Combinational Logic: Combinational Logic

Combinational Logic:

• Combinational Logic:

• Output depends only on current input
• Able to perform useful operations (add/subtract/multiply/…)
• Has no memory

Sequential Logic:

• Sequential Logic:

• Output depends not only on current input but also on past input values, e.g., design a counter
• Need some type of memory to remember the past input values

Sequential Logic circuits remember past inputs and past circuit state.

• Sequential Logic circuits remember past inputs and past circuit state.

• Outputs from the system are “fed back” as new inputs

• With gate delay and wire delay

• The storage elements are circuits that are capable of storing binary information: memory.

There are two types of sequential circuits:

• There are two types of sequential circuits:

• Synchronous sequential circuit: circuit output changes only at some discrete instants of time. This type of circuits achieves synchronization by using a timing signal called the clock.

• Asynchronous sequential circuit: circuit output can change at any time (clockless).

One way to eliminate the undesirable indeterminate state in the RS flip flop is to ensure that inputs S and R are never 1 simultaneously. This is done in the D latch:

• One way to eliminate the undesirable indeterminate state in the RS flip flop is to ensure that inputs S and R are never 1 simultaneously. This is done in the D latch:

Latches are “transparent” (= any change on the inputs is seen at the outputs immediately).

• Latches are “transparent” (= any change on the inputs is seen at the outputs immediately).

• This causes synchronization problems.

• Solution: use latches to create flip-flops that can respond (update) only on specific times (instead of any time).

• Types: RS flip-flop and D flip-flop

D-Type Positive Edge-Triggered Flip-Flop:

• D-Type Positive Edge-Triggered Flip-Flop:

Defines the logical properties of a flip-flop (such as a truth table does for a logic gate).

• Defines the logical properties of a flip-flop (such as a truth table does for a logic gate).

• Q(t) – present state at time t

• Q(t+1) – next state at time t+1

Analysis: Consists of obtaining a suitable description that demonstrates the time sequence of inputs, outputs, and states.

• Analysis: Consists of obtaining a suitable description that demonstrates the time sequence of inputs, outputs, and states.

• Logic diagram: Boolean gates, flip-flops (of any kind), and appropriate interconnections.

• The logic diagram is derived from any of the following:

• Boolean Equations (FF-Inputs, Outputs)
• State Table
• State Diagram

Input: x(t)

• Input: x(t)

• Output: y(t)

• State: (A(t), B(t))

• What is the Output Function?

• What is the Next State Function?

Boolean equations for the functions:

• Boolean equations for the functions:

• A(t+1) = A(t)x(t) + B(t)x(t)
• B(t+1) = A’(t)x(t)
• y(t) = x’(t)(B(t) + A(t))

State table – a multiple variable table with the following four sections:

• State table – a multiple variable table with the following four sections:

• Present State – the values of the state variables for each allowed state.
• Input – the input combinations allowed.
• Next-state – the value of the state at time (t+1) based on the present state and the input.
• Output – the value of the output as a function of the present state and (sometimes) the input.

• From the viewpoint of a truth table:

• the inputs are Input, Present State
• and the outputs are Output, Next State

The state table can be filled in using the next state and output equations:

• The state table can be filled in using the next state and output equations:

• A(t+1) = A(t)x(t) + B(t)x(t)
• B(t+1) =A (t)x(t);
• y(t) =x (t)(B(t) + A(t))

The sequential circuit function can be represented in graphical form as a state diagram with the following components:

• The sequential circuit function can be represented in graphical form as a state diagram with the following components:

• A circle with the state name in it for each state
• A directed arc from the Present State to the Next State for each state transition
• A label on each directed arc with the Input values which causes the state transition, and
• A label:
• On each circle with the output value produced, or
• On each directed arc with the output value produced.

Which type?

• Which type?

• Diagram gets confusing for large circuits

• For small circuits, usually easier to understand than the state table