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Combinational Circuits

Combinational Circuits. Alexander Titov 10/27/2012. Layers of Abstraction in Computes Science (CS). Application. Algorithms. Programming Language. Operating System. Instruction Set Architecture. Microarchitecture. Gates/Register-Transfer Level (RTL). More about logical circuits.

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Combinational Circuits

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  1. Combinational Circuits Alexander Titov 10/27/2012

  2. Layers of Abstraction in Computes Science (CS) Application Algorithms Programming Language Operating System Instruction Set Architecture Microarchitecture Gates/Register-Transfer Level (RTL) More about logical circuits Topics of this lecture Circuits Less about physics, wires and transistors… Physics

  3. Boolean Algebra • Boolean Algebra (BA) works with object that can take only two values (true and false, or 1 and 0). • Such object is called a Boolean object (term bit of information is used in CS) • BA defines operations on Boolean objects → Boolean operations and functions • It is convenient to represent these operations and functions via truth tables: a c b Boolean function Truth table

  4. Main Boolean operations inversion (NOT) • The simplest Boolean operation is … • AND (or Boolean multiplication) x !x x 0 x*y y 0 0 1

  5. Main Boolean operations • OR (or Boolean addition) x 0 y x+y 1 1 1 • XOR (or exclusive OR, or addition by module 1) 0 x x + y 1 y 1 1

  6. Boolean functions • Boolean operations can be combined into functions F(x, y, z) = x + !y*z x x + !y*z y z 1 0 1 0

  7. Main axioms • There about ten axioms that can be used to create new or to simplify existed functions • There are a lot of other useful equations: !(xy) = !x + !y or something more complex … n(n(x) + y) + n(n(x) + n(y)) = x (Huntington equation)

  8. Combinational logic • If the output of a function is completely defined by the current input then the function is called combinational. F = F(xt, yt, zt, Ft-1) Ft = F(xt, yt, zt) combinational sequentional • Combinational logic (scheme) is an implementation of a combinational function. • Ok, but how does this all connect with the microarchitecture?

  9. Half adder scheme • It is an adder, but it is not a full adder, because it does not have input carry • It is an adder, but it is not a full adder, because it does not have input carry 0 0 sum x 1 0 z0 y 1 0 0 1 carry half + sum x • It is called a half adder z1 carry y

  10. Full adder scheme cn-1 half + 0 1 half + sn xn 1 0 yn 0 1 1 1 Cn

  11. Wide adder cn-1 half + xn sn yn cn half + xn+1 sn+1 yn+1 cn+1

  12. Thank You

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