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Combinational Circuits
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Decoder Decoder : Takes n inputs Selects one of 2n output lines
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Truth Table Expression for each output has one term
O0 = I1' I0' O1 = I1 I0' O2 = I1' I0 O3 = I1 I0
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Decoder Implementation O0 = I1' I0' O1 = I1 I0' O2 = I1' I0 O3 = I1 I0
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Sample Use 3 8 decoder used to pick instruction
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Multiplexer Multiplexer : 2 input multiplexer 4 input multiplexer
N data lines lines Log2(N) select lines 2 input multiplexer 4 input multiplexer
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Decode to Mux Multiplexer selection logic = decoder
4 input multiplexer
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Shifter 1 bit shifter D4 goes to either S3 or S5
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Shifter 1 bit shifter If C = 1, goes to S5
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Shifter 1 bit shifter If C = 0, goes to S3
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Shifter S = direction 0: left, 1: right Logical shift:
Left bit always 0 on right shift
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Multibit Shift 4 bit wide 0-3bit shift S selects direction
Multiplexers select correct pattern
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Half Adder Half Adder : adds A + B, produce sum + carry 1 1 1
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Half Adder Truth Table Inputs a & b Carry : c Sum : s
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First Pass
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XOR Function XOR gate : exclusive OR Notation: A B OR A ^ B
AB' + A'B Notation: A B OR A ^ B
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Using XOR Half Adder Final
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Full Adder 1 1 1 Full Adder handles carry in Three inputs: Cin, A, B
Two out: Cout, S 1 1 1
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Full Adder Circuit Made from two half adders Add A & B
Add sum to carry
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Full Adder Abstracted
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Ripple Carry Adder Multibit adder : string of 1 bit full adders
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Ripple Carry Adder Implemented with carry out & signed overflow:
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Ripple Carry Adder Adding subtraction to adder Select line 0 : Normal
1 : Invert one pattern, add extra 1 via carry
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Ripple Carry Adder Fun Fact
Carry signal needs to propagate from one adder to next 3 gates delay per bit
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Carry Select Fun Fact Cary Select Adder
Calculate second block with both 0 and 1 carry Work in parallel with first Pick right answer when carry available
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ALU ALU : Arithmetic Logic Unit In: Out: 2 Operands Control signal
Result based on select operation
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ALU Harware to implement each desired op
Multiplexor to select desired one
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1-Bit Simple ALU Processes two 1-bit inputs A, B
3 functions: AND (0), OR (1), Add (2) Selected with function select bits
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1-Bit Simple ALU With selector to invert B input Can now also compute:
A OR ~B A AND ~B A + ~B
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4-Bit Simple ALU Four 1-bit ALU's chained together
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4-Bit Simple ALU AddOne allows for A + (~B) + 1 == A + (~B + 1) == A + (-B) == A - B
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