Appendix B The Basics of Logic Design Slides

Appendix B Table of Contents - Gates, Truth Tables, and Logic Equations Combinational Logic Using a Hardware Description Language Constructing a Basic Arithmetic Logic Unit Faster Addition: Carry Lookahead Clocks Memory Elelments: Flip-flops, Latches, and Registers Memory Elements: SRAMs and DRAMs Finite State Machines Timing Methodologies Field Programmable Devices

Gates, Truth Tables, and Logic Equations • Computer electronics are digital – only two voltages of interest • high = 1 = asserted = active = true • low = 0 = deasserted = inactive = false – really a single voltage level • all voltages above are high • all voltages below are low • Logic blocks – combinational logic = no memory elements – sequential logic = memory elements

Truth Tables • A table of all possible inputs and the associated outputs for a particular circuit • All combinational logic can be completely described using a truth table • Non-zero output truth tables only list the inputs that result in non-zero outputs • Text example: for all values of A, B, and C, let D be true if at least one input is true, let E be true if exactly two inputs are true, and let F be true only if all three inputs are true.

Boolean Algebra • binary OR is written + – logical sum – result is 1 if at least one variable is 1 • binary AND is written * – logical product – result is 1 only if both variables are 1 • unary NOT is written A

Boolean Algebra Laws Identity A+0=A A*1=A Zero and One A+1=1 A*0=0 Inverse A+A=1 A*A=0 A+B=B+A A*B=B*A A+(B+C)=(A+B)+C A*(B*C)=(A*B)*C Commutative Associative Distributive A*(B+C)=(A*B)+(B*C) A+(B*C)=(A+B)*(B+C)

De. Morgan’s Theorems ____ _ _ A + B = A * B ____ _ _ A * B = A + B Try to prove De. Morgan’s Theorems using a truth table!

De. Morgan’s Theorems Perhaps like this one!

Gates Definition: A gate is a device that implements basic logic functions, such as AND or OR. Logic blocks are built from gates that implement basic logic functions. Since AND is commutative and associative, an AND gate can have multiple inputs, with the output equal to the AND of all the inputs. The same is true of OR. The logical function NOT is implemented with an inverter that always has a single input.

AND gate OR gate inverter

A + B

NAND and NOR Gates In fact, all logic functions can be constructed with only a single gate type, if that gate is inverting. The two common inverting gates are called NOR and NAND and correspond to inverted OR and AND gates, respectively. NOR and NAND gates are called universal, since any logic function can be built using this one gate type.

Combinational Logic • • • Decoders Multiplexors Two-Level Logic and PLAs ROMs Don’t Cares Arrays of Logic Elements

Decoders Definition: A decoder is a logic block that has an n-bit input and 2 n outputs where only one output is asserted for each input combination.

A 3 -to-8 Decoder

The Truth Table For A 3 -to-8 Decoder

Multiplexors A basic logic function that is used quite often A multiplexor could be called a selector • its output is one of the inputs • the input which is to be output is selected by a special input Definition: A selector value (or control value) is the control signal that is used to select one of the input values of a multiplexor as the output of the multiplexor.

Two-Input Multiplexor How we draw it! The implementation with gates!

Two-Level Logic and PLAs • Any logic function can be implemented with only AND, OR, and NOT functions. • A canonical form is used, where every input is either a true or complemented (inverted) variable and there are only two levels of gates – one AND and the other OR – with a possible inversion on the final output • Definition: The sum of products is a logical representation that uses a logical sum (OR) of products (terms using AND).

Programmable Logic Array Definition: A programmable logic array (PLA) is a structured-logic element composed of a set of inputs and corresponding input complements and two stages of logic – the first stage generates product terms of the inputs and input complements – the second stage generates sum terms of the product terms Definition: Minterms (or product terms) are a set of logic inputs joined by conjunction (AND) Product terms form the first logic stage of a PLA.

Programmable Logic Array

ROMs Definition: A read-only memory (ROM) is a memory whose contents are designated at creation time, after which the contents can only be read ROMs can be used as structured logic to implement a set of logic functions – use the terms in the logic functions as address inputs – use the outputs as bits in each memory word Definition: A programmable ROM (PROM) is a form of ROM that can be programmed when a designer knows what to put into it

Don’t Cares Definition: A “don’t care” is a situation where we do not care what the value of some input or some output “Don’t cares” occur often in implementing some combinational logic We don’t care about an input or output either because another output is true or because a subset of the input combinations determines the values of the outputs

Arrays of Logic Elements • Many of the combinational operations to be performed on data have to be done to an entire word (32 bits) of data. • Definition: A bus is a collection of data lines that is treated together as a single logical signal (also, a shared collection of lines with multiple sources and uses)

A Multiplexor That Selects Between Two 32 -bit Buses

Constructing a Basic Arithmetic Logic Unit • A 1 -Bit ALU • A 32 -Bit ALU • Tailoring the 32 -Bit ALU to MIPS • Defining the MIPS ALU in Verilog

A 1 -Bit ALU The logical operations are easiest, because they map directly onto the hardware components: – AND gate – OR gate – Inverter

The 1 -bit Logical Unit for AND and OR

A 1 -bit Adder

Input and Output Specification for a 1 -bit Adder

Values of the Inputs When Carry. Out is a 1

Adder Hardware for the Carry. Out Signal

A 1 -bit ALU that Performs AND, OR, and Addition

A 32 -Bit ALU • Now that we have completed the 1 -bit ALU, the full 32 -bit ALU is created by connecting adjacent “black boxes. ”

A 32 -bit ALU Constructed from 32 1 -bit ALUs

1 -bit ALU that ANDs, ORs, or ADDs a and b or a and ~b

1 -bit ALU that ANDs, ORs, or ADDs a or ~a with b or ~b

Adding a Comparison Function to the 32 -Bit ALU • The four operations—add, subtract, AND, OR—are found in the ALU of almost every computer, and the operations of most instructions can be performed by this ALU. But the design of the ALU is incomplete. • One instruction that still needs support is the a comparison function: – set on less than (slt).

1 -bit ALU: ANDs, ORs, ADDs, and Compares a or ~a, b or ~b

1 -bit ALU for the Most Significant Bit

A 32 -bit ALU Constructed from 31 copies of the 4 -function 1 -bit ALU and one special 4 -function 1 -bit ALU for the MSB

The Final 32 -bit ALU

The values of the three ALU control lines Bnegate and Operation and the corresponding ALU operations

The Symbol Commonly Used to Represent an ALU

Faster Addition: Carry Lookahead • Fast Carry Using “Infinite” Hardware • Fast Carry Using the First Level of Abstraction: Propagate and Generate • Fast Carry Using the Second Level of Abstraction • Summary

Fast Carry Using “Infinite” Hardware c 2 = (b 1*c 1)+(a 1*b 1) c 1 = (b 0*c 0)+(a 0*b 0) c 2 = (a 1*a 0*b 0)+(a 1*a 0*c 0)+(a 1*b 0*c 0) +(b 1*a 0*b 0)+(b 1*a 0*c 0)+(b 1*b 0*c 0) +(a 1*b 1) Imagine how the equation expands as we get to higher bits in the adder! This complexity causes high hardware cost for fast carry, making this simple scheme prohibitively expensive for wide adders.

Fast Carry Using the First Level of Abstraction: Propagate and Generate Most fast carry schemes limit the complexity of the equations to simplify the hardware, while still making substantial speed improvements over ripple carry. One such scheme is a carry-lookahead adder There are two important factors called generate(gi) and propagate (pi): gi = ai*bi pi = ai+bi

Propagate and Generate The adder generates a Carry. Out (ci+1) independent of the value of Carry. In (ci). The adder propagates Carry. In to a Carry. Out. Putting the two together, Carry. Ini+1 is a 1 if either gi is 1 or both pi is 1 and Carry. Ini is 1

A plumbing analogy for carry lookahead for 1 bit, 2 bits, and 4 bits using water pipes and valves.

Fast Carry Using the Second Level of Abstraction Use a 4 -bit adder with its carry-lookahead logic as a single building block If we connect them in ripple carry fashion to form a 16 -bit adder, the add will be faster than the original with a little more hardware To go faster, we’ll need carry lookahead at a higher level To perform carry lookahead for 4 -bit adders, we need propagate and generate signals at this higher level

Propagate and Generate P 0 P 1 P 2 P 3 = = p 3 p 7 p 11 p 15 * * p 2 p 8 p 10 p 14 * * p 1 p 5 p 9 p 13 * * p 0 p 4 p 8 p 12 G 0 G 1 G 2 G 3 = = g 3 +(p 3 *g 2 )+(p 3 *p 2 *g 1 )+(p 3 *p 2 *p 1 *g 0 ) g 7 +(p 7 *g 6 )+(p 7 *p 6 *g 5 )+(p 7 *p 6 *p 5 *g 4 ) g 11+(p 11*g 10)+(p 11*p 10*g 9 )+(p 11*p 10*p 9 *g 8 ) g 15+(p 15*g 14)+(p 15*p 14*g 13)+(p 15*p 14*p 13*g 12)

A plumbing analogy for the next-level carrylookahead signals P 0 and G 0.

Final Carry Equations C 1 = G 0+(P 0*c 0) C 2 = G 1+(P 1*G 0)+(P 1*P 0*c 0) C 3 = G 2+(P 2*G 1)+(P 2*P 1*G 0)+(P 2*P 1*P 0*c 0) C 4 = G 3+(P 3*G 2)+(P 3*P 2*G 1)+(P 3*P 2*P 1*G 0) +(P 3*P 2*P 1*P 0*c 0)

Summary Carry lookahead adders are faster than ripplecarry adders. This speed is generated by two signals: • generate • propagate Generate creates a carry regardless of the carry input Propagate passes a carry along Carry lookahead is another example of how abstraction is important in computer design in order to cope with complexity

Speed of Ripple Carry vs. Carry Lookahead • Assume the propagation delay of a signal passing through each gate is the same time. • Time is estimated by simply counting the number of gates along the path through a piece of logic. • Compare the number of gate delays for paths of two 16 -bit adders, one using ripple carry and one using two-level carry lookahead.

Four 4 -bit ALUs Using Carry Lookahead To Form A 16 -bit Adder

Clocks • Definition: Edge-triggered clocking is a clocking scheme in which all state changes occur on a clock edge. • Definition: Clocking methodology is the approach used to determine when data is valid and stable relative to the clock. • Definition: State element is a memory element. • Definition: Synchronous system is a memory system that employs clocks and where data signals are read only when the clock indicates that the signal values are stable.

The Clock Signal

State Elements

Edge-Triggered Methodology

Register Files • Definition: A register file is a state element that consists of a set of registers that can be read and written by supplying a register number to be accessed.

Memory Elements Flip-flops, Latches, Registers • Flip-Flops and Latches • Register Files • Specifying Sequential Logic in Verilog

Cross-Coupled NOR Gates

Flip-Flops and Latches The Simplest Memory Elements • Definition: flip-flop is a memory element for which the output is equal to the value of the stored state inside the element and for which the internal state is changed only on a clock edge. • Definition: latch is a memory element in which the output is equal to the value of the stored state inside the element and the state is changed whenever the appropriate inputs change and the clock is asserted.

D Latch Logic Circuit

D Latch Timing Diagram

D Flip-flop Definition: A D flip-flop is a flip-flop with one data input that stores the value of that input signal in the internal memory when the clock edge occurs.

D Flip-flop Timing Diagram Falling-edge Trigger

Flip-Flops and Latches Timing Measurements • Definition: Set-up time is the minimum time that the input to a memory device must be valid before the clock edge. • Definition: Hold time is the minimum time during which the input must be valid after the clock edge.

D Flip-flop Set-up & Hold Falling-edge Trigger

Register Files

Two Read Ports

Write Port

Memory Elements: SRAMs and DRAMs • Definition: Static RAM (SRAM) is random access memory where data is stored in static circuits (as in flip-flops) and does not need to be refreshed. • Definition: Dynamic RAM (DRAM) is random access memory where data is stored in dynamic circuits (as in capacitors) and needs to be refreshed periodically to keep its values. • SRAMs are faster than DRAMs, but less dense and more expensive per bit.

SRAMs • SRAMs are simply integrated circuits that are memory arrays with (usually) a single access port that can provide either a read or a write. SRAMs have a fixed access time to any datum, though the read and write access characteristics often differ. • A SRAM chip has a specific configuration in terms of the number of addressable locations, as well as the width of each addressable location.

SRAMs A 32 K x 8 SRAM showing the fifteen address lines (32 K = 215) and eight data inputs, the three control lines, and the eight data outputs.

Four three-state buffers used to form a multiplexor

The basic structure of a 4 x 2 SRAM consists of a decoder that selects which pair of cells to activate.

Typical organization of a 4 M x 8 SRAM as an array of 4 K x 1024 arrays

DRAMs • In a Dynamic RAM (DRAM), the value kept in a cell is stored as a charge in a capacitor. • A single transistor is used to access this stored charge, either – to read the value, or – to overwrite the charge stored there

A Single-Transistor DRAM Cell A single-transistor DRAM cell contains • a capacitor that stores the cell contents • a transistor used to access the cell

A 4 M x 1 DRAM Built With a 2048 x 2048 Array

SDRAM Synchronous DRAM • Definition: Synchronous DRAM is high speed output DRAM, which changes the column address without changing the row address and clocks address inputs to increase speed and precision • Since 1999, SDRAM is the most used form of RAM in cache-based main memory • Since 2004, DDRRAM (Double Data Rate RAMs), which transfers data on both the rising and falling edge of the clock, is the most used form of SDRAM

Error Correction • Definition: An error-detecting code (EDC) is a code that enables the detection of an error in data, but not the precise location, and hence correction of the error. • Definition: An error-correcting code (ECC) is a code that enables the detection of an error in data and the determination of the precise location of the error, which allows correction of the error.

Error Detection vs. Error Correction • A 1 -bit parity code is a distance-2 code – No 1 -bit change can generate another legal combination of data plus parity – After any 2 -bit change in data plus parity, the parity will match the data and the error cannot be detected • A distance-3 code can detect more than one error or correct an error – Legal combinations of data plus ECC have at least 3 bits differing from any other legal combination – Two errors can be recognized, but we cannot correct the errors

A Distance-3 Error Correction Code Here are the data words and a distance-3 error correction code for a 4 -bit data item.

Finite State Machines • Definition: A finite state machine is a sequential logic function consisting of a set of inputs and outputs, a next-state function that maps the current state and the inputs to a new state, and an output function that maps the current state and possibly the inputs to a set of asserted outputs. • Definition: A next-state function is a combinational function that, given the inputs and the current state, determines the next state of a finite state machine.

A State Machine A State Element And Two Functions: Next-state and Output

Moore Machines vs. Mealy Machines • Definition: A Moore machine is a finite state machine whose output function depends on just the current state • Definition: A Mealy machine is a finite state machine whose output function depends on both the current state and the current input

Controlling a Traffic Light • Clock = 0. 033 Hz • Outputs (asserted=green, deasserted=red) – NSlite – EWlite • Inputs (from sensors embedded in road) – NScar – EWcar • States (indicates green light) – NSgreen – EWgreen

The Next-state Function

The Output Function

Graphical Representation of a Finite State Machine

Logical Representation of a Finite State Machine

The FSM Functions The Next-state Function ______ Next. State = (Current. State*EWcar) + (Current. State*NScar) The Output Functions ______ NSlite = Current. State EWlite = Current. State

Timing Methodologies Edge-triggered timing methodology is simpler than a level-triggered methodology. If all clocks arrive at circuits at the same time, a system with edge-triggered registers between blocks of combinational logic can operate correctly without races, if we simply make the clock long enough. A race occurs when the contents of a state element depend on the relative speed of different logic elements.

Clock Must Be Long Enough

Clock Skew Definition: Clock skew is the difference in absolute time between the times when two state elements see a clock edge.

Level-Sensitive Timing • In a level-sensitive timing methodology, the state changes occur at either high or low levels, but they are not instantaneous as they are in an edge-triggered methodology. • Because of the noninstantaneous change in state, races can easily occur. • To ensure that a level-sensitive design will also work correctly if the clock is slow enough, designers use two-phase clocking. • Two-phase clocking is a scheme that makes use of two nonoverlapping clock signals.

Two-Phase Clocking

Two-Phase Timing with Alternating Latches

Asynchronous Inputs and Synchronizers • Definition: Metastability is a situation that occurs if a signal is sampled when it is not stable for the required set-up and hold times, possibly causing the sampled value to fall in the indeterminate region between a high and low value. • Definition: Synchronizer failure is a situation in which a flip-flop enters a metastable state and where some logic blocks reading the output of the flip-flop see a 0 while others see a 1.

A “Synchronizer”

A Real Synchronizer