Hardware Implementations Gates and Circuits. Three Main Gates  AND  OR  NOT.

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Presentation transcript:

Hardware Implementations Gates and Circuits

Three Main Gates  AND  OR  NOT

Gate Diagrams  Example 1: [(today is Monday) AND (it is raining)] OR (it is snowing)

Gate Diagrams  Example: What does it represent?

Gate Diagrams  Example: {[(today is Monday) AND (it is raining)] OR (it is snowing)} AND {NOT [(it is raining) AND (it is snowing)]}

Truth Table to Gates  First, build the Boolean algebra expression that gives Z Z = AB + A’B’ Z = (A AND B) OR (NOT A AND NOT B) ABZ TTT TFF FTF FFT

Truth Table to Gates  Z = AB + A’B’  Next, build the circuit that goes with the Boolean algebra expression Z ABZ TTT TFF FTF FFT

Z = AB + A’B’

Binary Arithmetic  We can add binary numbers just like decimal numbers only using base two arithmetic.  For example:

Binary Addition  Notice in addition: FalseTrue False True Sum Carry ABSum (1) T (0) F (1) T(0) F(1) T (0) F(1) T (0) F

Sum and Carry ABCarry ABSum

Sum Circuit ABSum Sum = AB’ + A’B

Carry Circuit ABCarry Carry = AB

Half Adder - Sum and Carry

Half Adder  The sum digit is 0 if the sum is even.  The sum digit is 1 if the sum is odd.  The carry is 1 if the sum is greater than 1.  Handles the case where we add two binary digits with no inward carry.

Full Adder  Takes a carry in and produces the result and carry out.  So, we have 3 inputs and two outputs.  Combine two half-adders together with an OR gate to get a full adder for each binary digit.  How many half adders would we need to add two 8-digit binary numbers? How many gates?

Full Adder

Subtraction ABSub ABBorrow

Binary Subtraction  We do binary subtraction like decimal subtraction only the borrowing is done in 2’s instead of 10’s

Subtraction as Addition  If A = , B = , then using the twos-complement representation for –B, we have –B = = so

Binary Multiplication  Again, just like decimal except we add and multiply in binary. * x 7x

NAND Gates and NOT  This gate represents (A NAND NOT B).

NAND Truth Table ABA NAND B TTF TFT FTT FFT

NAND  Fact: All other gates (AND, OR, NOT) can be constructed using only NAND gates  Verification:

Exercises  Fill in a truth table and give a Boolean expression for the following circuits.

Exercises - How would you create a one binary digit multiplier? A two-digit by one-digit multiplier? A two-digit by two-digit multiplier? *