Topdown modular design Decoders nto2 n decoder logic

Top-down modular design

Decoders • n-to-2 n decoder: logic network with n inputs and 2 n outputs. • One output is active for each of the 2 n input combinations each minterm • Decoder minterm generator • Most common use: – Memory selection

Decoders • Parallel decoders (a) active-high (b) active-low

Decoders • Parallel is expensive • Problem extending for larger n • 2 n decoders require n-input ANDs: fan-in • Alternative: (c) alternate structure

Decoders: (a) parallel (b) tree

Decoders • Tree decoders – constant fan-in: 2 -input gates throughout • Compare the number of gates required for (a) and (b) • Dual tree decoders – n inputs divided into 2 groups j and k; • n = j+k – j-input decoder 2 j outputs – k-input decoder 2 k outputs – Array of 2 j * 2 k = 2 n 2 -input ANDs

Decoders • Why decision tree: @ a a a @ @ @a @a @a

Decoders: (c) dual tree • n=4 • j=k=2

Implementing logic functions using Decoders Since maxterm. , each output can be considered a

Logic functions using Decoders

Logic functions using Decoders • Active-high decoder with OR gate (a) • Active-low decoder with NAND gate (b)

Logic functions using Decoders • Active-high decoder with NOR gate (c) • Active-low decoder with AND gate (d)

Decoders Active-high decoder with enable (a) Symbol (b) • Function:

Decoders • 3 8 decoder using two 2 4 decoders with enable

Decoders • 4 16 decoder using 2 4 decoders with enable

Encoders • Opposite of decoder: one output code (binary) for each input; assumes one input active at a time • n inputs; s outputs • n ≤ 2 s • s ≥ log 2 n; usually s = log 2 n • 4: 2 encoder with exclusive inputs: functional diagram

Encoders • K-Maps of the 4: 2 encoder outputs

Encoders • Logic diagram and Truth table of the 4: 2 encoder

Encoders • Functional diagram and Truth table of the 4: 3 encoder • Outputs Zero unless exactly one line is active high

Encoders • K-Maps of the outputs of the 4: 3 encoder

Encoders • Logic diagram of the 4: 3 encoder

Encoders • 4: 2 priority encoder • No input active EO = 1 • At least one input active GS = 1 • If more than one input active output the one with highest priority • 3>2>1>0 • Useful in resource management request / acknowledge circuits in computers

Encoders • K-Maps of the 4: 2 priority encoder outputs

Encoders • Logic diagram and Truth table of the 4: 2 priority encoder

Multiplexers • Connects one of the inputs to the output • Needs log 2 n select lines

Multiplexers • Easy to realize in CMOS

Multiplexers • Question: we want a 8: 1 MUX using 4: 1 s as building blocks

Demultiplexers • Connects the input to one of the outputs • Needs log 2 n select lines

Demultiplexers • Where is it useful?

Adders • Often realized as “bit slice”s – Bit slice: module that handles one bit position – Can be replicated (arrayed) – Array handles n bits (the whole number) • Bit slice: Half Adder – Given two input bits, produces sum and carry

Adders • Bit slice: Full Adder – Given two input bits and carry in, produces sum and carry

Adders • Pseudo parallel (ripple carry) Adder

Adders • Two-bit “parallel” Adder

Adders • Four-bit “parallel” Adder • What is the difficulty with nbit parallel adders?

Adders • Parallel Adder carry logic: growing impractically • Generate Propagate

Adders • Expanding the carry expressions yields or, generalized:

Adders • When there is carry-in, it becomes c 0, and renumbering the terms yields: • Can form groups of 4:

Adders • Using the group carries: • Generate and propagate are from one bit so far • Can extend the idea to groups of bit positions

Adders • The group carry, generate and propagate are: • The groups can be adjoining or overlapping e. g.

Adders • Subtraction is addition – Two-s complement – Bitwise invert B and set the carry-in c 0=1

Adders • Overflow detection

Adders • Overflow detection

Comparators

Comparators • Example: 2 -bit comparator

Comparators • K-Maps of the comparator outputs

Comparators • Example: MIS 4 -bit magnitude comparator

Comparators • 4 -bit magnitude comparator function table

Comparators • Cascading 4 -bit comparators to obtain a 16 -bit comparator