Binary parallel adder, decimal adder

Binary parallel adder, decimal adder

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  Digital Logic Design 1 CombinationalLogic Prof. Shehzad Ali
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  Introduction  Logic circuitsfor digital systems may be combinational or sequential.  Consists of logic gates whose outputs at any time are determined from only the present combination of inputs.  Performs an operation that can be specified logically by a set of Boolean functions. Combinational Circuit 2
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  Digital Circuits Combinational Circuits Logiccircuits for digital system ◦ Combinational circuits ◦ the outputs are a function of the current inputs ◦ Sequential circuits ◦ contain memory elements ◦ the outputs are a function of the current inputs and the state of the memory elements ◦ the outputs also depend on past inputs 3
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  Digital Circuits Combinational circuits Acombinational circuits ◦ 2 n possible combinations of input values ◦ Specific functions ◦ MSI (Medium-Scale Integration) circuits or standard cells 4 Combinational Logic Circuit n input variables m output variables  Adders, subtractors, comparators, decoders, encoders and multiplexers
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  Sequential Circuit 5  Employsstorage elements in addition to logic gates.  Their outputs are a function of the inputs and the state of the storage elements.  Because the state of the storage elements is a function of previous inputs, the outputs of a sequential circuit depend not only on present value of inputs, but also on past inputs. Sequential Circuit
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  Digital Circuits Design Procedure Thedesign procedure of combinational circuits ◦ State the problem (system spec.) ◦ determine the inputs and outputs ◦ the input and output variables are assigned symbols ◦ derive the truth table ◦ derive the simplified Boolean functions ◦ draw the logic diagram and verify the correctness 6
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  7 Half adder  Isa combinational circuit that performs the addition of two bits. 0 + 0 = 0 ; 0 + 1 = 1 ; 1 + 0 = 1 ; 1+ 1 = 10 Elementary Operations Truth Table  two input variables  x, y.  two output variables.  C (output carry), S (least significant bit of the sum). Binary Adder-Subtractor
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  8 Half adder  S= x'y+xy'  C = xy Simplified Boolean Function (Sum of Products) Logic Diagram (Sum of Products) Binary Adder-Subtractor
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  9 Half adder S=xy C =xy Simplified Boolean Function (XOR and AND gates) Logic Diagram (XOR and AND gates) Binary Adder
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  Digital Circuits Binary Adder ◦S = x'y+xy' ◦ C = xy For the SUM bit: SUM = A XOR B = A ⊕ B For the CARRY bit: CARRY = A AND B = A.B 10
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  Digital Circuits 11 FunctionalBlock: Full-Adder  It is a combinational circuit that performs the arithmetic sum of three bits (two significant bits and previous carry).  It is similar to a half adder, but includes a carry-in bit from lower stages.  Two half adders can be employed to implement a full adder. Inputs & Outputs  Three input bits:  x, y : two significant bits  Z : the carry bit from the previous lower significant bit.  Two output variables:  C (output carry), S (least significant bit in sum). Binary Adder-Subtractor
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  Digital Circuits  Fora carry-in (Z) of 0, it is the same as half-adder:  For a carry- in (Z) of 1: 12 Z 0 0 0 0 X 0 0 1 1 + Y + 0 + 1 + 0 + 1 C S 00 0 1 0 1 1 0 Z 1 1 1 1 X 0 0 1 1 + Y + 0 + 1 + 0 + 1 C S 0 1 1 0 1 0 11 Functional Block: Full-Adder Operations Binary Adder
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  Digital Circuits Full-Adder Full-Adder ◦ Thearithmetic sum of three input bits ◦ three input bits ◦ x, y: two significant bits ◦ z: the carry bit from the previous lower significant bit ◦ Two output bits: C, S x y z C S 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1
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  Digital Circuits 14 Full-Adder Thenthe Boolean expression for a full adder is as follows. For the SUM (S) bit: SUM = (A XOR B) XOR Cin = (A ⊕ B) ⊕ Cin For the CARRY-OUT (Cout) bit: CARRY-OUT = A AND B OR Cin(A XOR B) = A.B + Cin(A ⊕ B)