Absolutely! Let’s proceed step-by-step as requested. --- ### **Step 1: Introduction to the Concept** A **Finite State Machine (FSM)** is a computational model used to design sequential logic circuits. There are two main types: **Mealy** and **Moore**. In a **Mealy FSM**, the output depends on both the current state and the current input. Here, we are asked to design a Mealy FSM that monitors a serial binary stream (MSB to LSB) and sets the output (let’s call it \( Z \)) to 1 only when the number represented so far is divisible by 3. --- ### **Step 2: Understanding the Problem (Divisibility by 3 in Binary)** When receiving bits serially, each new bit updates the current value. If the current value is \( N \) and the next bit is \( b \), then the new value is \( N' = 2N + b \) (since bits are received MSB first). We need to keep track of the remainder when dividing the number by 3. There are **three possible remainders**: 0, 1, and 2. --- ### **Step 3: Determining the Minimum Number of States** Since only the remainder modulo 3 matters, we need **3 states**, each representing one of the remainders (0, 1, or 2). The output \( Z = 1 \) only when the remainder is 0. --- ### **Step 4: Constructing the State Diagram** Let’s define: - **S0:** Remainder = 0 (Divisible by 3), \( Z = 1 \) - **S1:** Remainder = 1, \( Z = 0 \) - **S2:** Remainder = 2, \( Z = 0 \) **Transitions (for input bit \( b \)):** - From **S0**: - Input 0 → Stay in S0 (0*2+0=0, 0 mod 3 = 0) - Input 1 → Go to S1 (0*2+1=1, 1 mod 3 = 1) - From **S1**: - Input 0 → Go to S2 (1*2+0=2, 2 mod 3 = 2) - Input 1 → Go to S0 (1*2+1=3, 3 mod 3 = 0) - From **S2**: - Input 0 → Go to S1 (2*2+0=4, 4 mod 3 = 1) - Input 1 → Stay in S2 (2*2+1=5, 5 mod 3 = 2) **Output:** - \( Z = 1 \) only in S0. --- ### **Step 5: Drawing the State Diagram** ``` 0|1 +-----+ | v +---+ 1 +---+ |S0 |-----------> |S1 | +---+ +---+ ^ | 0 0 | | 1 | +-------------+ v +----------------+ +---+ <-------------| |S2 | 1 +---+ ``` **(Arrows labeled as "input/output")** - S0 --0/1--> S0 - S0 --1/0--> S1 - S1 --0/0--> S2 - S1 --1/1--> S0 - S2 --0/0--> S1 - S2 --1/0--> S2 --- ### **Step 6: Final Summary in Few Sentences** We designed a Mealy FSM with three states, each representing the remainder modulo 3, and transitions defined by the incoming bit. The FSM sets the output to 1 only when the current number is divisible by 3, ensuring the minimum number of states are used. --- ### **Step 7: One-Sentence Step Summary** - **Step 1:** Introduced FSM and Mealy machine concepts. - **Step 2:** Explained how to track divisibility by 3 using remainders. - **Step 3:** Identified that only three states are needed. - **Step 4:** Defined state meanings and transitions. - **Step 5:** Presented the state diagram. - **Step 6:** Summarized the complete solution concisely.