# (h) Explain the functioning of the edge-triggered flip-flop with the help of a diagram. (i) Represent (-121.25)10 and (0.0625)10 in IEEE 754 single precision and double precision formats.

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(h) Edge-Triggered Flip-Flop:

An edge-triggered flip-flop is a type of flip-flop that changes its state only on the edge (transition) of a clock signal. There are two types of edges: rising edge (positive edge) and falling edge (negative edge). The most common edge-triggered flip-flops are the positive edge-triggered (or simply edge-triggered) D flip-flop and the JK flip-flop. Let's use the D flip-flop as an example.

Functioning:

The edge-triggered D flip-flop changes its state (stores the value of the D input) only when the clock signal transitions from one state to another. For example, in a positive edge-triggered D flip-flop, the state changes on the rising edge of the clock signal.

Diagram:



+-----+

D ---->|     |

|  D  |---- Q

CLK--->|  F  |

|  F  |---- Q'

|  L  |

+-----+



- $$D$$: Data input

- $$Q$$: Output

- $$Q'$$: Complement of output

- $$CLK$$: Clock input

(i) IEEE 754 Single Precision and Double Precision Formats:

IEEE 754 Single Precision (32 bits):

Representation of (-121.25)10:

- Sign bit (1 bit): 1 (negative)

- Exponent (8 bits): $$2^{127 - 1 + 8} = 2^{134}$$

- Mantissa (23 bits): The binary representation of the fractional part

The binary representation of 121.25 is $$1111001.01$$. The normalized form is $$1.11100101 \times 2^6$$.

Putting it all together:

$\text{Single Precision} = 1 \, | \, 10000110 \, | \, 11100101000000000000000$

Representation of (0.0625)10:

- Sign bit (1 bit): 0 (positive)

- Exponent (8 bits): $$2^{127 - 1 - 6} = 2^{120}$$

- Mantissa (23 bits): The binary representation of the fractional part

The binary representation of 0.0625 is $$0.0001$$. The normalized form is $$1.0 \times 2^{-4}$$.

Putting it all together:

$\text{Single Precision} = 0 \, | \, 01111010 \, | \, 00000000000000000000000$

IEEE 754 Double Precision (64 bits):

The format is similar to single precision but with a longer mantissa.

Representation of (-121.25)10:

$\text{Double Precision} = 1 \, | \, 10000000011 \, | \, 1110010100000000000000000000000000000000000000000000$

Representation of (0.0625)10:

$\text{Double Precision} = 0 \, | \, 01111111110 \, | \, 0000000000000000000000000000000000000000000000000000$

In both cases, the sign bit, exponent, and mantissa are formatted according to IEEE 754 standards for single and double precision.