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### EELE 367 – Logic Design

- Module 1 – Classic Digital Design
- Agenda
- Number Systems
- Combinational Logic
- Sequential Logic

Number Systems

- Base Notation We’ll Use

- We will use the same notation as the HC12 Assembler.Decimal: nothing ex) 11 Base 10 = 10 Symbols Binary: % ex) %1011 Base 2 = 2 Symbols Octal: @ ex) @13 Base 4 = 4 Symbols Hexadecimal: $ ex) $BB Base 16 = 16 Symbols ASCII: ‘ ex) ‘a Code for Text

Number Systems

- Base Conversion – Binary to Decimal - each digit has a “weight” of 2n that depends on the position of the digit - multiply each digit by its “weight” - sum the resultant products ex) Convert %1011 to decimal 23 22 21 20 (weight) %1 0 1 1 = 1•(23) + 0• (22) + 1• (21) + 1• (20) = 1•(8) + 0• (4) + 1• (2) + 1• (1) = 8 + 0 + 2 + 1= 11 Decimal

Number Systems

- Base Conversion – Binary to Decimal with Fractions - the weight of the binary digits have negative positions ex) Convert %1011.101 to decimal 23 22 21 20 2-1 2-2 2-3 %1 0 1 1 . 1 0 1= 1•(23) + 0• (22) + 1• (21) + 1• (20) + 1• (2-1) + 0• (2-2)+ 1• (2-3) = 1•(8) + 0• (4) + 1• (2) + 1• (1) + 1• (0.5)+ 0• (0.25)+ 1• (0.125) = 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125= 11.625 Decimal

Number Systems

- Base Conversion – Decimal to Binary - the decimal number is divided by 2, the remainder is recorded - the quotient is then divided by 2, the remainder is recorded - the process is repeated until the quotient is zero ex) Convert 11 decimal to binaryQuotient Remainder 2 11 5 1 LSB 2 5 2 1 2 2 1 0 2 1 0 1 MSB = %1011

Number Systems

- Base Conversion – Decimal to Binary with Fractions - the fraction is converted to binary separately - the fraction is multiplied by 2, the 0th position digit is recorded - the remaining fraction is multiplied by 2, the 0th digit is recorded - the process is repeated until the fractional part is zero ex) Convert 0.375 decimal to binaryProduct 0th Digit0.375•2 0.75 0 MSB 0.75•21.50 1 0.5•21.00 1 LSB 0.375 decimal = % .011

finished

Number Systems

- Base Conversion – Hex to Decimal - the same process as “binary to decimal” except the weights are now BASE 16 - NOTE ($A=10, $B=11, $C=12, $D=13, $E=14, $F=15) ex) Convert $2BC to decimal 162 161 160 (weight) $2 B C = 2• (162) + B• (161) + C• (160) = 2•(256) + 11• (16) + 12• (1) = 512 + 176 + 12= 700 Decimal

Number Systems

- Base Conversion – Hex to Decimal with Fractions - the fractional digits have negative weights (BASE 16) - NOTE ($A=10, $B=11, $C=12, $D=13, $E=14, $F=15) ex) Convert $2BC.F to decimal 162 161 160 16-1 (weight) $2 B C . F = 2• (162) + B• (161) + C• (160) + F• (16-1) = 2•(256) + 11• (16) + 12• (1) + 15• (0.0625) = 512 + 176 + 12 + 0.938= 700.938 Decimal

Number Systems

- Base Conversion – Decimal to Hex - the same procedure is used as before but with BASE 16 as the divisor/multiplier ex) Convert 420.625 decimal to hex 1st, convert the integer part…Quotient Remainder16 420 26 4 LSB 16 26 1 10 16 1 0 1 MSB= $1A4

2nd, convert the fractional part…Product 0th Digit0.625•16 10.00 10 MSB= $ .A

420.625 decimal = $1A4.A

Number Systems

- Base Conversion – Octal to Decimal / Decimal to Octal - the same procedure is used as before but with BASE 8 as the divisor/multiplier

Number Systems

- Base Conversion – Hex to Binary - each HEX digit is made up of four binary bits ex) Convert $ABC to binary $A B C = % 1010 1011 1100= %1010 1011 1100

Number Systems

- Base Conversion – Binary to Hex - every 4 binary bits for one HEX digit - begin the groups of four at the LSB - if necessary, fill the leading bits with 0’s ex) Convert $ABC to binary= % 11 0010 1111

$3 2 F

Number Systems

- Binary Addition - same as BASE 10 addition - need to keep track of the carry bit for a given system size (n), i.e., 4-bit, 8-bit, 16-bit,… ex) Add %1011 and $10011 1% 1 0 1 1 % 1 0 0 1

+________ 1 0 1 0 0 Carry Bit

Number Systems

- Two’s Complement - we need a way to represent negative numbers in a computer - this way we can use “adding” circuitry to perform subtraction - since the number of bits we have is fixed (i.e., 8), we use the MSB as a sign bit Positive #’s : MSB = 0(ex: %0000 1111, positive number) Negative #’s : MSB = 1 (ex: %1000 1111, negative number) - the range of #’s that a two’s compliment code can represent is:(-2n-1) < N < (2n-1 – 1) : n = number of bits

ex) What is the range of #’s that an 8-bit two’s compliment code can represent?

(-28-1) < N < (28-1 – 1)(-128) < N < (+127)

Number Systems

- Two’s Complement Negation - to take the two’s compliment of a positive number (i.e., find its negative equivalent)Nc= 2n – N Nc = Two’s Compliment N = Original Positive Number

ex) Find the 8-bit two’s compliment representation of –52dec

N = +52dec

Nc= 28 – 52 = 256 – 52 = 204 = %1100 1100

Note the Sign Bit

Number Systems

- Two’s Complement Negation (a second method) - to take the two’s compliment of a positive number (i.e., find its negative equivalent) 1) Invert all the bits of the original positive number (binary) 2) Add 1 to the result

ex) Find the 8-bit two’s compliment representation of –52dec

N = +52dec = %0011 0100

Invert: = %1100 1011 Add 1 = %1100 1011 + 1 -------------------- %1100 1100 (-52dec)

Number Systems

- Two’s Complement Addition - Addition of two’s compliment numbers is performed just like standard binary addition - However, the carry bit is ignored

Number Systems

- Two’s Compliment Subtraction - now we have a tool to do subtraction using the addition algorithm ex) Subtract 8dec from 15dec 15dec = %0000 1111 8dec = %0000 1000 -> two’s compliment -> invert %1111 0111 add1 %1111 0111 + 1 -----------------%1111 1000 = -8decNow Add: 15 + (-8) = %0000 1111 %1111 1000 +_________ % 1 0000 0111 = 7dec Disregard Carry

Number Systems

- Two’s Compliment Overflow - If a two’s compliment subtraction results in a number that is outside the range of representation (i.e., -128 < N < +127), an “overflow” has occurred.ex) -100dec – 100dec- = -200dec (can’t represent) - There are three cases when overflow occurs 1) Sum of like signs results in answer with opposite sign 2) Negative – Positive = Positive 3) Positive – Negative = Negative - Boolean logic can be used to detect these situations.

Number Systems

- Binary Coded Decimal - Sometimes we wish to represent an individual decimal digit as a binary representation (i.e., 7-segment display to eliminate a decoder) - We do this by using 4 binary digits.DecimalBCD 0 0000 1 0001 ex) Represent 17dec 2 0010 3 0011 Binary = %10111 4 0100 BCD = %0001 0111 5 0101 6 0110 7 0111 8 1000 9 1001

Number Systems

- ASCII - American Standard Code for Information Interchange - English Alphanumeric characters are represented with a 7-bit code ex) ‘A’ = $40 ‘a’ = $61

Number Systems

- Important Things to Remember

Know the Code : A computer just knows 1’s and 0’s. It is up to you to keep trace of whether the bits represent unsigned numbers, two’s complement, ASCII, etc…

Two’s Complement Size : 2’s Complement always need a “size” associated with it. We always say “n-bit, two’s complement”

Basic Gates

Combinational Logic

Combinational Logic Gates :

- Output depends on the logic value of the inputs

- no storage

Basic Gates

XNOR out = abf(a,b) = ab

Also remember about XOR Gates:

f(a,b) = ab= (a’b + b’a)

Also remember the priority of logic operations (without parenthesis) is:

NOT, AND, OR

Boolean Algebra

- Boolean Algebra- formulated by mathematician George Boole in 1854- basic relationships & manipulations for a two-value system
- Switching Algebra- adaptation of Boolean Logic to analyzer and describe behavior of relays- Claude Shannon of Bell Labs in 1938- this works for all switches (mechanical or electrical)- we generally use the terms "Boolean Algebra" & "Switching Algebra" interchangeably

Boolean Algebra

- What is Algebra- the basic set of rules that the elements and operators in a system follow- the ability to represent unknowns using variables- the set of theorems available to manipulate expressions
- Boolean- we limit our number set to two values (0, 1)- we limit our operators to AND, OR, INV

Boolean Algebra

- Axioms- also called "Postulates"- minimal set of basic definitions that we assume to be true- all other derivations are based on these truths- since we only have two values in our system, we typically define an axiom and then its complement (A1 & A1')

Boolean Algebra

- Axiom #1 "Identity"- a variable X can only take on 1 or 2 values (0 or 1)- if it isn't a 0, it must be a 1- if it isn't a 1, it must be a 0 (A1) X = 0, if X ≠ 1 (A1') X = 1, if X ≠ 0
- Axiom #2 "Complement"- a prime following a variable denotes an inversion function (A2) if X = 0, then X' = 1 (A2') if X = 1, then X' = 0

Boolean Algebra

- Axiom #3 "AND"- also called "Logical Multiplication"- a dot (·) is used to represent an AND function
- Axiom #4 "OR"- also called "Logical Addition"- a plus (+) is used to represent an OR function
- Axiom #5 "Precedence"- multiplication precedes addition (A3) 0·0 = 0 (A3') 1+1 = 1

(A4) 1·1 = 1 (A4') 0+0 = 0

(A5) 0·1 = 1·0 = 0 (A5') 0+1 = 1+0 = 1

Boolean Algebra

- Theorems- Theorems use our Axioms to formulate more meaningful relationships & manipulations- a theorem is a statement of TRUTH- these theorems can be proved using our Axioms- we can prove most theorems using "Perfect Induction" - this is the process of plugging in every possible input combination and observing the output

Boolean Algebra

- Theorem #1 "Identity" (T1) X+0 = X (T1') X·1 = X
- Theorem #2 "Null Element" (T2) X+1 = 1 (T2') X·0 = 0
- Theorem #3 "Idempotency" (T3) X+X = X (T3') X·X = X
- Theorem #4 "Involution" (T4) (X')' = X
- Theorem #5 "Complements" (T5) X+X' = 1 (T5') X·X' = 0

Boolean Algebra

- Theorem #6 "Commutative" (T6) X+Y = Y+X (T6') X·Y = Y·X
- Theorem #7 "Associative" (T7) (X+Y)+Z= X+(Y+Z) (T7') (X · Y) · Z= X · (Y · Z)
- Theorem #8 "Distributive" (T8) X·(Y+Z) = X·Y + X·Z (T8') (X+Y)·(X+Z) = X + Y·Z

Boolean Algebra

- Theorem #9 "Covering" (T9) X + X·Y = X (T9') X·(X+Y) = X
- Theorem #10 "Combining" (T10) X·Y + X·Y' = X (T10') (X+Y)·(X+Y') = X
- Theorem #11 "Consensus" (T11) X·Y + X'·Z + Y·Z= X·Y + X'·Z (T11') (X+Y)·(X'+Z)·(Y+Z) = (X+Y) ·(X'+Z)

Boolean Algebra

- Notes on the Theorems- T9/T9' and T10/T10' are used heavily in logic minimization- these theorems can be useful for making routing more reasonable- these theorems can reduce the number of gates in a circuit- they can also change the types of gates that are used

Boolean Algebra

- More Theorem's - there are more generalized theorems available for large number of variables T13, T14, T15- one of the most useful is called "DeMorgan's Theorem"
- DeMorgan's Theorem - this theorem states a method to convert between AND and OR gates using inversions on the input / output

Boolean Algebra

- DeMorgan's TheoremPart 1: an AND gate whose output is complemented is equivalent to an OR gate whose inputs are complemented

Part 2: an OR gate whose output is complemented is equivalent to an AND gate whose inputs are complemented

=

=

Boolean Algebra

- Complement- complementing a logic function will give outputs that are inverted versions of the original function

ex) A B F F'

0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0

- DeMorgan's Theorem also gives us a generic formula to complement any expression: - for a Logic function F, we can get F' by : 1) Swapping all + and · 2) Complementing all Variables - KEEP THE PARENTHESIS ORDER OF THE ORGINAL FUNCTION !!!

Boolean Algebra

- Complement- Example: Complement the Function F

ex) A B F F'

0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0

We know: F = A · BWe swap + and · first, then we complement all variables F' = A' + B'- This is the same as putting an inversion bubble on the output of the logic diagram

→

Boolean Algebra

- Duality- An Algorithm to switch between Positive Logic and Negative Logic- Duality means that the logic expression is exactly the same even though the circuitry has been altered to produce Complementary Logic

- The steps are: - for a Logic function F, we can get FD by : 1) Swapping all + and · 2) Swapping all 0's and 1's

- Ex) F = A · B (Positive Logic) We swap + and · first, then swap any 0's and 1'sFD = A + B (Negative Logic or "Dual")

Boolean Algebra

- Complement vs. Duality, What is the difference?

Minterms

- Truth Tables

RowA B CF 0 0 0 0 1 Row We assign a "Row Number" for each entry starting at 0 1 0 0 1 0 2 0 1 0 0 Variables We enter all input combinations in ascending order. 3 0 1 1 1 We use straight binary with the MSB on the left

4 1 0 0 1

5 1 0 1 0 Function We say the output is a function of the input variables

6 1 1 0 1 F(A,B,C)

7 1 1 1 1 n = the number of input variables 2n = the number of input combinations

Minterms

- Let's also define the following termsLiteral = a variable or the complement of a variable ex) A, B, C, A', B', C'Product Term = a single literal or Logical Product of two or more literals ex) A A·B B'·CSum or Products = (SOP), the Logical Sum of Product Terms ex) A + B A·B + B'·C

Minterms

- Minterm- a normal product term w/ n-literals- a Minterm is written for each ROW in the truth table- there are 2n Minterms for a given truth table- we write the literals as follows: - if the input variable is a 0 in the ROW, we complement the Minterm literal - if the input variable is a 1 in the ROW, we do not complement the Minterm literal - for each ROW, we use a Logical Product on all of the literals to create the Minterm

Minterms

- Minterm

RowA B CMinterm F 0 0 0 0 A'·B'·C' F(0,0,0) 1 0 0 1 A'·B'·C F(0,0,1) 2 0 1 0 A'·B·C' F(0,1,0) 3 0 1 1 A'·B·C F(0,1,1) 4 1 0 0 A·B'·C' F(1,0,0) 5 1 0 1 A·B'·C F(1,0,1) 6 1 1 0 A·B·C' F(1,1,0) 7 1 1 1 A·B·C F(1,1,1)

- Canonical Sum

- we Logically Sum all Minterms that correspond to a Logic 1 on the output- the Canonical Sum represents the entire Logic Expression when the Output is TRUE- this is called the "Sum of Products" or SOP

Minterms

- Minterm List- we can also describe the full logic expression using a list of Minterms corresponding to a Logic 1- we use the Σ symbol to indicate we are writing a Minterm list- we list the Row numbers corresponding to a Logic 1

RowA B CMinterm F 0 0 0 0 A'·B'·C' 0 1 0 0 1 A'·B'·C 1 2 0 1 0 A'·B·C' 1 3 0 1 1 A'·B·C 0 4 1 0 0 A·B'·C' 0 5 1 0 1 A·B'·C 0 6 1 1 0 A·B·C' 1 7 1 1 1 A·B·C 0 F = ΣA,B,C (1,2,6) = (A'·B'·C) + (A'·B·C') + (A·B·C') - this is also called the "ON-set"- this list is very verbose and NOT minimized using our Axioms and Theorems (more on this later…)

Maxterms

- Let's define the following termsSum Term = a single literal or a Logical Sum of two or more literals ex) A A + B'Product of Sums = (POS), the Logical Product of Sum Terms ex) (A+B)·(B'+C)Normal Term = a term in which no variable appears more than once ex) "Normal A·B A + B' ex) "Non-Normal" A·B·B' A + A'

Maxterms

- Maxterm - a Normal Sum Term w/ n-literals- a Maxterm is written for each ROW in the truth table- there are 2n Maxterms for a given truth table- we write the literals as follows: - if the input variable is a 0 in the ROW, we do not complement the Maxterm literal - if the input variable is a 1 in the ROW, we complement the Maxterm literal - for each ROW, we use a Logical Sum on all of the literals to create the Maxterm

Maxterms

- Maxterm

RowA B CMinterm Maxterm F 0 0 0 0 A'·B'·C' A+B+C F(0,0,0) 1 0 0 1 A'·B'·C A+B+C' F(0,0,1) 2 0 1 0 A'·B·C' A+B'+C F(0,1,0) 3 0 1 1 A'·B·C A+B'+C' F(0,1,1) 4 1 0 0 A·B'·C' A'+B+C F(1,0,0) 5 1 0 1 A·B'·C A'+B+C' F(1,0,1) 6 1 1 0 A·B·C' A'+B'+C F(1,1,0) 7 1 1 1 A·B·C A'+B'+C' F(1,1,1)

- Canonical Product

- we Logically Multiply all Maxterms that correspond to a Logic 0 on the output- the Canonical Product represents the entire Logic Expression when the Output is TRUE- this is called the "Product of Sums" or POS

Maxterms

- Maxterm List- we can also describe the full logic expression using a list of Maxterms corresponding to a Logic 0- we use the π symbol to indicate we are writing a Maxterm list- we list the Row numbers corresponding to a Logic 0

RowA B CMintermMaxtermF 0 0 0 0 A'·B'·C' A+B+C 0 1 0 0 1 A'·B'·C A+B+C' 1 2 0 1 0 A'·B·C' A+B'+C 1 3 0 1 1 A'·B·C A+B'+C' 0 4 1 0 0 A·B'·C' A'+B+C 0 5 1 0 1 A·B'·C A'+B+C' 0 6 1 1 0 A·B·C' A'+B'+C 1 7 1 1 1 A·B·C A'+B'+C' 0

F = πA,B,C (0,3,4,5,7) = (A+B+C) · (A+B'+C') · (A'+B+C) · (A'+B+C') · (A'+B'+C') - this is also called the "OFF-set”, this list is very verbose and NOT minimized

Maxterms

- Maxterm vs. Minterm

- a Maxterm is the Dual of a Minterm- this implies an inversion- however, by writing a POS for when the Maxterm is a Logic 0, we perform another inversion- these two inversions yield the original logic expression for when the function is 1- SOP = POS

Maxterms

- Minterms & Maxterm- we now have 5 ways to describe a Logic Expression 1) Truth Table 2) Minterm List 3) Canonical Sum 4) Maxterm List 5) Canonical Product- these all give the same information
- Converting Between Minterms and Maxterms- Minterms and Maxterms are Duals- this means we can convert between then easily using DeMorgan's duality theorem- converting a SOP to its dual gives Negative Logic- converting a POS to its dual gives Negative Logic

Circuit Synthesis

- Circuit Synthesis - there are 5 ways to describe a Logic Expression 1) Truth Table 2) Minterm List 3) Canonical Sum 4) Maxterm List 5) Canonical Product- we can directly synthesis circuits from SOP and POS expressions SOP = AND-OR structure POS = OR-AND structure

Circuit Synthesis

- Circuit Synthesis - For the given Truth Table, synthesize the SOP and POS Logic DiagramsRowA BMinterm MaxtermF 0 0 0 A'·B' A+B 1 1 0 1 A'·B A+B' 1 2 1 0 A·B' A'+B 0 3 1 1 A·B A'+B' 0Minterm List & SOPMaxterm List & POSF = ΣA,B (0,1) = A'·B' + A'·B F = πA,B (2,3) = (A'+B) · (A'+B')

Circuit Synthesis

- Circuit Manipulation - we can manipulate our Logic Diagrams to give the same logic expression but use different logic gates- this can be important when using technologies that: - only have certain gates (i.e., INV, NAND, NOR) - have certain gates that are faster than others (i.e., NAND-NAND, NOR-NOR)- we can convert a SOP/POS logic diagram into a NAND-NAND or NOR-NOR structure

Circuit Synthesis

- Common Circuit Manipulation "Moving Inversion Bubbles"

Circuit Synthesis

- Common Circuit Manipulation "Inserting Double Inversion Bubbles"

Circuit Synthesis

- Common Circuit Manipulation "Bubbles can be moved to either side of an Inverter"

Logic Minimization

- Logic Minimization- We've seen that we can directly translate a Truth Table into a SOP/POS and in turn a Logic Diagram- However, this type of expression is NOT minimizedex) RowA BMinterm MaxtermF 0 0 0 A'·B' A+B 1 1 0 1 A'·B A+B' 1 2 1 0 A·B' A'+B 0 3 1 1 A·B A'+B' 0Minterm List & SOPMaxterm List & POSF = ΣA,B (0,1) = A'·B' + A'·B F = πA,B (2,3) = (A'+B) · (A'+B')

Logic Minimization

- Logic Minimization- using our Axioms and Theorems, we can manually minimize the expressions…Minterm List & SOPMaxterm List & POS F = A'·B' + A'·B F = (A'+B) · (A'+B') F = A'·(B'+B) = A' F = A' + (B'·B) = A'- doing this by hand can be difficult and requires that we recognize patterns associated with our 5 Axioms and our 15+ Theorems
- Karnaugh Maps- a graphical technique to minimize a logic expression

Logic Minimization

- Karnaugh Map Creation- we create an array with 2n cells- each cell contain the value of F at a particular input combination- we label each Row and Column so that we can easily determine the input combinations- each cell only differs from its adjacent neighbors by 1-variable

List Variables Top-to-Bottom

List Input Combinations for the Variables for each Row/Column

A

B

0

1

0

1

Logic Minimization

- Karnaugh Map Creation

A

A

B

0

1

0

2

0

1

3

1

B

We can also put the Truth Table Row number so that copying the Truth Table values into the K-map is straight-forward

We can put redundant labeling for when a variable is TRUE. This helps when creating larger K-maps

Logic Minimization

- Karnaugh Map Creation- we can now copy in the Function values RowA BMinterm MaxtermF 0 0 0 A'·B' A+B 11 0 1 A'·B A+B' 12 1 0 A·B' A'+B 03 1 1 A·B A'+B' 0- at this point, the K-map is simply the same information as in the Truth Table- we could write a SOP or POS directly from the K-map if we wanted to

A

A

B

0

1

0

1

2

0

0

1

1

0

3

1

B

Logic Minimization

- Karnaugh Map Creation- we can create 3 variable K-Maps- notice the input combination numbering in order to achieve no more than 1-variable difference between cells

AB

A

C

00

01

11

10

0

2

6

4

0

1

3

7

5

1

C

B

Logic Minimization

- Karnaugh Map Creation- we can create 4 variable K-Maps

AB

A

CD

00

01

11

10

0

4

12

8

00

1

5

13

9

01

D

3

7

15

11

11

C

2

6

14

10

10

B

Logic Minimization

- Karnaugh Map Minimization- we can create a minimized SOP logic expression by performing the following: 1) Circle adjacent 1's in groups of power-of-2 - powers of 2 means 1,2,4,8,16,…. - adjacent means neighbors above, below, right, left (not diagonal) - we can wrap around the ends to form group - this is called "Combining Cells" 2) We then write a Product Term for each circle following: - if the circle covers areas where the variable is 0, we enter a complemented literal in the product term - if the circle covers areas where the variable is 1, we enter an non-complemented literal in the product term - if the circle covers areas where the variable is both a 0 and 1, we can exclude the variable from the product term 3) We then Sum the Product Terms

Logic Minimization

- Karnaugh Map Minimization- let's write a minimized SOP for the following K-map

A

A

B

0

1

1

0

2

0

0

1

1

3

0

1

B

- this circle covers 2 cells- the circle covers: - where A=0, so the literal is A' - where B=0 and 1, so the literal is excluded- our final SOP expression is F = A'

Logic Minimization

- Karnaugh Map Minimization

- this circle covers 1 cell - the circle covers: - where A=0, so the literal is A' - where B=1, so the literal is B - where C=0, so the literal is C' - The product term for this circle is : A'·B·C'

AB

A

C

00

01

11

10

0

0

2

1

0

6

0

4

0

1

1

3

0

1

7

1

5

1

C

- this circle covers 2 cells - the circle covers: - where A=1, so the literal is A - where B=0 and 1, exclude the literal - where C=1, so the literal is C- The product term for this circle is : A·C

B

- this circle covers 2 cells- the circle covers: - where A=0 and 1, exclude the literal - where B=0, so the literal is B' - where C=1, so the literal is C- The product term for this circle is : B'·C

- our final minimized SOP expression is F = A'·B·C' + A·C + B'·C

Logic Minimization

- Karnaugh Map Minimization- the original Canonical SOP for this Map would have been F = A'·B'·C + A'·B·C' + A·B'·C + A·B·C - our minimized SOP expression is now: F = A'·B·C' + A·C + B'·C- this minimization resulted in: - one less input on the OR gate - one less AND gate - two AND gates having 2 inputs instead of 3

AB

A

C

00

01

11

10

0

0

2

1

0

6

4

0

0

1

0

1

1

1

3

7

5

1

C

B

Logic Minimization

- 2-Variable K-Map Example- write a minimal SOP for the following truth table RowA BMinterm MaxtermF 0 0 0 A'·B' A+B 01 0 1 A'·B A+B' 12 1 0 A·B' A'+B 13 1 1 A·B A'+B' 1- we first copy in the output values into the K-map

A

A

B

0

1

0

0

2

1

0

1

1

1

3

1

B

Logic Minimization

- 2-Variable K-Map Example - we then circle groups of 1's in order to find the Product Term for each circle

A

A

- this circle covers 2 cells - the circle covers: - where A=1, so the literal is A - where B=0 and 1, exclude the literal - The product term for this circle is : A

B

0

1

0

0

1

2

0

1

1

3

1

1

B

- this circle covers 2 cells - the circle covers: - where A=0 and 1, exclude the literal - where B=1, so the literal is B - The product term for this circle is : B

Logic Minimization

- 2-Variable K-Map Example - we then write a SOP expression for each circles' Product Term

A

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The minimized SOP is : A + B

Logic Minimization

- 3-Variable K-Map Example - write a minimal SOP expression for the following truth tableRowA B CMinterm MaxtermF 0 0 0 0 A'·B'·C' A+B+C 0 1 0 0 1 A'·B'·C A+B+C' 1 2 0 1 0 A'·B·C' A+B'+C 1 3 0 1 1 A'·B·C A+B'+C' 0 4 1 0 0 A·B'·C' A'+B+C 1 5 1 0 1 A·B'·C A'+B+C' 1 6 1 1 0 A·B·C' A'+B'+C 1 7 1 1 1 A·B·C A'+B'+C' 1

AB

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Logic Minimization

- 3-Variable K-Map Example

- this circle covers 2 cells - the circle covers: - where A=0/1, so exclude literal - where B=1, so the literal is B - where C=0, so the literal is C' - The product term for this circle is : B·C'

AB

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- this circle covers 4 cells - the circle covers: - where A=1, so the literal is A - where B=0/1, so exclude the literal - where C=0/1, so exclude the literal - The product term for this circle is : A

B

- this circle covers 2 cells - the circle covers: - where A=0 and 1, exclude the literal - where B=0, so the literal is B' - where C=1, so the literal is C - The product term for this circle is : B'·C

- our final minimized SOP expression is F = A + B·C' + B'·C

Logic Minimization

- 4-Variable K-Map Example - write a minimal SOP expression for the following truth tableRowA B C DF 0 0 0 0 0 0 1 0 0 0 1 1 2 0 0 1 0 0 3 0 0 1 1 1 4 0 1 0 0 0 5 0 1 0 1 0 6 0 1 1 0 0 7 0 1 1 1 0 8 1 0 0 0 1 9 1 0 0 1 1 10 1 0 1 0 1 11 1 0 1 1 1 12 1 1 0 0 0 13 1 1 0 1 0 14 1 1 1 0 0 15 1 1 1 1 0

AB

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CD

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Logic Minimization

- this circle covers 4 cells - the circle covers: - where A=1, so the literal is A - where B=0, so the literal is B' - where C=0/1, so exclude the literal - where D=0/1, so exclude the literal - The product term for this circle is : A·B'

- 4-Variable K-Map Example

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- this circle covers 4 cells - the circle covers: - where A=0/1, so exclude the literal - where B=0, so the literal is B' - where C=0/1, so exclude the literal - where D=1, so the literal is D - The product term for this circle is : B'·D

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- our final minimized SOP expression is F = A·B' + B'·D

Logic Minimization

- K-Map Logic Minimization- we can use K-maps to write a minimal SOP (and POS)- however, we've seen that there is a potential for redundant Product Terms- we need to define what it is to be "Minimized"

AB

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Is this circle necessary?

Logic Minimization

- K-Map Logic MinimizationMinimal Sum - No other expression exists that has - fewer product terms - fewer literals

Imply - a logic function P "implies" a function F if - every input combination that causes P=1 - also causes F=1 - may also cause more 1's - we say that: - "F includes P" - "F covers P" - "F => P"

Logic Minimization

- K-Map Logic MinimizationPrime Implicant - a Normal Product Term of F (i.e., a P that implies F) where if any variable is removed from P, the resulting product does NOT imply F - K-maps: a circled set of 1's that cannot be larger without circling 1 or more 0'sPrime Implicant Theorem - a Minimal Sum is a sum of Prime Implicants BUT Does not need to include ALL prime Implicants

Logic Minimization

- K-Map Logic MinimizationComplete Sum - the product of all Prime Implicants (not minimized)Distinguished 1-Cell - an "input combination" that is covered by only ONE Prime ImplicantEssential Prime Implicant - a Prime Implicant that covers one or more "Distinguished 1-Cells" NOTE: - the sum of Essential Prime Implicants is the Minimal Sum - this means we're done minimizing

Logic Minimization

- K-Map Logic MinimizationSteps for Minimization 1) Identify All Prime Implicants 2) Identify the Distinguished 1-Cells 3) Identify the Essential Prime Implicants 4) Create the SOP using Essential Prime Implicants

Logic Minimization

- Don't Cares- sometimes it doesn't matter whether the output is a 1 or 0 for a certain input combination- we can take advantage of this during minimization by treating the don't care as a 1 or 0 depending on whether it makes finding Prime Implicants easier- We denote X = Don't CareRowA B CMinterm MaxtermF 0 0 0 0 A'·B'·C' A+B+C 0 1 0 0 1 A'·B'·C A+B+C' 0 2 0 1 0 A'·B·C' A+B'+C 0 3 0 1 1 A'·B·C A+B'+C' X 4 1 0 0 A·B'·C' A'+B+C 1 5 1 0 1 A·B'·C A'+B+C' 1 6 1 1 0 A·B·C' A'+B'+C 1 7 1 1 1 A·B·C A'+B'+C' X

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B

Timing Hazards

- Hazards- we've only considered the Static (or steady state) values of combination logic- in reality, there is delay present in the gates- this can cause different paths through the circuit which arrive at different times at the input to a gate- this delay can cause an unwanted transition or "glitch" on the output of the circuit- this behavior is known as a "Timing Hazard"- a Hazard is the possibility of an input combination causing a glitch

Timing Hazards

- Static-1- when we expect the output to produce a steady 1, but a 0-glitch occurs- this occurs in SOP (AND-OR) structuresDefinition A pair of input combinations that (a) differ in only one input variable (b) both input combinations produce a 1 There is a possibility that a change between these input combinations will cause a 0

Timing Hazards

- Static-0- when we expect the output to produce a steady 0, but a 1-glitch occurs- this occurs in POS (OR-AND) structuresDefinition A pair of input combinations that (a) differ in only one input variable (b) both input combinations produce a 0 There is a possibility that a change between these input combinations will cause a 1

Timing Hazards

- Hazards and K-maps- K-maps graphically show input combinations that vary by only one variable- it is easy to see when adjacent cells have 1's and have a potential Timing Hazard- this is a Minimal Sum, BUT, what about the transition from A·B·C to A·B·C'?- there is a Timing Hazard present!!!

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Timing Hazards

- Hazards and K-maps- the solution is to add an additional product term (Prime Implicant) to cover the transition- this ensures that the output is valid while transitioning between any input combination- this is NOT a Minimal Sum, but it is Hazard Free

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Timing Hazards

- Dynamic Hazards- when we undergo a transition on the output but multiple transitions occur- this is again due to multiple paths w/ different delays from input to output- typically is larger leveled logic- Solution: If the circuit is Static Hazard Free, then it is Dynamic Hazard Free

Timing Hazards

- Hazard Prevention- adding redundant Prime Implicants will prevent Hazards but can sometime add too much logic- we can also perform delay matching through the circuit by inserting buffers so that the delay is the same at each level of logic

Design Flow

- Combinational Logic Design Flow- We now have all the pieces for a complete design process

1) Design Specifications : description of what we want to do 2) Truth Table : listing the logical operation of the system 3) Describe using : creating the logic expression SOP/POS/Minterm/Maxterm 4) Logic Minimization : K-maps 5) Logic Manipulation : Convert to desired technology (NAND/NAND, …) 6) Hazard Prevention

Q

Q

Q

Sequential LogicSequential Logic

- Concept of “Storage Element”

- With Storage, logic functions can depend on current & past values of inputs

- Sequential State Machines can be created

D-Flip-Flop

- on timing event (i.e., edge of clock input), D input goes to Q output

CLK

D

Q

tc2q

Finite State Machines

- Synchronous- we now have a way to store information on an edge (i.e., a flip-flop)- we can use these storage elements to build "Synchronous Circuitry"- Synchronous means that events occur on the edge of a clock
- State Machine- a generic name given to sequential circuit design- it is sequential because the outputs depend on : 1) the current inputs 2) past inputs - state machines are the basis for all modern digital electronics

Finite State Machines

- State Memory- a set of flip-flops that store the CurrentState of the machine- the Current State is due to all of the input conditions that have existed since the machine started- if there are "n" flip-flops, there can be 2n states- a state contains everything we need to know about all past events- we define two unique states in a machine 1) Current State 2) Next State
- Current State - the state that the machine is currently in- this is found on the outputs of the flip-flops (i.e., Q)

Finite State Machines

- Next State- the state that the machine "will go to" upon a clock edge- this is found on the inputs of the flip-flops (i.e., D)- the Next State depends on - the Current State - any Inputs- we call the Next State Logic "F"

Finite State Machines

- State Transition- upon a clock edge, the machine changes from the "Current State" to the "Next State"- After the clock edge, we reassign back the names (i.e., Q=Current State, D= Next State)
- State Table- a table where we list which state the machine will transition to on a clock edge- this table depends on the "Current State" and the "Inputs"- we can use the following notation when describing Current and Next StatesCurrent Next S S* Scur Snxt SC SN Acur Bnxt QC QN

Finite State Machines

- State Table cont…- we typically give the states of our machine descriptive names (i.e., Reset, Start, Stop, S0)ex) State Table for a 4-state counter, no inputs Current State Next State S0 S1 S1 S2 S2 S3 S3 S0

Finite State Machines

- State Table cont…- when there are inputs, we list them in the tableex) State Table for a 4-state counter with directional input Current State Dir Next State S0 Up S1 Down S3 S1 Up S2 Down S0 S2 Up S3 Down S1 S3 Up S0 Down S2 - we don't need to exhaustively write all of the Current States for each Input combination (i.e., Direction) which makes the table more readable

Finite State Machines

- State Variables- remember that the State Memory is just a set of flip-flops- also remember that the state is just the current/next binary number on the in/out of the flip-flops- we use the term State Variable to describe the input/output for a particular flip flop- let's call our State Variables Q1, Q0, …. Current State Dir Next State Q1 Q0 Q1* Q0* 0 0 Up 0 1 Down 1 1 0 1 Up 1 0 Down 0 0 1 0 Up 1 1 Down 0 1 1 1 Up 0 0 Down 1 0

Finite State Machines

- State Variables- we call the assignment of a State name to a binary number "State Encoding"ex) S0 = 00 S1 = 01 S2 = 10 S3 = 11- this is arbitrary and up to use as designers- we can choose the encoding scheme based on our application and optimize for 1) Speed 2) Power 3) Area

Finite State Machines

- Next State Logic "F"- for each "Next State" variable, we need to create logic circuitry- this logic circuitry is combinational and has inputs: 1) Current State 2) any Inputs ex) Q1* = F(Current State, Inputs) = F(Q1, Q0, Dir) Q0* = F(Current State, Inputs) = F(Q1, Q0, Dir) - the logic expression for the Next State Variable is called the "Characteristic Equation" - this is a generic term that describes the logic for all flip-flops- when we write the logic for a specific flip-flop, we call this the "Excitation Equation" - this is a specific term that describes logic for your flip-flop (i.e., DFF)- there will be an Excitation Equation for each Next State Variable (i.e., each Flip-Flop)

Finite State Machines

- State Variables and Next State Logic "F"- we already know how to describe combinational logic (i.e., K-maps, SOP, SOP)- we simply apply this to our State Tableex) State Table for a 4-state counter, no inputs Current State Next State Q1 Q0 Q1* Q0* 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 0 - we put these inputs and outputs in a K-map and come up with Q1* = Q1 Q0

NextState Variable

F inputs

F outputs

Finite State Machines

- State Variables and Next State Logic "F"- we continue writing Excitation Equations for each Next State Variable in our Machine- let's now write an Excitation Equation for Q0*ex) State Table for a 4-state counter, no inputs Current State Next State Q1 Q0 Q1* Q0* 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 0 - we simply put these values in a K-map and come up with Q0* = Q0'

NextState Variable

F inputs

F outputs

Finite State Machines

- Logic Diagram- the state machine just described would be implemented like this:

Finite State Machines

- Excitation Equations- we designed this State Machine using D-Flip-Flops- this is the most common type of flip-flop and widely used in modern digital systems- we can also use other flip-flops- the difference is how we turn a "Characteristic Equation" into an "Excitation Equation"- we will look at State Memory using other types of Flip-Flops later

Finite State Machines (Mealy vs. Moore)

- Output Logic "G"- last time we learned about State Memory- we also learned about Next State Logic "F"- the last part of our State Machine is how we create the output circuitry- the outputs are determined by Combinational Logic that we call "G"- there are two basic structures of output types 1) Mealy = the outputs depend on the Current State and the Inputs 2) Moore = the outputs depend on the Current State only

Finite State Machines (Mealy vs. Moore)

- State Machines“Mealy Outputs” – outputs depend on the Current State and the Inputs- G(Current State, Inputs)

Finite State Machines (Mealy vs. Moore)

- State Machines“Moore Outputs” – outputs depend on the Current State only - G(Current State)

Finite State Machines (Mealy vs. Moore)

- Output Logic "G"- we can create an "Output Table" for our desired results- most often, we include the outputs in our State Table and form a hybrid "State/Output Table"ex) Current State In Next State Out Q1 Q0 Q1* Q0* 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1

Finite State Machines (Mealy vs. Moore)

- State Machine Tables- officially, we use the following terms:State Table - list of the descriptive state names and how they transitionTransition Table - using the explicitly state encoded variables and how they transition Output Table - listing of the outputs for all possible combinations of Current States and Inputs State/Output Table - combined table listing Current/Next states and corresponding outputs

Finite State Machines (Mealy vs. Moore)

- Output Logic "G"- we simply use the Current State and Inputs (if desired) as the inputs to G and form the logic expression (K-maps, SOP, POS) reflecting the output variable- this is the same process as creating the Excitation Equations for the Next State Variablesex) Current State In Next State Out Q1 Q0 Q1* Q0* 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1- plugging these inputs/outputs into a K-map, we get G=Q1·Q0·In

OutputVariable

G outputs

G inputs

Finite State Machines

- State Diagrams- a graphical way to describe how a state machine transitions states depending on : - Current State - Inputs- we use a "bubble" to represent a state, in which we write its descriptive name or code- we use a "directed arc" to represent a transition- we can write the Inputs next to a directed arc that causes a state transition- we can write the Output either near the directed arc or in the state bubble (typically in parenthesis)

Finite State Machines

- State DiagramsEx)

Finite State Machines

- Timing Diagrams- notice we don't show any information about the clock in the State Diagram- the clock is "implied" (i.e., we know transitions occur on the clock edge)- note that the outputs can change asynchronously in a Mealy machine

Clock

In

Found

S0

S1

S2

STATE

t

Finite State Machines

- State Machines- there is a basic structure for a Clocked, Synchronous State Machine 1) State Memory (i.e., flip-flops) 2) Next State Logic “F” (combinational logic) 3) Output Logic “G” (combinational logic)

Finite State Machines

- State Machines- the steps in a state machine design are: 1) Word Description of the Problem 2) State Diagram 3) State/Output Table 4) State Variable Assignment 5) Choose Flip-Flop type 6) Construct F 7) Construct G 8) Logic Diagram

Finite State Machines

- State Machine Example “Simple Gray Code Counter”1) Design a 2-bit Gray Code Counter. There are no inputs (other than the clock). Use Binary for the State Variable Encoding2) State Diagram

Finite State Machines

- State Machine Example “Simple Gray Code Counter”3) State/Output Table Current State Next State Out CNT0 CNT1 0 0 CNT1 CNT2 0 1 CNT2 CNT3 1 1 CNT3 CNT0 1 0

Finite State Machines

- State Machine Example “Simple Gray Code Counter”4) State Variable Assignment – binary Current State Next State Out Q1 Q0 Q1* Q0* 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 1 05) Choose Flip-Flops - let's use DFF's

Finite State Machines

- State Machine Example “Simple Gray Code Counter”6) Construct Next State Logic “F” Q1* = Q1 Q0 Q0* = Q0’

Q1

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Finite State Machines

- State Machine Example “Simple Gray Code Counter”7) Construct Output Logic “G” Out(1) = Q1 Out(0) = Q1 Q0

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Finite State Machines

- State Machine Example “Simple Gray Code Counter”8) Logic Diagram

Finite State Machines

- State Machines- there is a basic structure for a Clocked, Synchronous State Machine 1) State Memory (i.e., flip-flops) 2) Next State Logic “F (combinational logic) 3) Output Logic “G” (combinational logic)

Finite State Machines

- State Machines- the steps in a state machine design are: 1) Word Description of the Problem 2) State Diagram 3) State/Output Table 4) State Variable Assignment 5) Choose Flip-Flop type 6) Construct F 7) Construct G 8) Logic Diagram

Finite State Machines

- State Machine Example “Sequence Detector”1) Design a machine by hand that takes in a serial bit stream and looks for the pattern “1011”. When the pattern is found, a signal called “Found” is asserted2) State Diagram

Finite State Machines

- State Machine Example “Sequence Detector”3) State/Output Table Current State In Next State Out (Found) S0 0 S0 0 1 S1 0 S1 0 S2 0 1 S0 0 S2 0 S0 0 1 S3 0 S3 0 S0 0 1 S0 1

Finite State Machines

- State Machine Example “Sequence Detector”4) State Variable Assignment – let’s use binary Current State In Next State Out Q1 Q0 Q1* Q0* Found 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 15) Choose Flip-Flop Type - 99% of the time we use D-Flip-Flops

Finite State Machines

- State Machine Example “Sequence Detector”6) Construct Next State Logic “F” Q1* = Q1’∙Q0∙In’ + Q1∙Q0’∙In Q0* = Q0’∙In

Q1

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Finite State Machines

- State Machine Example “Sequence Detector”7) Construct Output Logic “G” Found = Q1∙Q0∙In8) Logic Diagram

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State Variable Encoding

- State Variable Encoding- we design State Machines using descriptive state names- when we’re ready to synthesize the circuit, we need to assign values to the states- we can arbitrarily assign codes to the states in order to optimize for our application

State Variable Encoding

- Binary- this is the simplest and most straight forward method- we simply assign binary counts to the states S0 = 00 S1 = 01 S2 = 10 S3 = 11- for N states, there will be log(N)/log(2) flip flops required to encode in binary- the advantages to Binary State Encoding are: 1) Efficient use of area (i.e., least amount of Flip-flops)- the disadvantages are: 1) Multiple bits switch at one time leading to increased Noise and Power 2) The next state logic can be multi-level which decreases speed

State Variable Encoding

- Gray Code- to avoid multiple bits switching, we can use a Gray Code S0 = 00 S1 = 01 S2 = 11 S3 = 10- for N states, there will be log(N)/log(2) flip flops required- the advantages of Gray Code Encoding are: 1) One-bit switching at a time which reduces Noise/Power - the disadvantages are: 1) Unless doing a counter, it is hard to guarantee the order of state execution 2) The next state logic can again be multi-level which decreases speed

State Variable Encoding

- One-Hot- this encoding technique uses one flip-flop for each state.- each flip-flop asserts when in its assigned state S0 = 0001 S1 = 0010 S2 = 0100 S3 = 1000- for N states, there will be N flip flops required- the advantages of One-Hot Encoding are: 1) Speed, the next state logic is one level (i.e., a Decoder structure) 2) Suited well for FPGA’s which have LUT’s & FF’s in each Cell - the disadvantages are: 1) Takes more area

01

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Sequential LogicState Machine Example: Design a 2-bit Gray Code Counter with “State Encoded Outputs”

1) Number of States? : 4

2) Number of bits to encode states? : 2n=4, n=2

3) Moore or Mealy? : Moore

For this counter, we can make the outputs be the state codes

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D

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Sequential LogicState Machine Example: Design a 2-bit Gray Code Counter

STATE

CurrentNext

Acur Bcur Anxt Bnxt

0 0 0 1

0 1 1 1

1 1 1 0

1 0 0 0

Anxt Logic

Bnxt Logic

Bcur0 1

Bcur0 1

Acur 0 1

Acur 0 1

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0

Bnxt = Acur’

Anxt = Bcur

A

counter

output

B

A

B

CLK

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