Solution Overview

The question involves constructing a convolutional encoder diagram and its corresponding tree diagram for the given generator functions and states. Let's break this into parts:

(a) Diagram of a (1/3; 3) Convolutional Encoder

Given:

• Generator functions:
  • $g_1 = [1, 0, 0]$
  • $g_2 = [1, 0, 1]$
  • $g_3 = [1, 1, 1]$

Encoder Representation:

1. Shift Registers:

   • Use three memory registers (denoted as $M_1, M_2, M_3$).
   • These store the current and previous input bits.

2. Outputs:

   • Each output is generated by taking the XOR of the bits corresponding to the generator polynomials:
     • $g_1: \text{Output 1} = M_1$
     • $g_2: \text{Output 2} = M_1 \oplus M_3$
     • $g_3: \text{Output 3} = M_1 \oplus M_2 \oplus M_3$

3. Diagram: The diagram includes:

   • A single input bit shifted through the registers.
   • Three outputs connected to the XOR gates as per the generator polynomials.

I can create a visual representation of this if you’d like.

(b) Tree Diagram for Four Possible States

Given:

• Possible states:
  • $a = 00$, $b = 01$, $c = 10$, $d = 11$

Procedure:

1. State Representation:

   • Each state represents the contents of the two shift registers $M_2, M_3$ (ignoring $M_1$ for simplicity).

2. Transitions:

   • An input bit (0 or 1) determines the transition between states.
   • For each transition:
     • Calculate the new state based on the input bit.
     • Determine the corresponding outputs using the generator functions.

3. Tree Construction:

   • Start from the initial state (e.g., $a = 00$).
   • Branch for input 0 and 1, and repeat for all subsequent states.

Outputs:

Each branch will output a 3-bit code as computed from the generator functions.

I can also illustrate this tree diagram step by step. Would you like me to provide a clear visualization?

Next Steps

Would you like a detailed drawing of the encoder or the tree diagram, or both? Let me know if you'd like the diagrams visualized!

Related Questions:

1. What is the significance of generator polynomials in a convolutional encoder?
2. How do we interpret the rate $(1/3)$ of the encoder?
3. What are the key differences between a trellis and a tree diagram in encoding?
4. How would the system handle error correction using this encoder?
5. Can we generalize this encoder design for higher rates?

Tip: When constructing convolutional encoders, ensure you understand the generator polynomial structure as it directly impacts output calculation.