Metzler matrix: Difference between revisions
→See also: Fixed capitalization of Stochastic matrix |
en>Benja Remove claim that there is always a nonnegative eigenvalue, because this is clearly false, consider a diagonal matrix with all diagonal entries negative. (The person who wrote that statement was presumably thinking of the Metzler matrix's *exponential*.) |
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A '''lookahead carry unit''' ('''LCU''') is a logical unit in [[digital circuit]] design used to decrease calculation time in [[adder (electronics)|adder]] units and used in conjunction with [[carry look-ahead adder]]s (CLAs). | |||
==4-bit adder== | |||
A single 4-bit CLA is shown below: | |||
[[File:4-bit carry lookahead adder.svg|none|framed|4-bit adder with Carry Look Ahead (CLA)]] | |||
==16-bit adder== | |||
By combining four 4-bit CLAs, a 16-bit adder can be created but additional logic is needed in the form of an LCU. | |||
The LCU accepts the group propagate (<math>P_G</math>) and group generate (<math>G_G</math>) from each of the four CLAs. <math>P_G</math> and <math>G_G</math> have the following expressions for each CLA adder:<ref>http://www.seas.upenn.edu/~ese171/lab/CarryLookAhead/CarryLookAheadF01.html</ref> | |||
:<math>P_G = P_3 \cdot P_2 \cdot P_1 \cdot P_0</math> | |||
:<math>G_G = G_3 + P_3 \cdot G_2 + P_3 \cdot P_2 \cdot G_1 + P_3 \cdot P_2 \cdot P_1 \cdot G_0</math> | |||
The LCU then generates the carry input for each CLA. | |||
Assume that <math>P_i</math> is <math>P_G</math> and <math>G_i</math> is <math>G_G</math> from the i<sup>th</sup> CLA then the output carry bits are | |||
:<math>C_{4} = G_0 + P_0 \cdot C_0</math> | |||
:<math>C_{8} = G_{4} + P_{4} \cdot C_{4}</math> | |||
:<math>C_{12} = G_{8} + P_{8} \cdot C_{8}</math> | |||
:<math>C_{16} = G_{12} + P_{12} \cdot C_{12}</math> | |||
Substituting <math>C_{4}</math> into <math>C_{8}</math>, then <math>C_{8}</math> into <math>C_{12}</math>, then <math>C_{12}</math> into <math>C_{16}</math> yields the expanded equations: | |||
:<math>C_{4} = G_0 + P_0 \cdot C_0</math> | |||
:<math>C_{8} = G_4 + G_0 \cdot P_4 + C_0 \cdot P_0 \cdot P_4</math> | |||
:<math>C_{12} = G_8 + G_4 \cdot P_8 + G_0 \cdot P_4 \cdot P_8 + C_0 \cdot P_0 \cdot P_4 \cdot P_8</math> | |||
:<math>C_{16} = G_{12} + G_8 \cdot P_{12} + G_4 \cdot P_8 \cdot P_{12} + G_0 \cdot P_4 \cdot P_8 \cdot P_{12} + C_0 \cdot P_0 \cdot P_4 \cdot P_8 \cdot P_{12}</math> | |||
<math>C_{4}</math> corresponds to the carry input into the second CLA; <math>C_{8}</math> to the third CLA; <math>C_{12}</math> to the fourth CLA; and <math>C_{16}</math> to overflow carry bit. | |||
In addition, the LCU can calculate its own propagate and generate: | |||
:<math>P_{LCU} = P_0 \cdot P_4 \cdot P_8 \cdot P_{12}</math> | |||
:<math>G_{LCU} = G_{12} + G_8 \cdot P_{12} + G_4 \cdot P_8 \cdot P_{12} + G_0 \cdot P_4 \cdot P_8 \cdot P_{12} + C_0 \cdot P_0 \cdot P_4 \cdot P_8 \cdot P_{12} = C_{16}</math> | |||
[[File:16-bit lookahead carry unit.svg|none|frame|16-bit adder with LCU]] | |||
==64-bit adder== | |||
By combining 4 CLAs and an LCU together creates a 16-bit adder. | |||
Four of these units can be combined to form a 64-bit adder. | |||
An additional (second-level) LCU is needed that accepts the propagate (<math>P_{LCU}</math>) and generate (<math>G_{LCU}</math>) from each LCU and the four carry outputs generated by the second-level LCU are fed into the first-level LCUs. | |||
[[File:64-bit lookahead carry unit.svg|none|frame|64-bit adders with a second-level LCU]] | |||
==References== | |||
{{Reflist}} | |||
* {{cite book |last=Katz |first=Randy |authorlink=Randy Katz |title=Contemporary Logic Design |publisher=The Benjamin/Cummings Publishing Company |year=1994 |isbn=0-8053-2703-7 |pages=249–256}} | |||
* {{cite book |last=Vahid | first=Frank |title=Digital Design |publisher=John Wiley and Sons Publishers |year=2006 |isbn=0-470-04437-3 |pages=296–316}} | |||
[[Category:Digital circuits]] | |||
[[Category:Adders]] | |||
[[de:Paralleladdierer_mit_Übertragsvorausberechnung]] |
Latest revision as of 20:01, 10 November 2013
A lookahead carry unit (LCU) is a logical unit in digital circuit design used to decrease calculation time in adder units and used in conjunction with carry look-ahead adders (CLAs).
4-bit adder
A single 4-bit CLA is shown below:
16-bit adder
By combining four 4-bit CLAs, a 16-bit adder can be created but additional logic is needed in the form of an LCU.
The LCU accepts the group propagate () and group generate () from each of the four CLAs. and have the following expressions for each CLA adder:[1]
The LCU then generates the carry input for each CLA.
Assume that is and is from the ith CLA then the output carry bits are
Substituting into , then into , then into yields the expanded equations:
corresponds to the carry input into the second CLA; to the third CLA; to the fourth CLA; and to overflow carry bit.
In addition, the LCU can calculate its own propagate and generate:
64-bit adder
By combining 4 CLAs and an LCU together creates a 16-bit adder. Four of these units can be combined to form a 64-bit adder. An additional (second-level) LCU is needed that accepts the propagate () and generate () from each LCU and the four carry outputs generated by the second-level LCU are fed into the first-level LCUs.
References
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- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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