These questions are to aid you in your studies. They are not to be turned in and they do not cover all the topics covered in class after Problem Set 5.

1. Using a residue number system with two moduli, represent all of the decimal values between 0 and 11 inclusive when the moduli are

   1. 4 and 3
   2. 6 and 2

2. Consider the following example used to explain Tomasulo's Algorithm:

   
   Format: Opcode Destination Source1 Source2
   MUL R3,  R1, R2
   ADD R5,  R3, R4
   ADD R7,  R2, R6
   ADD R10, R8, R9
   MUL R11, R7, R10
   ADD R5,  R5, R11
   MUL R10, R4, R10
   

   Construct the Data Flow Graph for this program.

3. Consider the following format for transmitting a byte of data using the ECC mechanism described in class (Hamming Code).

   
   D7 D6 D5 D4 P8 D3 D2 D1 P4 D0 P2 P1
   

   For the following bit pattern, indicate whether there is no error or a single error. If there is a single error, list the corrected bit pattern.

   • 110010010111

   In general, what happens when there are two bit errors with this mechanism?

4. Using the Booth Multiplication Algorithm, multiply the two unsigned 10-bit numbers 0011011110 and 0001110010. Show the intermediate results after each step.

5. From Tanenbaum, 4th edition, Appendix B, 4.

   The following binary floating-point number consists of a sign bit, an excess 63, radix 2 exponent, and a 16-bit fraction. Express the value of this number as a decimal number.

   0 0111111 0000001111111111

6. From Tanenbaum, 4th edition, Appendix B, 5.

   To add two floating point numbers, you must adjust the exponents (by shifting the fraction) to make them the same. Then you can add the fractions and normalize the result, if need be. Add the single precision IEEE floating-point numbers 3EE00000H and 3D800000H and express the normalized result in hexadecimal. ['H' is a notation indicating these numbers are in hexadecimal]

7. From Tanenbaum, 4th edition, Appendix B, 6.

   The Tightwad Computer Company has decided to come out with a machine having 16-bit floating-point numbers. The model 0.001 has a floating-point format with a sign bit, 7-bit, excess 63 exponent and 8-bit fraction. Model 0.002 has a sign bit, 5-bit, excess 15 exponent and a 10-bit fraction. Both use radix 2 exponentiation. What are the smallest and largest positive normalized numbers on both models? About how many decimal digits of precision does each have? Would you buy either one?

8. The following numbers are represented exactly with a 9-bit floating point representation, in the format of the IEEE Floating Point standard:

   -infinity, -1, 0, 5/16, 19.5, 48.

   1. How many bits are needed for the fraction?
   2. What is the bias?
   3. Write each number in in the 9-bit floating point representation, below:

      Value	Representation
      48
      19.5
      5/16
      0
      -1
      -infinity

9. We must compute the following expression:

       a*x^6 + b*x^5 + c*x^4 + d*x^3 + e*x^2 + f*x + g 
   

   • How many operations and time-steps will the computation take on a single processor system (Use the smallest number of operations possible)?
   • How many operations and time-steps will the computation take on a multiprocessor system with 4 processors? (Use the smallest number of operations possible)
   • What is the speedup of the multiprocessor system over a single processor?

10. In an Omega network as presented in class, assume that there are n inputs and n outputs. Let k be the size of each switch. For k taking the values 2, 4, 8, and 64, answer the following questions. (Assume the cost of each switch is k^2)

    1. What is the cost of the network as a function of n?
    2. What is the latency of the network?
    3. Assume that n=64. What k value would you choose? Why? State your assumptions and design point.