of Principles Digital Communications (EL 601)
Polytechnic University, Fall 2005
Instructor: Dr. Elza Erkip
November 29, 2005
Due: December 6, 2005 (in class)
Homework #10
1. Problem 5 from Homework 9. The constraint length in that problem was stated incorrectly.
Consider a rate 1/4, constraint length 3 convolutional code with the octal generators
(5,7,7,7).
(a) Draw and label the state transition diagram.
(b) Find the transfer function T (D).
(c) Find df ree , the minimum free distance of the code.
(d) Show the path corresponding to df ree on the trellis diagram. How many information bits does this path correspond to?
2. Problem 8.26 from Proakis, Digital Communications. Assume that the system starts
from all zero state and is padded by zero information bits.
3. Problem 8.27 from Proakis, Digital Communications. Assume that the system starts
from all zero state and is padded by zero information bits.
4. Problem 8.36 from Proakis.
5. Proakis Table 8.2-2 lists rate 1/3 maximum free distance codes. We will consider
constraint length 3 code whose generators in octal are (5,7,7).
(a) Draw the state diagram and trellis diagram for this code.
(b) Find the transfer function T (D, N, J ).
(c) Find minimum free distance df ree . Identify the corresponding path with weight
df ree on the trellis diagram. What is the length of this path? How many bit errors
does this correspond to?
6. Use the transfer function calculated in the previous problem to nd an upper bound
for the probability of bit error for an AWGN channel with
(a) Hard decision decoding (you can use the Cherno bound).
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(b) Soft decision decoding.
Now:
(c) Find the coding gain in dB.
(d) Compare the performances of soft and hard decision decoding by plotting the
results on the same graph. You can use Matlab or your favorite program. Your
horizontal axis should be SNR per (b bit = Eb /N0 ) in dB, your vertical axis
should be bit error probability in log scale. On the same plot show the bit error
probability versus SNR/bit performance of uncoded BPSK.
(e) Which one is better hard or soft, by how much? Compare your plot with Proakis
Figure 8.2-15. (Note the mislabeling in that gure. Solid line corresponds to soft
decision decoding, dashed line corresponds to hard.) From your plot, how much
is the soft decision decoding better than uncoded transmission? How does this
relate to the coding gain?
7. Consider the constraint length K = 3, rate 1/4 convolutional code C with generator
polynomials (5,7,7,7) in octal. Note that according Proakis Table 8-2.3, this code has
maximum free distance. The path corresponding to df ree is represented by the state
sequence S0 S2 S3 S1 S0 where index of the state represents the contents of the
shift registers in decimal.
(a) Find the coded bit stream corresponding to the information bit stream 101100.
Assume initial contents of the shift registers are zero.
(b) Consider the puncturing matrix
P1 =
1
1
0
0
1
1
0
0
.
What is the rate of the punctured code C1 ? Find the punctured stream corresponding to the information bit stream 101100.
(c) Note that the above punctured code C1 corresponds to using only two generators
out of four. The resulting code can be found in Proakis Table 8-2.1. What is the
free distance df ree,1 ?
(d) Consider an alternative puncturing matrix
P2 =
0
1
1
1
0
0
0
1
.
What is the rate of the punctured code C2 ? Find the punctured stream corresponding to the information bit stream 101100.
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(e) For C2 , nd the Hamming weight of the punctured codeword represented by the
path S0 S2 S3 S1 S0 S0 . . .. Compare with df ree,1 of part (c). Based
on this comparison state whether you would prefer C1 or C2 .
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