Combinational Circuits

Decoder Decoder : Takes n inputs Selects one of 2n output lines

Truth Table Expression for each output has one term O0 = I1' I0' O1 = I1 I0' O2 = I1' I0 O3 = I1 I0

Decoder Implementation O0 = I1' I0' O1 = I1 I0' O2 = I1' I0 O3 = I1 I0

Sample Use 3  8 decoder used to pick instruction

Multiplexer Multiplexer : 2 input multiplexer 4 input multiplexer N data lines lines Log2(N) select lines 2 input multiplexer 4 input multiplexer

Decode to Mux Multiplexer selection logic = decoder 4 input multiplexer

Shifter 1 bit shifter D4 goes to either S3 or S5

Shifter 1 bit shifter If C = 1, goes to S5

Shifter 1 bit shifter If C = 0, goes to S3

Shifter S = direction 0: left, 1: right Logical shift: Left bit always 0 on right shift

Multibit Shift 4 bit wide 0-3bit shift S selects direction Multiplexers select correct pattern

Half Adder Half Adder : adds A + B, produce sum + carry 1 1 1

Half Adder Truth Table Inputs a & b Carry : c Sum : s

First Pass

XOR Function XOR gate : exclusive OR Notation: A  B OR A ^ B AB' + A'B Notation: A  B OR A ^ B

Using XOR Half Adder Final

Full Adder 1 1 1 Full Adder handles carry in Three inputs: Cin, A, B Two out: Cout, S 1 1 1

Full Adder Circuit Made from two half adders Add A & B Add sum to carry

Full Adder Abstracted

Ripple Carry Adder Multibit adder : string of 1 bit full adders

Ripple Carry Adder Implemented with carry out & signed overflow:

Ripple Carry Adder Adding subtraction to adder Select line 0 : Normal 1 : Invert one pattern, add extra 1 via carry

Ripple Carry Adder Fun Fact Carry signal needs to propagate from one adder to next 3 gates delay per bit

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

ALU ALU : Arithmetic Logic Unit In: Out: 2 Operands Control signal Result based on select operation

ALU Harware to implement each desired op Multiplexor to select desired one

1-Bit Simple ALU Processes two 1-bit inputs A, B 3 functions: AND (0), OR (1), Add (2) Selected with function select bits

1-Bit Simple ALU With selector to invert B input Can now also compute: A OR ~B A AND ~B A + ~B

4-Bit Simple ALU Four 1-bit ALU's chained together

4-Bit Simple ALU AddOne allows for A + (~B) + 1 == A + (~B + 1) == A + (-B) == A - B