program written in C++ to generate optimal block sizes

O & B’s Method • First generate number of stages based on skip delay value • Computed by the function n <= m + ½ (mT) + ¼ (m^2 T) + 1/8 (1-(-1)^m)T • n is the number of bits that can be covered • m is the number of stages • Goal is to find the lowest m that will generate enough bits to cover the required bits

O & B’s Method (cont.) • After m has been found, generate each y(i) • Use formula: • min{1+iT, 1+(m+1-i)T}, where i ranges from 1 to the number of stages • This generates a block distribution that covers the required number of bits

O & B’s Method (cont.) • The previous formula actually generates too many bits • Need to reduce, implement bit chopping algorithm • Start at middle value, decrement size • Move to bit to left, decrement. • Move two positions to the right, decrement • Continue until out of range or necessary bits have been chopped

Timing Analysis • Have to run though sets of timing triplets to optimize • Format (i, j, k) • j is the internal delay though the block • i is the skip delay plus the previous stage’s worst i, j • k is the internal propagation delay of the current stage plus the previous stage’s worst i, j

Timing Analysis (cont.) • If the j value is greater than the i value for a given stage, can optimize • Match bits by subtracting from the size of current stage • Reallocate bit later, where j is much less than i • Tends to lower total delay in some cases

Basic Skip Circuit (from notes)

Power / Delay Analysis • Two cases of 4-bit carry-skip • Used Cadence with ami06 process, existing MSU libraries for ami06 (voltage = 3.3v) • First with breakdown of 2,2 • Average power of 127.851 uW • Fastest time = 1.13x10^-9 s • Slowest time = 4.14x10^-8 s

Power / Delay Analysis (cont.) • Second case with 1, 2, 1 • Average power of 88.2793 uW • Fastest time = 7.19x10^-10 s • Slowest time = 4.14x10^-10 s • To be expected, propagated all the way through • Less skip logic, with case this small, makes it faster

Considerations / Limitations • Wanted to implement Turrini’s multilevel skip algorithm • Very optimal up to 128 bits and 5-6 skip levels • Lack of different skip values for larger skips • Skews results a bit, not really realistic • Partly due to limitations of chosen algorithm

Limitations • Skip delay has to be an integer value • Intended to make non-integer values possible, not done due to choice of algorithm • Approximation can be done, still will near-optimal results

Conclusions • Carry-skip a very effective way to cheaply speed up adder design • Optimization of block size can be very difficult • Program works fairly well in most cases • A fully functional version of this program could be very useful for quick carry-skip optimization • Only a few changes and fixes will be necessary to address the issues listed previously

Block Size Optimization in Carry-Skip Adder