### Problem 8-1:
**Question:** A datapath similar to the one in Figure 8-1 has 64 registers. How many selection lines are needed for each set of multiplexers and for the decoder?

**Solution:**
- For a datapath with 64 registers, we need to determine the number of selection lines required for the multiplexers and the decoder.
- The number of selection lines for a multiplexer is given by \( \log_2(n) \), where \( n \) is the number of inputs.
- For 64 registers, \( \log_2(64) = 6 \) selection lines are needed for each multiplexer.
- The decoder also requires 6 selection lines to select one of the 64 registers.

**Answer:** 6 selection lines are needed for each set of multiplexers and for the decoder.

### Problem 8-2:
**Question:** Given an 8-bit ALU with outputs \( F_7 \) through \( F_0 \) and available carries \( C_8 \) and \( C_7 \), show the logic circuit for generating the signals for the four status bits \( N \) (sign), \( Z \) (zero), \( V \) (overflow), and \( C \) (carry).

**Solution:**
- **N (Sign):** The sign bit is the most significant bit (MSB) of the output, \( F_7 \).
- **Z (Zero):** The zero bit is 1 if all the output bits \( F_7 \) through \( F_0 \) are 0. This can be implemented using an 8-input NOR gate.
- **V (Overflow):** The overflow bit is determined by the carries \( C_8 \) and \( C_7 \). Overflow occurs if \( C_8 \neq C_7 \). This can be implemented using an XOR gate.
- **C (Carry):** The carry bit is simply \( C_8 \).

**Answer:**
- \( N = F_7 \)
- \( Z = \overline{F_7 + F_6 + \ldots + F_0} \)
- \( V = C_8 \oplus C_7 \)
- \( C = C_8 \)

### Problem 8-3:
**Question:** Design an arithmetic circuit with two selection variables \( S_1 \) and \( S_0 \) and two 4-bit data inputs \( A \) and \( B \). The circuit generates the following eight arithmetic operations in conjunction with carry \( C_{in} \):

| \( S_1 \) | \( S_0 \) | \( C_{in} = 0 \) | \( C_{in} = 1 \) |
|-----------|-----------|------------------|------------------|
| 0 | 0 | \( F = A + B \) (add) | \( F = A + \overline{B} + 1 \) (subtract A - B) |
| 0 | 1 | \( F = \overline{A} + B \) | \( F = \overline{A} + B + 1 \) (subtract B - A) |
| 1 | 0 | \( F = A - 1 \) (decrement) | \( F = A + 1 \) (increment) |
| 1 | 1 | \( F = \overline{A} \) (1s complement) | \( F = \overline{A} + 1 \) (2s complement) |

Draw the logic diagram for the two least significant bits of the arithmetic circuit.

**Solution:**
- The arithmetic circuit can be designed using multiplexers controlled by \( S_1 \) and \( S_0 \).
- For each bit position, a 4-to-1 multiplexer can be used to select the appropriate operation based on \( S_1 \) and \( S_0 \).
- The inputs to the multiplexer for each bit position would be the results of the arithmetic operations listed in the table.

**Answer:** The logic diagram would involve multiplexers controlled by \( S_1 \) and \( S_0 \) to select the appropriate operation for each bit position.

### Problem 8-4:
**Question:** Design a 4-bit arithmetic circuit with two selection variables \( S_1 \) and \( S_0 \) that generates the arithmetic operations in the following table. Draw the logic diagram for a typical single-bit stage and the LSB stage.

| \( S_1 \) | \( S_0 \) | \( C_{in} = 0 \) | \( C_{in} = 1 \) |
|-----------|-----------|------------------|------------------|
| 0 | 0 | \( F = A + B \) (add) | \( F = A + B + 1 \) |
| 0 | 1 | \( F = A \) (transfer) | \( F = A + 1 \) (increment) |
| 1 | 0 | \( F = \overline{B} \) (complement) | \( F = \overline{B} + 1 \) (negate) |
| 1 | 1 | \( F = A + \overline{B} \) | \( F = A + \overline{B} + 1 \) (subtract) |

**Solution:**
- Similar to Problem 8-3, use multiplexers controlled by \( S_1 \) and \( S_0 \) to select the appropriate operation.
- For each bit position, a 4-to-1 multiplexer can be used to select the appropriate operation based on \( S_1 \) and \( S_0 \).

**Answer:** The logic diagram would involve multiplexers controlled by \( S_1 \) and \( S_0 \) to select the appropriate operation for each bit position.

### Problem 8-5:
**Question:** Draw the diagram of an ALU in which there are three bits allocated for selecting the following operations for two operands. The operations are below. Draw a simple block diagram. Also, consider what should be done in the case an invalid combination of selection bits appears.

(a) Add
(b) Subtract
(c) Logical OR
(d) Logical AND
(e) Logical XOR

**Solution:**
- Use a 3-to-8 decoder to decode the three selection bits.
- Each output of the decoder can be used to enable the corresponding operation.
- For invalid combinations, ensure that the ALU defaults to a safe operation or indicates an error.

**Answer:** The block diagram would include a decoder to select the appropriate operation based on the selection bits.

### Problem 8-6:
**Question:** Design one bit of a digital circuit that performs the four logic operations of exclusive-OR, exclusive-NOR, NOR, and NAND on register operands \( A \) and \( B \) with the result to be loaded into register \( A \). Use two selection variables.

**Solution:**
- Use a 2-to-4 decoder to decode the two selection variables.
- Each output of the decoder can be used to enable the corresponding logic operation.
- The operations can be implemented using standard logic gates.

**Answer:** The logic diagram would involve a decoder and logic gates to perform the specified operations.

### Problem 8-7:
**Question:** Design an ALU that performs the following operations:

- \( A + B \)
- \( A + \overline{B} + 1 \)
- \( \overline{B} \)
- \( \overline{B} + 1 \)
- \( \text{sr } A \) (shift right A)
- \( A \vee B \)
- \( \text{sl } A \) (shift left A)
- \( A \wedge B \)

Give the result of your design as the logic diagram for a single stage of the ALU. Your design should have one carry line to the left and one carry line to the right between stages and three selection bits. If you have access to logic optimization software, apply it to the design to obtain reduced logic. Model your ALU in an HDL and verify its operation by simulation.

**Solution:**
- Use a 3-to-8 decoder to decode the three selection bits.
- Each output of the decoder can be used to enable the corresponding operation.
- Implement each operation using standard logic gates and multiplexers.

**Answer:** The logic diagram would involve a decoder, logic gates, and multiplexers to perform the specified operations.

### Problem 8-8:
**Question:** Find the output \( Y \) of the 4-bit barrel shifter in Figure 8-9 for each of the following bit patterns applied to \( S_1, S_0, D_3, D_2, D_1, \) and \( D_0 \):

(a) 110101
(b) 101011
(c) 011010
(d) 001101

**Solution:**
- The barrel shifter shifts the input bits based on the selection bits \( S_1 \) and \( S_0 \).
- For each pattern, determine the shift operation and apply it to the input bits.

**Answer:**
- (a) Shift left by 2: \( Y = 010100 \)
- (b) Shift right by 1: \( Y = 010101 \)
- (c) Shift left by 1: \( Y = 101000 \)
- (d) Shift right by 2: \( Y = 000011 \)

### Problem 8-9:
**Question:** Specify the 16-bit control word that must be applied to the datapath of Figure 8-1 to perform the following microoperation: \( R0 \leftarrow R3 + R2 \).

**Solution:**
- The control word specifies the operations for the datapath components.
- For \( R0 \leftarrow R3 + R2 \), the control word would enable the addition operation and select the appropriate registers.

**Answer:** The 16-bit control word would include bits to select \( R3 \) and \( R2 \) as inputs to the ALU, enable the addition operation, and load the result into \( R0 \).
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