15-SequentialDesign

Sequential circuit design:

Sequential circuit design 1 Sequential circuit design Now let’s reverse the process: In sequential circuit design , we turn some description into a working circuit. We first make a state table or diagram to express the computation. Then we can turn that table or diagram into a sequential circuit.

Sequence recognizers:

Sequential circuit design 2 Sequence recognizers A sequence recognizer is a special kind of sequential circuit that looks for a special bit pattern in some input. The recognizer circuit has only one input, X. One bit of input is supplied on every clock cycle. For example, it would take 20 cycles to scan a 20-bit input. This is an easy way to permit arbitrarily long input sequences. There is one output, Z, which is 1 when the desired pattern is found. Our example will detect the bit pattern “1001”: Inputs: 1 1 1 0 0 1 1 0 1 0 0 1 0 0 1 1 0 … Outputs: 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 … Here, one input and one output bit appear every clock cycle. This requires a sequential circuit because the circuit has to “remember” the inputs from previous clock cycles, in order to determine whether or not a match was found.

A basic state diagram:

Sequential circuit design 3 A basic state diagram What state do we need for the sequence recognizer? We have to “remember” inputs from previous clock cycles. For example, if the previous three inputs were 100 and the current input is 1, then the output should be 1. In general, we will have to remember occurrences of parts of the desired pattern—in this case, 1, 10, and 100. We’ll start with a basic state diagram (edges are labeled input/output ): A B C D 1/0 0/0 0/0

Step 1: Making a state table:

Sequential circuit design 4 Step 1: Making a state table The first thing you have to figure out is precisely how the use of state will help you solve the given problem. Make a state table based on the problem statement. The table should show the present states, inputs, next states and outputs. Sometimes it is easier to first find a state diagram and then convert that to a table. This is usually the most difficult step. Once you have the state table, the rest of the design procedure is the same for all sequential circuits. Sequence recognizers are especially hard! They’re the hardest example we’ll see in this class, so if you understand this you’re in good shape.

A basic state diagram:

Sequential circuit design 5 A basic state diagram What state do we need for the sequence recognizer? We have to “remember” inputs from previous clock cycles. For example, if the previous three inputs were 100 and the current input is 1, then the output should be 1. In general, we will have to remember occurrences of parts of the desired pattern—in this case, 1, 10, and 100. We’ll start with a basic state diagram: A B C D 1/0 0/0 0/0

Overlapping occurrences of the pattern:

Sequential circuit design 6 Overlapping occurrences of the pattern What happens if we’re in state D (the last three inputs were 100), and the current input is 1? The output should be a 1, because we’ve found the desired pattern. But this last 1 could also be the start of another occurrence of the pattern! For example, 100 1 001 contains two occurrences of 1001. To detect overlapping occurrences of the pattern, the next state should be B. A B C D 1/0 0/0 0/0 1/1

Filling in the other arrows:

Sequential circuit design 7 Filling in the other arrows Remember that we need two outgoing arrows for each node, to account for the possibilities of X=0 and X=1. The remaining arrows we need are shown in blue. They also allow for the correct detection of overlapping occurrences of 1001. A B C D 1/0 0/0 0/0 1/1 0/0 0/0 1/0 1/0

Finally, making the state table:

Sequential circuit design 8 Finally, making the state table A B C D 1/0 0/0 0/0 1/1 0/0 0/0 1/0 1/0 input/output present state next state Remember how the state diagram arrows correspond to rows of the state table:

Sequential circuit design procedure:

Sequential circuit design 9 Sequential circuit design procedure Step 1: Make a state table based on the problem statement. The table should show the present states, inputs, next states and outputs. (It may be easier to find a state diagram first, and then convert that to a table.) Step 2: Assign binary codes to the states in the state table, if you haven’t already. If you have n states, your binary codes will have at least  log 2 n  digits, and your circuit will have at least  log 2 n  flip-flops. Step 3: For each flip-flop and each row of your state table, find the flip-flop input values that are needed to generate the next state from the present state. You can use flip-flop excitation tables here. Step 4: Find simplified equations for the flip-flop inputs and the outputs. Step 5: Build the circuit!

Step 2: Assigning binary codes to states:

Sequential circuit design 10 Step 2: Assigning binary codes to states We have four states ABCD, so we need at least two flip-flops Q 1 Q 0 . The easiest thing to do is represent state A with Q 1 Q 0 = 00, B with 01, C with 10, and D with 11. The state assignment can have a big impact on circuit complexity, but we won’t worry about that too much in this class.

Step 3: Finding flip-flop input values:

Sequential circuit design 11 Step 3: Finding flip-flop input values Next we have to figure out how to actually make the flip-flops change from their present state into the desired next state. This depends on what kind of flip-flops you use! We’ll use two JKs. For each flip-flip Q i , look at its present and next states, and determine what the inputs J i and K i should be in order to make that state change.

Finding JK flip-flop input values:

Sequential circuit design 12 Finding JK flip-flop input values For JK flip-flops, this is a little tricky. Recall the characteristic table: If the present state of a JK flip-flop is 0 and we want the next state to be 1, then we have two choices for the JK inputs: We can use JK=10, to explicitly set the flip-flop’s next state to 1. We can also use JK=11, to complement the current state 0. So to change from 0 to 1, we must set J=1, but K could be either 0 or 1. Similarly, the other possible state transitions can all be done in two different ways as well.

JK excitation table:

Sequential circuit design 13 JK excitation table An excitation table shows what flip-flop inputs are required in order to make a desired state change. This is the same information that’s given in the characteristic table, but presented “backwards.”

Excitation tables for all flip-flops:

Sequential circuit design 14 Excitation tables for all flip-flops

Back to the example:

Sequential circuit design 15 Back to the example We can now use the JK excitation table on the right to find the correct values for each flip-flop’s inputs, based on its present and next states.

Back to the example:

Sequential circuit design 16 Back to the example We can now use the JK excitation table on the right to find the correct values for each flip-flop’s inputs, based on its present and next states.

Step 4: Find equations for the FF inputs and output:

Sequential circuit design 17 Now you can make K-maps and find equations for each of the four flip-flop inputs, as well as for the output Z. These equations are in terms of the present state and the inputs. The advantage of using JK flip-flops is that there are many don’t care conditions, which can result in simpler MSP equations. J 1 = X’ Q 0 K 1 = X + Q 0 J 0 = X + Q 1 K 0 = X’ Z = Q 1 Q 0 X Step 4: Find equations for the FF inputs and output

Step 5: Build the circuit:

Sequential circuit design 18 Step 5: Build the circuit Lastly, we use these simplified equations to build the completed circuit. J 1 = X’ Q 0 K 1 = X + Q 0 J 0 = X + Q 1 K 0 = X’ Z = Q 1 Q 0 X

Step 5: Build the circuit:

Sequential circuit design 19 Step 5: Build the circuit Lastly, we use these simplified equations to build the completed circuit. J 1 = X’ Q 0 K 1 = X + Q 0 J 0 = X + Q 1 K 0 = X’ Z = Q 1 Q 0 X

Timing diagram:

Sequential circuit design 20 Timing diagram Here is one example timing diagram for our sequence detector. The flip-flops Q 1 Q 0 start in the initial state, 00. On the first three positive clock edges, X is 1, 0, and 0. These inputs cause Q 1 Q 0 to change, so after the third edge Q 1 Q 0 = 11. Then when X=1, Z becomes 1 also, meaning that 1001 was found. The output Z does not have to change at positive clock edges. Instead, it may change whenever X changes, since Z = Q 1 Q 0 X. CLK Q 1 Q 0 X Z 1 2 3 4

Building the same circuit with D flip-flops:

Sequential circuit design 21 Building the same circuit with D flip-flops What if you want to build the circuit using D flip-flops instead? We already have the state table and state assignments, so we can just start from Step 3, finding the flip-flop input values. D flip-flops have only one input, so our table only needs two columns for D 1 and D 0 .

D flip-flop input values (Step 3):

Sequential circuit design 22 D flip-flop input values (Step 3) The D excitation table is pretty boring; set the D input to whatever the next state should be. You don’t even need to show separate columns for D 1 and D 0 ; you can just use the Next State columns.

Finding equations (Step 4):

Sequential circuit design 23 Finding equations (Step 4) You can do K-maps again, to find: D 1 = Q 1 Q 0 ’ X’ + Q 1 ’ Q 0 X’ D 0 = X + Q 1 Q 0 ’ Z = Q 1 Q 0 X

Building the circuit (Step 5):

Sequential circuit design 24 Building the circuit (Step 5) D1 = Q1 Q0’ X’ + Q1’ Q0 X’ D0 = X + Q1 Q0’

Building the circuit (Step 5):

Sequential circuit design 25 Building the circuit (Step 5)

Flip-flop comparison:

Sequential circuit design 26 Flip-flop comparison JK flip-flops are good because there are many don’t care values in the flip-flop inputs, which can lead to a simpler circuit. D flip-flops have the advantage that you don’t have to set up flip-flop inputs at all, since Q(t+1) = D. However, the D input equations are usually more complex than JK input equations In practice, D flip-flops are used more often. There is only one input for each flip-flop, not two. There are no excitation tables to worry about. D flip-flops can be implemented with slightly less hardware than JK flip-flops.

An Aside On State Encoding:

Sequential circuit design 27 An Aside On State Encoding You do not have to use the most compact encoding of states possible Here is another possibility: suppose you have four states A, B, C, D Create one flip flop for each of A, B, C, D Feed the Set and Reset lines of these flip-flops based on current state and inputs This uses more flip-flops (4 instead of the minimum 2) but may result in a much simpler circuit overall The encoding and decoding of states is entirely artificial; it depends completely on the choice of state encodings. A little thinking about the circuit may lead you to a state encoding that simplifies the circuit a lot.

Summary:

Sequential circuit design 28 Summary The basic sequential circuit design procedure: Make a state table and, if desired, a state diagram. This step is usually the hardest. Assign binary codes to the states if you didn’t already. Use the present states, next states, and flip-flop excitation tables to find the flip-flop input values. Write simplified equations for the flip-flop inputs and outputs and build the circuit. Next, we’ll look at common examples of sequential circuits, including different types of counters.