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**Introduction to Digital Logic Design Appendix A of CO&A Dr. Farag**

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**Outline Boolean Algebra Gates Combinational Circuits**

Simplifications of Boolean Functions Multiplexers, Decoders, PLA, ROM, Adders Sequential Circuits Flip-Flops Registers Counters

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Boolean Algebra Similar to traditional algebra but it defines a set of logical operations and variables Basic operations: AND, OR, NOT (., +, _) These operations are defined by their truth tables Other operations can be derived from the basic ones. Ex: NAND, NOR, XOR, XNOR

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**Boolean Algebra (Cont)**

It defines a set of postulates and another set of identities that can be derived from these postulates A NAND E = NOT(AE) = Ā OR Ē. Proof?

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Gates To implement any logic function we need a functionally complete set of gates. Ex (AND, OR, NOT), (AND, NOT), (OR, NOT) (NAND), (NOR) How to implement all basic functions by NAND?? Examine its truth table. Same for NOR From manufacturing point of view, using only one type of gates to implement the circuit is very advantageous. Why? Regular -> Simple -> easy to design -> cheap Gates are the basic building blocks of all digital systems. They are implemented using electronics components (transistors, diodes, resistors, etc.) Different families are TTL, CMOS, ECL, etc. Not our problem

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**Combinational Logic Circuits**

It is an interconnected set of gates whose output at any times depends solely on the input at that instant of time. (NOT on a previous output state) These circuits have n inputs & m outputs. They can be defined in terms of truth tables, circuits diagrams, or Boolean equations Implement the following truth by using SOP or POS. Check equivalency

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**Combinational Logic Circuits (simplification)**

Simplification: algebraic rules, karnaugh maps, or Quine-McKluskey tables The output of Table 3 can be expressed: F = A’BC’ + A’BC + ABC’ = B(A’+C’) ?? Problem: algebraic rules depend mainly upon observation & experience A more systematic way to simplify digital logic expressions is the use of karnaugh maps

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**Combinational Logic Circuits (Karnaugh)**

The map is an array of 2n squares (n # of inputs) How do you fill the map from a truth table?? To use an expression, it should be in canonical form General rules of using the map: Combine ones into groups of (1, 2, 4, 8, …) squares Form the largest group size Form the min number of groups Two group should not intersect unless this will enable a small group to be larger in size Some input combinations shall not occur and in this situation we call the outputs, “do not care” conditions These “ds” can be used as either 1 or 0 Example: designing an incrementer for a BCD number, see Table 4 and Figure 10 in the following slide

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**Combinational Logic Circuits (Karnaugh)**

Exercise: Design a 4-bit combinational circuit 2’s complementer (The output generates the 2’s complement of the input binary number)

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**Combinational Logic Circuits (QMA)**

Quine-Mckluskey Algorithm is a systematic method to obtain the minimum form of a Boolean expression The details of the algorithm are described in the handout The algorithm is best described by an example Minimize the following function F(A, B, C, D) = Σ (1, 5, 6, 7, 11, 12, 13, 15) This can be expresses as F = A’B’C’D + A’BC’D + A’BCD’ + A’BCD + AB’CD + ABC’D’ + ABC’D + ABCD The minimal expression is F = A’C’D + A’BC + ABC’ + ACD Quiz: Try it using the decimal approach

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**Applications of Combinational Logic Circuits**

NAND and NOR implementation Multiplexer: a circuit that has multiple inputs and only one output. At any time one of the input is selected as output based on the value on the select line(s) Below is the truth table for a 4-to-1 multiplexer Implementation?? Application ex: Inputs to PC

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**Applications of Combinational Logic (Cont)**

A decoder has n input lines and 2n output lines. Only one output line is selected based on the input Example: Instruction Op code decoding With an additional input line, a decoder can be used as a demultiplexer which connects its single input to one of its outputs based on the value on the address lines PLA (Programmable Logic Array) has the objective of developing a general purpose chip that can be readily adapted to specific purposes It can be implemented by making every possible connection through a fuse. Undesired connections are removed by blowing their fuses This kind is called field-PLA See next slide

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**Applications of Combinational Logic (Cont)**

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**Applications of Combinational Logic (Cont)**

ROM is considered a combinational circuit because the outputs are a function only of the current inputs

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**Applications of Combinational Logic (Cont)**

How to implement the function of ALU. The basic circuit is the binary adder Sum = A’B’C + A’BC’ + ABC + AB’C’ Carry = AB + AC + BC Implementation??

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**Applications of Combinational Logic (Cont)**

We can form n-bit adders by cascading n 1-bit adders The carry of a unit is fed to the next one (ripple adders) The problem with ripple adders is the increasing delay The solution is using carry lookahead technique C0 = A0B0 C1 = A1B1 + A1A0B0 + B1A0 C2, ….. etc. Usually a full adder (32-bit say) is constructed from a number of modules (8-bit) adders where the carries are rippled between modules but each module uses carry lookahead to derive internal carry signals. Quiz: Design an 8-bit carry lookahead adder using two 4-bit units (derive all internal and external carry signals and draw the final diagram)

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**Sequential Logic Circuits**

Circuits whose new output depends not only on the current input but also on the past history of that input (in other words on the current output too) Basic application: making memory units (Flip-Flops) A Flip-Flop is a bistable device that has two outputs (one is the complement of the other: Q and Q’) S-R latch: Implementation two NORs with feedback

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**Sequential Logic Circuits (Cont)**

S-R characteristic or transition table

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**Sequential Logic Circuits (Applications)**

Clocked S-R Flip-Flop: Using AND at the input D Flip-Flop: One input is negated and used as the second input. Implementation & characteristic table J-K Flip-Flop: it differs in that all input combinations are allowed. The last case (J=K=1) causes the output to toggle, a very important feature Implementation and table Flip-Flips are used to implement registers. There are two types of registers A parallel register is a set of 1-bit memories that can be read or written simultaneously. See the following slide A shift register implements the shits function. See the following slide

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**Sequential Logic Circuits (Applications)**

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**Sequential Logic Circuits (Applications)**

Counters are one of the most important applications A counter is a register whose value is incremented by 1 modulo its capacity upon the reception of the clock Counters come into two flavors: Ripple counters: straight forward but slow Synchronous counters: Faster but more involved

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**Sequential Logic Circuits (Applications)**

Synchronous counters design example. First derive the excitation table of the JK Flip-Flop Construct the truth table then design the input to each latch For each input, use combinational design to implement the required circuit

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