One way is shorting the two inputs of the NAND gate and passing the input.

truth table:

A B output

1 1 0

0 0 1

The second way is passing the input to only one input(say A) of the NAND gate.Since the other input(say B) is floating, it is always logic one.

truth table:

A B output

1 1 0

0 1 1

TRUTH TABLE FOR HALF ADDER:

A B SUM CARRY

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

IMPLEMENTATION:

- For SUM, The two inputs A and B are given to XOR gate.
- For Carry, The two inputs A and B are given to AND gate.

Sequence detector : A sequence detector gives an output of 1 on detecting the given sequence else the output is zero.

Ex : if the given sequence to be detected is 111

and input stream is 1 1 0 1 1 1 0 0 1 0 1 1 1 1 1

the output should be 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1.

Soln:

One of the different possible ways to detect a sequence is using a Mealy type FSM.

Using the following table the State machine can be designed.

Short the two inputs of the nand gate and give the same input to the common wire,the nand gate works as an inverter.

One way is shorting the two inputs of the NAND gate and passing the input.

truth table:

A B output

1 1 0

0 0 1

The second way is passing the input to only one input(say A) of the NAND gate.Since the other input(say B) is floating, it is always logic one.

truth table:

A B output

1 1 0

0 1 1

You can divide the frequency of a clock by just implementing T Flip flop.

Give clock as clock input and tie the T input to logic 1.

Adders are generally of five types:

**1) Ripple Carry Adder:** The Ripple carry adder(RCA) consists of a building block named Half Adder(HA) which is cascaded to form a Full Adder(FA). These building blocks HAs and FAs are also the building blocks of all types of adders.The n full adders are cascaded to form n bit RCA.

The full adder has three input pins(input Ai,input Bi,carryin Ci) and two output pins(Sum and Ci+1).Its equations are:

Sum=Ai^Bi^Ci

Ci+1=Ai.Bi+Bi.Ci+Ai.Ci

**2)Carry Lookahead Adder: **The Carry Lookahead Adder(CLA) reduces the delay as that in RCA. Let

Gi=Ai.Bi, and Pi=Ai^Bi, then Ci+1=Gi+Pi.Ci.

The expressions for Sum and Ci+1 is then defined completely in terms of input pins rather wait for input carry to appear.

**3)Carry Select Adder:** The carry select adder uses duplicate modules for each combination of input carry(i.e. 1 and 0).The multiplexers then select the appropriate sum and carry output according to the carry output of the preceding stages.

Suppose your flip-flop is positive edge triggered. time for which data should be stable prior to positive edge clock is called setup time constraint .

Time for which data should be stable after the positive edge of clock is called as hold time constraint.

if any of these constraints are violated then flip-flop will enter in meta stable state, in which we cannot determine the output of flip-flop.

there are two equation:

- Tcq + Tcomb> Tskew + Thold
- Tcq + Tcomb<Tskew +T - Tsetup

Tcq is time delay when data enters the flip-flop and data comes at output of flip flop.

Tcomb is the logic delay between two flip flop.

Tskew is the delay of clock to flip flop: suppose there are two flip flop ,if clock reaches first to source flip flop and then after some delay to destination flip flop ,it is positive skew and if vice versa then negative skew.

so if you take 2 eq you will see that setup time is the determining factor of clock's time period.

Use the concept of register-retiming.

divide the total combinatorial delay in two segments such that individually the delay is less the clock period.

this can be done by inserting a flip-flop in the combinational path.

e.g,

clock period --- 5 ns

total cominational delay ---- 7

then divide the 7ns path in two path of 4 or 3 (best results are obtained if delays are same for both path i.e 3.5ns) by inserting a flip-flop in between.

Sequence detector : A sequence detector gives an output of 1 on detecting the given sequence else the output is zero.

Ex : if the given sequence to be detected is 111

and input stream is 1 1 0 1 1 1 0 0 1 0 1 1 1 1 1

the output should be 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1.

Soln: One of the different possible ways to detect a sequence is using a Mealy type FSM.

Using the following table the State machine can be designed. since the number of bits in the sequence 1101 is 4 we have 4 states

------------------------------------------------------

|PS | Seq detected by the state | NS/output |

| | |---------------|

| | | X=0 | X=1 |

|-----------------------------------------------------

| S1 | - | S1/0 | S2/0 |

|----------------------------------------------------|

| S2 | 1 | S1/0 | S3/0 |

|----------------------------------------------------|

| S3 | 11 | S4/0 | S3/0 |

|----------------------------------------------------|

| S4 | 110 | S1/0 | S2/1 |

|----------------------------------------------------|

when in state S4 (PS),and input(X) from the sequence is 1,the sequence "1101" has been detected once and (to find the next state select the longest "seq identified by a state" column that matches part of the sequence 1101--ie.,1 or 01 or 101 ....)the NS is S2 since the sequence detected by the state S2 is 1(in 1101- 01 or 101 ,etc are not present in the seq identified by the state column ,)