• Represent A and B in Binary:

  • A = 46 (decimal) = 101110 (binary)
  • B = 35 (decimal) = 100011 (binary)

• Determine 1's Complement of B:

  • B = 100011
  • 1's Complement of B = 011100 (invert all bits)

• Perform Addition of A and 1's Complement of B:

    A          =  101110
    1's Comp B = +011100
    --------------------
    Sum        =  1001010
  

• Handle End-Around Carry:

  • The sum results in a 7-bit number, indicating a carry-out from the most significant bit.
  • The carry-out is '1'.
  • Add this carry-out back to the least significant bit of the 6-bit result:

      001010 (6-bit sum, ignoring carry)
    +      1 (End-around carry)
    --------------------
         001011
    

• Final Result:

  • The result in binary is 001011.
  • Converting back to decimal: 0*32 + 0*16 + 1*8 + 0*4 + 1*2 + 1*1 = 8 + 2 + 1 = 11.
  • Therefore, A - B = 46 - 35 = 11.

• State Diagram:
  A graphical representation of the behavior of a sequential circuit. It consists of:

  • States (nodes): Represent the stable conditions of the circuit.
  • Transitions (directed arcs): Indicate the change from one state to another based on input conditions.
  • Labels on transitions: Show the input that causes the transition and the corresponding output produced.
  • Example: A state diagram for a traffic light controller might have states like "Red", "Green", "Yellow", with transitions based on a timer or external sensor inputs.

• State Table:
  A tabular representation of the information contained in a state diagram. It lists all possible present states, the inputs, the resulting next states, and the outputs produced.

  • Columns: Typically include Present State, Next State (for each input combination), and Output (for each input combination or present state, depending on Mealy/Moore model).
  • Rows: Represent each unique state.
  • Example: For a 2-bit counter, a state table would list Present State (e.g., 00), and the Next State (e.g., 01) when an increment input is active, and the output (e.g., 00).

• State Reduction:
  The process of minimizing the number of states in a sequential circuit while preserving its original input-output behavior. This is done by identifying and merging equivalent states.

  • Purpose: To simplify the circuit design, reduce the number of flip-flops required, and minimize the combinational logic, leading to a more cost-effective and faster implementation.
  • Example: If two states S1 and S2, for every possible input, transition to the same next state and produce the same output, then S1 and S2 are equivalent and can be replaced by a single state.

• State Assignment:
  The process of assigning unique binary codes (bit patterns) to each state in a sequential circuit. The number of bits required for assignment is determined by the number of states (N), using ceil(log2(N)) bits.

  • Purpose: To convert the symbolic states into binary values that can be implemented using flip-flops. The choice of assignment can significantly impact the complexity of the combinational logic required between flip-flops.
  • Example: For a circuit with four states (S0, S1, S2, S3), 2 bits are needed. A simple sequential assignment might be S0=00, S1=01, S2=10, S3=11. Other assignments like Gray code or one-hot might be used to optimize logic.

Encoder

• A combinational logic circuit that converts n input lines into m output lines, where 2^m ≥ n.
• The primary function is to generate a binary code corresponding to the active input line.
• For a standard encoder, only one input line is active (high) at any given time, and its position determines the binary output code.
• Common applications include converting decimal or octal inputs into their binary equivalents and generating control signals in digital systems.
• Example: An 8-to-3 line encoder converts one of eight active input signals into a 3-bit binary output code.

Error Detection Codes

• Techniques used to detect errors that may occur in digital data during transmission or storage.
• These codes work by adding redundant bits (checksums, parity bits, CRC remainders) to the original data at the sender.
• The receiver processes both the data and the redundant bits to determine if any corruption has occurred during transit.
• Common types include:
  • Parity Check: Adds a single parity bit to make the total number of '1's in the data unit either even (even parity) or odd (odd parity). Primarily detects single-bit errors.
  • Checksum: Involves summing the data words and transmitting the complement of the sum as the checksum. Detects many multiple-bit errors.
  • Cyclic Redundancy Check (CRC): Utilizes polynomial division over a finite field to generate a sequence of redundant bits. Highly effective in detecting burst errors.
• Error detection codes primarily identify the presence of errors but do not inherently correct them.