Solution to CS522 F2004 Homework#5:  CDMA and SECC

Goal:

Assignment Date: 11/1/2004
Due Day:11/10/2004
Description:

  1. CDMA. Chapter 2.51. Page 181. Prove if S.T=0, then S.T=0 where T is the complement chip sequence of T.
    Ans: S•T = =0
    S•T == =-1*=-1*0=0.

  2. CDMA. Chapter 2.53. Page 181. A CDMA receiver gets the following chips: (-1 +1 -3 +1 -1 -3 +1 +1). Assume the chip sequences defined in Figure. 2-45 (b), which stations transmited, and which bits did each one send?
    Ans: Just compute the four normalized inner products between the signal and the four chip seuqnces:
    -1+1-3+1-1-3+1+1)•(-1-1-1+1+1-1+1+1)/8=1     (-1-1-1+1+1-1+1+1) is A's chip sequence.
    (-1 +1 -3 +1 -1 -3 +1 +1) • (-1 -1 +1 -1 +1 +1 +1 -1)/8 =-1    (-1 -1 +1 -1 +1 +1 +1 -1) is B's chip sequence.
    (-1 +1 -3 +1 -1 -3 +1 +1) • (-1 +1 -1 +1 +1 +1 -1 -1)/8 = 0   (-1 +1 -1 +1 +1 +1 -1 -1) is C's chip sequence.
    (-1+1-3+1-1-3+1+1) •(-1+1-1-1-1-1+1-1)/8=1      (-1+1-1-1-1-1+1-1) is D's chip sequence.
    The result is that A and D sent bit-1s, B sent bit-0, and C was silent.
  3. SECC encoding.
    Page 244. Chapter 3 problem 10.
    An 8-bit byte with binary value 10101111 is to be encoded using an even-parity Hamming code. What is the binary value after encoding?
    Ans: The binary value of the codeword is 101001001111. The detail of the encoding process is shown below.
    1 2 3 4 5 6 7 8 9 10 11 12 Bit poisition
    C0 C1 D1 C2 D2 D3 D4 C3 D5 D6 D7 D8 codeword pattern
        1   0 1 0   1 1 1 1 Data bits
    1 1 ¿                   D1 contributes check bit pattern 11 (bit position 3)
      1   1   ¿             D3 contributes check bit pattern 110 (bit  position 6)
    1             1 ¿       D5 contributes check bit pattern 1001 (bit position 9)
      1           1   ¿     D6 contributes check bit pattern 1010 (bit position 10)
    1 1           1     ¿   D7 contributes check bit pattern 1011 (bit position 11)
          1       1       ¿ D8 contributes check bit pattern 1100 (bit position 12)
    1 0   0       0         Calculate check bit using even parity bit
    1 0 1 0 0 1 0 0 1 1 1 1 codeword generated
  4. SECC decoding.
    Page 244. Chapter 3 problem 11. I added one additional question to be answered in terms of the data bits to be delivered.
    A 12-bit Hamming code whose hexadecimal value is OxE4F arrives at a receiver.
    Ans: The original codeword set by the sender is 0xA4F. The data bit delivered to the upper layer of the receiver is 10101111. The detail of decoding process is shown below.
     
    1 2 3 4 5 6 7 8 9 10 11 12 Bit position
    C0 C1 D1 C2 D2 D3 D4 C3 D5 D6 D7 D8 codeword pattern
          E       4       F receive bit patter in hexadecimal
    1 1 1 0 0 1 0 0 1 1 1 1 receives bit pattern
        1   0 1 0   1 1 1 1 set aside received check bits
    1 1 ¿                   D1 contributes check bit pattern 11 (bit position 3)
      1   1   ¿             D2 contributes check bit pattern 110 (bit position 6)
    1             1 ¿       D5 contributes check bit pattern 1001 (bit position 9)
      1           1   ¿     D6 contributes check bit pattern 1010 (bit position 10)
    1 1           1     ¿   D7 contributes check bit pattern 1011 (bit position 11)
          1       1       ¿ D7 contributes check bit pattern 1100 (bit position 12)
    1 0   0       0         Calculate check bit using even parity bit
      X                     positions sent and rcvd check bit not the same
      2                     sum of position weight=2 position 2 is garbled
      0                     correct the bit.
    1 0 1 0 0 1 0 0 1 1 1 1 correct codeword
          A       4       F original codeword sent in hexadecimal.
        1   0 1 0   1 1 1 1 Data bits delivered to upper layer
    1. What was the original codeword sent by the sender in hexadecimal? Assume that not more than I bit is in error.
      Ans:
      1 0 1 0 0 1 0 0 1 1 1 1

    2. What are the data bits that should be delivered to the upper layer of the receiver.
      Ans:
      1 0 1 0 1 1 1 1