Let's analyze and complete the table step by step. ### **Counter Initialization** - The counter starts at \(Z_3 Z_2 Z_1 Z_0 = 0010\) (i.e., \(Z_2=1\), \(Z_1=0\), \(Z_0=0\), \(Z_3=0\)). ### **Toggle-Cell (T Flip-Flop)** - Changes state (toggles) when the input (CE) is 1. ### **Control Logic** - The CE for each stage is the AND of the previous outputs (as per the diagram). Let’s analyze the circuit: #### **Stage CEs:** - **Z0 Toggle:** Always enabled (CE = 1). - **Z1 Toggle:** CE = Z0. - **Z2 Toggle:** CE = Z0 AND Z1. - **Z3 Toggle:** CE = Z0 AND Z1 AND Z2. ### **Cycle-by-Cycle Analysis** #### **Cycle 0: Initial State** - Z0 = 0, Z1 = 0, Z2 = 1, Z3 = 0 #### **Cycle 1:** - **Z0 toggles:** 0 → 1 - **Z1 toggles:** (CE = Z0 = 0), so no toggle (stays 0) - **Z2 toggles:** (CE = Z0 AND Z1 = 0), so no toggle (stays 1) - **Z3 toggles:** (CE = Z0 AND Z1 AND Z2 = 0), so no toggle (stays 0) State: Z0 = 1, Z1 = 0, Z2 = 1, Z3 = 0 #### **Cycle 2:** - **Z0 toggles:** 1 → 0 - **Z1 toggles:** (CE = Z0 = 1), so toggles 0 → 1 - **Z2 toggles:** (CE = Z0 AND Z1 = 1 AND 0 = 0), so no toggle (stays 1) - **Z3 toggles:** (CE = Z0 AND Z1 AND Z2 = 1 AND 0 AND 1 = 0), so no toggle (stays 0) New state: Z0 = 0, Z1 = 1, Z2 = 1, Z3 = 0 #### **Cycle 3:** - **Z0 toggles:** 0 → 1 - **Z1 toggles:** (CE = Z0 = 0), so no toggle (stays 1) - **Z2 toggles:** (CE = Z0 AND Z1 = 0 AND 1 = 0), so no toggle (stays 1) - **Z3 toggles:** (CE = Z0 AND Z1 AND Z2 = 0 AND 1 AND 1 = 0), so no toggle (stays 0) New state: Z0 = 1, Z1 = 1, Z2 = 1, Z3 = 0 ### **Filled Table:** | Cycle | Z0 | Z1 | Z2 | Z3 | |-------|----|----|----|----| | 0 | 0 | 0 | 1 | 0 | | 1 | 1 | 0 | 1 | 0 | | 2 | 0 | 1 | 1 | 0 | | 3 | 1 | 1 | 1 | 0 | Let me know if you'd like to continue further cycles!

1. Introduction: A synchronous counter is a sequential circuit that advances its state in synchrony with a clock signal, with all flip-flops triggered simultaneously. Modified synchronous counters often incorporate additional control logic to enable specific counting sequences or to implement certain functionalities, such as reset or toggle conditions. In the context of the given question, the counter is composed of D flip-flops, each associated with toggle cells and control signals, which determine when each flip-flop toggles state. Understanding how these toggle-conditions behave based on the outputs of the flip-flops and the associated control logic is crucial for determining the counter’s state progression over successive clock cycles. The initial state is specified (0010), and the task involves analyzing the behavior of each flip-flop at each cycle, considering the toggle conditions, to complete the state transition table. This requires knowledge of flip-flop operation, the role of toggle cells (T flip-flops), and the logic controlling their toggling, typically based on the outputs of other flip-flops, following the standard binary counting sequence. Explanation: The introduction sets the foundation by clarifying the nature of a modified synchronous counter, emphasizing the role of toggle-logic and control signals in state transitions. Recognizing that the circuit’s operation depends on the toggle conditions derived from previous states allows for systematic analysis of each cycle, ensuring the accurate determination of the counter’s state progression. --- 2. A Detailed Step-By-Step Solution: **Step 1: Analyze the initial state** - The counter starts at \(Z_3 Z_2 Z_1 Z_0 = 0\,0\,1\,0\) (binary 2). **Explanation:** Knowing the initial state is essential as it serves as the basis for subsequent state transitions and for analyzing toggle conditions at each cycle. --- **Step 2: Determine toggle conditions for each flip-flop** - Each flip-flop toggles based on the control logic signals, which are typically derived from the outputs of other flip-flops. - In a typical modified synchronous counter, the toggle condition for each flip-flop is determined as follows: - \(Z_0\): Always toggles on each clock cycle if its toggle cell is enabled. - \(Z_1\): Toggles when \(Z_0 = 1\). - \(Z_2\): Toggles when \(Z_0 = 1\) AND \(Z_1 = 1\). - \(Z_3\): Toggles when \(Z_0 = 1\) AND \(Z_1 = 1\) AND \(Z_2 = 1\). **Explanation:** These conditions reflect the typical ripple or Gray code-like toggle logic used in binary counters, where each higher-order flip-flop toggles when all lower-order flip-flops are high. --- **Step 3: Cycle 0 (initial state)** - State: \(Z_3=0, Z_2=1, Z_1=0, Z_0=0\) **Step 4: Determine toggle signals for Cycle 1** - \(Z_0\): toggles every cycle (since toggle cell is always enabled). - \(Z_0_{next} = \sim Z_0 = 1\) - \(Z_1\): toggles if \(Z_0\) was 1 in current cycle. - Current \(Z_0=0\), so no toggle: \(Z_1_{next} = Z_1 = 0\) - \(Z_2\): toggles if \(Z_0=1\) AND \(Z_1=1\). Current \(Z_0=0\), so no toggle. - \(Z_3\): toggles if \(Z_0=1\) AND \(Z_1=1\) AND \(Z_2=1\). Current \(Z_0=0\), so no toggle. **Step 5: Update states after Cycle 1** - \(Z_0\): 0 → 1 - \(Z_1\): remains 0 - \(Z_2\): remains 1 - \(Z_3\): remains 0 **State after cycle 1:** 0 0 1 1 --- **Step 6: Cycle 2** - Current state: \(Z_3=0, Z_2=1, Z_1=0, Z_0=1\) - Toggling conditions: - \(Z_0\): toggles every cycle; current 1 → next 0. - \(Z_1\): toggles if \(Z_0=1\). Since \(Z_0=1\), toggle \(Z_1\): 0 → 1. - \(Z_2\): toggles if \(Z_0=1\) AND \(Z_1=1\). Now both are 1, so toggle \(Z_2\): 1 → 0. - \(Z_3\): toggles if \(Z_0=1\) AND \(Z_1=1\) AND \(Z_2=1\). Currently, \(Z_2=0\), so no toggle: remains 0. - **Update states:** - \(Z_0\): 1 → 0 - \(Z_1\): 0 → 1 - \(Z_2\): 1 → 0 - \(Z_3\): 0 → 0 **State after cycle 2:** 0 1 0 0 --- **Step 7: Cycle 3** - Current state: \(Z_3=0, Z_2=0, Z_1=1, Z_0=0\) - Toggling conditions: - \(Z_0\): toggles every cycle: 0 → 1. - \(Z_1\): toggles if \(Z_0=1\). Since \(Z_0=0\), no toggle: remains 1. - \(Z_2\): toggles if \(Z_0=1\) AND \(Z_1=1\). \(Z_0=0\), so no toggle. - \(Z_3\): toggles if \(Z_0=1\) AND \(Z_1=1\) AND \(Z_2=1\). Currently, \(Z_2=0\), so no toggle. - **Update states:** - \(Z_0\): 0 → 1 - \(Z_1\): remains 1 - \(Z_2\): remains 0 - \(Z_3\): remains 0 **State after cycle 3:** 0 1 0 1 --- **Step 8: Summary of states** | Cycle | Z3 | Z2 | Z1 | Z0 | |--------|-----|-----|-----|-----| | 0 (initial) | 0 | 1 | 0 | 0 | | 1 | 0 | 1 | 0 | 1 | | 2 | 0 | 0 | 1 | 0 | | 3 | 0 | 0 | 1 | 1 | **Explanation:** Each step involves analyzing the toggle conditions derived from the current state, updating the flip-flops accordingly, and progressively mapping the state transition sequence. This systematic approach ensures the correctness of the counter’s operation over successive clock cycles. --- 3. Conclusion: The detailed analysis demonstrates the state progression of the modified synchronous counter over multiple cycles, beginning from the initial state 0010. Each cycle's flip-flop toggling decisions are based on the control logic signals, which depend on the outputs of other flip-flops, following the standard binary counting sequence. The process involves evaluating toggle conditions at each step, updating the flip-flop states accordingly, and compiling these results into a comprehensive state transition table. This approach confirms that the counter advances correctly through its states, illustrating the impact of the toggle logic on the overall counting operation.